OPTION PRICING UNDER STOCHASTIC VOLATILITY MODEL WITH JUMPS IN BOTH THE STOCK PRICE AND THE VARIANCE PROCESSES
OPTION PRICING UNDER STOCHASTIC VOLATILITY MODEL WITH JUMPS IN BOTH THE STOCK PRICE AND THE VARIANCE PROCESSES
The Pure and Applied Mathematics. 2014. Nov, 21(4): 295-305
• Received : August 26, 2014
• Accepted : October 07, 2014
• Published : November 30, 2014 PDF e-PUB PubReader PPT Export by style
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JU HONG, KIM

Abstract
Yan & Hanson  and Makate & Sattayatham  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(ℓ,ν,t; k,T) = exp(g(τ) + vh(τ) + ixl). We add the term of order ν 1/2 to the exponent in the above equation and seek the explicit solution of f .
Keywords
1. INTRODUCTION
The Heston model  is the following risk-neutral stock price processes PPT Slide
Lager Image PPT Slide
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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 PPT Slide
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and PPT Slide
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are two correlated Brownian motions under the risk-neutral measure with constant correlation coefficient ρ
The Bates  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 PPT Slide
Lager Image PPT Slide
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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 PPT Slide
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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.  extended Bates model to a stochastic volatility model with contemporaneous jumps in the stock price and its volatility PPT Slide
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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  provide a formal ’closed-form solution’ of the stochastic-volatility jump-diffiusion model.
Heston’s  ’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  and Makate & Sattayatham  conjecture that its solution is given by PPT Slide
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where β 1 ( τ ) = 0 and β 2 ( τ ) = . In this paper, we add the term of order ν 1/2 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 PPT Slide
Lager Image PPT Slide
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where St , νt , κ , θ , σ , PPT Slide
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, PPT Slide
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are the same ones defined as in Bates model (1.2), r is a risk-free interest rate, PPT Slide
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and PPT Slide
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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 PPT Slide
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and PPT Slide
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are independent of standard Brownian motions PPT Slide
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and PPT Slide
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with PPT Slide
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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 PPT Slide
Lager Image PPT Slide
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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 PPT Slide
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is a risk-neutral probability such that
• ST>K, EQ[ST|St,νt] =er(T−t)St.
P 2 ( St , νt , T ; K , T ) = ProbQ ( ST > K | St , νt ) is the risk-neutral in-the-money probability. Note that the complement of P 2 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 P 1 and P 2 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 PPT Slide
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The value C of a European-style option as a function of Lt becomes PPT Slide
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that is, we have PPT Slide
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The Dynkin’s theorem  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 PPT Slide
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( Lt , νt , t ; k , T ), then we obtain the following Partial Integro-Diffierential Equations (PIDE) PPT Slide
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where PPT Slide
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is defined as PPT Slide
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In the current state variables Lt = ℓ and νt = ν , the option value (3.1) becomes PPT Slide
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where PPT Slide
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for j = 1, 2.
Lemma 3.1 (  ). The functions PPT Slide
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in (3.3) satisfies the following PIDEs PPT Slide
Lager Image PPT Slide
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with the boundary condition at expiration time t = T PPT Slide
Lager Image PPT Slide
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in (3.3) also satisfies the following PIDEs PPT Slide
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with the boundary condition at expiration time t = T PPT Slide
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A 1 and A 2 in Lemma 3.1 are respectively defined as PPT Slide
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and PPT Slide
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For j = 1, 2 the characteristic functions for PPT Slide
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with respect to the variable k are defined as PPT Slide
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in which a minus sign is given to account for the negativity of the measure PPT Slide
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For j = 1, 2, fj satisfies similar PIDEs as in (3.4) and (3.5) PPT Slide
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with the boundary conditions PPT Slide
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since PPT Slide
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Let’s find the characteristic functions fj for j = 1, 2. Let τ = T − t be the time to go. We seek the functions f 1 and f 2 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) fori= 1, 2 andj= 1, 2, 3, 4.
Lemma 3.2. The functions PPT Slide
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and PPT Slide
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can be computed by the inverse Fourier transforms of the characteristic function, e.g. , PPT Slide
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for j = 1, 2. Re [·] denote the real part of the complex number.
The characteristic function f 1 is given by
• f1(ℓ,ν,t;k,T) = exp(g1(τ) +ν1/2h1(τ) +νh2(τ) +ixℓ.
h 2 is given by PPT Slide
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where η 1 = ρσ ( ix + 1) − κ and PPT Slide
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h 1 is given by PPT Slide
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where γ 1 (0+) represents a small value factor which appears in the coefficient of ν 1/2 as the one of ν 3/2 . PPT Slide
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which is equal to the equations as in  if the coefficient h 1 ( τ ) of order ν 1/2 is zero .
The characteristic function f 2 is given by
• f2(ℓ,ν,T;k,T) = exp(g2(τ) +ν1/2h3(τ) +νh4(τ) +ixℓ+rτ).
h 4 is given by PPT Slide
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where η 2 = ρσix − κ and PPT Slide
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h 3 is given by PPT Slide
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where γ 2 (0+) represents a small value factor which appears in the coefficient of ν 1/2 as the one of ν 3/2 . PPT Slide
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which is equal to the equations as in  if the coefficient h 3 ( τ ) of order ν 1/2 is zero .
Theorem 3.3. The value of a European call option of (3.3) is PPT Slide
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where PPT Slide
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and PPT Slide
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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  . Let us compute PDE (3.6). First let’s calculate some diffierentials regarding to f 1 . PPT Slide
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• 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 PPT Slide
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in the following equation. PPT Slide
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If we substitute the above diffierentials and equations into the equation (3.6), then we have PPT Slide
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The coefficients of ν are PPT Slide
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The solution of h 2 ( τ ) is given by PPT Slide
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where η 1 = ρσ ( ix + 1) − κ and PPT Slide
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(See  for detail). The coefficients of ν 1/2 are PPT Slide
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where we denote γ 1 (0+) a small value factor which appears in the coefficient of ν 1/2 as the one of ν 3/2 . We seek h 1 ( τ ) as series solution such as PPT Slide
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h 2 can be written as PPT Slide
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where A 1 = σ −2 ( η 1 + Δ 1 ), B 1 = ( η 1 + Δ 1 ) / ( η 1 − Δ 1 ). Substituting (3.9) and (3.10) into (3.8), we obtain PPT Slide
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which can be solved in turn.
The constant terms are PPT Slide
Lager Image PPT Slide
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By integrating (3.11) from 0 to τ , we obtain PPT Slide
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Similarly, we can compute h 3 , h 4 and g 2 .                     □
References