Introduction to Affine Processes. Applications to Mathematical Finance

and Its Applications to Mathematical Finance Department of Mathematical Science, KAIST Workshop for Young Mathematicians in Korea, 2010

Outline Motivation 1 Motivation 2 Preliminary : Stochastic Calculus Definition of Transform Analysis 3 Affine Processes : General Theory Matrix

Option Pricing Motivation Call option : the right to buy a stock from the seller of the option at the maturity for the strike price Payoff of the call option: (S T K) + (S t ) t 0 : stock price process, K : strike price, T : maturity Is a stock price deterministic? No! We need a probabilistic model(or stochatic model) How to price the call option? Expectation of discounted payoff under a risk neutral measure E[D T (S T K) + ] (D t ) t 0 : discount factor process, i.e., D t = e R t 0 rudu, (r t ) t 0 : instantaneous interest rate process

Option Pricing : Continued In case the joint distribution function F ST,D T of S T and D T is known, ZZ E[D T (S T K) + ] = y(x K) + F ST,D T (dx, dy) R 2 In particular, under the Black-Scholes model, Z E[D T (S T K) + ] = e rt (x K) + 1 n 0 σ 2πT x exp 1 ` 1 (r 2σ 2 T 2 σ2 )T + log x 2o dx S 0 = S 0 N log(s 0/K) + (r + σ 2! /2)T σ Ke rt N log(s 0/K) + (r σ 2! /2)T T σ T Alternative methods PDE Theory(Feynman-Kac Theorem)!

Motivation Lemma For a nonnegative random variable Y with 0 < EY < and a random variable X, we define a finite measure µ by µ(b) = E[Y 1 {X B} ] for B B(R). The transform ψ of µ is given by ψ(w) = e wx µ(dx) = E[Ye wx ] for all w C R

: Continued Definition (Discounted transform) For a stock price process (S t ) t 0, the discounted transform is defined as for t 0, w C ψ(t, w) = E[e R t 0 rudu (S t ) w ] = E[e R t 0 rudu+wxt ] (X t ) t 0 : return process of the stock, i.e., S t = e Xt Distribution : µ(a, b, t; dx) µ(a, b, t; B) = E[e R t 0 rudu+axt 1 {bxt B}] for t 0, a, b R, and B B(R).

: Continued µ(a, b, t; R) = E[e R t 0 rudu+axt ] = ψ(t, a) Then, by lemma above, e iθx µ(a, b, t; dx) = E[e R t 0 rudu+axt e ibθxt ] R = E[e R t 0 rudu+(a+ibθ)xt ] = ψ(t, a + ibθ)

: Continued Theorem (Gil-Palaez Inversion Formula) Let µ be a finite measure on B(R). Then µ(, x] = µ(r) 2 for x with µ{x} = 0. In particular, µ(a, b, t; (, x]) = = µ(a, b, t; R) 2 ψ(t, a) 2 τ + lim δ 0 δ τ + lim δ 0 τ Z τ + lim δ 0 δ τ Z τ δ e iθx ψ( iθ) e iθx ψ(iθ) dθ 2πiθ e iθx ψ(t, a ibθ) e iθx ψ(t, a + ibθ) dθ 2πiθ e iθx ψ(t, a ibθ) e iθx ψ(t, a + ibθ) dθ 2πiθ

: Continued Therefore, price of the call option is E[e R T 0 ru du (ST K) + ] = E[e R T 0 ru du+x T 1{ XT log K} ] KE[e R T 0 ru du 1{ XT log K} ] = µ(1, 1, T ; (, log K]) Kµ(0, 1, T ; (, log K]) = ψ(t, 1) Kψ(T, 0) 2 Z τ + lim δ 0 δ τ e iθx (ψ(t, 1 + iθ) Kψ(t, iθ)) e iθx (ψ(t, 1 iθ) Kψ(t, iθ)) dθ 2πiθ Likewise, we can price various options including put options, bond derivatives, quatos, foreign bond options, chooser options, etc. Hence, in many cases, it suffices to find the discounted transform ψ(t, w) = E[e R t 0 rudu+wxt ]

Preliminary : Stochastic Calculus Definition of Transform Analysis Brownian Motions and Stochastic Integrals Definition (Brownian motion) An n-dimensional stochastic process (W t) t 0 is called a standard (F t) t 0 Brownian motion if satisfies the followings: t W t(ω) is continuous ω-a.s. W t W s is independent of F s W t W s is normally distributed with mean 0 and covariance matrix (t s)i Intuitively speaking, for an adapted LCRL process (H t) t 0, the stochastic integral is defined as follow: Z t 0 H sdw s = lim n k=1 nx H (k 1)t/n (W kt/n W (k 1)t/n )

Stochastic Differential Equations Preliminary : Stochastic Calculus Definition of Transform Analysis Stochastic differential equation(time-homogeneous case) : dx t = µ(x t )dt + σ(x t )dw t with X 0 = x Equivalent stochastic integral equation : t t X t = x + µ(x s )ds + σ(x s )dw s 0 0 Diffusion process : solution to time-homogeneous SDE Infinitesimal generator A of (X t ) t 0 : f (t, x) Af (t, x) = µ(x) + 1 ( x 2 tr σσ (x) 2 f (t, x) ) x 2

A Version of Feynman-Kac Theorems Preliminary : Stochastic Calculus Definition of Transform Analysis We consider continuous functions h, k : R n R and Cauchy problem f + kf = Af t with initial conditions f (0, x) = h(x). Suppose that the above Cauchy problem has a C 1,2 solution f on R + R n. Theorem Under some regularity condition on the solution f, the solution admits the stochastic representation: for all (t, x) R + R n. f (t, x) = E x [e R t 0 k(x u)du h(x t)]

Motivation Preliminary : Stochastic Calculus Definition of Transform Analysis Definition An R n -valued stochastic process (X t ) t 0 is called an affine diffusion if it solves the SDE: dx t = µ(x t )dt + σ(x t )dw t where (W t ) t 0 is a standard Brownian motion in R n, and µ, σσ are affine mappings on R n. Since µ and σσ are affine, they admit following representation: µ(x) = K 0 + K 1 x for some K 0 R n and K 1 reals n n σσ (x) = H 0 + n i=1 x ih1 i for some n n symmetric matrices H 0, H1 1,, Hn 1

: Examples Preliminary : Stochastic Calculus Definition of Transform Analysis Vasicek Model : dr t = (a br t )dt + σdw t Cox-Ingersoll-Ross Model : dr t = (a br t )dt + σ R t dw t Heston Model : dx t = (a + bv t )dt + V t dwt 1 dv t = κ(θ V t )dt + ρσ V t dwt 1 + 1 ρ 2 σ V t dwt 2

Transform Analysis Motivation Preliminary : Stochastic Calculus Definition of Transform Analysis Instantaneous interest rate: r t = R(X t ) where R is affine: R(x) = ρ 0 + ρ 1 x Discounted transform: Candidate for transform : ψ(t, x, w) = E x [e R t 0 R(Xu)du+w X t ] ψ(t, x, w) = e β(t,w)+α(t,w) x

Transform Analysis : Continued Preliminary : Stochastic Calculus Definition of Transform Analysis Partial differential equation: where R(x)ψ(t, x, w) + Aψ(t, x, w) = K 0 + K 1 ψ(t, x, w) t = Aψ(t, x, w) ψ(t, x, w) + 1 2 x 2 tr ψ(t, x, w) x 2 H 0 ψ(t, x, w) x 1 + x with initial condition ψ(0, x, w) = e w x 2 2 tr ψ(t, x, w) x 2 H 1 x

Transform Analysis : Continued Preliminary : Stochastic Calculus Definition of Transform Analysis With ψ(t, x, w) = e β(t,w)+α(t,w) x, Aψ = ψ`k 0 α + 1 2 tr(αα H 0 ) + ψ`k 1 α + 1 2 tr(αα H 1 ) x = ψ`k 0 α + 1 2 α H 0 α + ψ`k 1 α + 1 2 α H 1 α x Rψ + ψ t = ψ`ρ 0 + β + ψ`ρ 1 + α x since ψ t = ψ( β + α x), ψ x = ψα, 2 ψ = ψαα x 2 By the method of undetermined coefficients, α(t, w) = ρ 1 + K1 α(t, w) + 1 2 α(t, w) H 1 α(t, w) β(t, w) = ρ 0 + K0 α(t, w) + 1 2 α(t, w) H 0 α(t, w) with α(0, w) = w, β(0, w) = 0

Affine Processes Motivation Affine Processes : General Theory Matrix State space : D = R m + R n Transition kernel : p t (x, A) = P x (X t A) Definition A Markov process (X t) t 0 with state space D is called affine if for every (t, w) R + ir d there exist α(t, w) = (α Y (t, w), α Z (t, w)) C m C n and β(t, w) C such that E x [e w X t ] = e β(t,w)+α(t,w) x = e β(t,w)+αy (t,w) y+α Z (t,w) z for all x = (y, z) D. The affine process (X t) t 0 is said to be regular if it satisfies followings p t is stochastically continuous, i.e., p s(x, ) p t(x, ) weakly on D as s t d dt Ex [e w X t ] t=0 exists, and continuous at w = 0

Affine Processes : Characterization Affine Processes : General Theory Matrix Roughly speaking, a Markov process (X t ) t 0 is regular affine if and only if its infinitesimal generator A has the following form: Af (x) = dx (a kl + α I,kl, y ) 2 f (x) x k x l k,l=1 + b + βx, f (x) (c + γ, y )f (x) Z + (f (x + ξ) f (x) J f (x), χ J (ξ) )m(dξ) + D\{0} mx Z i=1 D\{0} (f (x + ξ) f (x) J (i) f (x), χ J (i) (ξ) )y i µ i (dξ) For detail, see Affine Processes and Applications in Finance by Duffie, et. al.

Matrix Affine Processes : General Theory Matrix Asset pricing model under Matrix : dx t = (r1 1 2 diag(yt))dt + Y tdw t dy t = (ΩΩ + MY t + Y tm )dt + Y tdb tq + Q db t Yt X t R n, Y t S n +, Ω, M R n n, and Q O n (B t ) t 0 : a standard n n matrix Brownian motion (Z t ) t 0 : a standard n-dimensional Brownian motion which is independent of (B t ) t 0 W t = B t ρ + 1 ρ ρz t, where ρ R n

Appendix References I D. Duffie, J. Pan, and K. Singleton, Transform Analysis and Asset Pricing For Affine Jump-Diffusions. Econometrica, 68(6):1342 1376, 2000 D. Duffie, D. Filipović, and W. Schachermayer, Affine Processes and Applications in Finance. Ann. Appl. Prob., 13(3):984 1053, 2003 C. Gourieroux, Continuous Time Wishart Process for Stochastic Risk. Econometric Reviews 25(2):177 217, 2006 M. Leippold and F. Trojani, Asset Pricing with Matrix Jump Diffusions. Preprint, 2010. Available Online at: http://ssrn.com/abstract=1572576 I. Karatzas and S. Shreve, Brownian Motion and Stochastic Calculus. Springer, 2nd edition, 1991