Option pricing with mean reversion and stochastic volatility

Many underlying assets of option contracts, such as currencies, commodities, energy, temperature and even some stocks, exhibit both mean reversion and stochastic volatility. This paper investigates the valuation of options when the underlying asset follows a mean-reverting lognormal process with stochastic volatility. A closed-form solution is derived for European options by means of Fourier transform. The proposed model allows the option pricing formula to capture both the term structure of futures prices and the market implied volatility smile within a unified framework. A bivariate trinomial lattice approach is introduced to value path-dependent options with the proposed model. Numerical examples using European options, American options and barrier options demonstrate the use of the model and the quality of the numerical scheme.     

Option pricing in incomplete markets

Expected utility maximization is a very useful approach for pricing options in an incomplete market. The results from this approach contain many important features observed by practitioners. However, under this approach, the option prices are determined by a set of coupled nonlinear partial differential equations in high dimensions. Thus, it represents numerous significant difficulties in both theoretical analysis and numerical computations. In this paper, we present accurate approximate solutions for this set of equations.     

Exact and approximate solutions for options with time-dependent stochastic volatility

In this paper it is shown how symmetry methods can be used to find exact solutions for European option pricing under a time-dependent 3/2-stochastic volatility model $\mathit{dv}=\mathit{kv}(A(t)-v)\mathit{dt}+{\mathit{bv}}^{\frac{3}{2}}\mathit{dZ}$. This model with $A(t)$ constant has been proven by many authors to outperform the Heston model in its ability to capture the behaviour of volatility and fit option prices. Further, singular perturbation techniques are used to derive a simple analytic approximation suitable for pricing options with short tenor, a common feature of most options traded in the market.     

Asymptotic expansion for pricing options for a mean-reverting asset with multiscale stochastic volatility

This work investigates the valuation of options when the underlying asset follows a mean-reverting log-normal process with a stochastic volatility that is driven by two stochastic processes with one persistent factor and one fast mean-reverting factor. Semi-analytical pricing formulas for European options are derived by means of multiscale asymptotic techniques. Numerical examples demonstrate the use of the model and the quality of the numerical scheme.     

A note on the reflected backward stochastic differential equations driven by a Lévy process with stochastic Lipschitz condition

In this note, we prove the existence and uniqueness of the solution for a class of reflected backward stochastic differential equations (RBSDEs in short) related to the subdifferential operator of a lower semi-continuous convex function, driven by Teugels martingales associated with a Lévy process. Some known results are generalized and improved.     

On the explicit evaluation of the Geometric Asian options in stochastic volatility models with jumps

In the present paper we provide a semiexplicit valuation formula for Geometric Asian options, with fixed and floating strike under continuous monitoring, when the underlying stock price process exhibits both stochastic volatility and jumps. More precisely, we shall work in the Barndorff-Nielsen and Shephard (BNS) model framework. We shall provide some numerical illustrations of the results obtained.     

带基差风险和交易费用的不完全市场下的期权定价方法

本文研究了同时带有基差风险和交易费用的不安全市场中的权证定价方法。把[1]的模型推广到了考虑基差风险的情况[2]。期权的价格以一个三维自由边界问题的解给出,并含有两个相关的股票价格变量的相关系数。     

Option pricing with regime switching by trinomial tree method

We present a fast and simple tree model to price simple and exotic options in Markov Regime Switching Model (MRSM) with multi-regime. We modify the trinomial tree model of Boyle (1986) [12] by controlling the risk neutral probability measure in different regime states to ensure that the tree model can accommodate the data of all different regimes at the same time preserving its combining tree structure. In MRSM, the market might not be complete, therefore we provide some ideas and discussions on managing the regime switching risk in support of our results.     

Large deviation principles for 2-D stochastic Navier-Stokes equations driven by Lévy processes

In this paper, we establish a large deviation principle for the two-dimensional stochastic Navier-Stokes equations driven by Lévy processes, which involves the study of the Lévy noise and the investigation of the effect of the highly nonlinear, unbounded drifts.     

Almost automorphic solutions for stochastic differential equations driven by Lévy noise

We introduce the concepts of Poisson square-mean almost automorphy and almost automorphy in distribution. Under suitable conditions on the coefficients, we establish the existence of solutions which are almost automorphic in distribution for some semilinear stochastic differential equations with infinite dimensional Lévy noise. We further discuss the global asymptotic stability of these solutions. Finally, to illustrate the theoretical results obtained in this paper, we give several examples.