Wenli Zhu

Option pricing using the fast Fourier transform under the double exponential jump model with stochastic volatility and stochastic intensity

This paper is based on the FFT (Fast Fourier Transform) approach for the valuation of options when the underlying asset follows the double exponential jump process with stochastic volatility and stochastic intensity. Our model captures three terms structure of stock prices, the market implied volatility smile, and jump behavior. Via the FFT method, numerical examples using European call options show effectiveness of the proposed model. Meanwhile, numerical results prove that the FFT approach is considerably correct, fast and competent.

Option Pricing under Risk-Minimization Criterion in an Incomplete Market with the Finite Difference Method

We study option pricing with risk-minimization criterion in an incomplete market where the dynamics of the risky underlying asset is governed by a jump diffusion equation with stochastic volatility. We obtain the Radon-Nikodym derivative for the minimal martingale measure and a partial integro-differential equation (PIDE) of European option. The finite difference method is employed to compute the European option valuation of PIDE.

Exponential Stability of Stochastic Nonlinear Dynamical Price System with Delay

Based on Lyapunov stability theory, Ito formula, stochastic analysis, and matrix theory, we study the exponential stability of the stochastic nonlinear dynamical price system. Using Taylors theorem, the stochastic nonlinear system with delay is reduced to an -dimensional semilinear stochastic differential equation with delay. Some sufficient conditions of exponential stability and corollaries for such price system are established by virtue of Lyapunov function. The time delay upper limit is solved by using our theoretical results when the system is exponentially stable. Our theoretical results show that if the classical price Rayleigh equation is exponentially stable, so is its perturbed system with delay provided that both the time delay and the intensity of perturbations are small enough. Two examples are presented to illustrate our results.

Exponential Stability of Stochastic Differential Equation with Mixed Delay

This paper focuses on a class of stochastic differential equations with mixed delay based on Lyapunov stability theory, Ito formula, stochastic analysis, and inequality technique. A sufficient condition for existence and uniqueness of the adapted solution to such systems is established by employing fixed point theorem. Some sufficient conditions of exponential stability and corollaries for such systems are obtained by using Lyapunov function. By utilizing Doob’s martingale inequality and Borel-Cantelli lemma, it is shown that the exponentially stable in the mean square of such systems implies the almost surely exponentially stable. In particular, our theoretical results show that if stochastic differential equation is exponentially stable and the time delay is sufficiently small, then the corresponding stochastic differential equation with mixed delay will remain exponentially stable. Moreover, time delay upper limit is solved by using our theoretical results when the system is exponentially stable, and they are more easily verified and applied in practice.

Optimal portfolio and consumption with habit formation in a jump diffusion market

This paper studies optimal portfolio and consumption selection with habit formation in a jump diffusions incomplete market in continuous-time. The stochastic maximum principle for jump processes is applied to solve habit-forming utility maximization problem. We transform this problem into the case not involving habit formation in mechanically. Then the solution in the state feedback form is given. The relationship between maximum principle and dynamic programming is employed to get the expression of the relative risk aversion (RRA) coefficient and its distribution. Finally, for a special case, the stationary mean of the RRA coefficient is obtained and the numerical experiment indicates our model with jump diffusions is better than the model in [1] to resolve the equity premium puzzle in a way.

Fast Fourier Transform Based Power Option Pricing with Stochastic Interest Rate, Volatility, and Jump Intensity

Firstly, we present a more general and realistic double-exponential jump model with stochastic volatility, interest rate, and jump intensity. Using Feynman-Kac formula, we obtain a partial integrodifferential equation (PIDE), with respect to the moment generating function of log underlying asset price, which exists an affine solution. Then, we employ the fast Fourier Transform (FFT) method to obtain the approximate numerical solution of a power option which is conveniently designed with different risks or prices. Finally, we find the FFT method to compute that our option price has better stability, higher accuracy, and faster speed, compared to Monte Carlo approach.

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

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.

Exponential Stability of Stochastic Systems with Delay and Poisson Jumps

This paper focuses on the model of a class of nonlinear stochastic delay systems with Poisson jumps based on Lyapunov stability theory, stochastic analysis, and inequality technique. The existence and uniqueness of the adapted solution to such systems are proved by applying the fixed point theorem. By constructing a Lyapunov function and using Doob’s martingale inequality and Borel-Cantelli lemma, sufficient conditions are given to establish the exponential stability in the mean square of such systems, and we prove that the exponentially stable in the mean square of such systems implies the almost surely exponentially stable. The obtained results show that if stochastic systems is exponentially stable and the time delay is sufficiently small, then the corresponding stochastic delay systems with Poisson jumps will remain exponentially stable, and time delay upper limit is solved by using the obtained results when the system is exponentially stable, and they are more easily verified and applied in practice.

Errata corrige optimal portfolio and consumption with habit formation in a jump diffusion market

According to errata corrige of [2] in [3], we find (17), (19), (20), (21), (46) and (48) in [1] are incorrect. Now we present corrected formulas in (1)–(7) as follows, respectively. dp ðtÞ 1⁄4 _ f ðtÞX ðtÞ 1 þ ðc 1Þf ðtÞX ðtÞ 2 rtXðtÞ þ ðlt rtÞu ðtÞ Ay ðtÞ þ 1 2 ðc 1Þðc 2Þf ðtÞX ðtÞ rt u ðtÞ 2 þ Z R0 f ðtÞ1⁄2ðX ðtÞ þ hðt; zÞu ðtÞÞ 1 X ðtÞ 1 ðc 1ÞX ðtÞ hðt; zÞu ðtÞ mðdzÞ dt þðc 1Þf ðtÞðX ðtÞÞ rtu ðtÞdBðtÞ þ Z R0 f ðtÞ ðX ðt Þ þ hðt; zÞu ðt ÞÞ 1 X ðt Þ 1 h ie Nðdt;dzÞ; ð1Þ k ðt; zÞ 1⁄4 f ðtÞX ðtÞ 1 ð1þ hðt; zÞu ðtÞX ðtÞ Þ c 1 1 h i ; ð2Þ _ f ðtÞ þWt f ðtÞ þ ð1 cÞðAedtÞ 1 c f ðtÞ c c 1 1⁄4 0; f ðTÞ 1⁄4 0; ( ð3Þ where wt 1⁄4 crt þ ðc 1Þðlt rtÞu ðtÞX ðtÞ 1 þ 1 2 ðc 1Þðc 2ÞX ðtÞ rt u ðtÞ 2 þ Z R0 ð1þ hðt; zÞu ðtÞX ðtÞ Þ c 1 1þ ðc 1Þhðt; zÞu ðtÞX ðtÞ 1 h i mðdzÞ; 236 X. Ruan et al. / Applied Mathematics and Computation 232 (2014) 235–236 Fðm ðtÞÞ 1⁄4 lt rt þ ðc 1Þrt m ðtÞ þ Z R0 hðt; zÞ1⁄2ð1þ hðt; zÞm ðtÞÞ 1 1 mðdzÞ 1⁄4 0; ð4Þ Fðm Þ 1⁄4 l r þ ðc 1Þrm þ k Z R0 z1⁄2ð1þ zm Þ 1 1 f ðzÞdz 1⁄4 0; ð5Þ W 1⁄4 cr þ ðc 1Þðl rÞm þ 1 2 ðc 1Þðc 2Þrt m 2 þ k Z R0 1⁄2ð1þ zm Þ 1 1 ðc 1Þzm f ðzÞdz; ð6Þ References [1] X. Ruan, W. Zhu, J. Hu, J. Huang, Optimal portfolio and consumption with habit formation in a jump diffusion market, Appl. Math. Comput. 222 (2013) 391–401. [2] N.C. Framstad, B. Øksendal, A. Sulem, Sufficient stochastic maximum principle for the optimal control of jump diffusions and applications to finance, J. Optim. Theory Appl. 121 (1) (2004) 77–98. [3] N.C. Framstad, B. Øksendal, A. Sulem, Errata corrige sufficient stochastic maximum principle for the optimal control of jump diffusions and applications to finance, J. Optim. Theory Appl. 124 (2) (2005) 511–512.

Continuous-Time Portfolio Selection and Option Pricing under Risk-Minimization Criterion in an Incomplete Market

We study option pricing with risk-minimization criterion in an incomplete market where the dynamics of the risky underlying asset are governed by a jump diffusion equation. We obtain the Radon-Nikodym derivative in the minimal martingale measure and a partial integrodifferential equation (PIDE) of European call option. In a special case, we get the exact solution for European call option by Fourier transformation methods. Finally, we employ the pricing kernel to calculate the optimal portfolio selection by martingale methods.