Generalized Affine Transform Formulae and Exact Simulation of the WMSV Model

Transcription

1 On of Affine Processes on S + d Generalized Affine and Exact Simulation of the WMSV Model Department of Mathematical Science, KAIST, Republic of Korea 2012 SIAM Financial Math and Engineering joint work with Wanmo Kang

2 On of Affine Processes on S + d Outline 1 On of Affine Processes on S + d 2

3 On of Affine Processes on S + d Table of Contents 1 On of Affine Processes on S + d 2

4 On of Affine Processes on S + d Literature Affine processes have received increasing interests in the literature of stochastic processes and computational finance Duffie et al. (2003) gives the mathematical foundation of affine processes on R m + R n Cuchiero et al. (2011) complements their results, namely they provides complete parametric characterization of affine processes on S + d S + d : the cone of d d symmetric positive semidefinite matrices

5 On of Affine Processes on S + d Applications of Affine Processes on S + d Stochastic covariance modeling for multivariate option pricing (Gourieroux and Sufana 2010, Da Fonseca et al. 2007, Barndorff-Nielsen and Stelzer 2011) Multifactor stochastic volatility modeling (Da Fonseca et al. 2008) Optimal portfolio choice with correlation risk (Buraschi et al. 2010)

6 On of Affine Processes on S + d Reason for Popularity Natural class of stochastic processes for covariance modeling (S + d -valued) Flexible enough to capture stylized facts in financial markets (state-dependent diffusion, jumps) Still computationally tractable (affine property)

7 On of Affine Processes on S + d Definition of Affine Processes on S + d Definition (Cuchiero et al. 2011) A time-homogeneous Markov process X with state space S + d 1 it is stochastically continuous, is called affine if 2 its Laplace transform has exponential-affine dependence on the state variable E x [e tr(ux T ) F t] = e φ(t t,u) tr(xt ψ(t t,u)), (Affine Transform Formula) for all t R + and u, x S + d, for some functions φ : R+ S+ d ψ : R + S + d S+ d. R+ and We call X conservative if it does neither explode nor be killed, i.e., X is conservative if and only if X t S + d for all t 0 with probability 1. We confine ourselves to the class of conservative affine processes on S + d.

8 On of Affine Processes on S + d Properties & Notions (Cuchiero et al. 2011) Affine transform formula E x [e tr(ux T ) φ(t t,u) tr(xt ψ(t t,u)) F t] = e Functional characteristics: F : S d R and R : S d S d F (u) = t φ(t, u) t=0, R(u) = t ψ(t, u) t=0. Generalized Riccati differential equations φ(t, u) t = F (ψ(t, u)), ψ(t, u) t = R(ψ(t, u)), with initial values φ(0, u) = 0 and ψ(0, u) = u Conservative affine processes on S + d are semimartingales

9 On of Affine Processes on S + d Motivation The affine transform formula provides a method for computing Laplace transform of marginal distributions of affine processes, and it gives a connection between such transforms and the generalized Riccati differential equations. In some cases, we are required to compute Laplace transforms of more general functionals of affine processes For example, E x [ e tr( ux T ) tr ( v T t X sds ) F t ], E x [ e tr( T t g(s)x sds) Ft ] More generally, E x [ e tr( T t X sκ(ds) ) F t ]

10 On of Affine Processes on S + d The Main Question For a conservative affine process X and an S + d -valued measure κ(ds) on (0, T ], how can we compute the following transform E x [ e tr( T t X sκ(ds) ) F t ]? Is there an equation which governs the above transforms? If so, does the equation have a solution?

11 On of Affine Processes on S + d The Answer for Squared Bessel Processes δ-dimensional squared Bessel process on S + 1 = R+ dx t = δdt + 2 X tdw t, X 0 = x R + Theorem (Pitman-Yor 1982) For a δ-dimensional squared Bessel process X and a positive Radon measure κ on (0, ), the following holds [ E x e X ] ( 0 tκ(dt) = φ( ) δ/2 x ) exp 2 φ +(0), where φ is the unique solution (in the distribution sense) of: φ = 2κφ on (0, ), φ(0) = 1, 0 φ 1.

12 On of Affine Processes on S + d Main Theorem Theorem (Kang and Kang 2012 A) Let X be a conservative affine process on S + d. Then, for every S + d -valued measure κ on (0, T ], we have E x [ e tr( T t X sκ(ds) ) F t ] = e φ(t,κ) tr(x t ψ(t,κ)), where ( φ(, κ), ψ(, κ)) is a bounded R + S + d -valued solution on [0, T ] to the following integral equation φ(t, κ) = T t T F ( ψ(s, κ))ds, ψ(t, κ) = κ(t, T ] + R( ψ(s, κ))ds, t where F and R are the functional characteristics of X.

13 On of Affine Processes on S + d Analogy with Affine Transform Formula Affine Transform Formula [ E x e tr(ux T ) ] φ(t t,u) tr(xt ψ(t t,u)) Ft = e T T φ(t t, u) = F (ψ(t s, u))ds, ψ(t t, u) = u + R(ψ(T s, u))ds t t Our Transform Formula E x [ e tr( T t X sκ(ds) ) Ft ] = e φ(t,κ) tr(x t ψ(t,κ)) T T φ(t, κ) = F ( ψ(s, κ))ds, ψ(t, κ) = κ(t, T ] + R( ψ(s, κ))ds t t In particular, these equations coincide if κ = uε T

14 On of Affine Processes on S + d Idea of Proof We take, for 0 t T, Z κ t { = exp tr ( X t ψ(t, κ) ) +tr ( x ψ(0, κ) ) + φ(0, κ) φ(t, κ) t 0 tr( X s κ(ds) )}. Z κ t Z κ t Z κ t is a local martingale by Itô s formula is bounded because ( φ(, κ), ψ(, κ)) is R + S + d -valued is a martingale Take expectation to prove our theorem Does the equation have solution ( φ(, κ), ψ(, κ))?

15 On of Affine Processes on S + d Existence Result Theorem (Kang and Kang 2012 A) For every S + d -valued measure κ on (0, T ], the system of equations T T φ(t, κ) = F ( ψ(s, κ))ds, ψ(t, κ) = κ(t, T ] + R( ψ(s, κ))ds, t t has a bounded R + S + d -valued solution.

16 On of Affine Processes on S + d Question about Bridges Can we extend our transform formula to the bridges of affine processes? [ E x e tr( T 0 Xsκ(ds)) ] X T = y

17 On of Affine Processes on S + d Change of Measure As we have shown before, Z κ t is a martingale and Z κ 0 = 1 We define an equivalent probability P κ x on F T by dpκ x dp x = Z κ T We can compute the differential characteristic of X under P κ x theorem or our transform formula X is a time-inhomogeneous affine process under P κ x by Girsanov

18 On of Affine Processes on S + d Formula for Bridge PROPOSITION (Kang and Kang 2012 A) Let X be a conservative affine process on S + d. Then for all S+ d -valued measures κ on (0, T ] and for all x S + d we have [ E x e tr( T 0 Xsκ(ds)) ] XT = y = e φ(0,κ) tr( ψ(0,κ)x) pκ 0,T (x, dy) p 0,T (x, dy), p 0,T (x, dy)-a.s., where pκ 0,T (x,dy) is the Radon-Nikodym derivative of the transition p 0,T (x,dy) kernel p κ (x, dy) of X under 0,T Pκ x with respect to the transition kernel p0,t (x, dy) of X under P x.

19 On of Affine Processes on S + d A Remark The transition kernel is not known in general But there is an important class of affine processes with well-known transition kernel(wishart processes)!

20 On of Affine Processes on S + d Wishart Processes Wishart process : a weak solution to the SDE dx t = (δσ Σ + HX t + X th )dt + X tdw tσ + Σ dwt Xt, Wishart processes are typical affine diffusion processes on S + d Wishart processes have noncentral Wishart transition distributions Laplace transform and probability density functions are known in closed forms They were introduced by Bru (1991)

21 On of Affine Processes on S + d Example 1 Let X be a Wishart process such that δ > d 1, Σ Σ S ++ d, HΣ Σ = Σ ΣH. Then, for any λ S + d, the formula holds [ E x exp { 1 2 tr( λ 2 ] T 0 Xtdt)} XT = y ( ) det(ξcsch (T ξ)) δ/2 = det(ζcsch (T ζ)) { exp 1 ((Σ 2 tr 1 ) (x + y)σ 1( ζ coth(t ζ) ξ coth(t ξ) ))} 0 F 1 ( 1 2 δ; 1 4 ξcsch (T ξ)(σ 1 ) xσ 1 csch (T ξ)ξ(σ 1 ) yσ 1) 0F 1 ( 1 2 δ; 1 4 ζcsch (T ζ)(σ 1 ) xσ 1 csch (T ζ)ζ(σ 1 ) yσ 1), where ξ = Σ(λ 2 + H (Σ Σ) 1 H)Σ and ζ = ΣH (Σ Σ) 1 HΣ.

22 On of Affine Processes on S + d Example 2 Let X be a Wishart process with Σ = I d, H = 0. Then, for any u S + d 0 < T 0 < T, the following formula holds and E x [e tr(ux T 0 ) ] XT = y = (T d det ( U(T 0 ) )) δ/2 ( ) exp { (U(T 1 T tr 0 )u ( 1 0F 1 (T T 0 ) 2 x + T0 2y))} 2 δ; 1 4 U(T 0)xU(T 0 )y ( ), 1 0F 1 2 δ; 1 4T 2 xy where U(T 0 ) = (TI d + 2(T T 0 )T 0 u) 1.

23 On of Affine Processes on S + d Table of Contents 1 On of Affine Processes on S + d 2

24 On of Affine Processes on S + d Wishart Multidimensional Stochastic Volatility Model A single asset multifactor stochastic volatility model A generalization of Heston s stochastic volatility model Flexibility due to multifactor nature Still computationally tractable due to its affine property Introduced by Da Fonseca et al. (2008)

25 On of Affine Processes on S + d Model Dynamics The asset price S t = e Yt is described by the stochastic differential equations dy t = (r 1 ) [ ] 2 Xt dt + tr XtdBt, dx t = (δσ Σ + HX t + X th )dt + X tdw tσ + Σ dw t db t = dw tr + dz t Id RR Xt, r: a constant which represents the risk neutral drift B t, Z t: independent d d matrix Brownian motions

26 On of Affine Processes on S + d Model Dynamics The asset price S t = e Yt is described by the stochastic differential equations dy t = (r 1 ) [ ] 2 Xt dt + tr XtdBt, dx t = (δσ Σ + HX t + X th )dt + X tdw tσ + Σ dw t db t = dw tr + dz t Id RR Xt, r: a constant which represents the risk neutral drift B t, Z t: independent d d matrix Brownian motions

27 On of Affine Processes on S + d Model Dynamics The asset price S t = e Yt is described by the stochastic differential equations dy t = (r 1 ) [ ] 2 Xt dt + tr XtdBt, dx t = (δσ Σ + HX t + X th )dt + X tdw tσ + Σ dw t db t = dw tr + dz t Id RR Xt, r: a constant which represents the risk neutral drift B t, Z t: independent d d matrix Brownian motions

28 On of Affine Processes on S + d Model Dynamics The asset price S t = e Yt is described by the stochastic differential equations dy t = (r 1 ) [ ] 2 Xt dt + tr XtdBt, dx t = (δσ Σ + HX t + X th )dt + X tdw tσ + Σ dw t db t = dw tr + dz t Id RR Xt, r: a constant which represents the risk neutral drift B t, Z t: independent d d matrix Brownian motions

29 On of Affine Processes on S + d Motivation The monte-carlo simulation is the only viable method to price derivatives with complicated payoff structure We want to devise a simulation method of WMSV model which does not suffer from bias error (Exact simulation) Since the model has time-homogeneous Markov property, it suffices to devise a method to simulate the state variable for a single period Hence, our question is How can we simulate (X T, Y T ) from its distribution?

30 On of Affine Processes on S + d Exact Simulation Methods 1 Generate X T from the distribution of X T 2 Generate Y T from the conditional distribution of Y T given X T

31 On of Affine Processes on S + d How to generate X T? It is well-known that X T has noncentral Wishart distribution There are many ways to simulate noncentral Wishart distribution for δ N Recently, Ahdida and Alfonsi (2010) developed a method to simulate noncentral Wishart distribution for δ N

32 On of Affine Processes on S + d Conditional Laplace Transform of Y T given X T There were NO previous results on the conditional distribution of Y T given X T We computed the following conditional Laplace transform in a semi-analytic form ] E [e uy T X 0, X T

33 On of Affine Processes on S + d Theorem (Kang and Kang 2012 B) The conditional Laplace transform of log-price Y T given X T S + d satisfies ( ) [ E e uy ] δ/2 T det[v (0, 0)] { } X 0, X T = exp φ(0, u) det[v (0, u)] { exp 2 1 tr[ (2ψ(0, u) + Ψ(0, u)v (0, u) 1 Ψ(0, u) Ψ(0, 0)V (0, 0) 1 Ψ(0, 0) ] } )X 0 { exp 2 1 tr[ (V (0, u) 1 V (0, 0) 1 ] } )X T ( ) 12 δ; 1 4 V (0, u) 1 Ψ(0, u) X 0 Ψ(0, u)v (0, u) 1 X T 0 F 1 0F 1 ( 12 δ; 1 4 V (0, 0) 1 Ψ(0, 0) X 0 Ψ(0, 0)V (0, 0) 1 X T ), where the matrix-valued functions ψ, Ψ, V, and the real-valued function φ are the solution of the system of ordinary differential equations: t ψ(t, u) = 2ψ(t, u)σ Σψ(t, u) (H urσ)ψ(t, u) ψ(t, u)(h uσ R ) + u(u+1) I 2 d, t φ(t, u) = δtr[ψ(s, u)σ Σ] ur, t Ψ(t, u) = (H urσ 2ψ(t, u)σ Σ)Ψ(t, u), t V (t, u) = Ψ(t, u) Σ ΣΨ(t, u), for 0 t T, with terminal values ψ(t, u) = V (T, u) = 0, Ψ(T, u) = I d, and φ(t, u) = 0.

34 On of Affine Processes on S + d How to conditionally generate Y T given X T? By taking u = iλ in the conditional Laplace transform, we found the conditional characteristic function Y T given X T ] ϕ(λ; X 0, X T ) = E [e iλy T X 0, X T ) Conditional distribution function: F (y; X 0, X T ) = P (Y T y X 0, X T The distribution function can be obtained by inverting the characteristic function F (y; X 0, X T ) = F (y ɛ; X 0, X T ) + 1 π 0 [ ] Im ϕ(λ; X 0, X T )(e iλyɛ e iλy dλ ) λ

35 On of Affine Processes on S + d How to conditionally generate Y T given X T? We can conditionally generate Y T given X T Y T = F 1 (U; X 0, X T ) where U is a uniform random variate between 0 and 1

36 On of Affine Processes on S + d Euler Discretization Equally spaced time grids 0 = t 0 < t 1 < < t N = T, t i = it N, t = T N Discretized model, ˆX t0 = ˆX 0 = X 0 ( ˆX ti = ˆXti 1 + (δσ Σ + H ˆX + ˆX ti 1 ti 1 H ) t + ˆX ti 1 Wt Σ + Σ ( ) ) +, i W ti ˆX ti 1 Ŷ ti = ( Ŷ ti 1 + r 1 ) ] 2 tr[ ˆX ti 1 ] t + tr[ ˆX ti 1 Bt i, To prevent ˆX ti S + d, we take the positive part at each time grid

37 On of Affine Processes on S + d Call Option Prices We compute the call option prices using our exact simulation method and Euler discretization method Theoretical price: (by transform method) Methods Exact Euler No. of No. of Time MC estimates std. errors time steps simulation runs (sec) N/A

38 On of Affine Processes on S + d Call Option Prices Theoretical price: (by transform method) RED numbers are those for which the theoretical price is outside of the 95% confidence interval. Methods Exact Euler No. of No. of Time MC estimates std. errors time steps simulation runs (sec) N/A

39 On of Affine Processes on S + d Call Option Prices Theoretical price: (by transform method) RED numbers are those for which the theoretical price is outside of the 95% confidence interval. Methods Exact Euler No. of No. of Time MC estimates std. errors time steps simulation runs (sec) N/A

40 On of Affine Processes on S + d Summary 1 Transform formulae for affine processes on S + d We provide a general recipe for computing Laplace transforms of linear functional of affine processes and their bridges on S + d In particular, we establish the relationship between such transforms and certain integral equations We prove the existence of the solutions of such integral equations Using our method, we derive some explicit transform formulae for Wishart process 2 Exact simulation of WMSV model We devise an exact simulation method for WMSV model Our method is superior to the standard Euler discretization method in terms of accuracy and performance

41 On of Affine Processes on S + d Thank you for your attention!