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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
Journal of the Korean Society of Mathematical Education Series B The Pure and Applied Mathematics. 2014. Nov, 21(4): 295-305
Copyright © 2014, Korean Society of Mathematical Education
  • Received : August 26, 2014
  • Accepted : October 07, 2014
  • Published : November 30, 2014
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JU HONG KIM

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(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 [5] is the following risk-neutral stock price processes
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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
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and
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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
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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
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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. [3] extended Bates model to a stochastic volatility model with contemporaneous jumps in the stock price and its volatility
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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 [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
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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
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where St , νt , κ , θ , σ ,
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,
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are the same ones defined as in Bates model (1.2), r is a risk-free interest rate,
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and
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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
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and
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are independent of standard Brownian motions
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and
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with
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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
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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
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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
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The value C of a European-style option as a function of Lt becomes
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that is, we have
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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
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( Lt , νt , t ; k , T ), then we obtain the following Partial Integro-Diffierential Equations (PIDE)
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where
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is defined as
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In the current state variables Lt = ℓ and νt = ν , the option value (3.1) becomes
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where
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for j = 1, 2.
Lemma 3.1 ( [6] ). The functions
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in (3.3) satisfies the following PIDEs
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with the boundary condition at expiration time t = T
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in (3.3) also satisfies the following PIDEs
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with the boundary condition at expiration time t = T
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A 1 and A 2 in Lemma 3.1 are respectively defined as
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and
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For j = 1, 2 the characteristic functions for
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with respect to the variable k are defined as
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in which a minus sign is given to account for the negativity of the measure
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For j = 1, 2, fj satisfies similar PIDEs as in (3.4) and (3.5)
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with the boundary conditions
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since
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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
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and
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can be computed by the inverse Fourier transforms of the characteristic function, e.g. ,
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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
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where η 1 = ρσ ( ix + 1) − κ and
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h 1 is given by
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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 .
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which is equal to the equations as in [6] 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
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where η 2 = ρσix − κ and
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h 3 is given by
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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 .
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which is equal to the equations as in [6] 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
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where
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and
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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 [6] . Let us compute PDE (3.6). First let’s calculate some diffierentials regarding to f 1 .
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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
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in the following equation.
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If we substitute the above diffierentials and equations into the equation (3.6), then we have
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The coefficients of ν are
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The solution of h 2 ( τ ) is given by
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where η 1 = ρσ ( ix + 1) − κ and
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(See [6] for detail). The coefficients of ν 1/2 are
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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
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h 2 can be written as
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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
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which can be solved in turn.
The constant terms are
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By integrating (3.11) from 0 to τ , we obtain
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Similarly, we can compute h 3 , h 4 and g 2 .                     □
References
Bates D. 1996 Jump and Stochastic Volatility: Exchange Rate Processes Implict in Deutche Mark in Options Review of Financial Studies 9 69 - 107    DOI : 10.1093/rfs/9.1.69
Cont R. , Tankov P. 2004 Financial Modeling with Jump Processes CRC Press Boca Raton
Eraker B. , Johannes M. , Polson N. 2003 The impact of jumps in volatility and returns The Journal of Finance 58 1269 - 1300    DOI : 10.1111/1540-6261.00566
Hanson F.B. 2007 Applied Stochastic Process and Control for Jump Diffusions: Modeling, Analysis and Computation Society for Industrial and Applied Mathematics Philadelphia
Heston S. 1993 A Closed-Form Solution For Option with Stochastic Volatility with Applications to Bond and Currency Options Review of Financial Study 6 337 - 343
Makate N. , Sattayatham P. 2011 Stochastic Volatility Jump-Diffusion Model for Option Pricing J. Math. Finance 3 90 - 97
Karatzas I. , Shreve S.E. 1991 Brownian Motion and Stochastic Calculus Springer-Verlag New York
Yan G. , Hanson F.B. 2006 Option Pricing for a Stochastic-Volatility Jump-Diffusion Model with Log-Uniform Jump-Amplitudes Proc. Amer. Control Conference 1 - 7