International Journal of Theoretical and Applied Finance

We construct fast and accurate methods for (a) approximate Laplace inversion, (b) approximate calculation of the Wiener-Hopf factors for wide classes of Lévy processes with exponentially decaying Lévy densities, and (c) approximate pricing of lookback options. In all cases, we use appropriate conformal change-of-variable techniques, which allow us to apply the simplified trapezoid rule with a small number of terms (the changes of variables in the outer and inner integrals and in the formulas for the Wiener-Hopf factors must be compatible in a certain sense). The efficiency of the method stems from the properties of functions analytic in a strip (these properties were explicitly used in finance by Feng and Linetsky 2008). The same technique is applicable to the calculation of the pdfs of supremum and infimum processes, and to the calculation of the prices and sensitivities of options with lookback and barrier features.

We perform a Laplace transform inversion in the time parameter on the two Wiener-Hopf factors for a spectrally negative tempered stable Lévy process. This yields the issuing price of continuously monitored lookback options. We also propose a simulation technique for the purpose of Monte-Carlo valuation and discuss the convergence rate to continuous prices when the number of discretization steps (i.e. monitoring dates) goes to infinity.

Current CVA modeling framework has ignored the impact of stochastic recovery rate. Due to the possible negative correlation between default and recovery rate, stochastic recovery rate could have a doubling effect on wrong-way risk. In the case of a payer CDS, when counterparty defaults, the CDS value could be higher due to default contagion while the recovery rate may also be lower if the economy is in a downturn. Using our recently proposed model of correlated stochastic recovery in the default time Gaussian copula framework, we demonstrate this double impact on wrong-way risk in the CVA calculation for a payer CDS. We also present a new form of Gaussian copula that correlates both default time and recovery rate.

The transition probability of a Cox–Ingersoll–Ross process can be represented by a non-central chi-square density. First, we establish a new representation for the central chi-square density based on sums of powers of generalized Gaussian random variables. Second, we show that Marsaglia's polar method extends to this distribution, providing a simple, exact, robust and efficient acceptance–rejection method for generalized Gaussian sampling and thus central chi-square sampling. Third, we derive a simple, high-accuracy, robust and efficient direct inversion method for generalized Gaussian sampling based on the Beasley–Springer–Moro method. Indeed the accuracy of the approximation to the inverse cumulative distribution function is to the tenth decimal place. We then apply our methods to non-central chi-square variance sampling in the Heston model. We focus on the case when the number of degrees of freedom is small and the zero boundary is attracting and attainable, typical in foreign exchange markets. Using the additivity property of the chi-square distribution, our methods apply in all parameter regimes.

We present an acceleration technique, effective for explicit finite difference schemes describing diffusive processes with nearly symmetric operators, called Super-Time-Stepping (STS). The technique is applied to the two-factor problem of option pricing under stochastic volatility. It is shown to significantly reduce the severity of the stability constraint known as the Courant-Friedrichs-Lewy condition whilst retaining the simplicity of the chosen underlying explicit method.

For European and American put options under Heston's stochastic volatility model we demonstrate degrees of acceleration over standard explicit methods sufficient to achieve comparable, or superior, efficiency to benchmark implicit schemes. We conclude that STS accelerated methods are powerful numerical tools for the pricing of options which inherit the simplicity of explicit methods whilst achieving high accuracy at low computational cost and offer a compelling alternative to conventional implicit techniques.

In this article, we study the price monotonicity in the parameters of the Heston model for a contract with a convex pay-off function; in particular we consider European put options. We show that the price is increasing in the constant term in the drift of the variance process and decreasing in the coefficient of the linear term in the drift of variance process. We also show that the price is increasing in the correlation for small values of the stock and decreasing for the large values.

While the proportional hazard model is recognized to be statistically meaningful for analyzing and estimating financial event risks, the existing literature that analytically deals with the valuation problems is very limited. In this paper, adopting the proportional hazard model in continuous time setting, we provide an analytical treatment for the valuation problems. The derived formulas, which are based on the generalized Edgeworth expansion and give approximate solutions to the valuation problems, are widely useful for evaluating a variety of financial products such as corporate bonds, credit derivatives, mortgage-backed securities, saving accounts and time deposits. Furthermore, the formulas are applicable to the proportional hazard model having not only continuous processes (e.g., Gaussian, affine, and quadratic Gaussian processes) but also discontinuous processes (e.g., Lévy and time-changed Lévy processes) as stochastic covariates. Through numerical examples, it is demonstrated that very accurate values can be quickly obtained by the formulas such as a closed-form formula.