VIEWS: 20 PAGES: 39 CATEGORY: Financial Models POSTED ON: 6/5/2009 Public Domain
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