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 . The Heston model [5] is the following risk-neutral stock price processes where S_t is a stock process, r is the riskless rate of return, v_t 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 S_t satisfying where the volatility process ν_t 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 Y_t 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 f_j explicitly, Yan & Hanson [8] and Makate & Sattayatham [6] conjecture that its solution is given by where β ₁ ( τ ) = 0 and β ₂ ( τ ) = rτ . 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 f_j . 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. We assume that a risk-neutral probability measure Q exists. We also assume that the asset price S_t under Q follows a jump- diffiusion process, and the volatility ν_t follows a pure mean-reverting and square root diffiusion process with jump, e.g ., our model is governed by the following dynamics where S_t , ν_t , 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 λ^S and λ^ν respectively. Y_t is the jump size of the asset price return with density ϕ_Y ( y ) and E [ Y_t ] = m , and Z_t is the jump size of the volatility with density ϕ_Z ( z ). Moreover, we assume that the Poisson processes and are independent of standard Brownian motions and with Let C denote the price at time t of a European style call option on S_t with strike price K and expiration time T . The terminal payoff of a European call option on the underlying stock S_t is 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 E_Q is the expectation with respect to the risk-neutral probability measure Q and P_Q ( S_T | S_t , ν_t ) is the corresponding conditional density function given ( S_t , ν_t ). Since is a risk-neutral probability such that P ₂ ( S_t , ν_t , T ; K , T ) = Prob_Q ( S_T > K | S_t , ν_t ) 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 S_t to L_t = ln S_t . Let k = ln K . By the jump-diffiusion chain rule, ln S_t satisfies the SDE The value C of a European-style option as a function of L_t 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 ν_t in (2.1b) to ( L_t , ν_t , t ; k , T ), then we obtain the following Partial Integro-Diffierential Equations (PIDE) where is defined as In the current state variables L_t = ℓ and ν_t = 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 A ₁ and A ₂ 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, f_j satisfies similar PIDEs as in (3.4) and (3.5) with the boundary conditions since Let’s find the characteristic functions f_j for j = 1, 2. Let τ = T − t be the time to go. We seek the functions f ₁ and f ₂ satisfying respectively with the boundary conditions 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 f ₁ is given by h ₂ is given by where η ₁ = ρσ ( ix + 1) − κ and h ₁ is given by 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 . 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 h ₄ is given by where η ₂ = ρσix − κ and h ₃ is given by 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 . 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 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 f ₁ . 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 h ₂ ( τ ) is given by where η ₁ = ρσ ( ix + 1) − κ and (See [6] for detail). The coefficients of ν ^1/2 are 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 h ₂ can be written as 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 which can be solved in turn. The constant terms are By integrating (3.11) from 0 to τ , we obtain Similarly, we can compute h ₃ , h ₄ and g ₂ .                     □