This paper follows the framework of P. Klein (1996) to price vulnerable options when the market is incomplete. Vulnerable options, which are usually traded in the over-the-counter market, may not only face the risk of default but also the risk of illiquidity. Thus, pricing such options under the assumption of market completeness, as was done by H. Johnson and R. Stulz (1987) and P. Klein (1996), seems to be a mistake. Accordingly, the proposed model uses the methodology proposed by J. H. Cochrane and J. Saá-Requejo (2000) to price vulnerable options under both deterministic and stochastic interest rates in an incomplete market. The model is found to perform well when the interest rate is stochastic.

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.

This article investigates the valuation of a foreign equity option whose value depends on the exchange rate and foreign equity prices. Assuming that these underlying price processes are correlated and driven by a multidimensional Lévy process, a method suitable for solving the complex valuation problem is developed. First, to reduce the number of dimensions of the problem, the probability measure is changed to embed some dimensions of the Lévy process into the pricing measure. Second, to simplify the integral complexity of the discounted terminal payoff, the valuation problem is transformed to Fourier space. The main contribution of this study is that by combining these two methods, the multivariate valuation problem is significantly simplified, and very accurate results are obtained relatively quickly. This powerful method can also be applied to other multivariate pricing problems involving Lévy processes.

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