Estimating Fractional Stochastic Volatility 1

Int. J. Contemp. Math. Sciences, Vol. 7, 2012, no. 38, 1861 - 1869

Hoang Thi Phuong Thao

Hanoi University of Natural Sciences Vietnam National University 334 Nguyen Trai Str. Thanh Xuan, Hanoi, Vietnam [email protected]

Tran Hung Thao

Institute of Mathematics Vietnamese Academy of Science and Technology No 18 Hoang-Quoc-Viet Road, Hanoi, Vietnam. [email protected]

Abstract

The aim of this paper is to ﬁnd the best state estimation for fractional stochastic volatility from point process observations by a method of semimartingale approximation.

Mathematics Subjects Classiﬁcation: 60H30, 91B28

Keywords: State estimation, Fractional stochastic volatility, L2-approximation, Point process

1 Introduction

It is known that stochastic volatility models for options and derivatives were developed out of a need to modify the Black-Scholes model, which failed to eﬀectively take the volatility in the price of underlying security into account. The Black-Scholes model assumed that the volatility of the underlying security was constant, while stochastic volatility models categorized the price of the underlying security as a random variable or more general, a stochastic process. In its turn, the dynamics of this stochastic process can be driven by some other process (commonly by a Brownian motion). Various models have been created for stochastic volatility such as Heston model, Constant Elasticity of 1This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2011.12. 1862 H. T. P. Thao and T. H. Thao

Variance model (CEV model), SABR volatility model, GARCH model, 3/2- model, Chen model,. . . In many situations, the dynamics of the volatility has a long-range eﬀect, for instance changes of very strong bluechips in a stock market. One thinks of a kind of stochastic volatility perturbed by a process of long memory such as a fractional Brownian motion. In this paper we consider the problem of state estimation of a stochastic volatility Vt given by the fractional stochastic diﬀerential equation

H dVt = −bVtdt + σdBt ,

H where Bt is a fractional Brownian motion of Liouville form, from a point process observation Yt. H Recall that a fractional Brownian motion Wt with Hurst index H is a centered H H Gaussian process such that its covariance function R(s, t)=EWs Wt is given by 1 R(s, t)= (t2H + s2H −|t − s|2H ) where 0

t H 1 α Wt = [Zt + (t − s) dWs], Γ(1 + α) 0 W α H − 1 where is a standard Brownian motion, = 2 and

0 α α Zt = [(t − s) − (−s) ]dWs. −∞

Since Zt has absolutely continuous trajectories [1] one can see that the term BH t t − s αdW ρ t := 0 ( ) s has long memory in the following sense (see [4]). If n = H H H ∞ E(B1 (Bn+1 − Bn )) then the series n=0 ρn is either divergent or convergent H with very late rate. Bt is not a semimartingale as well and it is known as a fractional Brownian motion of Liouville form. Every stochastic processes here are supposed to be adapted to a ﬁltration (Ft) in a probability space (Ω, F,P). Now we try to estimate the stochastic volatility Vt satisfying

H dVt = −bVtdt + σdBt , 0 ≤ t ≤ T, (1.1.1) Estimating fractional stochastic volatility 1863

2 V0 ∼ N(μ0,ν0 ),b and σ are positive constants from an observation given by a point process of intensity λt t Yt = λsds + Mt, (1.1.2) 0 λ E t λ2ds < ∞ M F where t is a progressive process such that 0 s , t is a t- mar- H tingale and V0,Bt and Mt are independent. We will use an approximation approach given in [7] to solve this problem.

H 2 Approximation for the fractional process Bt

In this section we will make a brief recall on a principal result for L2-approximation of the fractional Brownian motion (refer to [7]). For every >0 we deﬁne t H, H− 1 Bt = (t − s + ) 2 dWs, (2.2.1) 0 H 1 ,

H, H 2 Theorem 2.1. (See[7]) Bt converges to Bt in L (Ω) when tends to 0. This convergence is uniform with respect to t ∈ [0,T]. Moreover we have H, H α+ 1 ||Bt − Bt ||L2(Ω) ≤ C(α) 2 where C(α) depends only on α. By this theorem we can see now any fractional stochastic dynamical system driven by a fractional Brownian notion can be approximated in L2 by a system governed by a semimartingale Bt , where the traditional stochastic calculus can be applied. Then the remaining work is to prove the convergence of approximate solutions to the exact solution. Notice that the exact solution of (1.1) can be deﬁned as an adapted process Vt satifying V −b t V ds BH ,V ∼ N μ ,ν2 . t = 0 s + t 0 ( 0 0 ) And our method of estimating stochastic volatility Vt is included in this con- text. 1864 H. T. P. Thao and T. H. Thao

3 Approximate Fractional Stochastic Volatil- ity

Instead of (1.1) and (1.2) we consider the following estimating problem: The system dynamics for volatility is given by

H, dVt = −bVt dt + σdBt , 0 ≤ t ≤ T (3.3.1)

2 V0 = V0 ∼ N(μ0,ν0 ).

And the observation dynamics is

t Yt = λsds + Mt, 0 ≤ t ≤ T. (3.3.2) 0

H, By replacing Bt in (3.1) by its expression in (2.2) we have

α dVt =[−bVt + σαϕt]dt + σdWt. (3.3.3)

We can see that (3.3) can be splitted into two equations

,1 ,1 α dVt = −bVt dt + σdWt, (3.3.4)

,2 ,2 dVt =[−bVt + σαϕt]dt. (3.3.5)

Where (3.4) is a traditional stochastic Langevin equation and (3.5) is in fact an ordinary diﬀerential equation. And the solution Vt of (3.3) should be ,1 ,2 Vt = Vt + Vt . So the state estimation problem is splitted into two problems as well

Y ,1 Y ,2 Y E(Vt |Ft )=E(Vt |Ft )+E(Vt |Ft ).

Notation:

Y (i) The state estimation of a process Xt based on the information Ft given by an observation process Yt will be denoted by Xˆt or πt(X). More Y general, the notation πt(f) stands for the state estimation E(f(Xt)|Ft ), where f is a bounded and continuous function, f ∈ Cb(R). Recall that Y Ft = σ(Ys,s ≤ t) is the σ-algebra generated by all random variable Ys with 0 ≤ s ≤ t.

(ii) Suppose that the probability P is obtained from a probability Q by an absolutely continuous change of measure Q → P such that nt = Yt − t Estimating fractional stochastic volatility 1865

Y is a (Q, Ft )- martingale. A Bayes formula gives us

Y Y EQ(XtLt|Ft ) EP (Xt|Ft )= , EQ(Lt)

where dP t L E |F Y λ Y − λ ds. t = [dQ t ]= s s exp (1 s) 0≤s≤t 0

Denote by σ(Xt)orX˜t the unnormalized state estimation of Xt, given by

Y X˜t = σ(Xt)=EQ(LtXt|Ft ),

or more general

Y σ(f)=EQ(Ltf(Xt)|Ft ), for f ∈ Cb(R).

,1 4 Estimating Vt

,1 Y Theorem 4.1. (i) The state estimation πt(f)=E[f(Vt )|Ft ] for the system dynamics (3.4) with the point process observation (3.2) is given by

t α 2 ,1 ,1 ( σ) ,1 πt(f)=π0(f)+ πs[−bVt f (Vt )+ f (Vt )]ds 0 2b

t −1 + πs (λ)[πs−(λf) − πs(f)πs−(λ)]dms, (4.4.1) 0 m Y − t λ ds, f ∈ C2 R . where t = t 0 s b ( )

,1 Y (ii) The unnormalized state estimation σt(f)=EQ[Ltf(Vt )|Ft ] is given by

t α 2 ,1 ,1 ( σ) ,1 σt(f)=σ0(f)+ σs[−bVt f (Vt )+ f (Vt )]ds 0 2b

t + [σs−(λf) − σs(f)]dμs, (4.4.2) 0

2 where μt = Yt − t, f ∈ Cb (R). 1866 H. T. P. Thao and T. H. Thao

,1 Proof. It is known that the solution Vt is in fact an Ornstein-Uhlenbeck process that is a Markov process and its semigroup is deﬁned by a family (Pt,t≥ 0) of operations on bounded Borelian function f. ασ 2 −bt ( ) −2bt (Ptf)(x)= f[e x + 1 − e y]μ(dy), (4.4.3)

2b Ê where μ is a standard Gaussian measure on R

1 −y2 μ(dy)=√ exp ( )dy. 2π 2

,1 It is obvious that lim(Ptf)(x)=f(x) then Vt is a Feller process. The corre- t0 sponding inﬁnitesimal operator At is given by

α 2 1 ( σ) (Atf)(x) = lim (Ptf − f)(x)=−bxf (x)+ f (x). (4.4.4) t→0 t 2b

According to a result from [5], the state estimation πt(f) for a Fellerian system Y t λ ds M process from a point process observation t = 0 s + t is given by

t t −1 πt(f)=π0(f)+ πs(Af)ds + πs (λ)[πs−(λf) − πs(f)πs−(λ)]dms, 0 0 m Y − t π λ ds, π−1 where t = t 0 s( ) provided t = 0 and the unnormalized state estimation σt(f) is given by

t t σt(f)=σ0(f)+ σs(Af)ds + [σs−(λf) − σs(f)]dμs, 0 0 where μt = Yt − t. By replacing Af in these equations by its expression (4.4) we get equations (4.1) and (4.2) as required.

Corollary 4.1. By taking f = I (identiﬁcation function) we have

t t ,1 ,1 ,1 1 ,1 ,1 Vt = V0 − bVs ds + [λsVs − λˆs−Vs ](dYs − λˆsds). (4.4.5) 0 0 λˆs

t t ,1 ,1 ,1 ,1 Vt := σt(I)=V0 − bVs ds + [λ˜s− − Vs ](dYs − ds). (4.4.6) 0 0 Estimating fractional stochastic volatility 1867

,2 5 Estimating Vt and Vt .

It is clear from (3.5) that

t ,2 ,2 ,2 Vt = V0 + [−bVs + σαϕˆs]ds (5.5.1) 0 and t ,2 ,2 ,2 Vt = V0 + [−bVs + σαϕ˜s]ds. (5.5.2) 0 Remark 5.1. These equations have explicit solutions as follows:

t ,2 ,2 −bt −b(t−s) Vt = V0 e − σα e ϕˆsds (5.5.3) 0 and t ,2 ,2 −bt −b(t−s) Vt = V0 e − σα e ϕ˜sds. (5.5.4) 0 Combining (4.5), (4.6) and (5.1), (5.2) we get Corollary 5.1.

,1 ,2 Vt = Vt + Vt (5.5.5) t t t V − bV ds σαϕ ds 1 λV ,1 − λˆ V,1 dY − λˆ ds = 0 s + s + [ s s s− s ]( s s ) 0 0 0 λs (5.5.6)

,1 ,2 Vt = Vt + Vt (5.5.7) t t t ,1 = V0 − b Vs ds + σα ϕsds + [λ˜s− − Vs ](dYs − ds), (5.5.8) 0 0 0

,1 ,2 ,1 ,2 where we assume that V0 + V0 = V0 and V0 + V0 = V0 .

6 Convergence of Vt to the solution Vt

Assume that Vt and Vt are solutions of (1.1) and (3.1), as in [5] we can establish

2 Proposition 6.1. Vt converges to Vt in L (Ω) uniformly with respect to t ∈ [0,T]. 1868 H. T. P. Thao and T. H. Thao

Proof. We have

t H H, Vt − Vt = −b (Vs − Vs )ds + σ(Bt − Bt ) (6.6.1) 0 so

t H H, ||Vt − Vt || ≤ ||b (Vs − Vs )ds|| + σ||Bt − Bt || (6.6.2) 0

It follows from (2.1) that

t α+ 1 ||Vt − Vt || ≤ b ||Vs − Vs ||ds + σC(α) 2 , (6.6.3) 0 where ||.|| stands for the norm in L2(Ω). Applying Gronwall’s Lemma to (6.3) we will get

α+ 1 bt ||Vt − Vt || ≤ σC(α) 2 e . (6.6.4)

And we see that

α+1 bT sup ||Vt − Vt || ≤ σC(α) e . (6.6.5) 0≤t≤T

2 So Vt −→ Vt in L (Ω) uniformly with respect to t ∈ [0,T].

7 Estimating volatility Vt

Proposition 7.1. The state estimation Vˆt of the stochastic volatility Vt is the 2 ,1 ,2 L -limit of Vˆt = Vt + Vt as → 0 and the convergence is uniform with respect to t ∈ [0,T].

Proof. The assertion of this proposition follows from a result on the conver- L2 2 gence of condition expectation (refer, for instance, to [3]). If Vt −→ Vt ∈ L Y Y as → 0 then E(Vt |Ft ) −→ E(Vt|Ft )as → 0. And the uniform convergence is deduced from the estimation (6.5).

References

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[5] T. H. Thao, Optimal State Estimation of A Markov Process From Point Process Observations, Annales Scientiﬁques de l’Universit´e Blaise Pascal, Clermont Ferrand II, Fasc. 9, 1991, 1-10.

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Received: April, 2012