generalized affine transform formulae and exact simulation ... · exact simulation of wmsv model...

On Transform Formulae of Affine Processes on S+d

Exact Simulation of WMSV Model

Generalized Affine Transform Formulae

and Exact Simulation of the WMSV Model

Chulmin Kang

Department of Mathematical Science, KAIST, Republic of Korea

2012 SIAM Financial Math and Engineering

joint work with Wanmo Kang

Chulmin Kang Transform Formulae



Outline

1 On Transform Formulae of Affine Processes on S+d

Literature Review

Main Results

2 Exact Simulation of WMSV Model

Introduction to WMSV Model

Exact Simulation Technique

Numerical Results



Exact Simulation of WMSV ModelLiterature ReviewMain Results

Table of Contents


Literature Review

Main Results




Numerical Results




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 Rm+ × Rn

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




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)




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)




Definition of Affine Processes on S+d

Definition (Cuchiero et al. 2011)

A time-homogeneous Markov process X with state space S+d is called affine if

1 it is stochastically continuous,

2 its Laplace transform has exponential-affine dependence on the state variable

Ex [e−tr(uXT )|Ft ] = e−φ(T−t,u)−tr(Xtψ(T−t,u)), (Affine Transform Formula)

for all t ∈ R+ and u, x ∈ S+d , for some functions φ : R+ × S+

d → R+ and

ψ : R+ × S+d → S+

d .

We call X conservative if it does neither explode nor be killed, i.e., X is conservative if

and only if Xt ∈ S+d for all t ≥ 0 with probability 1. We confine ourselves to the class

of conservative affine processes on S+d .




Properties & Notions (Cuchiero et al. 2011)

Affine transform formula

Ex [e−tr(uXT )|Ft ] = e−φ(T−t,u)−tr(Xtψ(T−t,u))

Functional characteristics: F : Sd → R and R : Sd → Sd

F (u) =∂

∂tφ(t, u)

∣∣∣t=0

, R(u) =∂

∂tψ(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




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,

Ex

[e−tr

(uXT )− tr

(v∫ T

t Xs ds)|Ft

], Ex

[e−tr

( ∫ Tt g(s)Xs ds)|Ft

]More generally,

Ex

[e−tr

( ∫ Tt Xsκ(ds)

)|Ft

]




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

Ex

[e−tr

(∫ T

tXsκ(ds)

)|Ft

]?

Is there an equation which governs the above transforms?

If so, does the equation have a solution?




The Answer for Squared Bessel Processes

δ-dimensional squared Bessel process on S+1 = R+

dXt = δdt + 2√

XtdWt , X0 = 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

Ex

[e−∫∞

0Xtκ(dt)

]= φ(∞)δ/2 exp

(x

2φ′+(0)

),

where φ is the unique solution (in the distribution sense) of:

φ′′ = 2κφ on (0,∞), φ(0) = 1, 0 ≤ φ ≤ 1.




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

Ex

[e−tr

(∫ T

tXsκ(ds)

)∣∣Ft

]= e−φ(t,κ)−tr(Xt 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

F (ψ(s, κ))ds, ψ(t, κ) = κ(t,T ] +

∫ T

t

R(ψ(s, κ))ds,

where F and R are the functional characteristics of X .




Analogy with Affine Transform Formula

Affine Transform Formula

Ex

[e−tr(uXT )

∣∣Ft

]= e−φ(T−t,u)−tr(Xtψ(T−t,u))

φ(T−t, u) =

∫ T

tF (ψ(T−s, u))ds, ψ(T−t, u) = u +

∫ T

tR(ψ(T−s, u))ds

Our Transform Formula

Ex

[e−tr

(∫ Tt Xsκ(ds)

)∣∣Ft

]= e−φ(t,κ)−tr(Xt ψ(t,κ))

φ(t, κ) =

∫ T

tF (ψ(s, κ))ds, ψ(t, κ) = κ(t,T ] +

∫ T

tR(ψ(s, κ))ds

In particular, these equations coincide if κ = uεT




Idea of Proof

We take, for 0 ≤ t ≤ T ,

Zκt = exp{−tr(Xt ψ(t, κ)

)+tr(xψ(0, κ)

)+φ(0, κ)−φ(t, κ)−

∫ t0 tr(Xs−κ(ds)

)}.

Zκt is a local martingale by Ito’s formula

Zκt is bounded because (φ(·, κ), ψ(·, κ)) is R+ × S+d -valued

Zκt is a martingale

Take expectation to prove our theorem

Does the equation have solution (φ(·, κ), ψ(·, κ))?




Existence Result

Theorem (Kang and Kang 2012 A)

For every S+d -valued measure κ on (0,T ], the system of equations

φ(t, κ) =

∫ T

tF (ψ(s, κ))ds, ψ(t, κ) = κ(t,T ] +

∫ T

tR(ψ(s, κ))ds,

has a bounded R+ × S+d -valued solution.




Question about Bridges

Can we extend our transform formula to the bridges of affine processes?

Ex

[e−tr

(∫ T0 Xsκ(ds)

)∣∣XT = y]




Change of Measure

As we have shown before, Zκt is a martingale and Zκ0 = 1

We define an equivalent probability Pκx on FT bydPκxdPx

= ZκT

We can compute the differential characteristic of X under Pκx by Girsanov

theorem or our transform formula

X is a time-inhomogeneous affine process under Pκx




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

Ex

[e−tr

(∫ T0 Xsκ(ds)

)∣∣XT = y]

= e−φ(0,κ)−tr(ψ(0,κ)x) pκ0,T (x , dy)

p0,T (x , dy),

p0,T (x , dy)-a.s., wherepκ0,T (x,dy)

p0,T (x,dy)is the Radon-Nikodym derivative of the transition

kernel pκ0,T (x , dy) of X under Pκx with respect to the transition kernel p0,T (x , dy) of X

under Px .




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)!




Wishart Processes

Wishart process : a weak solution to the SDE

dXt = (δΣ>Σ + HXt + Xt H>)dt +√

Xt dWt Σ + Σ>dW>t√

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)




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

Ex

[exp

{− 1

2tr(λ2∫ T

0 Xt dt)}∣∣∣XT = y

]=

(det(ξcsch (Tξ))

det(ζcsch (Tζ))

)δ/2

× exp{

12

tr(

(Σ−1)>(x + y)Σ−1(ζ coth(Tζ)− ξ coth(Tξ)

))}× 0F1

(12δ; 1

4ξcsch (Tξ)(Σ−1)>xΣ−1csch (Tξ)ξ(Σ−1)>yΣ−1

)0F1

(12δ; 1

4ζcsch (Tζ)(Σ−1)>xΣ−1csch (Tζ)ζ(Σ−1)>yΣ−1

) ,where ξ =

√Σ(λ2 + H>(Σ>Σ)−1H)Σ> and ζ =

√ΣH>(Σ>Σ)−1HΣ>.




Example 2

Let X be a Wishart process with Σ = Id , H = 0. Then, for any u ∈ S+d and

0 < T0 < T , the following formula holds

Ex

[e−tr(uXT0

)∣∣XT = y]

=(

T d det(U(T0)

))δ/2

× exp{− 1

Ttr(

U(T0)u((T − T0)2x + T 2

0 y))} 0F1

(12δ; 1

4U(T0)xU(T0)y

)0F1

(12δ; 1

4T 2 xy) ,

where U(T0) = (TId + 2(T − T0)T0u)−1.




Introduction to WMSV ModelExact Simulation TechniqueNumerical Results

Table of Contents


Literature Review

Main Results




Numerical Results





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)





Model Dynamics

The asset price St = eYt is described by the stochastic differential equations

dYt =(

r −1

2Xt

)dt + tr

[√Xt dBt

],

dXt = (δΣ>Σ + HXt + Xt H>)dt +√

Xt dWt Σ + Σ>dW>t√

Xt ,

dBt = dWt R> + dZt

√Id − RR>

r : a constant which represents the risk neutral drift

Bt , Zt : independent d × d matrix Brownian motions





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 (XT ,YT ) from its distribution?”





Exact Simulation Methods

1 Generate XT from the distribution of XT

2 Generate YT from the conditional distribution of YT given XT





How to generate XT?

It is well-known that XT 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 6∈ N





Conditional Laplace Transform of YT given XT

There were NO previous results on the conditional distribution of YT

given XT

We computed the following conditional Laplace transform in a

semi-analytic form

E[e−uYT |X0,XT

]





Theorem (Kang and Kang 2012 B)

The conditional Laplace transform of log-price YT given XT ∈ S+d

satisfies

E[

e−uYT∣∣∣X0, XT

]=

(det[V (0, 0)]

det[V (0, u)]

)δ/2

exp{− φ(0, u)

}× exp

{− 1

2tr[(2ψ(0, u) + Ψ(0, u)V (0, u)−1Ψ(0, u)> − Ψ(0, 0)V (0, 0)−1Ψ(0, 0)>)X0

]}× exp

{− 1

2tr[(V (0, u)−1 − V (0, 0)−1)XT

]}×

0F1

(12δ; 1

4V (0, u)−1Ψ(0, u)>X0Ψ(0, u)V (0, u)−1XT

)0F1

(12δ; 1

4V (0, 0)−1Ψ(0, 0)>X0Ψ(0, 0)V (0, 0)−1XT

) ,where the matrix-valued functions ψ, Ψ, V , and the real-valued function φ are the solution of the system of

ordinary differential equations:

∂tψ(t, u) = 2ψ(t, u)Σ>Σψ(t, u)

−(H> − uRΣ)ψ(t, u)− ψ(t, u)(H − uΣ>R>) +u(u+1)

2Id ,

∂tφ(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) = Id , and φ(T , u) = 0.





How to conditionally generate YT given XT?

By taking u = −iλ in the conditional Laplace transform, we found the

conditional characteristic function YT given XT

ϕ(λ; X0,XT ) = E[e iλYT |X0,XT

]Conditional distribution function: F (y ; X0,XT ) = P

(YT ≤ y |X0,XT

)The distribution function can be obtained by inverting the characteristic

function

F (y ; X0,XT )

= F (yε; X0,XT ) +1

π

∫ ∞0

Im[ϕ(λ; X0,XT )(e−iλyε − e−iλy )

]dλ

λ





How to conditionally generate YT given XT?

We can conditionally generate YT given XT

YT = F−1(U; X0,XT )

where U is a uniform random variate between 0 and 1





Euler Discretization

Equally spaced time grids 0 = t0 < t1 < · · · < tN = T , ti = iTN

, ∆t = TN

Discretized model, Xt0 = X0 = X0

Xti =(

Xti−1 + (δΣ>Σ + HXti−1 + Xti−1 H>)∆t

+√

Xti−1 ∆Wti Σ + Σ>(∆Wti

)>√Xti−1

)+

,

Yti = Yti−1 +(

r − 1

2tr[Xti−1 ]

)∆t + tr

[√Xti−1 ∆Bti

],

To prevent Xti 6∈ S+d , we take the positive part at each time grid





Call Option Prices

We compute the call option prices using our exact simulation method and Euler

discretization method

Theoretical price: 0.191575 (by transform method)

MethodsNo. of No. of

MC estimates std. errorsTime

time steps simulation runs (sec)

Exact N/A

50000 0.192415 0.1430× 10−2 12.14

100000 0.191451 0.1009× 10−2 24.28

500000 0.192143 0.4535× 10−3 121.4

1000000 0.191513 0.3202× 10−3 242.8

Euler

50

50000 0.195202 0.1456× 10−2 179.4

100000 0.194829 0.1034× 10−2 358.7

500000 0.193827 0.4603× 10−3 1793.5

1000000 0.194197 0.3259× 10−3 3587.0

100

50000 0.194016 0.1461× 10−2 358.4

100000 0.193264 0.1027× 10−2 716.8

500000 0.193008 0.4570× 10−3 3584.2

1000000 0.193073 0.3234× 10−3 7168.5





Call Option Prices

Theoretical price: 0.191575 (by transform method)

RED numbers are those for which the theoretical price is outside of the 95%

confidence interval.

MethodsNo. of No. of

MC estimates std. errorsTime

time steps simulation runs (sec)

Exact N/A

50000 0.192415 0.1430× 10−2 12.14

100000 0.191451 0.1009× 10−2 24.28

500000 0.192143 0.4535× 10−3 121.4

1000000 0.191513 0.3202× 10−3 242.8

Euler

50

50000 0.195202 0.1456× 10−2 179.4

100000 0.194829 0.1034× 10−2 358.7

500000 0.193827 0.4603× 10−3 1793.5

1000000 0.194197 0.3259× 10−3 3587.0

100

50000 0.194016 0.1461× 10−2 358.4

100000 0.193264 0.1027× 10−2 716.8

500000 0.193008 0.4570× 10−3 3584.2

1000000 0.193073 0.3234× 10−3 7168.5





Summary

1 Transform formulae for affine processes on S+d

I We provide a general recipe for computing Laplace transforms of linear

functional of affine processes and their bridges on S+d

I In particular, we establish the relationship between such transforms and

certain integral equations

I We prove the existence of the solutions of such integral equations

I Using our method, we derive some explicit transform formulae for Wishart

process

2 Exact simulation of WMSV model

I We devise an exact simulation method for WMSV model

I Our method is superior to the standard Euler discretization method in

terms of accuracy and performance





Thank you for your attention!


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