A Short Course on Affine Financial Models Part 1 by Yoursismine

VIEWS: 20 PAGES: 39

									A Short Course on Affine Financial Models
                 Part 1


                     Darrell Duffie
               Lausanne, January 2008
        Swiss Finance Institute Doctoral Course




                           1
                               General Outline
 1. An Introduction to Affine Processes

 2. The Riccati Equation for Affine Transform Coefficients

 3. Multifactor Affine Models

 4. The Heston Volatility Model

 5. Multifactor Affine Term Structure Models

 6. Affine Jump Diffusions

 7. General Option Pricing Methods for Affine Models

 8. Wishart Processes and Stochastic Return Covariances

 9. Linear-Quadratic Affine Models

10. Cumulant and other Basket Option Approaches

11. Affine Correlated Default Models and CDS index tranche pricing.


                                         2
          1. An Introduction to Affine Processes
• Gaussian Ornstein-Uhlenbeck Process
• Feller’s Continuous Branching Process
• The Basic Affine Process
• The Key Equivalent Affine Properties
• General Definition of an Affine Process




                                    3
               Startup Model: Ornstein-Uhlenbeck

                         dXt = κ(θ − Xt ) dt + v dBt ,                      (1)

for a standard Brownian motion B


 • Short rate model of Vasicek [1977]
 • Mean reversion parameter κ.
 • Long-run (stationary) mean θ .
 • Volatility parameter v .
 • Xt given Xs is Gaussian with a mean and variance that are easily calculated.



                                        4
           Feller’s Continuous Branching Process


                   dXt = κ(θ − Xt ) dt + c        Xt dBt ,   (2)



• Short rate model of Cox, Ingersoll, and Ross [1985].
• Stochastic volatility model of Heston [1993].
• Default intensity model of Lando [1994].
• Xt given Xs is non-central χ2 .




                                       5
                         Basic Affine Process
                dXt = κ(θ − Xt ) dt + c Xt dBt + dJt ,                         (3)


                                    N (t)
• Compound Poisson: J(t) =          i=1     Zi .
• The process N counting jumps is Poisson with constant intensity.
• The iid jumps sizes Z1 , Z2 , . . . are exponential.
• Xt given Xs has an explicit characteristic function
• Used by Duffie and Garleanu [2001] to model correlated default intensities.
                     ˆ

• We can allow N to have intensity ℓ0 + ℓ1 Xt , and other models for Zi .




                                        6
                   Affine Characteristic Exponents

For any of the above models, Xt given Xs has the characteristic function



                E eizX(t) | X(s) = eα(t−s,z)+β(t−s,z)X(s) ,                   (4)



showing affine dependence on X(s) of the characteristic exponent

                         α(t − s, z) + β(t − s, z)X(s).


We will see how to compute the coefficients α(t, z) and β(t, z) as the solution of
an ordinary differential equation in t.




                                          7
                 Key Property of an Affine Process
Under mild regularity conditions on a d-dimensional Markov process X , the
following are equivalent:


 • Whenever s < t, the characteristic exponents of Xt given Xs are affine
   (constant-plus-linear) with respect to Xs .


 • There is an affine dependence on the current state Xt of the drift, of the
    diffusive covariance matrix, and of the jump intensity (or, more generally, the
    jump measure).




                                         8
                    Definition of an Affine Process

A d-dimensional time-homogeneous Markov process X is defined to be affine if
its conditional characteristic function is of the form, for any z   ∈ Rd ,




             E eiz·X(t) | X(s) = eα(t−s,z)+β(t−s,z)·X(s),                    (5)



for some coefficients α(t − s, z) and β(t − s, z).




                                           9
2. The Riccati Equation for Affine Transform Coefficients

• Objective: Computing Affine Transforms
• Feynman-Kac: Find the Martingale to get a PDE
• Reduce the PDE to an Riccati ODE by Separation of Variables
• Apply the same Riccati equation to get Characteristic Exponents
• The Discounted Transform




                                     10
        Starter Case: Transforms of Feller’s Diffusion

                           √
For dXt = (a + bXt ) dt + c Xt dBt , we will see that

            f (Xt , t) = E ewX(T )           Xt = eα(T −t)+β(T −t)x ,            (6)


In order to see why, take the Feynman-Kac approach:


Applying Ito’s Formula to Yt
           ˆ                     = f (Xt , t), we have
                            t                        t
          Yt = Y0 +             γ(Xs , s) ds +           fx (Xs , s)c Xs dBs ,   (7)
                        0                        0

where
                                                   1
          γ(x, t) = ft (x, t) + fx (x, t)(a + bx) + fxx (x, t)c2 x.
                                                   2


                                           11
                  Find the Martingale to get a PDE

By the law of iterated expectations, for any s and t   ≥ s,

              Ys = Es (ewX(T ) ) = Es (Et (ewX(T ) )) = Es (Yt ).

Thus, Y is a martingale, so has no drift, so for all x,

                                                 1
              0 = ft (x, t) + fx (x, t)(a + bx) + fxx (x, t)c2 x,   (8)
                                                 2
with the boundary condition f (x, T )   = ewx .
Now we need to solve the PDE (8).




                                          12
                    Prepare to Separate Variables

Substitution of f (x, t)   = eα(T −t)+β(T −t)x into the PDE leaves
                                                       1
eα(T −t)+β(T −t)x
                       −α (t) − β (t)x + β(t)(a + bx) + β(t)2 c2 x
                             ′       ′
                                                                     = 0,
                                                       2
                                                                       (9)

Dividing by eα(T −t)+β(T −t)x , and collecting terms in x,

                                 u(t)x + v(t) = 0,                    (10)

where
                                             1
                    u(t) = −β ′ (t) + β(t)b + β(t)2 c2                (11)
                                             2
                    v(t) = −α′ (t) + β(t)a.                           (12)




                                          13
             Riccati Equation for Affine Transforms
Separation of variables leaves the ordinary differential equation (ODE)

                                           1
                          ′
                        β (t)     = β(t)b + β(t)2 c2                         (13)
                                           2
                        α′ (t)    = β(t)a.                                   (14)

 • Because f (x, T ) = ewx for all x, we have the boundary conditions
   α(0) = 0 and β(0) = w.
 • Given a solution for β , we calculate that
                                                 s
                                 α(s) = a            β(u) du.
                                             0

 • Explicit solutions for α(t) and β(t) are available. (See Appendix of Notes.)



                                        14
            The ODE for Characteristic Exponents
• At a real number z , let

                       f (Xt , t; z) = E eizX(T )     Xt .

• f ( · , · ; z) solves the same PDE (8), with the boundary condition
  f (x, T ; z) = eiz .
• We again conjecture the exponential-affine solution form

                         f (x, t; z) = eα(T −t)+β(T −t)x ,

  with the boundary condition α(0)   = 0 and β(0) = iz .
• Applying separation of variables gives the same ODE (13). The solutions for
  α(s) and β(s), shown explicitly in the Appendix of the Notes, are in general
  complex numbers.



                                      15
              Inversion of Characteristic Function
• To calculate: P(XT ≤ y | Xt ), use the characteristic function ϕ( · ) of XT
  given Xt = x, defined by ϕ(z) = f (x, t; z).

• Under mild regularity, we can apply the Levy Inversion Formula
                                           ´
                                                   ∞
                             ϕ(0)   1                  1
           P(XT ≤ y | Xt ) =      −                      Im(ϕ(z)eizy ) dy,
                              2     π          0       z
  where Im(c) denotes the purely imaginary part of a complex number c.

• Alternatively, again under regularity, we can calculate the density p( · ) of XT
  via the inverse Fourier transform
                                         ∞
                                1
                        p(y) =               e−izy ϕ(z) dz.
                               2π      −∞

• Quadrature and fast Fourier transform are both in use for this calculation.


                                       16
                   The Discounted Transform
• For R(x) = ρ0 + ρ1 x, let
                                    T
                                        −R(X(u)) du + wX(T )
             f (Xt , t) = E e      t                                Xt .              (15)

                                                    t
                                                        R(Xs ) ds
• Extending the Feynman-Kac pattern: Yt = e−        0               f (Xt , t) is a
  martingale, leading to the PDE
                                                            1
  0 = −(ρ0 + ρ1 x)f (x, t) + ft (x, t) + fx (x, t)(a + bx) + fxx (x, t)c2 x.
                                                            2
• By the same separation-of-variables argument,
                                     1
                    β ′ (t) = β(t)b + β(t)2 c2 − ρ1                                   (16)
                                     2
                    α′ (t) = β(t)a − ρ0 ,                                             (17)

  with the boundary conditions α(0) = 0 and β(0) = w. Given a solution for
                                 t
  β , we calculate that α(t) = a 0 β(u) du − tρ0 .

                                         17
          Homework: One of Exercises 2.2, 2.3, or 2.4
Exercise 2.4. Suppose that X solves the stochastic differential equation

                      dXt = κ(θ − Xt ) dt + c          Xt dBt .

There are assets, with no dividends until after T , with price processes
St = ea(t)+b(t)X(t) and Ut = ef (t)+g(t)·X(t) , where a( · ), b( · ), f ( · ), and
g( · ) are continuously differentiable. A derivative security pays max(ST , UT ) at
some fixed time T > 0. In the absence of arbitrage trading strategies whose
market values are non-negative, represent the initial price of the derivative
                   ´
security with the Levy Inversion Formula. In this case, the characteristic
exponents have coefficients that are not explicit, but (under technical regularity
conditions) solve Ricatti equations with time-dependent coefficients. State the
Riccati equations. Hint: Use Girsanov’s Theorem, in a form stated as Proposition
1 of Appendix E. Adopt technical conditions on a( · ), b( · ), f ( · ), and g( · ) as
needed.


                                          18
                      Multifactor Affine Models
1. Multifactor Affine Diffusions

2. Multifactor Affine Riccati Equation

3. Explicit and Numerical Solution of Riccati Equations.




                                        19
                    Multifactor Affine Diffusions

                       dXt = µ(Xt ) dt + σ(Xt ) dBt ,              (18)

where B is a standard Brownian motion in Rd .

 • µ(x) = K0 + K1 x for some K0 ∈ Rd and K1 ∈ Rd×d .
 • (σ(x)σ(x)⊤ )ij = H0ij + H1ij · x for some H0ij ∈ R and H1ij ∈ Rd .
 • We will write σ(x)σ(x)⊤ = H0 + H1 x.
 • Restriction on H and the state space D : H0ii + H1ii · x ≥ 0.
 • Resriction on D and K : No drift toward the “exterior” of D .
 • Example, for the state space D = R+ , we must have K0 ≥ 0.



                                        20
            Multifactor Affine Discounted Transform
Consider, for w   ∈ C,
                                    t
                                        −(ρ0 +ρ1 ·Xu ) du + w·Xt
               f (Xs , s) = E e     s                              Xs ,

for some discount-rate coefficients ρ in R and ρ1       ∈ Rd , which is solved by the
PDE:

       0   =      −(ρ0 + ρ1 x)f (x, t) + ft (x, t) + fx (x, t)(K0 + K1 x) (19)
                                                                                  (20)
                      1         ∂f (x, t)
                    +                     (H0ij + H1ij · x).
                      2   i,j
                                ∂xi ∂xj

Separation of variables confirms that

                           f (x, s) = eα(t−s)+β(t−s)·x ,                          (21)

for some coefficient functions α( · ) and β( · ) to be calculated.


                                             21
               Multifactor Affine Riccati Equation
Let β(s)⊤ H1 β(s) denote the d-dimensional vector whose k -th element is

                                 βi (s)H1,ijk βj (s).
                           i,j


Then

                ′            ⊤   1
               β (s) =          + β(s)⊤ H1 β(s) − ρ1
                            K1 β(s)                                        (22)
                                 2
                                 1
                         ⊤
               α′ (s) = K0 β(s) + β(s)⊤ H0 β(s) − ρ0                       (23)
                                 2
with the boundary condition β(0)   = w and α(0) = 0.




                                       22
              Solving Multifactor Riccati Equation

• A general analytical framework for explicit solutions of a sub-class of affine
  models is provided by Grasselli and Tebaldi [2007].

• Otherwise, one can solve the ODE numerically, for example by a Runge-Kutta
  method.

• An explicit fourth-order Runge-Kutta method is often used in financial
  applications.

• For cases in which the ODE is “stiff,” an implicit Runge-Kutta method may be
  more effective. See Huang and Yu (2005).

• Standard software packages, for example in Matlab, are available.



                                       23
                       Heston’s Volatility Model

1. Heston’s One-Factor Diffusion Model of Stochastic Volatiltity

2. Explicit Transform for Heston’s Model

3. Example fit to S&P500 index options.

4. The Heston model is missing jumps (Pan (2003)) and multiple volatility
   factors, to which we return later in the course.




                                        24
                     Heston’s Volatility Model

                                                   √
                                        ¯                         ¯
• Underlying asset price process: dSt = rSt dt + St Vt dB1t where r is a
  constant interest rate and

                              v
                      dVt = κ(¯ − Vt ) dt + c      Vt dZt ,                 (24)

• For “instantaneous” correlation ρ between vol and returns, we take

                           Z = ρB1 +       1 − ρ2 B2 .

• For index options, ρ < 0 corresponds to the typical “skew” of the smile curve.




                                      25
                                24

                                                                                  17 days
                                22                                                45 days
                                                                                  80 days
                                                                                  136 days
                                20
                                                                                  227 days
Black-Scholes Implied Vol (%)




                                                                                  318 days
                                18

                                16

                                14

                                12

                                10

                                8

                                6
                                    0.6    0.7      0.8        0.9           1   1.1         1.2
                                                    Moneyness = Strike/Futures


                                Figure 1: Smile Curves for S&P500 Index Options November 2, 1993.

                                                                  26
                      Heston’s Model is Affine

• Letting Yt = log St , Ito’s Formula implies that
                          ˆ

                                  1
                     dYt =     ¯
                               r − Vt       dt +     Vt dB1t ,             (25)
                                  2

• This means that (V, Y ) is indeed affine, with state space D = R+ × R.
• In order to price and risk manage equity-linked annuities or other hybrid
                             ¯
  products, we can replace r with an affine term-structure model with short rate

                               rt = ρ0 + ρ1 · Zt ,

  where X   = (V, Y, Z) is affine.



                                       27
            Explicit Transform for Heston’s Model
• For options on St , we need only the characteristic function of Y (t):

                          ϕ(u) = E[euY (t) ],        u ∈ C.

• Solving the Riccati equation (22) for this case,

                        ϕ(u) = eα(t,u)+uY (0)+β(t,u)V (0) ,

  where, letting b   = uσv ρ − κ, a = u(1 − u), and γ =                2
                                                                b2 + aσv ,

                     a (1 − e−γt )
    β(t, u) = −                    −γt )
                                         ,
                2γ − (γ + b) (1 − e
                    γ+b         2          γ+b
    α(t, u) = −κv      2
                          t + 2 log 1 −        1 − e−γt                      .
                      σv       σv           2γ



                                       28
                                   24

                                                                                                              17 days
                                   22                                                                         45 days
                                                                                                              80 days
                                                                                                              136 days
                                   20
                                                                                                              227 days
                                                                                                              318 days
   Black-Scholes Implied Vol (%)




                                   18


                                   16


                                   14


                                   12


                                   10


                                   8


                                   6
                                       0.6             0.7           0.8         0.9           1             1.1
                                                                      Moneyness = Strike/Futures

                                                                           √
Figure 2:                               Calibration:   (r, ρ, κ, v, σv ,       V0 ) = (3.19%, −0.66, 19.7, 0.017, 1.52, 0.094).
                                                                                   30
          Multifactor Affine Term Structure Models
1. Affine yield curves.

2. Fitting a Term Structure Model to Data

3. Observable Macroeconomic State Variables

4. Unspanned Stochastic Volatility

                                  ¨
5. Worked Example: Lando and Feldhutter (2006): Swap Spreads.

6. Worked Example: Almeida, Graveline, and Joslin (2006).




                                      31
                Multifactor Term-Structure Model
• For a d-dimensional term-structure model, let

                        dXt = µ(Xt ) dt + σ(Xt ) dBt ,                     (26)

  where B is a standard Brownian motion in Rd .

• Take the short rate R(Xt ), and, under risk-neutral probabilities, the
  zero-coupon bond price for maturity at T is

                                      T
                                 −        R(Xs ) ds
                           E e       t                Xt .

• For an affine model with parameters (H, K, ρ), let

                                µ(x) = K0 + K1 x
                         σ(x)σ(x)⊤          = H0 + H1 x
                                R(x) = ρ0 + ρ1 · x.


                                          32
                     The Affine Term Structure
• Fixing risk-neutral parameters (H, K, ρ), the price at time t of a zero-coupon
  bond with maturity at time T , is

                     T
                         R(X(u)) du
          E e   −   t                 X(t)   = eα(T −t)+β(T −t)·X(t) ,

  where α( · ) and β( · ) solve the associated Riccati equation (22).

• The corresponding continuously-compounding yield to maturity, denoted
  yt,T −t , is defined by

                    e−yt,T −t (T −t) = eα(T −t)+β(T −t)·X(t) .

• We therefore have
                               α(T − t) β(T − t)
                  yt,T −t   =−         −         · X(t).                   (27)
                                T −t     T −t


                                       33
                        Rotational Invariance
• Different choices for the coefficients (H, K, ρ) can lead to the same affine
  term-structure model, that is, to the same dynamic model for the yield curve
  yt = {yt,m : m ≥ 0}.(Joslin [2007] and Cheridito, Filipovic, and Kimmel
  [2006])

• Example, let A be any invertible d × d matrix, and consider the state vector
  Yt = AXt .
• The original term structure model is equivalent to a term structure model with
                                                ˆ ˆ ˆ
  affine state process Y and with coefficients (H, K, ρ), where
   – ρ0 = ρ0 and ρ1 = A−1 ρ1 .
     ˆ           ˆ
      ˆ            ˆ
   – K0 = K0 and K1 = AK1 A−1 .
      ˆ                ˆ
   – H0 = AH0 A⊤ and H1 Yt = AH1 A−1 Yt A⊤ .
                                                        ˆ
  It is a good exercise to calculate the (i, j)-element H1,ij           ˆ
                                                                ∈ Rd of H1 .
• More generally, one can take Yt = a + AXt for a in Rd .

                                      34
     Calibrating Latent-State Term-Structure Models
• If we know (H, K, ρ), then we can typically calculate Xt by inverting from
  (27) for selected yields yt,T1 , . . . , yt,Td .

• One usually uses treasury yields or swap rates; the same idea works. We can
  also use swaptions and caps. See, for example, Joslin [2007].

• In financial industry practice, (H, K, ρ) and Xt are typically calibrated
  simultaneously to current yields and option prices, ignoring historical data.
  Allowing time dependence of (Ht , Kt , ρt ) gives a “perfect fit.”

• This practice is a poor discipline on model specification and leads to unstable
  parameter estimates. (But it is easy.)

• It is recommended to fit (H, K, ρ) to historical data, and then to make
                                   ˆ ˆ ˆ
  minimal calibration adjustments (Ht , Kt , ρt ) to match liquid current prices.
  (But this is not easy.) See, for example, Joslin [2007].


                                       35
           Time-Series Estimation of Affine Models
1. Of the risk-neutral parameters (H, K, ρ), only the drift parameters
   K A = K need to be separately specified under the “actual” probability
   measure.

2. For given risk-neutral parameters θ    = (H, K, ρ) and selected yields and
   option prices Zt = (yt,T1 , . . . , Yt,Tm , pt,1 , . . . , pt,d−m ), for each past
                                                        θ      θ          θ
   date t, we can infer the implied state history X1 , X2 , . . . , XT .
         θ    θ            θ
3. From X1 , X2 , . . . , XT , and the parameters, (H, K, K A , ρ), we can
   calculate the actual likelihood of the observed data Z1 , Z2 , . . . , ZT .

4. We choose (H, K, K A , ρ) to maximize the observed likelihood.

5. State of the art examples: Almeida, Graveline, and Joslin (2006), Joslin
   [2007], and Cheridito, Filpovic, and Kimmel (2006).



                                          36
         Observable Macroeconomic State Variables

Stimulated by Piazzesi (2001) and Ang and Piazzesi [2003], new classes of affine
term-structure models take some of the state variables to be macroeconomic
variables such as inflation and unemployment.


With the benefit of such a model, one can use yields and interest-rate options
prices to forecast macro-economic conditions, or to estimate the risk premia that
are charged by investors for bearing specific types of macroeconomic risk.




                                        37
                 Unspanned Stochastic Volatility

• Collin-Dufresne and Goldstein (2002, 2006) and Andersen and Benzoni
  (2006) have questioned whether it makes sense to assume that the state
  vector Xt can be observed through a selection yt,m1 , . . . , yt,md of bond
  yields of various maturities m1 , . . . , md .

• From (27), this can be done if and only if the associated “loading vectors”
  β(m1 )/m1 , . . . , β(md )/md are linearly independent.
• Collin-Dufresne and Goldstein (2002) proposed Unspanned Stochastic
  Volatility (USV), restricting the coefficients (H, K, ρ) of an affine model so
  that yield volatilities are not properties of yields alone, but have additional
  sources of risk of their own.

• Bikbov and Chernov (2005) and Joslin (2007) show that if one includes
  interest-rate option data when fitting an affine term-structure model, then the
  USV hypothesis is soundly rejected.


                                         38
                        Change of Measure

• Suppose that X is an d-dimensional solution to

                     dXt = µ(Xt , t) dt + σ(Xt , t) dBt                       (28)

  where B is a standard Brownian motion in Rd under P.

• For a given measurable a : Rd × [0, ∞) → Rd , our objective is to find
  conditions for an equivalent probability measure Q under which there is a
  standard Brownian motion B ∗ in Rd such that

                                                      ∗
                     dXt = a(Xt , t) dt + σ(Xt , t) dBt .                     (29)




                                      39
                   Girsanov’s Theorem For SDEs

Consider the existence of c     : Rd × [0, T ] → Rd solving the linear equations
         σ(x, t)c(x, t) = µ(x, t) − a(x, t), (x, t) ∈ Rd × [0, T ].                    (30)

Proposition (Lipster-Shiryaev). Suppose that X satisfies (28) and Y solves


                      dYt = a(Yt , t) dt + σ(Yt , t) dBt ,
with the same initial condition X0  = Y0 . If there is some measurable c satisfying
                T                                 T
(30) such that 0 c(Xt , t) · c(Xt , t) dt and 0 c(Yt , t) · c(Yt , t) dt are finite
almost surely (with respect to P), then there is a probability measure Q equivalent
to P and a standard Brownian motion B ∗ in Rd under Q satisfying (29).
Moreover, the density process for Q is the martingale ξ defined by
                          t                           t
                                              1
     ξt = exp −               c(Xs , s) dBs −             c(Xs , s) · c(Xs , s) ds .
                      0                       2   0



                                           40

								
To top