Efficient L-stable method for parabolic problems with application to pricing American options under stochastic volatility

Efficient L-stable numerical method for semilinear parabolic problems with nonsmooth initial data is proposed and implemented to solve Heston’s stochastic volatility model based PDE for pricing American options under stochastic volatility. The proposed new method is also used to solve two asset American options pricing problem. Cox and Matthews [S.M. Cox, P.C. Matthews, Exponential time differencing for stiff systems, Journal of Computational Physics 176 (2002) 430-455] developed a class of exponential time differencing Runge-Kutta schemes (ETDRK) for nonlinear parabolic problems. Kassam and Trefethen [A.K. Kassam, L.N. Trefethen, Fourth-order time stepping for stiff PDEs, SIAM Journal on Scientific Computing 26 (4) (2005) 1214-1233] showed that while computing certain functions involved in the Cox-Matthews schemes, severe cancelation errors can occur which affect the accuracy and stability of the schemes. Kassam and Trefethen proposed complex contour integration technique to implement these schemes in a way that avoids these cancelation errors. But this approach creates new difficulties in choosing and evaluating the contour integrals for larger problems. We modify the ETDRK schemes using positivity preserving Padé approximations of the matrix exponential functions and construct computationally efficient parallel version using splitting technique. As a result of this approach it is required only to solve several backward Euler linear problems in serial or parallel.     

Semiparametric bounds of mean and variance for exotic options

Finding semiparametric bounds for option prices is a widely studied pricing technique. We obtain closed-form semiparametric bounds of the mean and variance for the pay-off of two exotic (Collar and Gap) call options given mean and variance information on the underlying asset price. Mathematically, we extended domination technique by quadratic functions to bound mean and variances. This work was supported by National Science Foundation of the United States (Grant Nos. DMS-0720977 and DMS-0805929)     

有红利美式看跌期权定价树图模型的自适应性改进

本文基于支付固定红利美式看跌期权的三叉树图定价模型,对其进行了自适应性改进,从而解决了树图模型所存在的因为时间离散、状态不连续而产生的"非线性误差"问题.最后给出了实证分析,并与二叉树图和三叉树图进行了比较,结果表明进行自适应性改进后可以得到更加精确、有效的数值解.     

随机利率下服从分数O-U过程的欧式幂期权定价

该文考虑了利率和标的资产价格的随机性和均值回复行为,把扩展的Vasick模型和分数O-U过程进行组合,在随机利率环境下,研究了标的资产价格服从分数O-U过程的两类欧式幂期权定价问题,得到相应的定价公式,并给出了欧式幂期权的看涨.看跌平价关系.     

美式回望期权定价的有限元超收敛分析

考虑美式回望看跌期权的有限元方法．在把原问题转化成等价的变分不等式的基础上,研究了半离散格式在L^2和L^∞范数意义下的最优误差估计．此外,为了进一步提高逼近解的精度,借助超收敛分析技术和插值后处理方法,研究了H^1范数意义下的整体超收敛以及后验误差估计。     

FINITE DIFFERENCE APPROXIMATION FOR PRICING THE AMERICAN LOOKBACK OPTION

In this paper we are concerned with the pricing of lookback options with American type constrains. Based on the differential linear complementary formula associated with the pricing problem, an implicit difference scheme is constructed and analyzed. We show that there exists a unique difference solution which is unconditionally stable. Using the notion of viscosity solutions, we also prove that the finite difference solution converges uniformly to the viscosity solution of the continuous problem. Furthermore, by means of the variational inequality analysis method, the O（△t ＋ △x^2）-order error estimate is derived in the discrete L2-norm provided that the continuous problem is sufficiently regular. In addition, a numerical example is provided to illustrate the theoretical results.     

Optimal static-dynamic hedges for exotic options under convex risk measures

We study the problem of optimally hedging exotic derivatives positions using a combination of dynamic trading strategies in underlying stocks and static positions in vanilla options when the performance is quantified by a convex risk measure. We establish conditions for the existence of an optimal static position for general convex risk measures, and then analyze in detail the case of shortfall risk with a power loss function. Here we find conditions for uniqueness of the static hedge. We illustrate the computational challenge of computing the market-adjusted risk measure in a simple diffusion model for an option on a non-traded asset.     

Valuing the flexibility of investing in security process innovations

In this paper, we develop a decision model of a firm’s optimal strategy for investment in security process innovations (SPIs) when confronted with a sequence of malicious attacks. The model incorporates real options as a methodology to capture the flexibility embedded in such investment decisions. SPIs, when seamlessly integrated with the organization’s overall business dynamics, induce organizational learning and provide the flexibility of switching to more suitable technologies as the environment of malicious attacks changes. The theoretical contribution of this paper is a mathematical model of the invest-to-learn and switching options generated upon early investment in flexible SPIs. The practical significance of the paper is the application of a binomial lattice model to approximate the continuous-time model, resulting in an easy to use decision aid for managers.     

The determinant of production entry and exit model on financing behavior

This study seeks for equity/debt values and the relevant potential firm value with financing or not when the real options approach is assessed. The paper deals with the following relative problems: (1) the assessment rule of decision whether to stop production or not; (2) whether the (dis) investment cost or salvage could reflect the production scale; (3) whether the rate of capital cost or the rate of return in different stages could reflect the suitable risk premium; (4) when the investment cost, exit cost, and salvage are the linear functions of production volume and follow the geometric Brownian motion to analyze the optimum external financing behavior and to decide the production thresholds of production entry and exit.     

Applying the maximum NPV rule with discounted/growth factors to a flexible production scale model

This study examines the maximum net present value of the market entry and exit thresholds derived by the traditional net present value method and combines the real options approach for the project investment or disinvestment. The discounted and growth factors are incorporated into the proposed entry and exit models, facilitating the complicated calculations required to identify the discounted and growth rates so as to assess and determine the expected present value of uncertain cash flow streams. Consequently, this investigation successfully combines applying the maximum net present value method and the real options approach to decision-making with the simple consideration of the discounted and growth factors in the flexible production scale model.