OPTION PRICING UNDER STOCHASTIC VOLATILITY MODEL WITH JUMPS IN BOTH THE STOCK PRICE AND THE VARIANCE PROCESSES

Abstract

Yan & Hanson [8] and Makate & Sattayatham [6] extended Bates' model to the stochastic volatility model with jumps in both the stock price and the variance processes. As the solution processes of finding the characteristic function, they sought such a function f satisfying $$f({\ell},{\nu},t;k,T)=exp\;(g({\tau})+{\nu}h({\tau})+ix{\ell})$$. We add the term of order ${\nu}^{1/2}$ to the exponent in the above equation and seek the explicit solution of f.

1. INTRODUCTION

The Heston model [5] is the following risk-neutral stock price processes

where St is a stock process, r is the riskless rate of return, vt is the volatility of asset returns, κ > 0 is a mean-reverting rate, θ is the long term variance, σ > 0 is the volatility of volatility, and and are two correlated Brownian motions under the risk-neutral measure with constant correlation coefficient ρ

The Bates [1] extended the Heston model (1.1) to include jumps in the stock price process. The model has the following dynamics which define the evolution of St satisfying

where the volatility process νt is the same as one in the Heston model and the driving Brownian motions in the two processes have an instantaneous correlation coefficient ρ, the process represents a Poisson process under the risk-neutral measure, with jump intensity λ. The Poisson process is independent of the two Brownian motions in the stock price and the variance processes. The percentage jump size of the stock price is denoted by the random variable Yt with log-normal distribution.

Eraker et al. [3] extended Bates model to a stochastic volatility model with contemporaneous jumps in the stock price and its volatility

Eraker et al. tested their model with empirical data and showed that the models with jumps performed better than those without jumps in volatility. Makate and Sattayatham [6] provide a formal ’closed-form solution’ of the stochastic-volatility jump-diffiusion model.

Heston’s [5] ’closed-form solution’ for risk-neutral pricing of European options is given by first converting the problem into characteristic functions, then using the Fourier inversion formula for probability distribution functions to find a more numerically robust form which everyone won’t call it closed. To solve for the characteristic function fj explicitly, Yan & Hanson [8] and Makate & Sattayatham [6] conjecture that its solution is given by

where β1(τ ) = 0 and β2(τ ) = rτ. In this paper, we add the term of order ν1/2 to the exponent in (1.3) for the exploit of nonlinearity and seek the explicit solution of fj .

This paper is structured as follows. The introduction is given in Section 1. The stochastic-volatility jump-diffiusion model is explained in detail in Section 2. The formulation for European call option pricing is given in Section 3.

 

2. STOCHASTIC-VOLATILITY JUMP-DIFFUSION MODEL

We assume that a risk-neutral probability measure Q exists. We also assume that the asset price St under Q follows a jump- diffiusion process, and the volatility νt follows a pure mean-reverting and square root diffiusion process with jump, e.g., our model is governed by the following dynamics

where St, νt, κ, θ, σ, , are the same ones defined as in Bates model (1.2), r is a risk-free interest rate, and are independent Poisson processes with constant intensities λS and λν respectively. Yt is the jump size of the asset price return with density ϕY (y) and E[Yt] = m, and Zt is the jump size of the volatility with density ϕZ(z). Moreover, we assume that the Poisson processes and are independent of standard Brownian motions and with

 

3. FORMULATION FOR EUROPEAN CALL OPTION PRICING

Let C denote the price at time t of a European style call option on St with strike price K and expiration time T. The terminal payoff of a European call option on the underlying stock St is

max {ST − K, 0 }

Assume that the short-term risk-free interest rate r is constant over the lifetime of the option. The price of the European call at time t equals the discounted and conditional expected payoff

where EQ is the expectation with respect to the risk-neutral probability measure Q and PQ(ST |St, νt) is the corresponding conditional density function given (St, νt).

Since

is a risk-neutral probability such that

ST > K, EQ[ST | St, νt] = er(T−t)St.

P2(St, νt, T; K, T) = ProbQ(ST > K|St, νt) is the risk-neutral in-the-money probability. Note that the complement of P2 is a risk-neutral distribution function. It is difficult to find the cumulative distribution function in European option pricing. The main job is to evaluate P1 and P2 under the distribution assumptions embedded in the risk-neutral probability measure.

We make a change of variable from St to Lt = ln St. Let k = ln K. By the jump-diffiusion chain rule, ln St satisfies the SDE

The value C of a European-style option as a function of Lt becomes

that is, we have

The Dynkin’s theorem [4] shows a relationship between stochastic diffierential equations and partial diffierential equations. If we apply two-dimensional Dynkin’s theorem for the price dynamics (3.2) and volatility νt in (2.1b) to (Lt, νt, t; k, T), then we obtain the following Partial Integro-Diffierential Equations (PIDE)

where is defined as

In the current state variables Lt = ℓ and νt = ν, the option value (3.1) becomes

where for j = 1, 2.

Lemma 3.1 ([6]). The functions in (3.3) satisfies the following PIDEs

with the boundary condition at expiration time t = T

in (3.3) also satisfies the following PIDEs

with the boundary condition at expiration time t = T

A1 and A2 in Lemma 3.1 are respectively defined as

and

For j = 1, 2 the characteristic functions for with respect to the variable k are defined as

in which a minus sign is given to account for the negativity of the measure For j = 1, 2, fj satisfies similar PIDEs as in (3.4) and (3.5)

with the boundary conditions

since

Let’s find the characteristic functions fj for j = 1, 2. Let τ = T − t be the time to go. We seek the functions f1 and f2 satisfying

f1(ℓ, ν, t; k, T) = exp(g1(τ ) + ν1/2h1(τ ) + (ν1/2)2h2(τ) + ixℓ), f2(ℓ, ν, t; k, T) = exp(g2(τ ) + ν1/2h3(τ ) + (ν1/2)2h4(τ) + ixℓ+ rτ).

respectively with the boundary conditions

gi(0) = 0 = hj(0) for i = 1, 2 and j = 1, 2, 3, 4.

Lemma 3.2. The functions and can be computed by the inverse Fourier transforms of the characteristic function, e.g.,

for j = 1, 2. Re[·] denote the real part of the complex number.

The characteristic function f1 is given by

f1(ℓ, ν, t; k, T) = exp(g1(τ ) + ν1/2h1(τ ) + νh2(τ) + ixℓ.

h2 is given by

where η1 = ρσ(ix + 1) − κ and h1 is given by

where γ1 (0+) represents a small value factor which appears in the coefficient of ν1/2 as the one of ν3/2.

which is equal to the equations as in [6] if the coefficient h1(τ ) of order ν1/2 is zero.

The characteristic function f2 is given by

f2(ℓ, ν, T; k, T) = exp(g2(τ ) + ν1/2h3(τ ) + νh4(τ) + ixℓ + rτ).

h4 is given by

where η2 = ρσix − κ and

h3 is given by

where γ2 (0+) represents a small value factor which appears in the coefficient of ν1/2 as the one of ν3/2.

which is equal to the equations as in [6] if the coefficient h3(τ) of order ν1/2 is zero.

Theorem 3.3. The value of a European call option of (3.3) is

where and are given in Lemma 3.2.

Now we prove Lemma 3.2.

Proof. For the derivation of the equation (3.7), refer to the paper [6]. Let us compute PDE (3.6). First let’s calculate some diffierentials regarding to f1.

f1(ℓ + y, ν, t; x, t + τ) − f1(ℓ, ν, t; x, t + τ) = (eixy − 1) f1(ℓ, ν, t; x, t + τ).

We use the series expansion, which is valid only when |z| < ν

in the following equation.

If we substitute the above diffierentials and equations into the equation (3.6), then we have

The coefficients of ν are

The solution of h2(τ) is given by

where η1 = ρσ(ix + 1) − κ and (See [6] for detail). The coefficients of ν1/2 are

where we denote γ1(0+) a small value factor which appears in the coefficient of ν1/2 as the one of ν3/2. We seek h1(τ) as series solution such as

h2 can be written as

where A1 = σ−2 (η1 + Δ1), B1 = (η1 + Δ1) / (η1 − Δ1). Substituting (3.9) and (3.10) into (3.8), we obtain

which can be solved in turn.

The constant terms are

By integrating (3.11) from 0 to τ , we obtain

Similarly, we can compute h3, h4 and g2.                     □