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

Proposition 1. In general equilibrium framework, the equilibrium equity premium is given by

where

Proof. From the optimal control problem (5), we get the Bellman equation as follows:

where

Equating the derivatives of the Bellman equation (9) with respect to ω to zero, we have following equation:

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):

Then Bellman equation (9) can be written as follows: From (6), we conjecture that Then, substituting (11) into (14), we obtain This leads to a system of two ordinary differential equations (ODEs): where

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

we obtain the second-order differential equation A general solution has the form where Thus, Then, the solution of the second ODE (17) is 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].

Proposition 3. In general equilibrium framework, the pricing kernel is given in differential form by

and the integration is given by 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:

Proof. To satisfy the martingale condition, π_tS_t = E_t[π_TS_T], from (3) and (26), we have

Substituting (7) into (28), we have Using the property of Poisson process, $$, 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:

since, for any assets P_t at time t, we have which means Q is a risk-neutral probability measure.

Lemma 6. Define a new probability measure, Q^*, by the following Radon-Nikodym derivative:

then the following relation is true.

Proof. The change of probability measure formula gives

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

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.

Proposition 8. In general equilibrium framework, the price of European call option satisfies the following PDE:

where

Proof. First, we rewrite the stock with continue part and jump part:

where Similarly, where where From (40) and (42), we get where The martingale condition E[d(πc)] = 0 requires Using Lemma 6, we have 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 If we denote κ^Q = κ + (γ − ερA)ερ and θ^Q = (κθ/(κ + (γ − ερA)ερ)), then (48) can be written as

Remark 9. The stock process (2) in a risk-neutral measure Q can be written as

where 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

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].

Proposition 11. In general equilibrium framework, the pricing formula is given by

where

Proof. We denote X = ln S and g(X, V, t): = c(e^X, V, t) = c(S, V, t); then, the PDE (36) can be rewritten as

Let G(k, V, t) be Fourier transform of g(X, V, t): with G(k, V, T) = −(K^ik+1/(k² − ik)).

Denote k = k_r + ik_i; the inverse Fourier transform is given by

Then, the PDE (55) can be rewritten as

Denote h ≡ e^(1+ik)rτG; the PDE (58) can be rewritten as

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:

and the option price satisfies

Denoting H(V, τ): = h(k, V, τ), then, the PDE (61) can be rewritten as

We guess that a solution of (63) is as the following form:

with Γ^Q(0) = 0 and Λ^Q(0) = 0.

This leads to a system of two ODEs:

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

we obtain the second-order differential equation

A general solution of (68) has the form

where Thus,

Then, the solution of the second ODE (66) is

Proposition 12. The risk-neutral density of the continuously compounded return within τ = T − t, R_τ = ln S_T/S_t, S_t given in (38), is given by

The first moment, second, third, and fourth central moments of the continuously compounded return in the risk-neutral measure are given by where $$,  $$ is the first moment, $$, $$, and $$ are the second, third, and fourth central moments in the risk-neutral measure of random number x.

The physical density is given by

where The first moment, second, third, and fourth central moments of the continuously compounded return in the physical measure are given by where $$,  $$ is the first moment, $$, $$, and $$ 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 $$. Under the Fourier transform, this boundary condition becomes (K^ik/ik). Solving the PDE (36) under this new boundary condition, we have

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

then

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

Then, the first moment of the continuously compounded return in the physical measure can be easy to be obtained:

Define the first central moment, second, third, and fourth central moments in the physical measure of random number x as $$, $$,  $$, and $$.

From (2) we have

The integration is given by Then Using (81), we have The results of risk-neutral moments can be obtained with the same procedure in the risk-neutral probability measure.

Remark 13. If $$, where σ² is constant, our results will degenerate into Proposition 3.19 by Zhang et al. (2012) [6].

We denote $$, $$ 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

And the skewness and kurtosis in risk-neutral measure are given by

Corollary 15. For jump size x, the first moment and the second, third, and fourth central moments in the physical measure are given by

The first moment and the second, third, and fourth central moments in the risk-neutral are given by where $$,

Proof. To compute central moments in the physical measure, we have $$, $$, $$, $$, and $$.

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:

Proof. From Corollary 15, for small risk aversion coefficient x, we have