Equilibrium Asset and Option Pricing under Jump-Diffusion Model with Stochastic Volatility

Abstract

We study the equity premium and option pricing under jump-diffusion model with stochastic volatility based on the model in Zhang et al. 2012. We obtain the pricing kernel which acts like the physical and risk-neutral densities and the moments in the economy. Moreover, the exact expression of option valuation is derived by the Fourier transformation method. We also discuss the relationship of central moments between the physical measure and the risk-neutral measure. Our numerical results show that our model is more realistic than the previous model.

1. Introduction

Option pricing problem is one of the predominant concerns in the financial market. Since the advent of the Black-Scholes option pricing formula in [1], there has been an increasing amount of literature describing the theory and its practice. Due to drawbacks of the Black-Scholes model which cannot explain numerous empirical facts such as large and sudden movements in prices, heavy tails, volatility clustering, the incompleteness of markets, and the concentration of losses in a few large downward moves, many option valuation models have been proposed and tested to fit those empirical facts. Jump-diffusion models with stochastic volatility could overcome these drawbacks of the Black-Scholes model in [2–21]. Based on those advantages, in this paper, we focus on studying the jump-diffusion model with stochastic volatility.

Different from the Black-Scholes framework, we use jump diffusion to describe the price dynamics of underlying asset. The market of our model is incomplete; that is, it is not possible to replicate the payoff of every contingent claim by a portfolio, and there are several equivalent martingale measures. How to choose a consistent pricing measure from the set of equivalent martingale measures becomes an important problem. This means that we need to find some criteria to determine one from the set of equivalent martingale measures in some economically or mathematically motivated fashion. A unique martingale measure was found by various researchers via using optimal criteria, for instance, minimal martingale criterion, minimal entropy martingale criterion, and utility maximization criterion [22–31].

General equilibrium framework method is also a popular method to deal with the option pricing in an incomplete market. General equilibrium framework is initially introduced by Lucas Jr. (1978) [32], Cox et al. (1985) [33] and developed by Vasanttilak and Lee (1990) [34], Pan (2002) [35], Liu and Pan (2003) [36], Liu et al. (2005) [37], Bates (2008) [38], Santa-Clara and Yan (2004) [12], and Zhang et al. (2012) [6]. They assumed that there is a representative investor who wants to maximize an objective function in a rational expectations economy where there are one risk-free asset and one risky asset. When the market is clear, the representative investor takes all money into the risky asset. In this paper, we build a general equilibrium model which is the same as that due to Santa-Clara and Yan (2004) [12]. Under this model, we obtain an exact expression of the equity premium and the pricing kernel in a general equilibrium economy. This can be regarded as a great contribution to the literature.

The pricing kernel which acts like the physical and risk-neutral densities and moments in the economy is also a vitally important problem in mathematical finance. In some constant volatility models with jump diffusions, Pan (2002), Liu and Pan (2003), and Liu et al. (2005) [35–37] derived the pricing kernel with some restrictions of jump sizes in a general equilibrium setting. Recently, Zhang et al. (2012) [6] presented an analytical form for the pricing kernel without any distributional assumption on the jumps. In this paper, we extend the results of Zhang et al. (2012) to the pricing kernel with stochastic volatility.

Duffie et al. (2000) [10] and Chacko and Das (2002) [39] presented a transform analysis to price the valuation of options for affine jump diffusions with stochastic volatility. Lorig and Lozano-Carbasse (2013) [21] studied option pricing in exponential Lévy-type models with stochastic volatility and stochastic jump intensity. Lewis (2008) [8] used Fourier transformation methods to obtain the transform-based solution of option price. In this paper, we employ the Fourier transformation method to get the exact expression of European options.

Finally, we get the relationship of central moments between the physical measure and the risk-neutral measure which can help us to study the negative variance risk premium, the implied volatility smirk, and the prediction of realized skewness. Some relevant work has been done by Bakshi et al. (2003) [13], Carr and Wu (2009) [14], and Neuberger (2012) [15]. However, to the best of our knowledge, except for Zhang et al. (2012) [6], there is no literature studying this relationship. In this paper, we extend it to a stochastic volatility case. This can be regarded as another contribution to the literature.

The rest of the paper is organized as follows. In Section 2, we present our jump-diffusion model with stochastic volatility. In Section 3, we study the equity premium in a general equilibrium economy. The option pricing and the relationship of central moments between the physical measure and the risk-neutral measure are studied in Section 4. Numerical results and conclusions are shown in Sections 5 and 6, respectively.

2. Our Model

In this paper, we consider the financial market with the following two basic assets.(i)A Bond whose price 𝑃_𝑡 at time 𝑡 is given by 𝑑𝑃_𝑡=𝑟𝑃_𝑡𝑑𝑡,𝑃₀=1. (1)(ii)A Stock whose price 𝑆_𝑡 at time 𝑡 is given by 𝑑𝑆_𝑡𝑆_𝑡=𝜇𝑑𝑡+𝑉_𝑡𝑑𝐵_𝑡^𝑆+𝑒^𝑥−1𝑑𝑁_𝑡−𝜆𝐸𝑒^𝑥𝑆−1𝑑𝑡,₀>0,𝑑𝑉_𝑡=𝜅𝜃−𝑉_𝑡𝑑𝑡+𝜀𝑉_𝑡𝑑𝐵^𝑉_𝑡,𝑉₀>0, (2)where 𝑡∈[0,𝑇] and 𝑇>0; on the filtered complete space (Ω,ℱ,{ℱ_𝑡}_𝑡≥0,𝑃), there are 𝐵^𝑆_𝑡, 𝐵^𝑉_𝑡which both are 1-dimensional Brownian motions with 𝑑𝐵^𝑆_𝑡𝑑𝐵^𝑉_𝑡=𝜌𝑑𝑡, and 𝑑𝑁_𝑡 is a Poisson process with the constant jump intensity 𝜆. 𝐸[⋅] is the expectation under the physical measure. The jump size 𝑥 is stochastic and 𝑟, 𝜇, 𝜅,  and  𝜃 are constants.

The integration form of stock process in (2) is given by𝑆ln^𝑇𝑆_𝑡=^𝑇_𝑡1𝜇−2𝑉_𝜏𝑑𝜏−𝜆𝐸𝑒^𝑥𝜏+−1_𝑇𝑡𝑉_𝜏𝑑𝐵^𝑆_𝜏+^𝑁𝜏_𝑖=1𝑥_𝑖. (3)

We suppose that the portfolio is (𝜔,1−𝜔) which, respectively, means the fraction of wealth invested in the stock and money market; then the wealth process 𝑊(𝑡) and the initial wealth 𝑊(0)=𝑤₀>0 satisfy the equations as follows: 𝑑𝑊_𝑡𝑊_𝑡=𝑟+𝜔𝜙−𝜔𝜆𝐸𝑒^𝑥−1𝑑𝑡+𝜔𝑉_𝑡𝑑𝐵^𝑆_𝑡+𝜔𝑒^𝑥−1𝑑𝑁_𝑡,𝑑𝑉_𝑡=𝜅𝜃−𝑉_𝑡𝑑𝑡+𝜀𝑉_𝑡𝑑𝐵^𝑉_𝑡, (4) where 𝜙=𝜇−𝑟 is the equity premium.

The representative investor maximizes his/her expected utility: 𝐽(𝑊(𝑡),𝑉(𝑡),𝑡)=max𝐸_𝑡[𝑈(𝑊(𝑇))], (5) where 𝐸_𝑡[⋅] is the conditional expectation and equals 𝐸[⋅∣ℱ_𝑡].

For tractability, we concentrate our attention on the case of constant relative risk aversion (CRRA) utility function: 𝑈𝑊_𝑇=𝑊,𝑇^𝑇1−𝛾1−𝛾,𝛾>0,𝛾≠1, (6) where the constant 𝛾 is the relative risk aversion coefficient.

3. Equity Premium

The equity premium is very important for option pricing in general equilibrium framework. Following the idea of Santa-Clara and Yan (2004) [12] and Zhang et al. (2012) [6], we obtain the equilibrium equity premium by modeling general equilibrium economy in the following proposition.

Proposition 1. In general equilibrium framework, the equilibrium equity premium is given by 𝜙=𝑉_𝑡𝛾−𝜀𝜌𝐴(𝑡)+𝜆𝐸1−𝑒^−𝛾𝑥𝑒^𝑥−1, (7) where 𝐴(𝑡)=𝛾1−𝛾1−𝑒^{−𝛼(𝑡−𝑇)}𝛽₋−𝛽₊𝑒^{−𝛼(𝑡−𝑇)},𝛽_±=𝜅±𝛼,𝛼=𝜅²−𝜀²𝛾1−𝛾. (8)

Proof. From the optimal control problem (5), we get the Bellman equation as follows: 0=𝐽_𝑡+max_𝜔{ℒ(𝐽)}, (9) where ℒ(𝐽)=𝑟+𝜔𝜙−𝜔𝜆𝐸𝑒^𝑥−1𝑊𝐽_𝑊+12𝜔²𝑊²𝑉𝐽_𝑊𝑊+𝜅(𝜃−𝑉)𝐽_𝑉+12𝜀²𝑉𝐽_𝑉𝑉+𝜀𝜔𝑊𝑉𝜌𝐽_𝑊𝑉+𝜆𝐸𝐽𝑊1+𝜔𝑒^𝑥−1,𝑉,𝑡−𝐽. (10)
Equating the derivatives of the Bellman equation (9) with respect to 𝜔 to zero, we have following equation: 0=𝜙−𝜆𝐸𝑒^𝑥−1𝑊𝐽_𝑊+𝜔𝑊²𝑉𝐽_𝑊𝑊+𝜀𝑊𝑉𝜌𝐽_𝑊𝑉+𝜆𝐸𝐽_𝑊𝑊1+𝜔𝑒^𝑥−1,𝑉,𝑡⋅𝑊𝑒^𝑥−1. (11)
In equilibrium, the money market is in zero net supply. Therefore, the representative investor holds all the wealth in the stock market; that is, 𝜔=1. Then we can get the expression of 𝜙 from (11): 𝜙=−𝑉𝑊𝐽_𝑊𝑊𝐽_𝑊+𝜆𝐸𝑒^𝑥𝐽−1−𝜀𝑉𝜌^𝑊𝑉𝐽_𝑊−𝜆𝐽_𝑊𝐸𝐽_𝑊𝑊𝑒^𝑥,𝑉,𝑡⋅𝑒^𝑥−1. (12) Then Bellman equation (9) can be written as follows: 0=𝐽_𝑡+𝑟𝑊𝐽_𝑊−12𝑊²𝑉𝐽_𝑊𝑊+𝜅(𝜃−𝑉)𝐽_𝑉+12𝜀²𝑉𝐽_𝑉𝑉−𝜆𝑊𝐸𝐽_𝑊𝑊𝑒^𝑥,𝑉,𝑡⋅𝑒^𝑥𝐽−1+𝜆𝐸𝑊𝑒^𝑥,𝑉,𝑡−𝜆𝐽. (13) From (6), we conjecture that 𝐽𝑊_𝑡,𝑉_𝑡,𝑡=exp𝐴(𝑡)𝑉_𝑡(𝑡)𝑊+𝐵^𝑡1−𝛾.1−𝛾 (14) Then, substituting (11) into (14), we obtain 0=𝐴̇(𝑡)𝑉_𝑡+𝐵̇(𝑡)+𝑟1−𝛾+12𝛾𝑉()𝐴(𝑡)+11−𝛾+𝜅𝜃−𝑉2𝜀²𝑉𝐴²(𝑡)+𝜆𝛾𝐸𝑒^{(1−𝛾)𝑥}+𝜆1−𝛾𝐸𝑒^−𝛾𝑥−𝜆. (15) This leads to a system of two ordinary differential equations (ODEs): 𝐴̇(𝑡)+12𝜀²𝐴²(𝑡)(𝑡)+1−𝜅𝐴2𝛾𝐴(𝑇)𝐵̇(𝑡)(𝑡)𝐵(𝑇)1−𝛾=0,=0,+𝜅𝜃𝐴+𝑎=0,=0, (16)(17) where 𝑎=𝑟1−𝛾+𝜆𝛾𝐸𝑒^{(1−𝛾)𝑥}+𝜆1−𝛾𝐸𝑒^−𝛾𝑥−𝜆. (18)
This system (16) and (17) can be solved explicitly. First, we solve the first ODE (16), which is the Riccati differential equation. Making the substitution 𝐴(𝑡)=2𝜀²𝐶^′(𝑡)𝐶(𝑡), (19) we obtain the second-order differential equation 𝐶^′′(𝑡)−𝜅𝐶^′(𝑡)+𝜀²4𝛾𝐶(𝑡)𝐶1−𝛾=0,_′(𝑇)=0. (20) A general solution has the form 𝐶(𝑡)=𝑑₊𝑒^{(1/2)𝛽+𝑡}+𝑑₋𝑒^{(1/2)𝛽−𝑡}, (21) where 𝛽_±=𝜅±𝛼,𝛼=𝜅²−𝜀²𝛾,𝑑1−𝛾₊=𝛽₋𝐶(𝑇)𝛽₋−𝛽₊𝑒^{(1/2)𝛽+𝑇},𝑑₋=𝛽₊𝐶(𝑇)𝛽₊−𝛽₋𝑒^{(1/2)𝛽−𝑇}. (22) Thus, 𝐴(𝑡)=𝛾1−𝛾1−𝑒^{−𝛼(𝑡−𝑇)}𝛽₋−𝛽₊𝑒^{−𝛼(𝑡−𝑇)}. (23) Then, the solution of the second ODE (17) is 𝐵(𝑡)=2𝜅𝜃𝜀²12𝛽₊()𝛽𝑡−𝑇+ln⁺𝑒^{−𝛼(𝑡−𝑇)}−𝛽₋().2𝛼+𝑎𝑡−𝑇 (24) Substituting (14) into (12), we get Proposition 1.

Remark 2. In the special case where there is no stochastic volatility and jumps, 𝑉_𝑡=𝜎² and 𝜀=𝜅=0, and consequently 𝜙=𝜎²𝛾 which is constant in Merton (1976) [2]. In the special case where there is no stochastic volatility, 𝜀=0, and consequently 𝜙=𝜎²𝛾+𝜆𝐸[(1−𝑒^−𝛾𝑥)(𝑒^𝑥−1)] which is constant in Zhang et al. (2012) [6].

4. Option Pricing

In this section, we will study the pricing kernel and the option pricing in general equilibrium framework. We first derive the pricing kernel which acts like the physical and risk-neutral densities in the economy and is the key to obtain the PDE of option price as follows.

Proposition 3. In general equilibrium framework, the pricing kernel is given in differential form by 𝑑𝜋_𝑡𝜋_𝑡=−𝑟𝑑𝑡−𝛾−𝜀𝜌𝐴𝑉_𝑡𝑑𝐵^𝑆_𝑡+𝑒^𝑦−1𝑑𝑁_𝑡−𝜆𝐸𝑒^𝑦−1𝑑𝑡, (25) and the integration is given by 𝜋_𝑇𝜋_𝑡=exp−𝑟−𝜆𝐸𝑒^𝑦𝜏−1−12^𝑇_𝑡𝛾−𝜀𝜌𝐴_𝜏²𝑉_𝜏−𝑑𝜏_𝑇𝑡𝛾−𝜀𝜌𝐴_𝜏𝑉_𝜏𝑑𝐵^𝑆_𝜏+^𝑁𝜏_𝑖=1𝑦_𝑖, (26) where 𝜏=𝑇−𝑡.
The martingale condition, 𝜋_𝑡𝑆_𝑡=𝐸_𝑡[𝜋_𝑇𝑆_𝑇], requires that the jump size 𝑦 satisfies the following restriction: 𝐸𝑒^𝑦−𝑒^−𝛾𝑥𝑒^𝑥−1=0. (27)

Proof. To satisfy the martingale condition, 𝜋_𝑡𝑆_𝑡=𝐸_𝑡[𝜋_𝑇𝑆_𝑇], from (3) and (26), we have 𝐸exp^𝑇_𝑡𝜙_𝜏−12𝑉_𝜏−12𝛾−𝜀𝜌𝐴_𝜏²𝑉_𝜏𝐸𝑒𝑑𝜏−𝜆^𝑦−1𝜏+𝐸𝑒^𝑥𝜏+−1_𝑇𝑡1−𝛾−𝜀𝜌𝐴_𝜏𝑉_𝜏𝑑𝐵^𝑆_𝜏+^𝑁𝜏_𝑖=1𝑥_𝑖+^𝑁𝜏_𝑖=1𝑦_𝑖=1. (28) Substituting (7) into (28), we have 𝐸exp^𝑇_𝑡𝜆𝐸1−𝑒^−𝛾𝑥𝑒^𝑥𝜏𝐸𝑒−1−𝜆^𝑦−1𝜏+𝐸𝑒^𝑥𝜏+−1_{𝑁𝜏𝑖=1}𝑥_𝑖+^𝑁𝜏_𝑖=1𝑦_𝑖=1. (29) Using the property of Poisson process, 𝐸[exp(∑^𝑁𝜏_𝑖=1𝑥_𝑖)]=exp[𝜆𝐸(𝑒^𝑥−1)𝜏], it is easy to obtain Proposition 3.

Remark 4. In this market, there is only one tradable asset, a stock with price 𝑆_𝑡, but there are at least two dimensions of risk, diffusive risk, and jump risk. Therefore, the market is incomplete and the pricing kernel is not unique. The nonuniqueness of the pricing kernel can be justified by the fact that the distribution of jump size 𝑦 in the pricing kernel can be arbitrary as long as it satisfies the martingale restriction (27). In a special case, we can choose 𝑦=−𝛾𝑥, as in Liu et al. (2005) [37].

Remark 5. With Proposition 3, we define a new probability measure 𝑄: 𝑑𝑄𝑑𝑃=𝑍_𝑇=𝑒^𝑟𝑇𝜋_𝑇=exp−𝜆𝐸𝑒^𝑦1−1𝑇−2^𝑇₀𝛾−𝜀𝜌𝐴_𝜏²𝑉_𝜏−𝑑𝜏_𝑇0𝛾−𝜀𝜌𝐴_𝜏𝑉_𝜏𝑑𝐵^𝑠_𝜏+^𝑁𝑇_𝑖=1𝑦_𝑖, (30) since, for any assets 𝑃_𝑡 at time 𝑡, we have 𝐸^𝑄_𝑡𝑃_𝑇=1𝑍_𝑡𝐸_𝑡𝑍_𝑇𝑃_𝑇=𝑒^{𝑟(𝑇−𝑡)}𝐸_𝑡𝜋_𝑇𝜋_𝑡𝑃_𝑇=𝑒^{𝑟(𝑇−𝑡)}𝑃_𝑡, (31) which means 𝑄 is a risk-neutral probability measure.

Lemma 6. Define a new probability measure, 𝑄^∗, by the following Radon-Nikodym derivative: 𝑑𝑄^∗𝑒𝑑𝑃=exp−𝜆𝐸^𝑦−1𝑇+^𝑁𝑇_𝑖=1𝑦_𝑖; (32) then the following relation 𝐸𝑒^𝑦𝑓(𝑥)=𝐸𝑒^𝑦𝐸^𝑄∗𝑓(𝑥) (33) is true.

Proof. The change of probability measure formula gives 𝐸^𝑄∗𝑓(𝑥)=𝐸𝑑𝑄^∗𝑓(𝑥𝑒𝑑𝑃)=exp−𝜆𝐸^𝑦−1𝑇𝐸𝑓(𝑥)^𝑁𝑇_𝑖=1𝑒^𝑦𝑖=exp−𝜆𝐸𝑒^𝑦𝑇×−1_+∞𝑛=1𝑒^−𝜆𝑇(𝜆𝑇)^𝑛𝐸𝑓(𝑥)𝑛!^𝑛_𝑖=1𝑒^𝑦𝑖. (34)
Since 𝑦_𝑖, 𝑖=1,2,…𝑖,…,𝑛, is i.i.d. and 𝑦 and 𝑥 are correlated, this means that only one of the 𝑦_𝑖 is correlated with 𝑥. Without loss of generality, we assume that 𝑦_𝑛 is correlated with 𝑥 and other 𝑦_𝑖𝑠 are independent of 𝑥. Then we have 𝐸^𝑄∗𝑓(𝑥)=𝐸𝑑𝑄^∗𝑓(𝑥𝑒𝑑𝑃)=exp−𝜆𝐸^𝑦−1𝑇𝐸𝑓(𝑥)^𝑁𝑇_𝑖=1𝑒^𝑦𝑖=𝐸𝑓(𝑥)𝑒^𝑦exp−𝜆𝐸𝑒^𝑦𝑇×−1_+∞𝑛=1𝑒^−𝜆𝑇(𝜆𝑇)^𝑛𝐸𝑓(𝑥)𝑛!^𝑛−1_𝑖=1𝑒^𝑦𝑖=𝐸𝑒^𝑦𝑓(𝑥)𝐸(𝑒^𝑦). (35)

Remark 7. These results are also true in 𝑄 measure, because the difference between 𝑄 and 𝑄^∗ is the Brownian motion that is independent of the jumps.

Now, we consider a European call written on the stock price 𝑆_𝑡 at time 𝑡. The option has a payoff function (𝑆_𝑇−𝐾)⁺ at time 𝑇. Its price is denoted as 𝑐(𝑆_𝑡,𝑉_𝑡,𝑡) at time 𝑡. We derive a PDE which 𝑐(𝑆_𝑡,𝑉_𝑡,𝑡) has to satisfy in the following proposition.

Proposition 8. In general equilibrium framework, the price of European call option satisfies the following PDE: 0=𝑐_𝑡+12𝑉𝑆²𝑐_𝑆𝑆+12𝜀²𝑉𝑐_𝑉𝑉+𝜀𝜌𝑉𝑆𝑐_𝑆𝑉+𝑟−𝜆^𝑄𝐸^𝑄𝑒^𝑥−1𝑆𝑐_𝑆−𝑟𝑐+𝜅^𝑄𝜃^𝑄−𝑉𝑐_𝑉+𝜆^𝑄𝐸^𝑄𝑐𝑆𝑒^𝑥(,𝑐()=(),𝑉,𝑡−𝑐𝑆,𝑉,𝑡)𝑆,𝑉,𝑇𝑆−𝐾⁺, (36) where 𝐸^𝑄𝑓(𝑥=𝐸𝑒)^𝑦𝑓(𝑥)𝐸(𝑒^𝑦),𝜆^𝑄=𝜆𝐸𝑒^𝑦,𝜅^𝑄=𝜅+𝛾−𝜀𝜌𝐴𝜀𝜌,𝜃^𝑄=𝜅𝜃.𝜅+𝛾−𝜀𝜌𝐴𝜀𝜌 (37)

Proof. First, we rewrite the stock with continue part and jump part: 𝑑𝑆=𝑑^𝐶𝑆+𝑆𝑒^𝑥−1𝑑𝑁_𝑡, (38) where 𝑑^𝐶𝑆=𝑟+𝜙−𝜆𝐸𝑒^𝑥−1𝑆𝑑𝑡+√𝑉𝑆𝑑𝐵^𝑆𝑑𝑉=𝜅(𝜃−𝑉)𝑑𝑡+𝜀√𝑉𝑑𝐵^𝑉_𝑡. (39) Similarly, 𝑑𝜋=𝑑^𝐶𝜋+𝑒^𝑦−1𝜋𝑑𝑁_𝑡, (40) where 𝑑^𝐶𝜋=−𝑟−𝜆𝐸𝑒^𝑦−1𝜋𝑑𝑡−𝛾−𝜀𝜌𝐴√𝑉𝜋𝑑𝐵^𝑆_𝑡,𝑑𝑐=𝑑^𝐶𝑐+𝑐𝑆𝑒^𝑥,𝑉,𝑡−𝑐(𝑆,𝑉,𝑡)𝑑𝑁_𝑡, (41)(42) where 𝑑^𝐶𝑐=𝑐_𝑡+𝑟+𝜙−𝜆𝐸𝑒^𝑥−1𝑆𝑐_𝑆+12𝑉𝑆²𝑐_𝑆𝑆+𝜅(𝜃−𝑉)𝑐_𝑉+12𝜀²𝑉𝑐_𝑉𝑉+𝑐_𝑆𝑉+√𝜀𝜌𝑆𝑉𝑑𝑡𝑉𝑆𝑐_𝑆𝑑𝐵^𝑆+𝜀√𝑉𝑐_𝑉𝑑𝐵^𝑉_𝑡. (43) From (40) and (42), we get 𝑑(𝜋𝑐)=𝑑^𝐶(𝜋𝑐)+𝜋𝑒^𝑦𝑐𝑆𝑒^𝑥,𝑉,𝑡−𝜋𝑐(𝑆,𝑉,𝑡)𝑑𝑁_𝑡, (44) where 𝑑^𝐶(𝜋𝑐)=𝑐−𝑟−𝜆𝐸𝑒^𝑦√−1𝜋𝑑𝑡−𝑐𝛾−𝜀𝜌𝐴𝑉𝜋𝑑𝐵^𝑆_𝑡+𝜋𝑐_𝑡+𝑟+𝜙−𝜆𝐸𝑒^𝑥−1𝑆𝑐_𝑆+12𝑉𝑆²𝑐_𝑆𝑆+𝜅(𝜃−𝑉)𝑐_𝑉+12𝜀²𝑉𝑐_𝑉𝑉+𝑐_𝑆𝑉√𝜀𝜌𝑆𝑉𝑑𝑡+𝜋𝑉𝑆𝑐_𝑆𝑑𝐵^𝑆+𝜋𝜀√𝑉𝑐_𝑉𝑑𝐵^𝑉_𝑡−𝑉𝜋𝛾−𝜀𝜌𝐴𝑆𝑐_𝑆𝑑𝑡+𝜀𝜌𝑐_𝑉𝑑𝑡. (45) The martingale condition 𝐸[𝑑(𝜋𝑐)]=0 requires 0=𝑐_𝑡+12𝑉𝑆²𝑐_𝑆𝑆+12𝜀²𝑉𝑐_𝑉𝑉+𝑟+𝜙−𝑉𝛾−𝜀𝜌𝐴−𝜆𝐸𝑒^𝑥−1𝑆𝑐_𝑆−𝑟+𝜆𝐸𝑒^𝑦𝑐+𝜅()𝑐−1𝜃−𝑉−𝑉𝛾−𝜀𝜌𝐴𝜀𝜌_𝑉+𝜀𝜌𝑆𝑉𝑐_𝑆𝑉+𝜆𝐸𝑒^𝑦𝑐𝑆𝑒^𝑥,𝑉,𝑡−𝜆𝑐(𝑆,𝑉,𝑡). (46) Using Lemma 6, we have 𝐸^𝑄𝑓(𝑥=𝐸𝑒)^𝑦𝑓(𝑥)𝐸(𝑒^𝑦). (47) Denoting 𝜆^𝑄=𝜆𝐸(𝑒^𝑦) and using the restriction condition 𝐸[(𝑒^𝑦−𝑒^−𝛾𝑥)(𝑒^𝑥−1)]=0, 𝜙=𝑉[𝛾−𝜀𝜌𝐴]+𝜆𝐸[(1−𝑒^−𝛾𝑥)(𝑒^𝑥−1)] in Proposition 1 and the terminal payoff function 𝑐(𝑆,𝑉,𝑇)=(𝑆−𝐾)⁺, we obtain 0=𝑐_𝑡+12𝑉𝑆²𝑐_𝑆𝑆+12𝜀²𝑉𝑐_𝑉𝑉+𝜀𝜌𝑉𝑆𝑐_𝑆𝑉+𝑟−𝜆^𝑄𝐸^𝑄𝑒^𝑥−1𝑆𝑐_𝑆+𝜅()𝑐−𝑟𝑐𝜃−𝑉−𝑉𝛾−𝜀𝜌𝐴𝜀𝜌_𝑉+𝜆^𝑄𝐸^𝑄𝑐𝑆𝑒^𝑥(𝑐()=(),𝑉,𝑡−𝑐𝑆,𝑉,𝑡)𝑆,𝑉,𝑇𝑆−𝐾⁺. (48) If we denote 𝜅^𝑄=𝜅+(𝛾−𝜀𝜌𝐴)𝜀𝜌 and 𝜃^𝑄=(𝜅𝜃/(𝜅+(𝛾−𝜀𝜌𝐴)𝜀𝜌)), then (48) can be written as 0=𝑐_𝑡+12𝑉𝑆²𝑐_𝑆𝑆+12𝜀²𝑉𝑐_𝑉𝑉+𝜀𝜌𝑉𝑆𝑐_𝑆𝑉+𝑟−𝜆^𝑄𝐸^𝑄𝑒^𝑥−1𝑆𝑐_𝑆−𝑟𝑐+𝜅^𝑄𝜃^𝑄−𝑉𝑐_𝑉+𝜆^𝑄𝐸^𝑄𝑐𝑆𝑒^𝑥(𝑐()=(),𝑉,𝑡−𝑐𝑆,𝑉,𝑡)𝑆,𝑉,𝑇𝑆−𝐾⁺. (49)

Remark 9. The stock process (2) in a risk-neutral measure 𝑄 can be written as 𝑑𝑆_𝑡𝑆_𝑡=𝑟𝑑𝑡+𝑉_𝑡𝑑𝐵^𝑆_𝑡+𝑒^𝑥−1𝑑𝑁_𝑡−𝜆^𝑄𝐸^𝑄𝑒^𝑥𝑆−1𝑑𝑡,₀>0,𝑑𝑉_𝑡=𝜅^𝑄𝜃^𝑄−𝑉_𝑡𝑑𝑡+𝜀𝑉_𝑡𝑑𝐵^𝑉_𝑡,𝑉₀>0, (50) where 𝑑𝐵^𝑆_𝑡=𝑑𝐵^𝑆_𝑡+𝛾−𝜀𝜌𝐴𝑉_𝑡𝑑𝐵𝑑𝑡,_𝑉𝑡=𝑑𝐵^𝑉_𝑡+𝜌𝛾−𝜀𝜌𝐴𝑉_𝑡𝜆𝑑𝑡,_𝑄=𝜆𝐸𝑒^𝑦,𝜅^𝑄𝜃=𝜅+𝛾−𝜀𝜌𝐴𝜀𝜌,_𝑄=𝜅𝜃𝜅+𝛾−𝜀𝜌𝐴𝜀𝜌,𝐸^𝑄𝑓(𝑥=𝐸𝑒)^𝑦𝑓(𝑥)𝐸(𝑒^𝑦). (51) The proof is very easy. Substituting all equations in Remark 9 to (50) and using the equilibrium equity premium (7) and the restriction (27), we will get (2).
Furthermore, we also can understand 𝜆^𝑄=𝜆𝐸(𝑒^𝑦) by 𝐸^𝑄₀𝑁_𝑇=𝐸₀𝑍_𝑇𝑁_𝑇=𝐸₀𝑒^{−𝜆𝐸(𝑒𝑦−1)𝑇+∑𝑁𝑇𝑖=1𝑦𝑖}𝑁_𝑇=^+∞_𝑛=0𝑒^{−𝜆𝐸(𝑒𝑦)𝑇}𝜆𝐸𝑒^𝑦𝑇^𝑛𝑒𝑛!𝑛=𝜆𝐸^𝑦𝑇∶=𝜆^𝑄𝑇. (52) Employing the Feynman-Kac theorem to (38), we also can obtain PDE (36) in Proposition 8.

Remark 10. 𝜅^𝑄=𝜅+(𝛾−𝜀𝜌𝐴)𝜀𝜌 and 𝜃^𝑄=(𝜅𝜃/(𝜅+(𝛾−𝜀𝜌𝐴)𝜀𝜌)) are similar forms in Heston (1993) [40].

Following the approach of Lewis (2000) [8], we will derive the pricing formula in a risk-neutral measure 𝑄 from the PDE.

Proposition 11. In general equilibrium framework, the pricing formula is given by 𝑐()𝑒𝑆,𝑉,𝜏=𝑆−^−𝑟𝜏2𝜋₍₎^(𝑖/2)+∞_𝑖/2−∞𝐾^𝑖𝑘+1𝑘²𝑒−𝑖𝑘^{−𝑖𝑘(𝑟𝜏+ln𝑆)+Γ𝑄(𝜏)𝑉+Λ𝑄(𝜏)}𝑑𝑘, (53) where Γ^𝑄(𝜏)=𝑘²−𝑖𝑘1−𝑒^{−𝜁𝑄𝜏}𝜉^𝑄₋−𝜉^𝑄₊𝑒^{−𝜁𝑄𝜏},Λ^𝑄(𝜏)=−2𝜅^𝑄𝜃^𝑄𝜀²12𝜉^𝑄₊𝜉𝜏+ln^𝑄+𝑒^{−𝜁𝑄𝜏}−𝜉^𝑄₋2𝜁^𝑄+𝑏^𝑄𝜉𝜏,_𝑄±=𝑖𝑘𝜀𝜌+𝜅^𝑄±𝜁^𝑄,𝜁^𝑄=𝑖𝑘𝜀𝜌+𝜅^𝑄2+𝜀²𝑘²,𝑏−𝑖𝑘_𝑄=𝑖𝑘𝜆^𝑄𝐸^𝑄𝑒^𝑥−1+𝜆^𝑄𝐸^𝑄𝑒^{−𝑖𝑘𝑥}−1. (54)

Proof. We denote 𝑋=ln𝑆 and 𝑔(𝑋,𝑉,𝑡)∶=𝑐(𝑒^𝑋,𝑉,𝑡)=𝑐(𝑆,𝑉,𝑡); then, the PDE (36) can be rewritten as −𝑔_𝑡1=−𝑟𝑔+2𝑉𝑔_𝑋𝑋+12𝜀²𝑉𝑔_𝑉𝑉+𝜀𝜌𝑉𝑔_𝑋𝑉+𝑟−𝜆^𝑄𝐸^𝑄𝑒^𝑥−1−12𝑉𝑔_𝑋+𝜅^𝑄𝜃^𝑄−𝑉𝑔_𝑉+𝜆^𝑄𝐸^𝑄𝑔((,𝑔()=𝑒𝑋+𝑥,𝑉,𝑡)−𝑔𝑋,𝑉,𝑡)𝑋,𝑉,𝑇^𝑋−𝐾⁺. (55) Let 𝐺(𝑘,𝑉,𝑡) be Fourier transform of 𝑔(𝑋,𝑉,𝑡): 𝐺(𝑘,𝑉,𝑡)=^+∞_−∞𝑒^𝑖𝑘𝑋𝑔(𝑋,𝑉,𝑡)𝑑𝑋 (56) with 𝐺(𝑘,𝑉,𝑇)=−(𝐾^𝑖𝑘+1/(𝑘²−𝑖𝑘)).
Denote 𝑘=𝑘_𝑟+𝑖𝑘_𝑖; the inverse Fourier transform is given by 𝑔()=1𝑋,𝑉,𝑡2𝜋^{𝑖𝑘𝑖+∞}_{𝑖𝑘𝑖−∞}𝑒^{−𝑖𝑘𝑋}𝐺(𝑘,𝑉,𝑡)𝑑𝑘. (57) Then, the PDE (55) can be rewritten as −𝐺_𝑡=−(1+𝑖𝑘)𝑟𝐺+𝑖𝑘𝜆^𝑄𝐸^𝑄𝑒^𝑥1−1𝐺−2𝑉𝑘²𝐺+1−𝑖𝑘2𝜀²𝑉𝐺_𝑉𝑉−𝑖𝑘𝜀𝜌𝑉𝐺_𝑉+𝜅^𝑄𝜃^𝑄−𝑉𝐺_𝑉+𝜆^𝑄𝐸^𝑄𝑒^{−𝑖𝑘𝑥}𝐺()𝐾−1𝐺,𝑘,𝑉,𝑇=−^𝑖𝑘+1𝑘².−𝑖𝑘 (58)
Denote ℎ≡𝑒^{(1+𝑖𝑘)𝑟𝜏}𝐺; the PDE (58) can be rewritten as ℎ_𝜏=𝑖𝑘𝜆^𝑄𝐸^𝑄𝑒^𝑥−1−12𝑉𝑘²ℎ+1−𝑖𝑘2𝜀²𝑉ℎ_𝑉𝑉+𝜅^𝑄𝜃^𝑄−𝑉−𝑖𝑘𝜀𝜌𝑉ℎ_𝑉+𝜆^𝑄𝐸^𝑄𝑒^{−𝑖𝑘𝑥}ℎ()𝐾−1withℎ𝑘,𝑉,0=−^𝑖𝑘+1𝑘².−𝑖𝑘 (59)(60)
According to the study of Lewis (2000) [8], to solve the PDE (59) with the initial condition (60), it is enough to solve the same equation with the initial value being equal to one. We call this solution fundamental transform and denote it ℎ(𝑘,𝑉,𝜏) which satisfies following equation by: ℎ_𝜏=𝑖𝑘𝜆^𝑄𝐸^𝑄𝑒^𝑥−1−12𝑉𝑘²1−𝑖𝑘ℎ+2𝜀²𝑉ℎ_𝑉𝑉+𝜅^𝑄𝜃^𝑄−𝑉−𝑖𝑘𝜀𝜌𝑉ℎ_𝑉+𝜆^𝑄𝐸^𝑄𝑒^{−𝑖𝑘𝑥}ℎ()−1ℎ,𝑘,𝑉,0=1, (61) and the option price satisfies 𝑐()1𝑆,𝑉,𝜏=𝑆−2𝜋₍₎^(𝑖/2)+∞_𝑖/2−∞𝑒^{−𝑖𝑘ln𝑆}𝑒^{−(1+𝑖𝑘)𝑟𝜏}𝐾^𝑖𝑘+1𝑘²ℎ()−𝑖𝑘𝑘,𝑉,𝜏𝑑𝑘. (62)
Denoting 𝐻(𝑉,𝜏)∶=ℎ(𝑘,𝑉,𝜏), then, the PDE (61) can be rewritten as 𝐻_𝜏=𝑖𝑘𝜆^𝑄𝐸^𝑄𝑒^𝑥−1−12𝑉𝑘²1−𝑖𝑘𝐻+2𝜀²𝑉𝐻_𝑉𝑉+𝜅^𝑄𝜃^𝑄−𝑉−𝑖𝑘𝜀𝜌𝑉𝐻_𝑉+𝜆^𝑄𝐸^𝑄𝑒^{−𝑖𝑘𝑥}𝐻()−1𝐻,𝑉,0=1. (63)
We guess that a solution of (63) is as the following form: 𝐻(𝑉,𝜏)=𝑒^{Γ𝑄(𝜏)𝑉+Λ𝑄(𝜏)}, (64) with Γ^𝑄(0)=0 and Λ^𝑄(0)=0.
This leads to a system of two ODEs: Γ̇^𝑄(𝜏)−12𝜀²Γ^𝑄(𝜏)²+𝑖𝑘𝜀𝜌+𝜅^𝑄Γ^𝑄(𝜏)+12𝑘²Γ−𝑖𝑘=0,_𝑄(0)Λ̇=0,_𝑄(𝜏)−𝜅^𝑄𝜃^𝑄Γ^𝑄(𝜏)−𝑏^𝑄Λ=0,_𝑄(0)=0, (65)(66) where 𝑏^𝑄=𝑖𝑘𝜆^𝑄𝐸^𝑄(𝑒^𝑥−1)+𝜆^𝑄𝐸^𝑄(𝑒^{−𝑖𝑘𝑥}−1).
This system can be solved explicitly. First, we solve the first ODE (65), which is the Riccati differential equation. Making the substitution Γ^𝑄(𝜏)2=−𝜀²𝐼^′(𝜏)𝐼(𝜏), (67) we obtain the second-order differential equation 𝐼^′′(𝜏)+𝑖𝑘𝜀𝜌+𝜅^𝑄𝐼^′(𝜏)−𝜀²4𝑘²𝐼(𝜏)𝐼−𝑖𝑘=0,_′(0)=0. (68)
A general solution of (68) has the form 𝐼(𝜏)=𝐷^𝑄₊𝑒^{(1/2)𝜉𝑄+𝜏}+𝐷^𝑄₋𝑒^{(1/2)𝜉𝑄−𝜏}, (69) where 𝜉^𝑄_±=𝑖𝑘𝜀𝜌+𝜅^𝑄±𝜁^𝑄,𝜁^𝑄=𝑖𝑘𝜀𝜌+𝜅^𝑄2+𝜀²𝑘²𝐷−𝑖𝑘_𝑄+=𝜉^𝑄₋𝐼(0)𝜉^𝑄₋−𝜉^𝑄₊,𝐷^𝑄₋=𝜉^𝑄₊𝐼(0)𝜉^𝑄₊−𝜉^𝑄₋. (70) Thus, Γ^𝑄(𝜏)=𝑘²−𝑖𝑘1−𝑒^{−𝜁𝑄𝜏}𝜉^𝑄₋−𝜉^𝑄₊𝑒^{−𝜁𝑄𝜏}. (71)
Then, the solution of the second ODE (66) is Λ^𝑄(𝜏)=−2𝜅^𝑄𝜃^𝑄𝜀²12𝜉^𝑄₊𝜉𝜏+ln^𝑄+𝑒^{−𝜁𝑄𝜏}−𝜉^𝑄₋2𝜁^𝑄+𝑏^𝑄𝜏. (72)

Note. Although the pricing formula (40) contains a complex integral, the result is real.

With the European option price being well understood, we study the relations between the physical and risk-neutral densities and moments which can help us to study the negative variance risk premium, the implied volatility smirk, and the prediction of realized skewness. In this paper, we extend the results about those relations in [6] to a stochastic volatility case.

Proposition 12. The risk-neutral density of the continuously compounded return within 𝜏=𝑇−𝑡, 𝑅_𝜏=ln𝑆_𝑇/𝑆_𝑡, 𝑆_𝑡 given in (38), is given by 𝜑^𝑄𝑅_𝜏=𝑒∣𝑉,𝜏^−𝑟𝜏2𝜋^(𝑖/2)+∞_{𝑖(𝑖/2)−∞}𝑒^{−𝑖𝑘(𝑅−𝑟𝜏)+Γ𝑄(𝜏)𝑉+Λ𝑄(𝜏)}𝑑𝑘. (73) The first moment, second, third, and fourth central moments of the continuously compounded return in the risk-neutral measure are given by𝐸^𝑄_𝑡𝑅_𝜏=1𝜇−2𝐸^𝑄_𝑡𝑉_𝜏−𝜆^𝑄𝐸^𝑄𝑒^𝑥𝐸−1−𝑥𝜏,_𝑄𝑡𝑅_𝜏−𝐸^𝑄_𝑡𝑅_𝜏²=𝜏𝐸^𝑄_𝑡𝑉_𝜏+𝜆^𝑄𝜏𝜇^𝑄_𝑥²+Var^𝑄(𝑥,𝐸_𝑄𝑡)𝑅_𝜏−𝐸^𝑄_𝑡𝑅_𝜏³=𝜆^𝑄𝜏𝜇^𝑄_𝑥³+3𝜇^𝑄_𝑥Var^𝑄(𝑥)+3𝑟𝑑^𝑄(𝑥,𝐸_𝑄𝑡)𝑅_𝜏−𝐸^𝑄_𝑡𝑅_𝜏⁴=3𝜆^𝑄𝜏𝜇^𝑄_𝑥²+Var^𝑄(𝑥)+𝜏𝐸^𝑄_𝑡[𝑉_𝜏]²+𝜆^𝑄𝜏𝜇^𝑄_𝑥⁴+6𝜇^𝑄_𝑥²Var^𝑄(𝑥)+4𝜇^𝑄_𝑥3𝑟𝑑^𝑄(𝑥)+4𝑡ℎ^𝑄(𝑥), (74) where 𝐸^𝑄_𝑡(𝑉_𝜏)=𝑒^{−𝜅𝑄𝜏}𝑉₀+𝜃^𝑄(1−𝑒^{−𝜅𝑄𝜏}),  𝜇^𝑄_𝑥=𝐸^𝑄(𝑥) is the first moment, Var^𝑄(𝑥)=𝐸^𝑄[(𝑥−𝜇^𝑄_𝑥)²], 3𝑟𝑑^𝑄(𝑥)=𝐸^𝑄[(𝑥−𝜇^𝑄_𝑥)³], and 4𝑡ℎ^𝑄(𝑥)=𝐸^𝑄[(𝑥−𝜇^𝑄_𝑥)⁴] are the second, third, and fourth central moments in the risk-neutral measure of random number 𝑥.
The physical density is given by 𝜑𝑅_𝜏=𝑒∣𝑉,𝜏^−𝜇𝜏2𝜋^(𝑖/2)+∞_{(𝑖/2)−∞}𝑒^{−𝑖𝑘(𝑅−𝜇𝜏)+Γ(𝜏)𝑉+Λ(𝜏)}𝑑𝑘, (75) where Γ(𝜏)=𝑘²−𝑖𝑘1−𝑒^−𝜁𝜏𝜉₋−𝜉₊𝑒^−𝜁𝜏,Λ(𝜏)=−2𝜅𝜃𝜀²12𝜉₊𝜉𝜏+ln⁺𝑒^−𝜁𝜏−𝜉₋𝜉2𝜁+𝑏𝜏,_±=𝑖𝑘𝜀𝜌+𝜅±𝜁,𝜁=𝑖𝑘𝜀𝜌+𝜅²+𝜀²𝑘²,𝑒−𝑖𝑘𝑏=𝑖𝑘𝜆𝐸^𝑥−1+𝜆𝐸𝑒^{−𝑖𝑘𝑥}−1. (12) The first moment, second, third, and fourth central moments of the continuously compounded return in the physical measure are given by 𝐸_𝑡𝑅_𝜏=1𝜇−2𝐸_𝑡𝑉_𝜏−𝜆𝐸𝑒^𝑥𝐸−1−𝑥𝜏,_𝑡𝑅_𝜏−𝐸_𝑡𝑅_𝜏²=𝜏𝐸_𝑡𝑉_𝜏+𝜆𝜏𝜇²_𝑥(𝑥,𝐸+Var)_𝑡𝑅_𝜏−𝐸_𝑡𝑅_𝜏³=𝜆𝜏𝜇³_𝑥+3𝜇_𝑥(𝑥)(𝑥,𝐸Var+3𝑟𝑑)_𝑡𝑅_𝜏−𝐸_𝑡𝑅_𝜏⁴=3𝜆𝜏𝜇²_𝑥+Var(𝑥)+𝜏𝐸_𝑡𝑉_𝜏²+𝜆𝜏𝜇⁴_𝑥+6𝜇²_𝑥Var(𝑥)+4𝜇_𝑥3𝑟𝑑(𝑥)+4𝑡ℎ(𝑥), (77) where 𝐸_𝑡(𝑉_𝜏)=𝑒^−𝜅𝜏𝑉₀+𝜃(1−𝑒^−𝜅𝜏),  𝜇_𝑥=𝐸(𝑥) is the first moment, Var(𝑥)=𝐸[(𝑥−𝜇_𝑥)²], 3𝑟𝑑(𝑥)=𝐸[(𝑥−𝜇_𝑥)³], and 4𝑡ℎ(𝑥)=𝐸[(𝑥−𝜇_𝑥)⁴] are second, third, and fourth central moments in the physical measure of random number 𝑥.

Proof. We denote the corresponding probability function Φ=Prob(𝑆_𝑇≤𝐾), and Φ also satisfies the PDE (36) with a different boundary condition Φ|_𝑡=𝑇=1_{{𝑆𝑇≤𝐾}}. Under the Fourier transform, this boundary condition becomes (𝐾^𝑖𝑘/𝑖𝑘). Solving the PDE (36) under this new boundary condition, we have 𝑒Φ=^−𝑟𝜏2𝜋^(𝑖/2)+∞_{𝑖(𝑖/2)−∞}𝑒^{𝑖𝑘(ln(𝐾/𝑆)−𝑟𝜏)}𝑒𝑖𝑘^{Γ𝑄(𝜏)𝑉+Λ𝑄(𝜏)}𝑑𝑘. (78)
We denote the stock return between time 𝑡 and 𝑇 by 𝑅 so that 𝐾=𝑆𝑒^𝑅; then we can differentiate Φ with respect to 𝑅 to get the density function 𝜑^𝑄(𝑅_𝜏∣𝑉,𝜏). We find that the parameters of the physical stock price process in (2) are the same those of risk-neutral stock price process in (50); then the parameters of 𝜑(𝑅_𝜏∣𝑉,𝜏) are just removed by the superscript ⋅^𝑄.
Now we compute the first moment, second, third, and fourth central moments of the continuously compounded return in the physical measure. There are at least two methods to solve the problem of how to compute central moments.
One method is using the 𝑅_𝜏’s characteristic function to calculate the 𝑛th moment of 𝑅_𝜏. Since 𝑓(𝑘,𝑉,𝑡)=^+∞_−∞𝑒^𝑖𝑘𝑅𝜑()𝜕𝑅∣𝑉,𝜏𝑑𝑅,_𝑛𝑓𝜕𝑘^𝑛=(𝑖)^𝑛+∞_−∞𝑅^𝑛𝑒^𝑖𝑘𝑅𝜑()𝐸𝑅∣𝑉,𝜏𝑑𝑅,_𝑡𝑅^𝑛_𝜏=(−𝑖)^𝑛𝜕^𝑛𝑓𝜕𝑘^𝑛_𝑘=0, (79) then 𝐸_𝑡𝑅^𝑛_𝜏=^+∞_−∞𝑅^𝑛𝑒^𝑖𝑘𝑅𝜑(𝑅∣𝑉,𝜏)𝑑𝑅_𝑘=0. (80)
Although this method is very straightforward, it is too complicated to calculate the integration. We introduce another way to obtain the central moments by a direct calculation.
From (3), we have 𝑅_𝜏=𝜇−𝜆𝐸𝑒^𝑥1−1𝜏−2^𝑇_𝑡𝑉_𝜏𝑑𝜏+^𝑇_𝑡𝑉_𝜏𝑑𝐵^𝑆_𝜏+^𝑁𝜏_𝑖=1𝑥_𝑖. (81)
Then, the first moment of the continuously compounded return in the physical measure can be easy to be obtained: 𝐸_𝑡𝑅_𝜏=𝜇−𝜆𝐸𝑒^𝑥𝜏−1−1−𝑥2^𝑇_𝑡𝐸_𝑡𝑉_𝜏=1𝑑𝜏𝜇−2𝐸_𝑡𝑉_𝜏−𝜆𝐸𝑒^𝑥−1−𝑥𝜏. (82)
Define the first central moment, second, third, and fourth central moments in the physical measure of random number 𝑥 as 𝜇_𝑥=𝐸(𝑥), Var(𝑥)=𝐸[(𝑥−𝜇_𝑥)²],  3rd(𝑥)=𝐸[(𝑥−𝜇_𝑥)³], and 4th(𝑥)=𝐸[(𝑥−𝜇_𝑥)⁴].
From (2) we have 𝑑𝑒^𝜅𝜏𝑉_𝜏=𝜅𝜃𝑒^𝜅𝜏𝑑𝑡+𝜀𝑒^𝜅𝜏𝑉_𝜏𝑑𝐵^𝑉_𝜏. (83) The integration is given by 𝑒^𝜅𝜏𝑉_𝜏=𝑉₀+𝜃𝑒^𝜅𝜏−1𝜏+𝜀^𝜏₀𝑒^𝜅𝜏𝑉_𝜏𝑑𝐵^𝑉_𝜏. (84) Then 𝐸_𝑡𝑉_𝜏=𝑒^−𝜅𝜏𝑉₀+𝜃1−𝑒^−𝜅𝜏. (85) Using (81), we have 𝐸_𝑡𝑅_𝜏−𝐸_𝑡𝑅_𝜏²=𝐸_𝑡^𝑇_𝑡𝑉_𝜏𝑑𝐵^𝑆_𝜏⋅^𝑇_𝑡𝑉_𝜏𝑑𝐵^𝑆_𝜏+𝐸_𝑡^𝑁𝜏_𝑖=1𝑥_𝑖−𝜆𝜏𝜇_𝑥²=^𝑇_𝑡𝐸_𝑡𝑉_𝜏𝑑𝜏+𝐸𝑁_𝜏𝐸𝑥−𝜇_𝑥²+2𝐸𝑁_𝜏𝑁_𝜏−𝜆𝜏𝐸𝑥−𝜇_𝑥𝜇_𝑥+𝐸𝑥−𝜇_𝑥²𝜇²_𝑥=𝐸_𝑡𝑉_𝜏𝜏+𝜆𝜏𝜇²_𝑥(𝑥,𝐸+Var)_𝑡𝑅_𝜏−𝐸_𝑡𝑅_𝜏³=𝐸_𝑡^𝑁𝜏_𝑖=1𝑥_𝑖−𝜆𝜏𝜇_𝑥³=𝜆𝜏𝜇³_𝑥+3𝜇_𝑥(𝑥)(𝑥,𝐸Var+3rd)_𝑡𝑅_𝜏−𝐸_𝑡𝑅_𝜏⁴=3!!𝐸_𝑡^𝑇_𝑡𝑉_𝜏𝑑𝜏²+6^𝑇_𝑡𝐸_𝑡𝑉_𝜏𝑑𝜏⋅𝐸_𝑡^𝑁𝜏_𝑖=1𝑥_𝑖−𝜆𝜏𝜇_𝑥²+𝐸_𝑡^𝑁𝜏_𝑖=1𝑥_𝑖−𝜆𝜏𝜇_𝑥⁴=3𝜏²𝐸_𝑡𝑉_𝜏²+6𝜆𝜏𝐸_𝑡𝑉_𝜏𝜇²_𝑥(𝑥)𝜇+Var+𝜆𝜏⁴_𝑥+6𝜇²_𝑥Var(𝑥)+4𝜇_𝑥3rd(𝑥)+4th(𝑥)+3𝜆²𝜏²𝜇²_𝑥+Var(𝑥)²=3𝜆𝜏[𝜇²_𝑥+Var(𝑥)]+𝜏𝐸_𝑡[𝑉_𝜏]²+𝜆𝜏𝜇⁴_𝑥+6𝜇²_𝑥Var(𝑥)+4𝜇_𝑥3rd(𝑥)+4th(𝑥). (86) The results of risk-neutral moments can be obtained with the same procedure in the risk-neutral probability measure.

Remark 13. If 𝐸_𝑡[𝑉_𝜏]=𝐸^𝑄_𝑡[𝑉_𝜏]=𝜎², where 𝜎² is constant, our results will degenerate into Proposition 3.19 by Zhang et al. (2012) [6].
We denote Skewness, Kurtosis as the skewness and kurtosis in risk-neutral measure, respectively. Skewness and kurtosis are very important for asset pricing. For example, Bakshi et al. (2003) [13] concluded that variations in the risk-neutral skew were instrumental in explaining the differential pricing of individual equity options and found that less negatively skewed stocks have flatter smiles. From Proposition 12, we easily obtain the following three corollaries about skewness and kurtosis.

Corollary 14. The skewness and kurtosis in physical measure are given by 𝐸Skewness=^𝑡𝑅_𝜏−𝐸_𝑡𝑅_𝜏³𝐸_𝑡𝑅_𝜏−𝐸_𝑡𝑅_𝜏^23/2=𝜆𝜏𝜇³_𝑥+3𝜇_𝑥Var(𝑥)+3𝑟𝑑(𝑥)𝐸_𝑡[𝑉_𝜏]𝜏+𝜆𝜏𝜇²_𝑥+Var(𝑥)^3/2,𝐸Kurtosis=^𝑡𝑅_𝜏−𝐸_𝑡𝑅_𝜏⁴𝐸_𝑡𝑅_𝜏−𝐸_𝑡𝑅_𝜏²²=3𝜇−3𝜆𝜏²_𝑥+Var(𝑥)+𝜏𝐸_𝑡[𝑉_𝜏]²+𝜆𝜏𝜇⁴_𝑥+6𝜇²_𝑥Var(𝑥)+4𝜇_𝑥(𝑥)(𝑥×𝐸3𝑟𝑑+4𝑡ℎ)_𝑡𝑉_𝜏𝜏+𝜆𝜏𝜇²_𝑥+Var(𝑥)²⁻¹−3. (87) And the skewness and kurtosis in risk-neutral measure are given by 𝐸Skewness=^𝑄𝑡𝑅_𝜏−𝐸^𝑄_𝑡𝑅_𝜏³𝐸^𝑄_𝑡𝑅_𝜏−𝐸^𝑄_𝑡𝑅_𝜏^23/2=𝜆^𝑄𝜏𝜇^𝑄_𝑥³+3𝜇^𝑄_𝑥Var^𝑄(𝑥)+3𝑟𝑑^𝑄(𝑥)𝜏𝐸^𝑄_𝑡𝑉_𝜏+𝜆^𝑄𝜏𝜇^𝑄_𝑥²+Var^𝑄(𝑥)^(3/2),𝐸Kurtosis=^𝑄𝑡𝑅_𝜏−𝐸^𝑄_𝑡𝑅_𝜏⁴𝐸^𝑄_𝑡𝑅_𝜏−𝐸^𝑄_𝑡𝑅_𝜏²²=3𝜇−3𝜆𝜏^𝑄_𝑥²+Var^𝑄(𝑥)+𝜏𝐸^𝑄_𝑡𝑉_𝜏²+𝜆^𝑄𝜏𝜇^𝑄_𝑥⁴+6𝜇^𝑄_𝑥²Var^𝑄(𝑥)+4𝜇^𝑄_𝑥3rd^𝑄(𝑥)+4th^𝑄(𝑥×)𝜏𝐸^𝑄_𝑡𝑉_𝜏+𝜆^𝑄𝜏𝜇^𝑄_𝑥²+Var^𝑄(𝑥)²⁻¹−3. (88)

Corollary 15. For jump size 𝑥, the first moment and the second, third, and fourth central moments in the physical measure are given by 𝐸_𝑡𝑅_𝜏1=𝜇−2𝐸_𝑡𝑉_𝜏−𝜆𝐸𝑒^𝑥,𝐸−1−𝑥_𝑡𝑅_𝜏−𝐸_𝑡𝑅_𝜏²=𝐸_𝑡𝑉_𝜏𝜏+𝜆𝜏𝐸𝑥²,𝐸_𝑡𝑅_𝜏−𝐸_𝑡𝑅_𝜏³=𝜆𝜏𝐸𝑥³,𝐸_𝑡𝑅_𝜏−𝐸_𝑡𝑅_𝜏⁴=𝜆𝜏𝐸𝑥⁴+3𝜆𝜏𝐸𝑥²+𝜏𝐸_𝑡𝑉_𝜏². (89) The first moment and the second, third, and fourth central moments in the risk-neutral are given by 𝐸^𝑄_𝑡𝑅_𝜏1=𝜇−2𝐸^𝑄_𝑡𝑉_𝜏−𝜆𝐸𝑒^−𝛾𝑥𝑒^𝑥,𝐸−1−𝑥_𝑄𝑡𝑅_𝜏−𝐸^𝑄_𝑡𝑅_𝜏²=𝜏𝐸^𝑄_𝑡𝑉_𝜏+𝜆𝜏𝐸𝑒^−𝛾𝑥𝑥²,𝐸^𝑄_𝑡𝑅_𝜏−𝐸^𝑄_𝑡𝑅_𝜏³=𝜆𝜏𝐸𝑒^−𝛾𝑥𝑥³,𝐸^𝑄_𝑡𝑅_𝜏−𝐸^𝑄_𝑡𝑅_𝜏⁴=𝜆𝜏𝐸𝑒^−𝛾𝑥𝑥⁴+3𝜆𝜏𝐸𝑒^−𝛾𝑥𝑥²+𝜏𝐸^𝑄_𝑡𝑉_𝜏², (90) where 𝐸_𝑡(𝑉_𝜏)=𝜃−𝑒^−𝜅𝜏(𝜃−𝑉₀), 𝐸^𝑄_𝑡𝑉_𝜏=𝜅𝜃𝜅+𝛾−𝜀𝜌𝐴𝜀𝜌−𝑒^{−[𝜅+(𝛾−𝜀𝜌𝐴)𝜀𝜌]𝜏}𝜅𝜃𝜅+𝛾−𝜀𝜌𝐴𝜀𝜌−𝑉₀. (91)

Proof. To compute central moments in the physical measure, we have 𝐸_𝑡(𝑉_𝜏)=𝜃−𝑒^−𝜅𝜏(𝜃−𝑉₀), 𝜇_𝑥=𝐸(𝑥), Var(𝑥)=𝐸[(𝑥−𝜇_𝑥)²]=𝐸[𝑥²]−𝜇²_𝑥, 3rd(𝑥)=𝐸[𝑥³]−3𝜇²_𝑥𝐸[𝑥²]+2𝜇³_𝑥, and 4th(𝑥)=𝐸[(𝑥−𝜇_𝑥)⁴]=𝐸[𝑥⁴]−4𝜇_𝑥𝐸[𝑥³]+6𝜇²_𝑥𝐸[𝑥²]−3𝜇⁴_𝑥.

Then, from Proposition 12, we have 𝐸_𝑡𝑅_𝜏1=𝜇−2𝐸_𝑡𝑉_𝜏−𝜆𝐸𝑒^𝑥,𝐸−1−𝑥_𝑡𝑅_𝜏−𝐸_𝑡𝑅_𝜏²=𝐸_𝑡𝑉_𝜏𝜏+𝜆𝜏𝐸𝑥²,𝐸_𝑡𝑅_𝜏−𝐸_𝑡𝑅_𝜏³=𝜆𝜏𝐸𝑥³,𝐸_𝑡𝑅_𝜏−𝐸_𝑡𝑅_𝜏⁴=𝜆𝜏𝐸𝑥⁴+3𝜆𝜏𝐸𝑥²+𝜏𝐸_𝑡𝑉_𝜏². (92) In order to compute central moments in the physical measure, from Proposition 8, we have 𝐸^𝑄_𝑡𝑉_𝜏=𝜅𝜃𝜅+𝛾−𝜀𝜌𝐴𝜀𝜌−𝑒^{−[𝜅+(𝛾−𝜀𝜌𝐴)𝜀𝜌]𝜏}𝜅𝜃𝜅+𝛾−𝜀𝜌𝐴𝜀𝜌−𝑉₀,𝜇^𝑄_𝑥=𝐸^𝑄(𝑥)=𝐸𝑒^−𝛾𝑥𝑥𝐸(𝑒^−𝛾𝑥),Var^𝑄(𝑥)=𝐸𝑒^−𝛾𝑥𝑥²𝐸(𝑒^−𝛾𝑥)−𝐸[𝑒^−𝛾𝑥𝑥]𝐸(𝑒^−𝛾𝑥)²3rd^𝑄(𝑥)=𝐸𝑒^−𝛾𝑥𝑥³𝐸(𝑒^−𝛾𝑥)𝐸𝑒−3^−𝛾𝑥𝑥𝐸(𝑒^−𝛾𝑥)𝐸𝑒^−𝛾𝑥𝑥²𝐸(𝑒^−𝛾𝑥)+2𝐸[𝑒^−𝛾𝑥𝑥]𝐸(𝑒^−𝛾𝑥)³,4th^𝑄(𝑥)=𝐸𝑒^−𝛾𝑥𝑥⁴𝐸(𝑒^−𝛾𝑥)𝐸𝑒−4^−𝛾𝑥𝑥³𝐸(𝑒^−𝛾𝑥)𝐸𝑒^−𝛾𝑥𝑥𝐸(𝑒^−𝛾𝑥)𝐸𝑒+6^−𝛾𝑥𝑥²𝐸(𝑒^−𝛾𝑥)𝐸𝑒^−𝛾𝑥𝑥𝐸(𝑒^−𝛾𝑥)²𝐸𝑒−3^−𝛾𝑥𝑥𝐸(𝑒^−𝛾𝑥)⁴. (93) Then, from Proposition 12, we have 𝐸^𝑄_𝑡𝑅_𝜏1=𝜇−2𝐸^𝑄_𝑡𝑉_𝜏−𝜆𝐸𝑒^−𝛾𝑥𝑒^𝑥,𝐸−1−𝑥_𝑄𝑡𝑅_𝜏−𝐸^𝑄_𝑡𝑅_𝜏²=𝐸^𝑄_𝑡𝑉_𝜏𝜏+𝜆𝜏𝐸𝑒^−𝛾𝑥𝑥²,𝐸^𝑄_𝑡𝑅_𝜏−𝐸^𝑄_𝑡𝑅_𝜏³=𝜆𝜏𝐸𝑒^−𝛾𝑥𝑥³,𝐸^𝑄_𝑡𝑅_𝜏−𝐸^𝑄_𝑡𝑅_𝜏⁴=𝜆𝜏𝐸𝑒^−𝛾𝑥𝑥⁴+3𝜆𝜏𝐸𝑒^−𝛾𝑥𝑥²+𝜏𝐸^𝑄_𝑡𝑉_𝜏². (94)

Corollary 16. For small risk aversion coefficient, one has the following relationship between the third central moments in the neutral-risk measure and in the physical measure: 𝐸^𝑄_𝑡𝑅_𝜏−𝐸^𝑄_𝑡𝑅_𝜏³≈𝐸_𝑡𝑅_𝜏−𝐸_𝑡𝑅_𝜏³−𝛾𝐸_𝑡𝑅_𝜏−𝐸_𝑡𝑅_𝜏⁴−3𝐸_𝑡𝑅_𝜏−𝐸_𝑡𝑅_𝜏²². (95)

Proof. From Corollary 15, for small risk aversion coefficient 𝑥, we have 𝐸^𝑄_𝑡𝑅_𝜏−𝐸^𝑄_𝑡𝑅_𝜏³=𝜆𝜏𝐸𝑒^−𝛾𝑥𝑥³≈𝜆𝜏𝐸1−𝛾𝑥𝑥³=𝜆𝜏𝐸𝑥³−𝛾𝜆𝜏𝐸𝑥⁴=𝐸_𝑡𝑅_𝜏−𝐸_𝑡𝑅_𝜏³−𝛾𝐸_𝑡𝑅_𝜏−𝐸_𝑡𝑅_𝜏⁴−3𝐸_𝑡𝑅_𝜏−𝐸_𝑡𝑅_𝜏²². (96)

5. Numerical Results

In this section, we use our model to solve the equity premium puzzle. First, we get the long equilibrium risk premium 𝜙 for 𝑡→𝑇, where 𝑇 is sufficiently large. In this case, 𝑉_𝑡→𝜃; 𝐴(𝑡)→0; then 𝜙=𝜇−𝑟=𝜃𝛾+𝜆1−𝑒^−𝛾𝑥𝑒^𝑥−1. (97)

To compare with the results from the model in [6], we take nonrandom constant jump size 𝑥=−0.08 and 𝑟=0.05,  𝜆=0.48, 𝜇=0.11, 𝜃=0.16². Then, we obtain the relative risk aversion coefficient 𝛾=2.13 which is the same as the numerical value in [6]. However, the risk premium 𝜙 is stochastic. We get the expectation of 𝜙 as follows: 𝐸𝜙=𝑒^−𝜅𝑡𝑉₀+𝜃1−𝑒^−𝜅𝑡×𝛾−𝜀𝜌𝐴(𝑡)+𝜆1−𝑒^−𝛾𝑥𝑒^𝑥−1. (98)

If we assume that other parameter values in our model are 𝑘=0.03, 𝑉₀=0.1, 𝜀=0.1, and 𝜌=−0.25, we will find that the expectation of 𝜙 tends to 0.6 from Figure 1, which is consistent with above analysis.

Figure 1 

The expectation of 𝜙.

As we know that the risk premium 𝜙 is stochastic and it has the same stochastic characteristic with volatility 𝑉_𝑡, the risk premium 𝜙 can be generated by a discrete scheme of system as follows: 𝑉(𝑡+Δ𝑡)=𝑉(𝑡)+𝜅(𝜃−𝑉(𝑡))Δ𝑡+𝜀𝑉(𝑡)𝜌𝜔√Δ𝑡,𝑉₀>0, (99) where Δ𝑡 is the time interval and 𝜔 is a sample from the standard normal distribution.

One path of 𝜙 is given in Figure 2.

Figure 2 

One path of 𝜙.

In realistic world, the risk premium is stochastic. For example, Figure 1a and Figure 5 in Graham and Harvey (2012) show that risk premium follows mean-reverting stochastic process in [41]. Thus, our model is more realistic than that in [6].

6. Conclusions

We successfully extend the model in [6] and obtain more general results. First, the equilibrium equity premium in general equilibrium framework links not only the jump risk but also the stochastic volatility risk. Moreover, the pricing kernel with stochastic volatility is shown in this paper. We employ the Fourier transformation method to obtain the exact expression of option price. Next, we get the relationship of central moments between the physical measure and risk-neutral measure. It is a vitally crucial problem in financial area. Our numerical results show that our model is more realistic than the one in [6].

Acknowledgment

This work is supported by the Fundamental Research Funds for the Central Universities (JBK130401).