This article introduces a general quadratic approximation scheme for pricing American options based on stochastic volatility and double jump processes. This quadratic approximation scheme is a generalization of the Barone-Adesi and Whaley approach and nests several option models. Numerical results show that this quadratic approximation scheme is efficient and useful in pricing American options.

This article provides quasi-analytic pricing formulae for forward-start options under stochastic volatility, double jumps, and stochastic interest rates. Our methodology is a generalization of the Rubinstein approach and can be applied to several existing option models. Properties of a forward-start option may be very different from those of a plain vanilla option because the entire uncertainty of evolution of its price is cut off by the strike price at the time of determination. For instance, in contrast to the plain vanilla option, the value of a forward-start option may not always increase as the maturity increases. It depends on the current term structure of interest rates.

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By applying the Heath-Jarrow-Morton (HJM) framework, an analytical approximation for pricing American options on foreign currency under stochastic volatility and double jump is derived. This approximation is also applied to other existing models for the purpose of comparison. There is evidence that such types of jumps can have a critical impact on earlyexercise premiums that will be significant for deep out-of-the-money options with short maturities. Moreover, the importance of the term structure of interest rates to early-exercise premiums is demonstrated as is the sensitivity of these premiums to correlation-related parameters.

Although quasi-analytic formulas can be derived for European-style financial claims in Heston's stochastic volatility model, the inverse Fourier integration involved makes the calculation somewhat complicated. This challenge has puzzled practitioners for many years because most implementations of Heston's formula are not robust, even for customarily-used Heston parameters, as time to maturity is increased. In this article, a simplified approach is proposed to solve the numerical instability problem inherent to the fundamental solution of the Heston model. Specifically, the solution does not require any additional function or a particular mechanism for most software packages or programming library routines to correctly evaluate Heston's analytics.Stochastic volatility model, Heston, discountinuity, options,