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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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