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Pricing and Hedging Embedded Mortgage Options

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					Technical document:




      Pricing and Hedging Embedded
             Mortgage Options



                         Ridha Mahfoudhi
                           Chief Analyst
                         ALM Department
                      National Bank of Canada




                            March 2007
I.     Interest Rate Models: Brief Description
The interest rate models we employ in our analysis are the Hull & White (1990), HW, and the Black &
Karasinski (1991), BK, models. Under these models, the term structure of interest rates is generated from
the short rate process (rt )t≥0 , defined here as the 1 month-CDOR (Canadian Deposits Offering Rate), as
follows:
                                            P (t, T ) = Et [D(t, T )] ,                                     (1)

where P (t, T ) is time t-value of a discount bond with maturity date T and unit face value, and
                                                       Ã Z         !
                                                                   T
                                        D(t, T ) = exp −               ru du ,                              (2)
                                                               t

is the stochastic discount factor (SDF) used to discount back at time t a payoff to be received at time T .
The conditional expectation Et [.] is defined under the risk-neutral (martingale) measure.

     The general form of any short rate model could be described by,

                                        drt = µQ (t, rt )dt + υ(t, rt )dzt ,                                (3)

where µQ (.) is a defined drift function of the short rate; υ(.) is the process’s instantaneous volatility and
(zt )t≥0 is a standard Q-Brownian motion. Namely, under the HW and BK models, the most well-known and
popular short rate models, the short rate dynamics exhibit mean-reversion patterns of the form,
                                      ⎧
                                      ⎨ µQ (t, r ) = θ(t) − ar
                                                 t            t
                             HW :                                                                           (4)
                                      ⎩ υ(t, r ) =        σ
                                               t
                                      ⎧                 h                   i
                                      ⎨ µQ (t, r ) = rt θ(t) + σ2 − a ln rt
                                                 t              2
                             BK :                                                                           (5)
                                      ⎩ υ(t, r ) =             σr
                                                  t                         t

where a is the reversion speed (constant parameter) of the process, θ(t) is a time-deterministic drift term
capturing the equilibrium level of interest rates to be inferred from the implied forward curve; and σ is the
instantenous volatility of: rt under the Gaussian HW model and ln rt under the lognormal BK model

     The HW and BK models are said arbitrage-free short rate models in the sense that they exactly replicate
the observed market prices of discount bonds. To ensure this arbitrage-free condition, we solve for the drift
term such that the model’s bond prices respect the observed market yield curve. The numerical procedure
behind the calibration of short rate models to the yield curve is possible thanks to the trinomial trees method.
Further, before being used to price options and other contingent assets, the reversion speed and volatility
parameters of the short rate models above are calibrated to the market volatility structure composed of the
implied volatility of forward swap rates contained in the volatility surface of swaptions and the forwards
volatilities implied in the prices of ‘at-the-money’ caps.

     Because we have to deal with mortgage options and the path-dependency they may involve, we have
to implement the interest rate models above using the Monte Carlo method. Indeed, once the models are

                                                         2
calibrated to the market yield curve and volatility structure using the trinomial tree method, we generate
N paths (i = 1, ...N ) of a pure mean-reverting process (xt )t≥0 using the transition density technique,
                                                             µ               ¶1/2
                                                                 1 − e−2a∆
                                 x(j+1)∆,i = e−a∆ xt,i + σ                          εj∆,i ,
                                                                     2a
              1
where ∆ =    120   is the time-increment used to discretize the interest rate processes and (εj∆,i )i=1,...N ∼iid
N (0, 1) is the sample of random Gaussian numbers for time t = j∆. In our all subsequent analysis, the Monte
Carlo paths sample {(εj∆,i )i=1,...N , j = ∆, 2∆, ...} is held unchanged and it contains N = 10240 paths. The
simulation time-horizon is set equal to 20 years. The uniformity of the Gaussian numbers is enhanced using
the stratified sample technique. Moreover, we use the moments matching technique to minimize the Monte
Carlo’s standard errors. The short rate paths are then obtained from the generated paths of (xt )t≥0 as
follows:
                                               g(rt,i ) = xt,i + α(t),                                       (6)

with g(rt ) = rt for the HW model and g(rt ) = ln rt for the BK model. The time-deterministic function
α(t) represents the expected short rate E0 [rt ] and is inferred from the initial yield curve using the HW-tree
method in the same way we infer the drift term θ(t).

   Once the short rate paths are simulated, we obtain the paths of discount bonds, (Pi (t, t + η))i=1,..N , over
key points of the term structure (i.e., η = 6, 12, 18, ....120 months) upon which we are able to construct the
N paths of the spot swap rates, (Si (t, t + η))i=1,..N for η = 12, 24, 36, 48, 60 months. The path-conditional
values of the discount bonds are obtained analytically under the HW model, while they are inferred under
the BK model by regressing the simulated paths of the SDF on a simple affine model equation. The discount
bond values and swap rates are computed at each time instant t = j∆ over our simulation horizon of 20
years. To price mortgages options, we need to store observed swap rates and SDF at the end of each month
or week depending on the type of the option considered.
   The interest rate models are calibrated to the CDOR-swap curve. The Monte Carlo paths of the future
funding cost (COF) of the bank, (Yi (t, t + η))i=1,..N , are obtained by adding the credit spread observed at
time zero to the simulated swap rate with the same term. This is consistent with the risk management policy
in place, according to which the initial COF-swap spread must be held constant over simulations. The paths
of mortgage rates, (Ri (t, t + η))i=1,..N , for the key balloon terms η = 12, 24, 36, 48, 60 months are recovered
from the COF-par yields paths as follows:

                                                                        ˆ
                                        Ri (t, t + η) = Yi (t, t + η) + φ(η),                                (7)

     ˆ
with φ(η) denotes the mortgage rate spread over the COF par yield as observed at time zero for the key
mortgage balloon terms. The whole term structure of this mortgage spread is obtained for any term point
by interpolating the mortgage spreads over the key points using the cubic splines technique. Notice that we
have already developed rate index models describing the dynamics of the mortgage rate spread in function
of the COF-par yields and some seasonality factors. These models provide a very good fit of empirical

                                                         3
observations. We have chosen, however, to not make use of these mortgage margin models in our subsequent
analysis in order to keep our results insensitive to mortgage margin risk.

   We conclude our description of the term structure environment by explaining the reasons that motivate
our choice of the class of short rate models. First, market models like the Libor model (Brace, Gatareck
and Musiela (1997)) build swap rates based on the set of market observable forward rates, which results in
largely spaced simulated swap rates over the time horizon. Since mortgage options we aim to price and study
depend on the swap rates at monthly and even daily/weekly frequencies, the granularity shortcoming becomes
a critical disadvantage. Of course, we are able to create some ‘stochasticity’ between resetting dates over
the time calendar using for example Brownian bridge technique, but this turns to be irrelevant solution for
mortgage options with relatively short terms (less then one year) like mortgage rate commitments. Second,
the short rate models used here are those already employed to evaluate the risk metrics of our mortgages
portfolio. Pricing mortgage options by the mean of the same models would be helpful for some extensions
and advanced analysis.




II.    Mortgage Pricing: Basic Elements
The time horizon is divided into equal time periods with a length of 1 month each one, and the current time
is set equal to zero. The subsequent analysis ignores the default risk of mortgagors. I consider a fixed-rate
Canadian balloon mortgage newly issued with an amortization horizon of n months and a balloon term of
T months (with (n − T ) ≥ 0 is the residual amortization period at the balloon maturity date T ). Let K be
the contractual mortgage rate and B0 is the initial mortgage balance amount.

i. Plain-Vanilla Mortgage:
   By ignoring prepayment risk, the market value at time zero of the mortgage described above, called in
this case by a plain-vanilla mortgage (PVM), is given by
                                          " T          #
                                           X
                                     V0 =     P (0, t)A + P (0, T )BT ,                                  (8)
                                            t=1

with BT is the residual balance at the balloon maturity date T and A is the monthly contractual payment
amount (annuity):
                                                  ∙            1        ¸
                                                              12 K
                                        A = B0                   1        .
                                                      1−   (1 + 12 K)−n
The PVM is a pure theoretical case. Real-world mortgages and mortgages-backed securities are more difficult
to price because of the embedded prepayment option.

ii. Prepayment Risk:




                                                           4
   Let CF (t, (R(t, t + η)η ) be the mortgage rates-sensitive cash flow promised by the mortgage at month t
(t = 1, 2...T months) after allowing for prepayment risk. Therefore, the market value of the risky mortgage
at time zero is given by,
                                         T
                                         X
                                  V0 =         E0 [D(0, t) CF (t, (R(t, t + η)η ))] .                     (9)
                                         t=1

Remark that the (random) residual balance at the balloon maturity T is by construction included in the
terminal cash flow CF (T ).

iii. Prepayment Option:
   Under the rational option pricing theory, mortgagors exercise their option only to benefit form lower
mortgage rates. Prepayment events are thus fully explained by interest rate movements. The prepayment
option we mean here could be defined as follows: the option the mortgagor holds to refinance at a lower
mortgage rate at any time during the life of the loan. More precisely, this option has the following features
(in the context of Canadian market):
   - Underlying asset: The original mortgage (the mortgage we want to price);
   - Strike level: The market value of a new mortgage at the prepayment (refinancing) time, given the
mortgage rates observed at that time;
   - Exercise style: American-style (at least Bermudian, if we recognize discretization of exercise-times);
   - Type: Call (buy-back) option;
   - Term: The term of the original mortgage (both the underlying mortgage and the prepayment option
has the same origination date);
   - Holder: The mortgagor (long position);
   - Seller: The originator bank (short position).

iv. Statistical Prepayment Rate Function:
   Using some actuarial calculus, mortgages could be valued based on some prepayment rate function
calibrated to empirical mortgages prepayment data. Let Smm(t) denotes the monthly prepayment rate
function inferred from historical mortgages prepayment data (more precisely, Smm stands for Single Monthly
Mortality. The relationship between the monthly prepayment rate, Smm, and the annualized Conditional
Prepayment Rate, CPR, is given by: CP R = 1 − (1 − Smm)12 ), which can be either time-deterministic or
a function of stochastic mortgage rates. Note that for a constant or time-determinstic Smm, the schedule
of contractual cash flows will be perturbed by the introduced prepayment flows, but since these flows are
known at time 0 the mortgage will simplify to a fixed-income asset with no embedded option risk as the
theoretical PVM does. To capture the economic prepayment risk banks bear in real-world, the prepayment
rate Smm(t) must be function of the stochastic mortgage rates to be observed over the market during the
future. This could be achieved by formulating Smm(t) in the same way than an option payoff to yield the
asymmetric relationship between prepayment flows and interest rates capturing the refinancing incentive of
mortgagors.

                                                            5
v. Mortgage Rate Spread and Mortgage Value:
     By letting the mortgage rate K equal to the time zero-COF par yield Y (0, T ) (the credit-spread-adjusted
par swap rate), one can easily verify that the plain-vanilla mortgage (PVM) value we obtain under the
assumption of no prepayment risk will simply collapse to the par-value given by the initial balance B0 .
Whenever we increase the mortgage rate K above Y (0, T ), we immediately observe that the PVM value
(always under the assumption of no prepayment risk) will shift upward thus exceeding the par-value. This
means that by clipping the mortgage rate spread (K − Y (0, T )) at the origination date (t = 0), the bank
will add economic value above the par-value B0 . The PVM market value in excess of the par value reflects
the present value of promised mortgage rate spreads to be earned over the balloon period of the mortgage.
As we will see later, the mortgage rate spread does not affect the PVM value only, but may has a significant
impact on the evaluation of prepayment risk as well.




III.       Pricing Mortgage Prepayment Options
By allowing mortgagors the right to prepay their mortgage loan early for various reasons or to refinance
it at lower mortgage rates, the originator bank bears the risk of loss of residual contractual interests at
the prepayment/refinancing date. Focusing on interest rate-driven prepayments only, this loss of residual
scheduled interests implies a reinvestment risk for the bank, since refinancing events most likely occur at
lower rates environments. Of course, if we take into account the possibility that the bank charges a penalty
for clients when refinancing their loans, this loss of interests would be attenuated by collected penalties. In
our analysis below of interests’ loss, we shall assume that there is no penalty mechanism in place. Later in
the document, we will introduce the penalty component in our analysis and examine its effects.
     The value of the prepayment option must therefore reflect the opportunity cost the bank bears due
to the loss of residual interests at refinancing dates. In this section, we introduce two different measures
of prepayment risk. We describe the formal development behind each measure and then implement the
two measures based on observed historical data and analyze the numerical results obtained based on each
measure.



1.     Rational Option Pricing Model

Let VtOrig and VtNew denotes the market value of the original mortgage (underlying asset) and which of
the new mortgage at month t = 1, 2...(T − 1), respectively. Therefore, the exercise value of the prepayment
option at any month t = 1, 2...(T − 1) is given by:
                                                h                  i
                                       EVt = max 0, VtOrig − VtN ew .                                    (10)



                                                       6
One typical feature of prepayment options (independently from the underlying asset) is that the exercise
value at the origination date as well as at the maturity date is worth nothing. This is due to the simple fact:

                                           V0Orig    = V0N ew =⇒ EV0 = 0,
                                            Orig
                                           VT           N
                                                     = VT ew =⇒ EVT = 0.

This mean that the mortgagor, the holder of the prepayment option, has to benefit from the allowed option
before the underlying mortgage comes at maturity. Again, at the origination date, because market conditions
have not yet changed, the prepayment option will always be at-the-money. Between the origination and the
maturity dates, changes of market rates will be observed, such that there is a strict positive probability that
the prepayment option will be in-the-money at intermediate dates.
   The exercise value of the prepayment option captures the differential of mortgage value the client will
realize by refinancing the loan at a lower mortgage rate. The economic argument behind this formulation
of the option payoff is that mortgagors are assumed to minimize the value of their liability given by the
mortgage market value.

   To price this prepayment option, we simply need to project the future mortgages values, VtOrig and VtNew ,
for different scenarios of interest rates paths. Both tree method and Monte Carlo method could be used.
Although the tree method is mostly used to price similar American-style options, we prefer to implement
here the Monte Carlo simulation method. The reason is that we shall use Monte Carlo method to evaluate
a second measure of prepayment risk (described in the following subsection) for which we cannot apply the
tree method. For comparison purposes, we prefer to use the same pricing tool for these both measures.

   The market value of the original mortgage at month t = 1, 2...(T − 1) observed at the interest rate path
i (i = 1, ...N ) is computed as follows:
                                            T
                                            X
                              Orig
                            Vt,i = A               e−φ(s−t) Pi (t, s) + e−φ(T −t) Pi (t, T )BT ,          (11)
                                           s=t+1

where Pi (t, s) is the path i-conditional discount bond.
   As one may observe, we have added the premium φ(s − t) to discount back future payments. The
premium φ(η) reflects the mortgage rate spread over the COF of the bank for the associated term of η months
(η = 1, 2...T ). This assumes that the mortgage spread curve observed at time 0 will still be unchanged over
the simulation horizon of T months. Notice that we use the following formula,

                                                             ˆ      τ
                                                    φ(τ ) := φ(τ ) × ,
                                                                    12

to account for the lengh of the time interval τ expressed in number of months.
   In fact, the quantity e−φ(s−t) Pi (t, s) must be viewed as the discount factor (at month t conditional on
the rate path i) associated with the mortgage rate curve. This discount factor reflects the opportunity cost
of the mortgagor and should be therefore used to price the prepayment option held by the later. Remark

                                                              7
that in the case of traded interest rate options like caps or swaptions, we do not face a difference between
the discount factor that reflects the opportunity cost of the buyer of the option and which of the seller. This
divergence takes place only when dealing with non-traded options embedded in banks’ products. We think
that the choice of using the mortgage rate curve to construct the discount factor should make sense, since the
exercise decision will be taken by the mortgagor rather than the bank. Nevertheless, as it is unambiguously
illustrated by our numerical tests presented later, the impact of adding or not the premium φ(.) (that is,
the difference between using the pure discount factor Pi (t, t + u) or the mortgage spread-adjusted discount
factor e−φ(u) Pi (t, t + u)) on the value of the prepayment option is negligible.

   Since the interest rate model used here is arbitrage-free, one can easily deduce that the value of the new
mortgage that would be issued at any month t = 1, 2...(T − 1), given the mortgage rates observed at that
date, will be close to the par-value provided by the residual mortgage balance at month t, i.e., Bt . That is,
we have:
                                       t+T
                                       X        −φ(s−t)
                        N
                      Vt,i ew = At,i           et           Pi (t, s) + e−φ(T ) Pi (t, t + T )Bt+T = Bt ,   (12)
                                       s=t+1

with                                                ∙                                       ¸
                                                                   1
                                                                  12 Ri (t, t + T )
                                  At,i = Bt                         1                         ,
                                                        1 − (1   + 12 Ri (t, t + T ))−(n−t)
is the new monthly contractual payment reset in function of the new mortgage rate Ri (t, t + T ) (remember
                                                                                                      ˆ
that mortgage rates are constructed from COF par yields as follows: Ri (t, t + T ) = Yi (t, t + T ) + φ(T ))
observed at time t, conditional on the generated path i. Of course, because time-discretization, the equality
above will hold approximately. However, a high computation precision permits to ensure a good convergence.



   Figure 1 illustrates the probability distributions of the par ratio (the par ratio is the market value-to-
par value where the par value is given by the residual balance amount) of the original mortgage value at 30
months from the origination date (that is at the middle of the mortgage balloon term), as inferred form the
HW and BK models. It shows how the market value is distributed around the par value given by the residual
balance. We see that because of the lognormal distribution of the short rate, the par ratio distribution is
more skewed above the 100% level under the BK model.

   Figure 2 shows the Monte Carlo paths-average of the exercise value EVt of the prepayment option over
the mortgage’s balloon term horizon, as obtained form both the HW and BK models. We see that the
exercise value exhibits a humped shape: it increases very fast during the first year, reaches a pick and then
starts to decline slowly until collapse to zero at maturity. Whatever the underlying asset, this is a typical
pattern of prepayment options. Interest economies (in the case of loans) or interest gains (in the case of
cashable saving/deposit products) due to favorable movements of interest rates are higher when the residual
time to maturity is long enough. Whenever this residual term decreases, the potential economies/gains start
to decline. Only the time increasing dispersion of interest rates around the initial forward curve (i.e., the

                                                                     8
volatility effect) mitigates this decline in such a way the exercise value does not drop rapidly after reaching its
maximum level. At the maturity date, the option is worth nothing, since the underlying comes at maturity
too.



1..1      The LSM-Monte Carlo Pricing Model

The Monte Carlo algorithm we implement is based on the least-squares method (LSM) developed by Longstaff
& Schwartz (2001). This advanced Monte Carlo method is now largely popular, since it permits to price
American-style options (like the mortgage prepayment option described above) efficiently as tree methods
do. It has also the advantage to price more complex exotic and path-dependent options that tree methods
do not permit to price. Our basic Monte Carlo simulations of swap and mortgage rates paths that will serve
to price the prepayment option are drawn from the interest rate models described in Section 1. We simply
here add the LSM component to our basic algorithm in order to be able to value the prepayment option.
       Now, given the expression of the option’s exercise value at each path i,
                                  " "      T
                                                                                           #    #
                                          X
                                                −φ(s−t)               −φ(T −t)
                     EVt,i = max 0, A         e         Pi (t, s) + e          Pi (t, T )BT − Bt ,           (13)
                                            s=t+1

we are able to price the prepayment option using our Monte Carlo-LSM algorithm. The LSM consists of
exploiting the cross-sectional information contained through the generated N paths by regressing the path-
conditional approximation of the continuation value against relevant path-conditional instrumental variables,
i.e., the basis functions of the LSM algorithm. More precisely, the method proceeds iteratively as follows:
i. At any stopping-time t < T , we start by identifying the ‘in-the-money’ region (not to confuse with the
optimal early exercise region), we denote by ΩIT M (that is, for any i ∈ ΩIT M , we should have EVt,i > 0).
                                              t                           t
                                                            c
ii. We determine the path-conditional continuation value CVt,i at the stopping-time t by discounting back
the prepayment option payoff at month (t + 1) ≤ T as conditional on the path i, we denote by CFt+1,i . That
is,
                                                       c
                                                    CVt,i = Ut,i CFt+1,i ,                                   (14)

with Ut,i := e−φ(1) Di (t, t + 1) = e−φ(1) DD(0,t+1) is the mortgage spread-adjusted stochastic discount factor
                                            i
                                              i (0,t)

observed at month t as conditional on the path i for a term of 1 month exactly the length of the time-
increment used in our option pricing algorithm.
iii.    After defining a set of relevant path-conditional instrumental variables, {Xt,i }, we proceed to the
                                                                               c
regression (using OLS technique) of the path-conditional continuation value CVt,i on the LSM basis function
Xt,i in order to infer the fair continuation value of the option CVt,i from the cross-sectional information
contained through the generated N paths. Indeed, we have,

                                  c
                               CVt,i   → OLS [cons tan t, Xt,i ; β OLS ] ,   i ∈ ΩIT M ,
                                                                                  t                          (15)

                               CVt,i    :     ˆ
                                            = β OLS × Xt,i .                                                 (16)



                                                              9
   Based on their analysis of the convergence of the LSM algorithm, Longstaff and Schwartz recommend
to perform the regression over the ‘in-the-money’ region ΩIT M only to ensure more efficiency. The choice
                                                          t

of the basis function is also important, since it is determinant for the accuracy of the pricing results. We
describe later the basis function considered.
iv. Knowing the fair continuation value of the option CVt,i we are able to determine the time t-option
payoff CFt,i based on the early-exercise indicator function: EDt,i = 1 if EVt,i > CVt,i for i ∈ ΩIT M and 0
                                                                                                t

otherwise. Namely,

                                  CFt,i   = EDt,i EVt,i , t = 1, 2...(T − 1),                                (17)

                                  CFT,i   = EVT,i = 0.                                                       (18)

Afterward, we need to adjust the subsequent option payoffs such that the option is exercises only at once
during its life. To do this for each path where early exercise is optimal as inferred from the time t-decision
rule, we have simply to multiply all the subsequent payoffs (CFs,i )s>t by the same early-exercise indicator
that takes the value of zero if EDt,i = 1.
v. We repeat steps i-iv iteratively starting from the maturity date T and going backward until month 1.
vi. Finally, define the optimal exercise time θ as the first time the mortgagor finds optimal to exercise the
prepayment option. More precisely, let θi denotes the optimal stopping-time associated with the path i,

                                       θi := inf {t > 0 : EVt,i > CVt,i } .                                  (19)

This leads to the following Monte Carlo-LSM averaging pricing formula of the prepayment option:
                                                N
                                             1 X −φ(θi )
                                      C0 =         e     Di (0, θi ) CFθi ,i .                               (20)
                                             N i=1

In contrast to the tree pricing method that consists of discounting backward the option value max [EVt,i , CVt,i ]
until time zero, the LSM method prices early-exercise options by discounting back the optimal early-exercise
payoffs in the same manner we price Arrow-Debreu contingent assets. The two approaches must lead to the
same faire option price, but the backward induction is not suited with the LSM Monte Carlo.

   What we described just above is the LSM algorithm serving to price early-exercise-style options. The LSM
method finds more other applications. For instance, the application of the LSM method to estimate Monte
Carlo path-conditional values of discount bonds (forward values of discount bonds) under the lognormal BK
model is one of these applications that could be viewed as a linear-asset LSM pricing. The LSM method
could be also very useful to price path-conditional options (with or without early-exercise-style feature) as
we will see later in this document when we investigate the valuation of mortgage rate commitment options.

   To implement the LSM pricing algorithm described above, we need to specify the basis functions to be
used as instrumental variables in the LSM regression. We have conducted numerical tests of the smoothness
and the stability of both the N paths-average option value max [EVt,i , CVt,i ] over the time horizon and

                                                        10
which of the early exercise intensity for many basis function candidates. The basis functions {Xt,i } we have
selected based on these tests are the 5 years swap rate, Si (t, t + 60), and a quadratic function of the slope
of the swap curve captured by the spread between the 1 month CDOR rate (i.e, the short rate) and the 3
years swap rate, [Si (t, t + 36) − ri (t)]. The choice of the 3 years swap rate to define the swap curve slope
rather than the 5 years swap rate is motivated by two reasons. First, the 5 years swap rate is used as an
independent regressor. Using the 5 years rate to define the curve slope will over-load the model with respect
to the cross-sectional information contained through the paths of this swap rate. Second, the horizon of 3
years seems a very good proxy of the average life of 5 years-balloon mortgages with allowed prepayment
flows, as documented based on the empirical prepayments data. Therefore, the chosen swap curve slope
captures well the pricing horizon of the considered mortgage.



1..2   Risky Mortgage Pricing

Now, we show how determining the market value of the risky mortgage, we denote by V0Pr ep . In doing so, we
have to define the discount factor curve to be used to discount back the expected cash flows. As explained
above, the prepayment option is priced from the mortgagor’s point of view by using the mortgage rate curve
as a discount factor curve. The market value of the plain-vanilla mortgage (PVM), we denote by V0PVM ,
under this setup will be therefore given at time 0 by:
                                      N
                                         "Ã T                      !                        #
                                   1 X      X
                     V0PVM =                     e−φ(t) Di (0, t) A + e−φ(T ) Di (0, T ) BT               (21)
                                  N i=1     t=1
                                  " T                  #
                                    X
                                        −φ(t)
                             =         e      P (0, t)A + e−φ(T ) P (0, T )BT = B0 .                      (22)
                                    t=1

Using the argument of absence of arbitrage, we must have:

                                              V0Pr ep = V0PVM − C0 .                                      (23)

This could be intuitively interpreted as follows: To replicate a long position over a risky mortgage, one needs
to hold an equivalent PVM and going short over the prepayment option.



2.     Behavioral Model of Prepayment Risk

Beyond the reaction of prepayment flows to changes of interest rates, we have to recognize that the exercise of
the prepayment option by mortgagors may result from exogenous factors that are not explained by refinancing
incentives (see Mansukhani & Srinivasan (2001)). In addition, although they would be completely informed
and rational decision-makers, real-world mortgagors do not usually behave as predicted by the rational
option theory. Many factors, such as market frictions and transaction costs, may lead to behavioral patterns
that cannot be captured by the standard option pricing models (see Richard & Roll (1989), Schwartz &
Torous (1989) and Stanton (1995)). This leads us to experiment the evaluation of prepayment risk from a

                                                        11
different perspective than the standard rational option pricing theory. Supposing that we dispose of reliable
behavioral model that replicates well the average prepayment exercise decisions made by mortgagors, we
can price prepayment risk in function of (historically) observed prepayment patterns. This kind of pricing
model leads to two main deviations from the prepayment flows distribution predicted by the pure rational
option theory. In one hand, mortgagors’ rationality in terms of optimal exercise decisions as implied by this
kind of behavioral models will imply a slower prepayment activity than which implied by the pure rational
option theory. In another hand, prepayment flows under a behavioral model are not completely determined
by interest rate movements. Rather, some of the prepayment activity will be due to non-rate-based factors.
The main implication of this is that prepayment flows will partially exhibit linear trends. That is, because of
these non-rate-based factors, we will partially observe the same prepayment activity for both high and low
interest rate environments. As a result, a fraction of predicted prepayment flows will exhibit zero correlation
with interest rate changes. As rates may be either higher or lower with comparison to the rates observed at
the mortgage’s origination date, this linear prepayment activity will either attenuate or amplify the overall
required reward for prepayment risk.

   In contrast to the prepayment option value, the measure of prepayment risk proposed here captures the
economic opportunity cost of prepayment based on empirical analysis of prepayment rates. More precisely,
we build a measure of the expected economic loss the bank bears due to the allowance for prepayments. In
doing so, we use econometric models that explain variations of historical prepayment rates in function of
the mortgage features and market rates. The measure we build here should not be viewed as a prepayment
option premium, since prepayment decisions predicted from the underlying econometric models are not based
on the rational option pricing arguments (i.e., the optimal early exercise policy illustrated earlier), but rather
inferred from empirical data of mortgages prepayment rates.



2..1   The Expected Economic Loss (EEL)

The method proceeds as follows. In a first step, using an econometric model of prepayment risk, we project
the future monthly prepayment rates, Smm(t), over the balloon life horizon of the mortgage (t = 1, 2, ...T ).
The forecasted prepayment rates are stochastic, since the used models make prepayment rates conditional
on the mortgage rates observed over the market in a manner to capture the “optionality” of the prepayment
decision. Once the interest rate paths and the prepayment rates associated with these paths are projected,
the cash flows to be received by the bank are computed using a formula involving the Smm function and
the scheduled contractual payments. Thereafter, we discount back the forecasted cash flows to obtain the
market value of the risky mortgage as shown by equation (20). In the last step, the expected economic loss
(EEL) is computed as the gap between the plain-vanilla mortgage (PVM) value and the market value of
the risky mortgage. This definition follows from the same arbitrage argument used earlier to price the risk
mortgage within the rational option pricing model. The reason is that under the assumption that the used


                                                       12
econometric model perfectly describes the distribution of prepayment events, we would be able to replicate
the risky mortgage position by trading market securities reproducing the same profile of prepayment flows.
As we will see below, this assumption is very realistic, given the simple functional form of the econometric
models used to describe prepayment flows.

   Formally, using the COF curve as a discount factor curve, the EEL is calculated under our Monte Carlo
model as follows:
                                              EEL0 = V0PVM − V0Pr ep ,                                 (24)

with the PVM value, V0PVM , and the risky mortgage value, V0Pr ep , are given by,
                                           " T           #
                                            X
                                 PVM
                               V0       =       P (0, t)A + P (0, T )BT ,                              (25)
                                                   t=1
                                                     N
                                                       " T                   #
                                                  1 X X
                                    V0Pr ep   =             Di (0, t) CFi (t) ,                        (26)
                                                  N i=1 t=1

where CFi (t) is the forecasted path i-dependent cash flow taking into account prior prepayment flows that
have occurred over the time interval [s = 1, s = t − 1] as well as the current prepayment flow observed at
month t. Each path-dependent stochastic cash flow CFi (t) is projected over time based on the observed
mortgage rate (R(s, s + T ))s=1,...t .
   As mentioned above, prepayment flows are forecasted using an econometric model. The functional form
of this econometric model could be roughly described by the simple function:

                                 Smmi (t) = λ(t) + β max [0, K − Ri (t, t + T )] ,                     (27)

where K is the original contractual mortgage rate, λ(t) is a time-deterministic function and β is a model
parameter capturing the non-linear sensitivity of prepayment flows to observed market rates. Refinancing
incentives are captured through the non-linear component max [0, K − Ri (t, t + T )], which represents the
exercise value of the prepayment option. Notice that the Smm function above could be approximately
replicated by building a dynamic-rebalanced portfolio combining standard linear instruments like bonds and
interest rate options such as swaptions, which justifies the arbitrage-pricing argument used to define the
EEL.

   We have developed econometric prepayment models for each key mortgage balloon term. Key terms are
1, 3, 5, and 10 years (around 80% of volume), with the term of 5 years is the dominant term (more that
50% of volume). For other irregular terms, prepayment rates are determined by interpolating the Smm
values predicted by the two models of the nearest low and high key terms. For each key balloon term, two
econometric models are calibrated for both the high LTV and low LTV categories. The time-deterministic
function λ(t) captures seasonality and seasoning factors. Beyond the refinancing incentive associated with
the mortgage rates of the same original balloon term, other refinancing factors were modeled to capture the
willing of mortgagors to switch for another balloon term. These factors permit to take into account the

                                                         13
option on the term structure of mortgage rates allowed by Canadian balloon mortgages. Empirical results
show that these refinancing factors are significant only for mortgages with the lowest and highest balloon
terms of the term structure; that is1 year and 10 years.



2..2    Differences between the EEL and the Prepayment Option

The main differences between the behavioral (econometric) model above and the rational option pricing
model illustrated earlier are two:
     i. Prepayment Rationality: Under the option pricing model, prepayments are motivated by interest rates
moves. Rationality of mortgagors is described by the standard optimal stopping rule (optimal early exercise
policy) associated with early-exercise-style options. Under the behavioral model, refinancing incentives are
captured through the option payoff max [0, K − Ri (t, t + T )]. However, this component affects prepayment
flows depending on the regression parameter β. That is, the clients rationality assumed here is inferred
from historical data and does not reflect the same rationality implied by the option pricing model. Since
rational option pricing models do not capture market frictions, one must expect that they will yield higher
mortgagor’s sensitivity to interest rates with comparison to behavioral models. As a result, the rationality
argument makes us expecting that EEL0 < C0 .

     ii. Linear Prepayment Component: The prepayment rate function involves a linear component, λ(t),
which captures seasonality and seasoning factors the rational option pricing model cannot account for. This
has a main consequence on the distribution of prepayment flows. The linear component will generate pre-
payment flows in both low and high interest rate environments, which reduces the skew of prepayments
distribution with comparison to the rational option pricing model where prepayments are exclusively in-
terest rate-driven events. Most importantly, as we will see later, these linear prepayment flows will not
systematically act against the interest of the bank. Prepayment flows received in high rates environments
will permit the bank to reinvest funds at attractive market rates and to not wait until the mortgage maturity.
This positive effect will be captured through our pricing formula of V0Pr ep via the stochastic discount factor
          ˜
function (Di (0, t))t . As a consequence, depending on the shape of the initial COF curve, and thus which
of the implied forward COF curve, the argument of linear prepayment component makes us expecting that:
EEL0 ≶ C0 .



3.     Empirical Analysis

The numerical results shown in this section are obtained using the Monte Carlo pricing model described
earlier. The mortgage analyzed consists of a 5 years fixed-rate mortgage (i.e., T = 60 months) with an
amortization horizon of 25 years (n = 300 months). As mentioned in Section 1, mortgage rates are projected
over time by adding the mortgage rate spread observed at time 0 to the simulated bank’s COF curve. In


                                                     14
                                                                       ˆ
other words, we assume that the term structure of the mortgage spread, φ(η)η=1,2,...T , observed at time 0
will be maintained the same over the time horizon [t = 1, t = T ]. The prepayment option and the EEL
values illustrated in these results are obtained for each quarter over the time window March 1998−March
2007. At the last trading day of each quarter, we calibrate the HW and BK models to the CDOR-swap/COF
curve and then store the observed term structures of both the COF spread and the mortgage rate spread
that are used to simulate both COF par yields and mortgage rates based on the generated par (spot) swap
rates. Based on the empirical data of market volatilities, the reversion speed and the volatility parameters
of the short rate models were calibrated to the joint volatility structure of caps-swaptions observed at the
end of each quarter. With regard to the EEL, two different econometric models were implemented: one for
the high LTV (loan-to-value) ratio 5Y-mortgages (HR) and the other is for the low LTV ratio 5Y-mortgages
(LR). Detailed documentation of these econometric models of mortgage prepayment is available.

      Figure 3 illustrates the obtained historical values of the prepayment option and the EEL. The most
important points are:

 1. The prepayment option value is not necessarily equal to the EEL. The average option value over the
        time interval considered is about 0.56$ (0.47$) per 100$ of notional under the HW (BK) model, while
        the EEL measures are around 0.64$ (0.53$) per 100$ of notional under the HW (BK) model. Both
        the rationality argument and the argument of linear prepayment component raised earlier may explain
        this finding.


 2.    Both the prepayment option and the EEL values are unstable over time. The current term structures of
        interest rates and volatilities explain this time-dynamics. Another important pattern we notice is that
        the prepayment option value and the EEL measure do not seem to co-move together in an important
        fashion. The rationale behind is mainly due to the linear prepayment component priced by the EEL
        model. Indeed, while the prepayment option model accounts for the non-linear prepayment risk only,
        the EEL model allows for prepayment flows under high interest rates environments. Depending on the
        shape and the level of the initial curve, this would give raise to important deviations between the two
        prepayment risk measures.


 3.    The EEL is highly volatile over time with comparison to the time-variability of the prepayment option
        value. This is mainly due to the fact that, in contrast to the prepayment option value, the EEL is
        very sensitive to the mortgage rate spread. We recall that we use the historical mortgage spread in our
        pricing procedure. Thus, in addition to be affected by the initial COF curve and market volatility, the
        EEL is affected by the highly volatile mortgage rate spread. This also gives additional explanation of
        the low co-variation intensity between the two risk measures pointed out earlier.




                                                       15
 4. The two EEL values obtained for the high LTV ratio (HR) and the low LTV ratio (LR) mortgages
        are very similar. This is due to the fact that the econometric models describing historical prepayment
        rates associated with these two mortgage categories are very similar.


 5.     The EEL may take negative values. This striking fact could be explained by the linear prepayment
        component involved in the historical prepayment model. When the initial forward curve is very steep,
        predicted future interest rates are high. This implies that prepayment flows due to the refinancing
        incentive will be low, while linear prepayment flows received at high interest rates environments will
        be very high. Combined together, these two implications result in negative EEL values. Nevertheless,
        based on our empirical sample, this occurred for very few occasions when the initial curve has exhibited
        a very pronounced and abnormal steepness.


      To get an idea on how the factors of risk mentioned above affect differently the two prepayment risk
measures, we have regressed the prepayment option and EEL values obtained for each end of quarter (those
illustrated in Figure 3) against the 5Y-mortgage rate spread and the curve slope (5 years COF − 1 month
CDOR) observed at the end of the same quarter. Table 1 summarizes these regression results. We see that
the prepayment option value is overall stable over time (significant regression intercept), slightly affected by
the initial curve slope, but insensitive to the mortgage rate spread. In contrast, the EEL is largely explained
by both the mortgage rate spread and the curve slope (the squared R of the regression is about 67%).

      Based on the industry practice, it is recommended to use the EEL as a prepayment risk measure for risk
management purposes. Indeed, while the rational option model has rigorous arbitrage pricing foundations,
the mortgage prepayment literature suggests that pure rational models may lead to overestimate the non-
linear interest rate risk (the gamma risk reflecting the hedging risk) exposure of the bank, while underestimate
prepayment flows due to exogenous factors (seasoning factors, seasonality patterns and housing market
turnover) that do not adversely affect the bank revenue in a systematic manner. Nevertheless, the option
pricing model is useful to price the full financial risk associated with mortgage products. This feature is
appealing for funds transfer pricing (FTP) considerations. It is a common practice in the industry to refer
to the rational option model as a pricing tool for FTP, although ALM risk management decisions are based
on econometric models of mortgage prepayment.

      Finally, notice that another family of models exist, which we can qualify them as market models of
mortgage prepayment. These models use rational option pricing models to infer implicit intensity of mortgages
prepayment risk through the quoted yields of mortgages-backed securities (MBS). These models have recently
met some popularity among practitioners and proven to be efficient to price and hedge MBS. Unfortunately,
because of the absence of a free and liquid Canadian secondary market of MBS like which of the U.S., we
are not allowed to implement such type of models.



                                                       16
4.     Prepayment Penalty

So far, we have ignored in our analysis the fact that Canadian banks may charge a penalty to mortgagors
upon prepayment. We adjust here our mortgage pricing models by incorporating the fact that the bank will
charge a penalty upon prepayment. To conduct our analysis, we adopt exactly the penalty formula used
by NBC. Broadly, this formula entails that the dollar amount of penalty will be given by multiplying the
average residual balance upon maturity by some penalty rate. This penalty rate is determined in function of
the original mortgage rate and the market mortgage rates observed at prepayment. Before moving further,
an important point needs to be clarified. Prepayments in real-world are not exclusively due to interest
rate-incentives (i.e., refinancing). In many cases, the bank is inclined to forego the penalty that would have
been charged. In addition, the intensity by which branch units charge this penalty depends on their latitude
to protect market shares from competition. Overall, this means that penalty will be never charged in a
systematic way. In the analysis below, we abstract from this fact. The results we generate are based on
the assumption that the bank will charge systematically the required amount of penalty to clients upon
prepayment.

     To introduce penalty into the option pricing model, we simply need to readjust the formula of the path
i’s conditional exercise value of the prepayment option at month t by the amount of penalty charged to the
mortgagor, we denote by pt,i , as follows:
                                                h                        i
                                                      Orig   N
                                     EVt,i = max 0, Vt,i − Vt,i ew − pt,i .                                 (28)

Baed on the penalty formula used by NBC and many other Canadian banks, the penalty amount is fixed as
follows:

                              pt,i                                        ¯
                                     = max [0, K − Ri (t, t + τ (t, T ))] B(t, T )                          (29)
                                                                          Bt + BT
                                     ' max [0, K − Ri (t, t + τ (t, T ))]          ,                        (30)
                                                                              2

where τ (t, T ) = T − t is the residual balloon term in months, Ri (t, t + τ (t, T )) is the market mortgage rate
observed at the prepayment date t with a term equal to the mortgage’s residual balloon term τ (t, T ), and
¯
B(t, T ) ' (Bt + BT )/2 is the average mortgage balance over the residual balloon term as computed at the
prepayment date t.

     Figure 4 illustrates the impact of introducing penalty on the prepayment option value. The figure
compares the prepayment option value where there is no penalty with which obtained after considering
penalty, as computed for each end of quarter over the time window March 1998−March 2007. As we can
see, the prepayment option value drops significantly when taking into account the charged penalty amount.
For instance, the average option value over the time interval considered has dropped from 0.56$ per 100$
of notional in the case of zero penalty to 0.22$ under the HW model after introducing penalty. This means
that as aimed, penalty attenuates the bank’s exposure to prepayment risk by reducing refinancing incentives

                                                        17
of mortgagors. But on the other hand, we notice that the prepayment option value does not collapse to zero
after introducing penalty. One may expect that facing a well-specified penalty mechanism, mortgagors will
never find optimal to refinance their loan at the lower market rates, since the realized economies of interests
will be completely offset by the amount of penalty to be paid. This, however, is not true. Mortgagors may
find optimal to refinance their loan if the realized interest economy, thanks to a significant decline of interest
rates, sufficiently exceeds the amount of penalty to be paid. This implies that there is always a minimum
level of prepayment risk, due to large movements of interest rates, the ex-post penalty mechanism cannot
offset.

   The empirical mortgage prepayment model offers more flexibility to account for the penalty effect and how
the competition considerations as well as behavioral patterns attenuate the impact of penalties. In addition,
mortgagors in real-world have the opportunity to amortize the penalty by accepting a higher mortgage rate at
refinancing than the observed mortgage rate, but not higher enough to offset any significant economies due to
refinancing. Indeed, taking in consideration this amortization option, which clients make often use according
to empirical observations, a mortgagor is able to lock-in lower interest rate and avoid to pay therefore the
penalty amount upfront. We model this by formulating the penalty as a fraction of the mortgage rate taking
into account the amortization of this penalty over the residual balloon term (after simple analysis, one can
arrive at the conclusion that the residual balloon term at refinancing corresponds to the amortization period
of the penalty). To consider the fact that banks do not charge in a systematic way the penalty to clients,
we multiply the penalty rate by a scale parameter α ∈ (0, 1). The value of this parameter is determined
endogenously from empirical observations, such that the average Smm in function of the moneyness rate, as
predicted from the empirical prepayment model, reproduces well the switching point of options payoffs around
the ‘at-the-money’ region after varying the moneyness rate (for more details, a detailed documentation of the
econometric models of mortgage prepayment rates is available). Thus, according to the described empirical
prepayment model, the Smm function used in our Monte Carlo simulations is transformed into:

                          Smmi (t) = λ(t) + β max [0, K − Ri (t, t + T ) − pt,i ] ,
                                                                              ˆ                           (31)
                                                                         τ (t, T )
                              ˆ
                             pt,i = α max [0, K − Ri (t, t + τ (t, T ))]           .                      (32)
                                                                            T

To price the risky mortgage, we need to discount back the risky cash flows function of the prepayment rate
function described above. As described in Section 2, cash flows include principal and interest components
as well as a principal prepayment flow. Some actuarial calculus of mortgage amortization with prepayment
hypothesis is used to projected these flows dynamically over time. However, when introducing penalty, we
also need to account for charged penalties in our EEL-pricing model. Indeed, penalty flows will be added to
cash flows. As penalty reduces the exercise value of the prepayment option under the rational option model
thus leading to higher risky mortgage value (lower option value), the penalty flows added to cash flows
under the EEL model will contribute to increase the risky mortgage value. The incremental value added to
the risky mortgage market value could be viewed as the market value of a contingent Arrow-Debreu asset


                                                      18
representing the present value of penalty flows.




IV.      Mortgage Rate commitment (Lock-in) Options: Valuation &
         Hedging
A mortgage rate commitment (MRC), also known as a mortgage rate lock-in, is a commitment to a mortgage
rate fixed in advance for a mortgage loan to be originated in the future offered by the bank to a client.
While the bank has the obligation (moral but not legal obligation) to respect her commitment, clients are
not forced to take the offer by engaging themselves in a mortgage loan transaction. A non negligible fraction
of MRC is not followed by mortgage origination trades. Nevertheless, MRC positions banks bear are so
important that the economic risk behind requires an accurate and rigorous evaluation.
   MRCs could be viewed as a forward mortgage contract with an embedded option offered to clients to
leave the contract. This optionality to exercise or not the MRC by clients and entering into a mortgage
loan transaction is the focus of our subsequent analysis. We abstract from exogenous factors making clients
leaving the offer even though it is rational to take it.
   MRCs are characterized by three dates. The inception date, T0 , the maturity date, Tc , which corresponds
to the planned origination date of the mortgage and Tm is the maturity date of the mortgage so that the
chosen balloon term of the mortgage is given by

                                                  τ = Tm − Tc .

The MRC takes end at time Tc , and if exercised, a new mortgage is issued at the offered rate with a balloon
term of τ beginning form the date Tc . It is always the case to see clients constrained by the banker to choose
their balloon term before receiving a MRC for that term. For our subsequent analysis, this means that the
two dates, Tc and Tm , are fixed in time, so that no floating tenor similar to that occurring with Bermudan
swaptions is allowed here.

   MRCs contracts are in fact a particular form of forward options in the sense that the client -the holder of
the option- seeks to lock himself into a mortgage position at the best mortgage rate as possible through the
MRC, but without being forced to take the MRC offer. If the rate offered by the MRC is better (lower) than
the live mortgage rate observed over the market at the mortgage origination date Tc (which is by definition
the maturity date of the MRC), the client will take the offer and the contractual mortgage rate will be set
equal to the offered rate of the MRC. Elsewhere, the client will leave the MRC offer rate and contracts his
mortgage loan at the lowest mortgage rate observed live over the market in that case.




                                                       19
1.     The MRC Option Payoff and the Floating Strike Provision

Let V (t, K; (τ , x(.))) denotes the market value of a mortgage newly issued at time t with a contractual
interest rate K. The mortgage has a balloon term of τ months, and exhibits a prepayment risk intensity
described by a general function x(.). This prepayment rate function could be either equal to zero meaning
no prepayment risk (plain-vanilla mortgage), or either could take strictly positive values over the mortgage
balloon horizon. In the most complex case, yet the most realistic, the function x(.) could be given by a
stochastic Smm function capturing the optional (non-linear) prepayment risk.
     Therefore, we can express the maturity payoff of the MRC option as follows:

                 EVTc = max [0, V (Tc , R(Tc , Tc + τ ); (τ , x(.))) − V (Tc , KMRC (T0 , Tc ); (τ , x(.)))] ,   (33)

with R(Tc , Tc + τ ) is the mortgage rate observed over the market at time Tc for a balloon term of τ , while
KMRC (T0 , Tc ) denotes the mortgage rate as offered by the MRC issued at T0 and ending at Tc .
     We notice that the MRC option payoff is very similar to the payoff returned by swaptions. From the
perspective of the client (the originator bank), the option exercise value represents the time Tc -present value
of the gain (loss) taking the form of economies (opportunity cost) of future contractual interests occurring
when the MRC’s offer rate is below the market’s spot mortgage rate. We discuss later how this similarity
with swaptions could be exploited to hedge MRC options.

     MRCs could either offer the client a fixed offer rate that does not change until the maturity date Tc or
either offer him a first offer rate at the inception date as well as the possibility to recontact the bank in
order to lower the committed rate to the level of observed mortgage rates if these rates have subsequently
declined below the last offered rate. The first type is qualified as a flat-strike-MRC, while the second is called
a floating-strike-MRC. In practice, most commonly observed MRCs include a floating-strike provision.
         ¯                                                 ˆ
     Let K be the fixed offer rate of the flat-strike-MRC and K(T0 , t) be the floating strike observed at time
t ≤ Tc verifying,

                               ˆ
                               K(T0 , t) = min (R(s, s + τ ))T0 ≤s≤t ,            T0 < t ≤ Tc ,                  (34)
                             ˆ
                             K(T0 , T0 ) = R(T0 , T0 + τ ).                                                      (35)

We are implicitly assuming here that clients are rationale in the sense they will re-strike their MRC at
the observed mortgage rate whenever there is a gain in doing so. We discuss later the behavioral patterns
observed from empirical data and how can we adjust our rational option pricing model to take into account
these findings.



2.     The Pricing Model

To price MRC options, one faces two obstacles making valuation not possible to perform using the simple
tree method as we can do for standard swaptions. First, the prepayment option embedded in the underlying

                                                              20
mortgage makes the valuation of the underlying mortgage not allowed by a tree if (and only if) we formulate
the prepayment option risk by the mean of a monthly prepayment rate function Smm(.). Indeed, as we
have pointed out earlier when discussing the pricing of prepayment options, the Smm function makes by
construction the cash flows generated by the mortgage path-dependent. However, a rational option pricing
model could be easily solved using the HW/BK trinomial trees. Because we are implementing here a
statistical prepayment rate function to evaluate the risk of MRC positions of the bank, we have to resort to
the Monte Carlo method. Second, the floating-strike provision makes the MRC option payoff path-dependent
in the same manner than well-known exotic options like Asian options.
   To solve the valuation problem of MRC options in an elegant way, we employ again here the Monte
Carlo-LSM method. The need of making use of the LSM method is justified by the fact the option payoff
consists of a gap between two possible time Tc -market values of the underlying mortgage. To compute these
path-conditional market values, the only efficient solution that handles well both the path-dependency due
to the floating-strike provision and projected prepayment risk is the LSM method.

   The application of the LSM here is different and simpler than in the case early-exercise-style options.
The reason is that we do not need to perform any iterative decision rule. Only what we need is to get an
LSM estimation of the time Tc -mortgage values based on the cross-sectional information contained across the
generated Monte Carlo paths at the stopping-time Tc . To do this, we apply the same regression technique,
used earlier to infer the continuation value of early-exercise-style options, to estimate the mortgage forward
values V (Tc , Z(Tc ); (τ , x(.))) for Z(Tc ) = R(Tc , Tc + τ ) and KMRC (T0 , Tc ). The difference is that we are now
allowed to use the whole sample of generated Monte Carlo paths rather than limiting regression over the
‘in-the-money’ region as we have done with the mortgage prepayment option.

   The LSM estimation of the forward market value of the mortgage, Vi (Tc , Zi (Tc ); (τ , x(.))), for any path
i is formally as follows:

                  Vic (Tc , Zi (Tc ); (τ , x(.))) → OLS [cons tan t, Xt,i ; β OLS ] ,   i = 1, ...N,            (36)

                   Vi (Tc , Zi (Tc ); (τ , x(.)))   :     ˆ
                                                        = β OLS × Xt,i .                                        (37)

with the path i-conditional value Vic is determined by discounting back cash flows generated by the mortgage
over the balloon time horizon [Tc , Tm ]. These cash flows are projected based on the usual Smm-actuarial
calculus formula of mortgage amortization in the presence of prepayment flows. It is the same formula we
have used to estimate the EEL measure of prepayment risk, and also the same we are using to determine
the risk exposure of the mortgage portfolio of the bank.

   The basis functions {Xt,i } are chosen to provide the best accuracy of the LSM price estimators. The
numerical tests conducted lead us to conclude that a good specification of the LSM regressors is:
1) A quadratic function of the slope of the swap curve captured by the spread between the 1 month CDOR
rate (i..e, the short rate) and the 3 years swap rate, [Si (Tc , Tc + 36) − ri (Tc )], similarly to the case of the

                                                             21
prepayment option;
2) The path-dependent contractual mortgage rate Zi (Tc ) = Ri (Tc , Tc + τ ) or KMRC,i (T0 , Tc ) depending on
                                                                                    ¯
the regression we are placed in. Note that for flat-strike-MRCs, the constant strike K is skipped from the
regression;
3) The path i-average value, Avg_Smmi (Tc , Tm ), of the future prepayment rates projected over the mort-
gage balloon horizon [Tc , Tm ].
     As in the case of the prepayment option, the vector of discount bonds, (Pi (Tc , Tc + η))η=6,12,...60 , repre-
senting the generated time Tc -term structure was proven inefficient choice as a basis function for the same
reasons mentioned earlier.

                                                         ˆ
     The implementation of the floating-strike component, K(T0 , t), is easy via our LSM-Monte Carlo model.
We simply apply along each path the min(.) operator during the MRC’s time-to-maturity interval. In our
discrete Monte Carlo setup, the frequency by which we re-strike the MRC option is chosen equal to 1 week,
which slightly exceeds the time increment used in simulations. We will examine further the sensitivity of the
MRC option to the re-striking frequency.

     In contrast to the mortgage prepayment option, no early-exercise decision rule is involved in the exercise
of the MRC option. Hence, to price the MRC option, we do not longer need to discount back cash flows
using the adjustment factor e−φ(t) for the mortgage rate spread. The MRC value, as well as the involved
forward values of the underlying mortgages, is obtained by using the COF curve as a stochastic discount
factor to reflect the actual risk exposure of the bank.



3.     The Determinants of the MRC Option Value & Delta Risk

We examine here the sensitivity of the MRC option to some key parameters or factors of risk:
     1). The option strike provision: floating-strike vs. flat-strike;
     2). The projected prepayment risk of the underlying forward mortgages.

     Our numerical results we discuss below are based on the assumption of an underlying forward mortgage
with a balloon term of 5 years (i.e., τ = 60 months) and an amortization horizon of 25 years (n = 300
months). The HW and BK interest rate models used in simulations are calibrated to the CDOR-swap/COF
curve and the volatility structure of caps-swaptions of 31 March 2007. We vary the time-to-maturity (Tc −T0 )
of the MRC option over the range of 2, 3, 4, 5, 6 months. Without loss of generality, we consider a newly
issued MRC, which allows us to make the assumption in our numerical investigations that the strike of the
                            ¯   ˆ
flat-strike-MRC is given by: K = K(T0 , T0 ). It is important to note that in practice, the current value of
MRC options is determined based on the actual strike observed from the frequently updated data of the
MRCs position.



                                                        22
3..1   Flat-Strike vs. Floating-Strike

We define the moneyness rate of the MRC option at its maturity as follows:
                                         ⎧
                       R(Tc , Tc + τ )   ⎨ R(T , T + τ )/K(T , t) for a floating-strike-MRC,
                                                         ˆ 0
                                              c   c
        M oneyness :=                  =                                                                  (38)
                      KMRC (T0 , Tc ) ⎩      R(T , T + τ )/K
                                                      c    c
                                                            ¯         for a flat-strike-MRC.

When this ratio is above 1, the option could be viewed as ‘in-the-money’. However, this is not always true
because of the involved prepayment function x(.), that may adversely influence the option payoff when set
in function of the contractual mortgage rate. We shall discuss in details this relatively complex point just
below. .Abstracting form prepayment risk, we want to evaluate the impact of allowing for a floating-strike
provision: how much the MRC option is sensitive to that provision?

   Figure 5 plots the distribution of the moneyness rate as inferred from Monte Carlo simulations for MRC
options with tow different maturities of 2 and 6 months, respectively. We compare the floating-strike option
to the flat-strike benchmark case. Of course, we observe that the main consequence of the floating-strike
feature is that moneyness rate is always equal or more than 1, implying that the option will be never ‘out-
of-money’. The main premium we have to pay for this strike provision represents the eliminated downside
risk. Further, as expected from the volatility effect, we see that the floating-strike provision significantly
impacts MRC options with longer maturities. The impact on the moneyness of short lived options, however,
is quasi-negligible. Based on the case of the long maturity of 6 months, we also notice that the introduction
of the floating-strike feature implies a more skewed distribution of the moneyness rate, meaning that there
is higher chances the option will be exercised, but most of these opportunities are at lower realized payoffs.
In contrast, the moneyness of flat-strike-MRC exhibits as expected a perfect normal distribution.

   Table 2 reports the statistical parameters describing the moneyness distribution in function of the strike
provision and the option maturity. We see that the skewness of this distribution is around zero (that
is a normal distribution) for flat-strike-MRC options independently from their maturity. In contrast, the
skewness of the moneyness rate is significantly different from zero for floating-strike-MRC options and is
more pronounced with longer maturity intervals. Another expected fact to report is that the floating strike
effect seems more pronounced under the Gaussian HW model rather than the lognormal BK model. Indeed,
simulated lognormal interest rates tend to be higher than the projected Gaussian rates, which as a result
lowers (increases) the likelihood of observing high moneyness scenarios under the BK (HW) model. The
same table shows also that the frequency by which clients re-strike their MRC influences the moneyness of
their floating-strike option at maturity. The lower the re-striking frequency, the smaller is the floating-strike
effect captured through the skew of the moneyness distribution.

   Panel A of Table 3 compares the market value and the delta (DV01) of the MRC option for the two
strike provisions. To focus on the strike effect, results are obtained by ignoring prepayment risk (that is,
we assumed that x(.) = 0). The pricing model was adjusted to this assumption by skipping the projected

                                                      23
average Smm (Avg_Smmi (Tc , Tm )) from the LSM regression. We see that the floating-strike-MRCs worth
more than the flat-strike-MRC. This value gap is quite expected and it reflects the premium one has to pay
for the floating-strike feature eliminating the downside option risk (i.e., ‘out-of-money’ risk). In contrast,
the floating-strike-option exhibits a lower delta risk (DV01) than the flat-strike-option. This is also should
be expected, since the floating-strike provision allows the holder to increase the likelihood of exercising the
MRC option at maturity, and thus lower the option sensitivity to shifts of the initial interest rates curve.
Moreover, Panel B of Table 3 reports the same pricing results of the floating-strike-option for re-striking
frequencies of 2 and 4 weeks that are lower than the base case frequency of 1 week. We observe that in
conformity with the impact of the re-striking frequency on the moneyness distribution discusses earlier, the
decrease of the re-striking frequency lowers the floating-strike-option value and increases its delta risk, thus
making it more similar to the flat-strike-option.



3..2   Prepayment Risk

Certainly, taking into account the prepayment risk involved behind the underlying forward mortgages is the
most significant factor of risk that affects the MRC option values and its delta-sensitivity to interest rates.
Panel C of Table 3 reports the market values and deltas of the MRC options, for both the two strike
provisions (i.e., flat-strike and floating-strike), in the presence of projected prepayment flows. The first case
consists of a constant prepayment rate while the second case, more realistic, allows for stochastic prepayment
flows capturing the non-linear or optional prepayment risk. The stochastic prepayment flows are projected
based on the same econometric Smm function used in the EEL measure of prepayment risk. In the constant
prepayment rate case, the flat Smm used in simulations is chosen appropriately to reflect the Monte Carlo
paths-average of the prepayment rate Avg_Smmi (Tc , Tm ) projected from the Smm econometric model.
Because the prepayment rate is held constant in that case, the pricing model was adjusted by skipping the
path-conditional projected Smm, Avg_Smmi (Tc , Tm ), from the LSM regression.

   By comparing pricing results reported in Panel A of Table 3, where prepayment risk is ignored, with
those of Panel C of Table 3, where prepayment flows are considered, we see very significant deviations, that
reflect the impact of the underlying mortgages’ prepayment risk. Indeed, incorporating prepayment risk into
the MRC option pricing model has two main implications:

i. The Mortgage’s Average-Life Effect of Prepayment Risk (Underlying Tenor Effect):
   Prepayment flows reduces the average life of the underlying mortgages. Because the MRC option dy-
namics are tightly dependent on the term of the underlying mortgages in the same way swaptions are very
sensitive to the tenor of the underlying swap, the allowance for prepayment flows impacts significantly the
MRC option. This underlying mortgage’s average life effect is captured through the stochastic prepayment
rate model as well as the simple constant prepayment rate model. Comparing pricing results reported in
Panel A with those of Panel C-1, shows that the introduction of a constant prepayment rate lowers the MRC

                                                      24
option value and its delta; and this for both the flat-strike and floating-strike options. The case of a constant
prepayment rate illustrates well the pure underlying tenor effect. We conclude that prepayment makes the
underlying mortgages’ average life shorter, which as a consequence considerably decreases the value of the
MRC option and its delta risk with comparison to the benchmark case of no prepayment risk.

ii. The Compounded Option Effect of Prepayment Risk:
   The second implication of incorporating prepayment risk, however, is captured through the stochastic
(optional) prepayment rate model only. Remember that the stochastic prepayment rate model allows us to
capture the non-linear or optional prepayment risk reflecting the prepayment option. Therefore, projecting
prepayment flows based on this model makes the forward mortgages, representing the underlings of the MRC
option, incorporating by themselves an implicit option position. As a consequence, the MRC option could
be viewed in that case as a compounded option with each of the underling forward mortgage is a portfolio
composed of a long position into a plain-vanilla mortgage and a short position into the prepayment option.
The compounded option effect influences the MRC option dynamics in two ways.
   First, through the volatility of interest rates. The implicit prepayment option increases the sensitivity
of the MRC option to interest rates. Second, through the contractual mortgage rate as fixed at the MRC’s
maturity date. Independently form the strike provision of the MRC option (flat strike or floating-strike),
the implicit prepayment option makes the MRC option more sensitive to the contractual rate Zi (Tc ) =
Ri (Tc , Tc + τ ) or KMRC,i (T0 , Tc ) of the two underlying forward mortgages Vic (Tc , Zi (Tc ); (τ , x(.))) based on
which the MRC option payoff is determined. This underlying strike effect (not to confuse with the strike
provision effect discussed earlier) is crucial for hedging MRC options. In fact, when the MRC option is
deeply in-the-money, the contractual rate KMRC,i (T0 , Tc ) is very low and thus projected prepayment flows
are low, meaning a low compounded option risk. In contrast, when the MRC option is ‘out-of-money’, the
contractual rate Ri (Tc , Tc + τ ) could be either high or low depending on the strike provision of the MRC and
the level of interest rates at the MRC’s inception date. In such a case, no precise prediction could be made
about the intensity of the compounded option risk. In this sense, when the MRC option is exercised, there
is some trade-off to take into consideration between the moneyness of the MRC option and the prepayment
risk involved in the underlying mortgage.
   The compounded option risk due to stochastic prepayment flows could be captured by comparing pricing
results reported in Panel C-1 of Table 3 related to the constant prepayment case with those of Panel C-2
of Table 3 generated based on the stochastic prepayment model. We see that the allowance for stochastic
prepayment risk lowers the value of the MRC option value, but increases in general its delta risk (DV01).
The decrease of the MRC option value is essentially attributed to the strike effect. Indeed, introducing the
refinancing incentive effect through the stochastic prepayment model reduces the gap between the underlying
forward mortgages V (KMRC (T0 , Tc )) and V (R(Tc , Tc + τ )) representing the MRC option payoff. Further,
the increasing DV01 of the MRC option should make sense, since the underlying mortgages becomes more
sensitive to interest rates once prepayment flows are dependent on the interest rates paths.


                                                          25
     Interestingly, the strike provision of the MRC (flat-strike vs. floating-strike) does not influence the
intensity by which the compounded option risk described above affects the MRC option. We see that the
relative impact of the compounded option risk due to the introduction of stochastic prepayment risk (this
relative impact is quantified by expressing the option value and delta of Panel C-2 in percentage of those of
Panel C-1) is identically the same for both the floating-strike-MRC and the flat-strike-MRC.

iii. The Impact of Prepayment Risk on the MRC Option’s Payoff and Exercise Probability:
     Figure 6 plots the Monte Carlo distributions of the exercise value of a MRC option at its maturity date,
as inferred 6 months before. Only strictly positive payoffs are considered (we excluded the zero payoffs of
‘out-of-money’ scenarios). Both the flat-strike-MRC and the floating-strike-MRC are considered. As we see,
the allowance for projected prepayment risk behind the underlying forward mortgages significantly affects
the distribution of the MRC option’s payoff. When no prepayment risk is assumed, this distribution is
slightly humped, with extreme high payoffs are assigned a relatively high probability. When we introduce a
constant prepayment rate, we observed that this distribution becomes more humped, exhibiting an increased
skew meaning that low (high) payoffs have higher (lower) probabilities of occurrence with comparison to the
base case of no prepayment risk. Interestingly, once we introduce the stochastic prepayment rate model, the
distribution becomes extremely skewed, thus assigning very high (low) likelihood for very low (high) payoffs.
     This prepayment risk effect is already contained in the market value of the MRC options shown in
Table3. It means that the allowance for prepayment risk makes the MRC option less worthy. This could
be understood after recognizing that prepayments lower the expected present value of interest income to be
earned over the mortgage. Indeed, according to the compounded option effect described above, the chance of
observing high losses of future contractual interests over the mortgage upon the exercise of the MRC option
becomes very low. This is because by allowing for rate-sensitive prepayment flows (stochastic prepayments),
the average time period the bank will suffer this loss (i.e., the mortgage’s average-life) becomes shorter.
     From the perspective of ALM, recognizing stochastic prepayment risk will make the expected economic
loss the bank bears in the form of opportunity cost of future contractual interests low. But at the same time,
the delta risk of the MRC option, and thus its hedging risk, becomes higher.



4.     Hedging MRC Options

By letting the mortgage loan pays interests at the same frequency of swaps and ignoring prepayment risk, we
end exactly by the same swaption payoff after reducing mortgage rates to swap rates. This means that the
basic source of risk the bank bears when committing to forward mortgage rates is the same par value/market
value ratio risk a trader takes when shorting a swaption. From the ALM perspective, this means that MRC
options should be hedged to prevent the bank form an economic loss in the form of a shadow cost of future
interest income.

     As shown earlier, including stochastic prepayment risk and floating-strike feature has a considerable

                                                      26
impact on the MRC option risk. While the floating-strike provision affects the moneyness of the MRC
option and thus the intensity at which the bank bears the potential economic loss of future interest income,
the underlying tenor risk and the compounded option risk both involved in the projected prepayment flows of
the underlying forward mortgages influence the magnitude of this economic loss. Additionally, floating-strike
provision and prepayment risk impacts with a significant manner the MRC option delta risk, and thus are
determinant for the hedging effectiveness and hedging risk of MRC options.

   The pricing of MRC options we perform in practice is based on the following assumptions. First, historical
observations show that the re-striking frequency increases as long as the MRC option approaches the maturity
date. By seeking to reflect the historical average values, we have fixed this frequency at 1 week as long as
the time-to-maturity of MRC options is less than 3 months. This frequency is set at 4 weeks for MRC
options with a time-to-maturity between 3 and 6 months. A flat-strike equal to the commitment rate at
inception date is used as long as the time-to-maturity of the MRC option is more than 6 months. Second,
strike levels at the inception date of MRCs as well as current strike levels used to price MRC options are the
actual strikes observed from the MRCs position data. Finally, the same mortgages prepayment risk models
described earlier in the present document and serving to evaluate the mortgages portfolio risk position are
used to price and hedge MRC options.

   The hedging strategy we propose consists of solving for a hedge portfolio composed of market traded
payer swaptions with underlying tenors ranging from 1 year up to the balloon term of the forward mortgage
underlying the MRC option. For example, for a MRC on a mortgage with a balloon term of 5 years, the
hedge portfolio we aim to solve for it will be composed of market traded payer swaptions with underlying
tenors ranging from 1 to 5 years.
   The idea consists of constructing a hedge portfolio by trading the payer swaptions mimicking the risk
profile of the MRC options position. Of course, the hedge portfolio must be rebalanced over time to reflect
the actual and time-changing risk of the MRCs position. Before moving forward, it is worthwhile to note
that we have examined alternative hedge strategies using receiver swaptions or combining both payer and
receiver swaptions. The results we have obtained confirmed to us that building the hedge portfolio based on
payer swaptions only (i.e., without incorporating receiver swaptions) delivers the best hedge performances.
Of course, the choice of swaptions rather than caps or other Libor derivatives should make sense, since their
payoff is linked to the swap rates as MRC options.

   The optimal hedge strategy we propose here is based on the least-squares hedge or quadratic hedge, widely
used in hedging expected credit losses of credit portfolios. The method consists of solving for the optimal
hedge, the payer swaptions portfolio here, which promised cash flows fit well in the least-squares sense those
generated by the MRC option. Let P Sj (rT0 , T0 , Tc , Kj , τ j ) denotes the payer swaption with a tenor τ j and
a strike Kj = S(T0 , T0 + τ j ), as determined at time T0 . As one can notice, we only deal with swaptions that



                                                       27
are ‘at-the-money’ at time T0 and we impose here to the swaptions to exhibit the same maturity date Tc
than the MRC option we aim to hedge.
   Therefore, the hedge portfolio of payer swaptions resolves the following least-squares optimization prob-
lem,                                        ⎡Ã                                   !2 ⎤
                                                         P       PS
                            min          E0 ⎣ EVTc
                                                MRC
                                                    − µ − λj × EVTc j (Kj , τ j ) ⎦ ,
                       (µ,λj )j=1,...J                        j

        MRC                                                                           PS
where EVTc  is the MRC option’s exercise value given by equation (33) and EVTc j (Kj , τ j ) denotes the
swaption j’s payoff. The parameters µ and (λj )j are the least-squares model coefficients, which could be
viewed as OLS regressors.

   The quadratic hedge model above is interpreted as follows. The intercept µ of the quadratic equation
represents the average component of the total payoff generated by the MRC option in excess of the average
payoff of the payer swaptions’ hedge portfolio. The quadratic model means that we cannot mimic this excess
payoff by the mean of the payer swaptions. The higher this intercept, the lower is the quality of the protection
provided by the considered swaption. The coefficients (λj )j of the quadratic equation regrouped together
                           P
gives us the hedge leverage j λj . That is the notional we have to expend in buying the payer swaptions
considered per 1 dollar of notional of the MRC option position. Keeping the market price of the considered
payer swaptions unchanged, the higher this hedge-leverage, the more expensive is the hedge strategy. Indeed,
each coefficient λj yields the optimal hedge’s load on the payer swaption P Sj (Kj , τ j ).

   The optimal hedge portfolio is the swaptions mix (λj )j that provides the best least-squares fit; that is the
swaptions mix that gives the lowest level of the MRC’s unhedged payoff component, µ. From a statistical
point of view, this selection method is equivalent to solve for the highest squared-R for a zero-intercept OLS
regression. Of course, one can set the quadratic hedge model such that the intercept is exogenously fixed at
                                                           P
zero and focus on the best fit measured by the hedge ratio, j λj . But this zero-intercept model is useless
to appreciate and evaluate the unhedged MRC option risk. In this regard, we must expect that under our
quadratic hedge model, the hedge quality will be higher for flat-strike MRCs than for floating-strike MRCs.
The reason is that the floating-strike feature adds value to the MRC option in the form of skewed moneyness
distribution a standard payer swaption will never be able to replicate in full.

   Of course, both the MRC option and swaptions exhibits truncated payoffs. This should not affect the
quadratic hedge model significantly, since a high-quality hedge is expected to mimic the same truncated
payoff function of the MRC option, which leads to squared-errors close to zero over the ‘out-of-money’
region. In the opposite case, the lower the quality of the hedge, the larger is the gap between the two payoffs
functions, and thus the higher will be the squared-errors. Both these two extreme cases as well as moderated
cases will be captured and evaluated appropriately by the least-squares model.
   Because of the embedded prepayment risk and floating-strike feature, the least-squares hedge model could
not solved analytically. The hedge model described above is therefore solved using Monte Carlo simulations.

                                                         28
For each considered swaption, we estimate the expected squared-errors after generating a Monte Carlo sample
of both the MRC option payoff and the swaption payoff.

   Table 4 reports the optimal hedge portfolio for two MRC options, with maturities of 2 and 6 months, in
function of the initial moneyness spread [R(T0 , T0 + τ ) − KMRC (T0 , T0 )]. Both flat-strike-MRC and floating-
strike-MRC options are considered. The numerical results are based on the same assumption of an underlying
forward mortgage with a balloon term of 5 years (i.e., τ = 60 months) and an amortization horizon of 25
years (n = 300 months). The optimal hedge portfolio is summarized by two indicators: i) the inefficient
component given by the unhedged component µ normalized to 1 dollar of notional of MRC option position,
                          P
and ii) the hedge leverage j λj . We also report the market value of the hedge portfolio representing the
implementation cost of the optimal hedge strategy.

   In order to access the effectiveness of our hedging model, we have regressed the MRC option payoff on the
payoff generated by the optimal hedge portfolio. We generate a Monte Carlo sample of the payoffs of both
the MRC option and the swaptions at the maturity date Tc . The Monte Carlo sample is different from which
used to solve the quadratic hedge model above (that is which used to setup the hedge portfolio). Then, we
regress the MRC option’s payoff on the optimal hedge’s payoff. The squared-R of the regression is a good
indicator of the effectiveness of the quadratic hedge model. Table 4 also reports the results obtained for
the hedge effectiveness analysis; that is the squared-R and the standard-error of the regression

   The most important result to notice is that the extent to which the MRC option is in-the-money or
out-of-money initially at the implementation of the hedge strategy affects significantly the optimal hedge
portfolio and its future performances.
   We observe that the unhedged component, µ, is higher with MRC options that are in-the-money. This
means that the optimal hedge portfolio is less able to replicate the future MRC option payoff when the
moneyness rate of this option is initially high. This must make sense, since the initial moneyness of the MRC
option or its immediate exercise value is very close to its maturity date’s payoff. We need an exceptionally
high volatility of interest rates to offset this initial moneyness effect, which is not the case most of the time.
Rather, adding the floating-strike provision will increase this adverse impact of initial moneyness on the
quality of the optimal hedge. This is a common issue we have to face when hedging any option security.
   Interestingly, the higher (lower) is the hedge quality, the lower (higher) is the unhedged component as
predicted by the quadratic model and the lower (higher) will be the hedge cost. This means that when the
MRC option position is less (more) worthy due to low (high) initial moneyness rate, the success of the hedge
portfolio as predicted ex-ante by the hedge model is high (low), thus implying low (high) hedge investments
to reduce the position risk.

   The statistical results obtained form the hedge effectiveness regression also confirms this moneyness effect,
already signaled ex-ante by the optimal hedge model through the intercept µ. We see that squared-R of the


                                                      29
regression, indicator of the ex-post success of the hedge portfolio, decreases with the initial moneyness rate
of the MRC option, while the hedge’s standard-error increases with this initial moneyness.

   With respect to the MRC maturity, both the ex-ante model quality measure µ and the ex-post squared-R
indicator shows that the hedge portfolio performs better (less) for long (short) maturity MRC options. This
is intuitive, since the time-to-maturity effect tends to dominate the volatility effect of options whenever we
approach maturity date, thus making the initial moneyness effect described above stronger for short-lived
options.
   Interestingly, the strike provision of the MRC option seems to have very small impact on the hedge
performances. This is true for both ex-ante and ex-post measures of hedge quality. Caution must be made
here, however. The floating-strike feature cannot be captured in full by the mean-variance measures of
hedge performances reported here. We need third-order measures capturing the skewness of hedge errors to
illustrate this impact.

   Figure 7 plots the scatter formed by the ex-post Monte Carlo scenarios of the hedge portfolio’s payoffs,
as plotted against the MRC option’s payoff. The MRC’s payoff scenarios were ranked from the highest to
the lowest outcome to illustrate well the dispersion of the hedge portfolio’s payoffs around it. The Monte
Carlo scenarios are those generated for the hedge effectiveness’s regression. We illustrate this scatter for 6
months-floating-strike options at two different initial moneyness spreads.
   As we observed from Figure 7, the hedge portfolio’s payoffs are dispersed below the stream line repre-
senting the ranked payoff of the in-the-money MRC option. We see that for this option, there is a distance
between the scatter formed from the hedge portfolio’s payoffs and the MRC payoff line. This distance
captures the unhedged component µ signaled initially by the quadratic hedge model. However, for the
out-of-money MRC option, the scatter is well dispersed around the MRC payoff line. These two different
outcomes perfectly illustrate the ex-post effect of the initial moneyness of the MRC option.
   Interestingly, Figure 7 shows us how the hedge portfolio formed from payer swaptions fails to generate
the extreme high realizations of the MRC option payoff. As we can see, this failure is more pronounced
with MRC options with very high initial moneyness rate. This shortcoming is quite expected and must
be mitigated here because these extreme payoffs have relatively low probability of occurrence as illustrated
by Figure 6. In addition, the extent to which the hedge portfolio fails to replicate these extreme payoffs
tightly depends on the initial moneyness of the MRC option. This means that whenever we start hedging
MRC options earlier from their inception date, we minimize the initial moneyness effect and thus effectively
improve the success of the hedge strategy.




                                                     30
Cited References
Black, F and P. Karasinski, “Bond and Option Pricing when Short Rates are Lognormal”, Financial
    Analysts Journal, Vol. 47, 1991, 52−59.

Brace, A, D. Gatareck and M. Musiela, “The Market Model of Interest Rate Dynamics”, Mathematical
    Finance, Vol. 7, 1997, 127−155.

Hull, J. and A. White, “Pricing Interest Rate Derivative Securities”, Review of Financial Studies, Vol. 3,
    1990, 573−592.

Longstaff, F. and E. S. Schwartz, “Valuing American Options by Simulations: A Simple Least-Squares
    Approach”, Review of Financial Studies, Vol. 14, 2001, pp. 113−147.

Mansukhani, S. and V. Srinivasan, “GNMA ARM Prepayment Model”, The Handbook of Mortgage-Backed
    Securities, Edited by Frank Fabozzi, McGrw Hill 2001.

Richard, S. and R. Roll, “Prepayments on Fixed-Rate Mortgage-Backed Securities”, Journal of Portfolio
    Management, Vol 15, 1989, pp 73−82.

Schwartz, E. S. and W. N. Torous, “Prepayment and the Valuation of Mortgage-Backed Securities”, Journal
    of Finance, Vol. 44, 1989, pp. 375−392.

Stanton, R., “Rational Prepayment and the Valuation of Mortgage-Backed Securities”, Review of Financial
    Studies, Vol. 8, No. 3 (Autumn, 1995), pp. 677−708.




                                                   31
             Appendix A:


         Numerical Results for
The Mortgage Prepayment Option Analysis




                  32
                                              Figures 1–4 & Table 1

                          Fig. 1: Monte Carlo (smoothed) Distribution of the Par Ratio of the Original
                                    Mortgage at the Middle of the Prepayment Option Life
                                                     (at t = 30 months)
16%
                                                                                                         HW
14%

12%
                                                                                                         BK
10%

 8%

 6%

 4%
 2%

 0%
   93%                    94%     95%   96%   97%   98%    99% 100% 101% 102% 103% 104% 105% 106%

                                                            Par Ratio




                          Fig. 2: Monte Carlo Paths-Average of the Exercise Value of the Prepayment
                                        Option over the Balloon Mortgage's Life Horizon

                           0.90
                           0.80                                                                      HW
Average Option Payoff




                           0.70
(in $/100$ of Notional)




                           0.60                                                                      BK
                           0.50
                           0.40
                           0.30
                           0.20
                           0.10
                           0.00
                                  0            12             24             36             48                60

                                                           Time Horizon (months)




                                                              33
                  Fig. 3-A: Historical Values of Prepayment Risk Measures
                         (in $/100$ of Notional) under the HW Model



2.00

                                                                            Prep. Option
1.60


1.20
                                                                            EEL (High
                                                                            LTV)
0.80


0.40                                                                        EEL (Low
                                                                            LTV)

0.00
   Mar-98 Mar-99 Mar-00 Mar-01 Mar-02 Mar-03 Mar-04 Mar-05 Mar-06 Mar-07




                  Fig. 3-B: Historical Values of Prepayment Risk Measures
                          (in $/100$ of Notional) under the BK Model


1.60


                                                                            Prep. Option
1.20



0.80                                                                        EEL (High
                                                                            LTV)


0.40
                                                                            EEL (Low
                                                                            LTV)

0.00
   Mar-98 Mar-99 Mar-00 Mar-01 Mar-02 Mar-03 Mar-04 Mar-05 Mar-06 Mar-07




                                            34
                                           Fig. 4-A: Impact of Penalty on the Prepayment Option Value
                                                               under the HW Model

                             2.00
   (in $/100$ of Notional)



                             1.60


                             1.20
                                                                                                          Zero penalty

                             0.80
   Option Value




                                                                                                          With penalty
                             0.40


                             0.00
                                    Mar-   Mar-   Mar-   Mar-   Mar-   Mar-   Mar-   Mar-   Mar-   Mar-
                                     98     99     00     01     02     03     04     05     06     07




                                           Fig. 4-B: Impact of Penalty on the Prepayment Option Value
                                                                under the BK Model

                             1.60
(in $/100$ of Notional)




                             1.20


                                                                                                          Zero penalty
                             0.80

                                                                                                          With penalty
Option Value




                             0.40



                             0.00
                                Mar-98 Mar-99 Mar-00 Mar-01 Mar-02 Mar-03 Mar-04 Mar-05 Mar-06 Mar-07




                                                                        35
                      Table 1: Determinants of the Prepayment Risk Measures *
                                            Prep. Option       EEL (High LTV)       EEL (Low LTV)
              Squared R                         12%                  67%                676%
Variables                                                   Regression coefficients
      Intercept                                       0.610                   -
      5Y Mortgage Rate Spread                           -                  0.507               0.531
      Curve Slope (5Y -1m)                           (0.041)              (0.232)            (0.242)
* : Empirical values of the prepayment options and EEL values are determined from the HW Model




                                             36
        Appendix B:


    Numerical Results for
The Mortgage Rate Commitment
       Option Analysis




             37
                          Figures 5–7 & Tables 2–4

       Fig. 5-a: The Monte Carlo Distribution (under the HW Model) of the Moneyness
                            Rate in Function of the MRC's Strike:
                                    Maturity = 2 months
35%

30%                                                                         Floating-Strike

25%
                                                                            Flat-Strike
20%

15%

10%

 5%

 0%
      95.0%   97.5%   100.0%   102.5%   105.0%   107.5%   110.0%   112.5%   115.0%    117.5%

                                   Moneyness Rate




       Fig. 5-b: The Monte Carlo Distribution (under the HW Model) of the Moneyness
                            Rate in Function of the MRC's Strike:
                                     Maturity = 6 months
25%

                                                                            Floating-Strike
20%

                                                                            Flat-Strike
15%


10%


5%


0%
      95.0%   97.5%   100.0%   102.5%   105.0%   107.5%   110.0%   112.5%   115.0%    117.5%

                                   Moneyness Rate




                                                 38
       Fig. 5-c: The Monte Carlo Distribution (under the BK Model) of the Moneyness
                           Rate in Function of the MRC's Strike:
                                    Maturity = 2 months
40%

35%                                                                         Floating-Strike

30%
                                                                            Flat-Strike
25%

20%

15%

10%

5%

0%
      95.0%   97.5%   100.0%   102.5%   105.0%   107.5%   110.0%   112.5%   115.0%    117.5%

                                   Moneyness Rate




       Fig. 5-d: The Monte Carlo Distribution (under the BK Model) of the Moneyness
                           Rate in Function of the MRC's Strike:
                                    Maturity = 6 months
25%

                                                                            Floating-Strike
20%

                                                                            Flat-Strike
15%


10%


5%


0%
      95.0%   97.5%   100.0%   102.5%   105.0%   107.5%   110.0%   112.5%   115.0%    117.5%

                                   Moneyness Rate




                                                 39
          Fig. 6-a: The Monte Carlo Distribution (under the HW Model) of the Exercise
                     Value of a Flat-Strike-MRC Option (Maturity = 6 months)

28%

24%
                                                                    Prepayment Risk Model
20%                                                                    (Stochastic Smm)

16%
                                                                    Constant Prepayment Rate
12%

8%                                                                    No Prepayment Risk


4%

0%
      -      1.00       2.00        3.00        4.00        5.00       6.00     7.00        8.00

                               Exercise Value (in $ per 100$ of notional)




          Fig. 6-b: The Monte Carlo Distribution (under the HW Model) of the Exercise
                   Value of a Floating-Strike-MRC Option (Maturity = 6 months)

24%


20%
                                                                    Prepayment Risk Model
                                                                       (Stochastic Smm)
16%
                                                                    Constant Prepayment Rate
12%
                                                                      No Prepayment Risk
8%


4%


0%
      -      1.00       2.00        3.00        4.00        5.00       6.00     7.00        8.00

                               Exercise Value (in $ per 100$ of notional)




                                                       40
          Fig. 6-c The Monte Carlo Distributions (under the BK Model) of the Exercise
                    Value of a Flat-Strike-MRC Option (Maturity = 6 months)

36%


30%
                                                                    Prepayment Risk Model
                                                                       (Stochastic Smm)
24%
                                                                    Constant Prepayment Rate
18%

                                                                      No Prepayment Risk
12%


6%


0%
      -      1.00       2.00        3.00       4.00        5.00        6.00     7.00        8.00

                               Exercise Value (in $ per 100$ of notional)




          Fig. 6-d: The Monte Carlo Distributions (under the BK Model) of the Exercise
                   Value of a Floating-Strike-MRC Option (Maturity = 6 months)

36%


30%
                                                                    Prepayment Risk Model
                                                                       (Stochastic Smm)
24%
                                                                    Constant Prepayment Rate
18%
                                                                      No Prepayment Risk
12%


6%


0%
      -      1.00       2.00        3.00       4.00        5.00        6.00     7.00        8.00

                               Exercise Value (in $ per 100$ of notional)




                                                      41
                                             Fig. 7-a: Monte Carlo Scenarios of the MRC Option Payoff and Hedge Portfolio
                                                Payoff: 6 months-Floating-Strike MRC with 50 bps In-the-Money Spread

                                               0.12
         Payoff (in $ per 1$ of Underlying




                                               0.10                                                                   MRC Payoff
                Mortgage Notional)




                                                                                                                      (Ordered
                                               0.08
                                                                                                                      Payoff)
                                               0.06

                                               0.04                                                                   Hedge Portfolio
                                                                                                                      Payoff
                                               0.02

                                               0.00

                                              -0.02
                                                         0    512 1024 1536 2048 2560 3072 3584 4096 4608 5120

                                                               Ordered Monte Carlo Scenarios of the MRC payoff




                                             Fig. 7-b: Monte Carlo Scenarios of the MRC Option Payoff and Hedge Portfolio
                                                Payoff: 6 months-Floating-Strike MRC with 50 bps Out-of-Money Spread
                                             0.08
Payoff (in $ per 1$ of Underlying




                                             0.06                                                                    MRC Payoff
       Mortgage Notional)




                                                                                                                     (Ordered
                                                                                                                     Payoff)
                                             0.04


                                             0.02                                                                    Hedge Portfolio
                                                                                                                     Payoff

                                             0.00


                                             -0.02
                                                     0       512   1024 1536 2048 2560 3072 3584 4096 4608 5120

                                                              Ordered Monte Carlo Scenarios of the MRC payoff




                                                                                      42
             Table 2- Part 1: Distribution of the MRC Options' Moneyness Rate under the HW Model

Panel A: Flat-Strike                                              MRC Option Maturity (in months)
                                                              2          3           4           5        6
 Mean                                                     106%       106%       106%         106%    106%
 Median                                                   106%       106%       106%         106%    106%
 Minimum                                                   94%        93%        92%          90%      89%
 Maximum                                                  117%       121%       123%         124%    128%
 Skewness                                                 -0.02       0.01       0.03         0.03    -0.01


Panel B: Floating-Strike: Base Case                               MRC Option Maturity (in months)
(Re-Striking Frequency = 1 week)                              2          3           4           5       6
  Mean                                                    106%       106%       106%         107%    107%
  Median                                                  106%       106%       106%         106%    107%
  Minimum                                                 100%       100%       100%         100%    100%
  Maximum                                                 117%       121%       123%         124%    128%
  Skewness                                                 0.15       0.32       0.39         0.45    0.49


Panel C-1: Floating-Strike: Re-Striking Effect                    MRC Option Maturity (in months)
(Re-Striking Frequency = 2 weeks)                             2          3           4           5       6
  Mean                                                    106%       106%       106%         106%    107%
  Median                                                  106%       106%       106%         106%    107%
  Minimum                                                 100%       100%       100%         100%    100%
  Maximum                                                 117%       121%       123%         124%    128%
  Skewness                                                 0.14       0.30       0.37         0.43    0.47


Panel C-2: Floating-Strike: Re-Striking Effect                    MRC Option Maturity (in months)
(Re-Striking Frequency = 4 weeks)                             2          3           4           5       6
  Mean                                                    106%       106%       106%         106%    107%
  Median                                                  106%       106%       106%         106%    107%
  Minimum                                                 100%       100%       100%         100%    100%
  Maximum                                                 117%       121%       123%         124%    128%
  Skewness                                                 0.13       0.28       0.35         0.42    0.45




                                                     43
              Table 2- Part 2: Distribution of the MRC Options' Moneyness Rate under the BK Model

Panel A: Flat-Strike                                               MRC Option Maturity (in months)
                                                               2          3           4           5       6
 Mean                                                      106%       106%       106%         106%    106%
 Median                                                    106%       106%       106%         106%    106%
 Minimum                                                    97%        95%        94%          93%     93%
 Maximum                                                   117%       118%       120%         127%    128%
 Skewness                                                   0.19       0.21       0.18         0.25    0.27


Panel B: Floating-Strike: Base Case                                MRC Option Maturity (in months)
(Re-Striking Frequency = 1 week)                               2          3           4           5       6
  Mean                                                     106%       106%       106%         106%    107%
  Median                                                   106%       106%       106%         106%    106%
  Minimum                                                  100%       100%       100%         100%    100%
  Maximum                                                  117%       118%       120%         127%    128%
  Skewness                                                  0.28       0.36       0.40         0.53    0.60


Panel C-1: Floating-Strike: Re-Striking Effect                     MRC Option Maturity (in months)
(Re-Striking Frequency = 2 weeks)                              2          3           4           5       6
  Mean                                                     106%       106%       106%         106%    107%
  Median                                                   106%       106%       106%         106%    106%
  Minimum                                                  100%       100%       100%         100%    100%
  Maximum                                                  117%       118%       120%         127%    128%
  Skewness                                                  0.27       0.34       0.38         0.51    0.59


Panel C-2: Floating-Strike: Re-Striking Effect                     MRC Option Maturity (in months)
(Re-Striking Frequency = 4 weeks)                              2          3           4           5       6
  Mean                                                     106%       106%       106%         106%    107%
  Median                                                   106%       106%       106%         106%    106%
  Minimum                                                  100%       100%       100%         100%    100%
  Maximum                                                  117%       118%       120%         127%    128%
  Skewness                                                  0.26       0.33       0.36         0.49    0.57




                                                      44
                Table 3 - Part 1: MRC Options Values & Deltas under the HW Model

                              Panel A: Strike Effect: Flat-Strike vs. Floating-Strike
    MRC Option                                 Flat-Strike                         Floating-Strike*
 Maturity (in months)                         Value          Delta (DV01)            Value         Delta (DV01)
           2                                1.2121                (0.0564)         1.2187              (0.0562)
           3                                1.2778                (0.0392)         1.2902              (0.0378)
           4                                1.3257                (0.0387)         1.3453              (0.0364)
           5                                1.3729                (0.0383)         1.4031              (0.0352)
           6                                1.4232                (0.0375)         1.4638              (0.0336)
* : Based on a Re-Striking Frequency of 1 week

                Panel B: Re-Striking Frequency Effect under the Floating-Strike Provision
    MRC Option                        Re-Strike Freq = 2 weeks                Re-Strike Freq = 4 weeks
 Maturity (in months)                       Value         Delta (DV01)              Value         Delta (DV01)
           2                              1.2176               (0.0564)           1.2173              (0.0565)
           3                              1.2874               (0.0383)           1.2846              (0.0390)
           4                              1.3405               (0.0370)           1.3339              (0.0378)
           5                              1.3966               (0.0358)           1.3881              (0.0366)
           6                              1.4555               (0.0342)           1.4444              (0.0351)


                       Panel C-1: Prepayment Risk Effect: Constant Prepayment Rate
    MRC Option                                 Flat-Strike                         Floating-Strike*
 Maturity (in months)                         Value          Delta (DV01)            Value         Delta (DV01)
           2                                0.8926                (0.0297)         0.8972              (0.0292)
           3                                0.9407                (0.0281)         0.9496              (0.0270)
           4                                0.9810                (0.0282)         0.9957              (0.0265)
           5                                1.0131                (0.0280)         1.0353              (0.0257)
           6                                1.0516                (0.0275)         1.0815              (0.0246)


                Panel C-2: Prepayment Risk Effect: Stochastic Prepayment Rate Function
    MRC Option                                 Flat-Strike                         Floating-Strike*
 Maturity (in months)                         Value          Delta (DV01)            Value         Delta (DV01)
           2                                0.8563                (0.0241)         0.8582              (0.0236)
           3                                0.9143                (0.0313)         0.9205              (0.0302)
           4                                0.9579                (0.0311)         0.9708              (0.0292)
           5                                0.9962                (0.0292)         1.0190              (0.0267)
           6                                1.0338                (0.0294)         1.0651              (0.0262)



    All reported values and deltas are expressed in $ per 100$ of notional




                                                               45
                Table 3 - Part 2: MRC Options Values & Deltas under the BK Model

                              Panel A: Strike Effect: Flat-Strike vs. Floating-Strike
    MRC Option                                 Flat-Strike                         Floating-Strike*
 Maturity (in months)                         Value          Delta (DV01)            Value         Delta (DV01)
           2                                1.0886               (0.0487)          1.0913               (0.0484)
           3                                1.1355               (0.0357)          1.1398               (0.0351)
           4                                1.1690               (0.0371)          1.1781               (0.0356)
           5                                1.2032               (0.0357)          1.2184               (0.0338)
           6                                1.2396               (0.0352)          1.2616               (0.0328)
* : Based on a Re-Striking Frequency of 1 week

                Panel B: Re-Striking Frequency Effect under the Floating-Strike Provision
    MRC Option                        Re-Strike Freq = 2 weeks                Re-Strike Freq = 4 weeks
 Maturity (in months)                       Value         Delta (DV01)              Value         Delta (DV01)
           2                              1.0906              (0.0486)            1.0901               (0.0487)
           3                              1.1387              (0.0353)            1.1369               (0.0355)
           4                              1.1755              (0.0360)            1.1715               (0.0365)
           5                              1.2155              (0.0342)            1.2104               (0.0348)
           6                              1.2583              (0.0332)            1.2526               (0.0339)


                       Panel C-1: Prepayment Risk Effect: Constant Prepayment Rate
    MRC Option                                 Flat-Strike                         Floating-Strike*
 Maturity (in months)                         Value          Delta (DV01)            Value         Delta (DV01)
           2                                0.8232               (0.0295)          0.8252               (0.0292)
           3                                0.8509               (0.0278)          0.8541               (0.0273)
           4                                0.8765               (0.0278)          0.8835               (0.0266)
           5                                0.9033               (0.0270)          0.9148               (0.0255)
           6                                0.9301               (0.0267)          0.9466               (0.0249)


                Panel C-2: Prepayment Risk Effect: Stochastic Prepayment Rate Function
    MRC Option                                 Flat-Strike                         Floating-Strike*
 Maturity (in months)                         Value          Delta (DV01)            Value         Delta (DV01)
           2                                0.7841               (0.0292)          0.7849               (0.0290)
           3                                0.8258               (0.0295)          0.8282               (0.0289)
           4                                0.8556               (0.0269)          0.8628               (0.0258)
           5                                0.8858               (0.0267)          0.8962               (0.0252)
           6                                0.9168               (0.0265)          0.9312               (0.0244)



   All reported values and deltas are expressed in $ per 100$ of notional




                                                               46
       Table 4- Part 1:   Hedging 6 month-MRC Option: Hedge Portfolio & Hedge Effectiveness
                                        Moneyness Spread (in bps) of the MRC Option at Time To
MRC Type                            Out-of-Money              At-the-Money                 In-the-Money
                              -50                  -25              0                 25                  50
                                                            Hedge Leverage : Σ λj
 Floating-Strike                     -0.9               0.7                 1.2              -0.5                -2.1
 Flat-Strike                          5.7              4.1                 2.0                0.0                -1.7
                                 Unhedged Component : µ (in $ per 1$ of Notional of MRC Option Position)
 Floating-Strike                   0.427             0.541               0.644             0.714                0.765
 Flat-Strike                       0.384             0.541               0.645             0.715                0.766
                                                                Hedge Cost *
 Floating-Strike                   0.993             1.105               1.140             1.159                1.165
 Flat-Strike                       1.020             1.097               1.137             1.156                1.162
                                             Squared-R of the Hedge Effectiveness Regression
 Floating-Strike                    66%               59%                 51%               45%                  40%
 Flat-Strike                        68%               59%                 51%               45%                  40%
                                          Standard-Error of the Hedge Effectiveness Regression *
 Floating-Strike                  0.0079           0.0101               0.0122            0.0140               0.0155
 Flat-Strike                      0.0080           0.0102               0.0122            0.0139               0.0154




       Table 4- Part 2:   Hedging 2 month-MRC Option: Hedge Portfolio & Hedge Effectiveness
                                        Moneyness Spread (in bps) of the MRC Option at Time To
MRC Type                            Out-of-Money              At-the-Money                 In-the-Money
                              -50                  -25              0                 25                  50
                                                            Hedge Leverage : Σ λj
 Floating-Strike                    -10.1              -9.7                -9.8              -9.4                -9.1
 Flat-Strike                         -8.2              -9.5                -9.7              -9.4                -9.1
                                 Unhedged Component : µ (in $ per 1$ of Notional of MRC Option Position)
 Floating-Strike                   0.587             0.718               0.790             0.853                0.878
 Flat-Strike                       0.592             0.718               0.790             0.835                0.865
                                                                Hedge Cost *
 Floating-Strike                   0.615             0.636               0.645             0.647                0.646
 Flat-Strike                       0.606             0.635               0.645             0.647                0.646
                                             Squared-R of the Hedge Effectiveness Regression
 Floating-Strike                     53%              40%                 31%               25%                  21%
 Flat-Strike                         53%              40%                 31%               25%                  21%
                                          Standard-Error of the Hedge Effectiveness Regression *
 Floating-Strike                  0.0066           0.0089               0.0111            0.0130               0.0147
 Flat-Strike                      0.0066           0.0089               0.0111            0.0130               0.0146



(*): In $ per 100$ of Underlying Mortgage Notional.




                                                         47

				
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