Abstracts

As a consequence of the recent financial crisis, quantities like Expected Exposure (EE), the expected amount of money that may be lost in the case of a counterparty default, and Potential Future Exposure (PFE) which measures the loss given a fixed confidence interval are now very important. Both EE and PFE can be deduced from the distribution of the option price at future time points. In this research, we show that this distribution can be obtained efficiently for a broad class of payoffs and stochastic models, by combining two popular methods in finance, namely: the Monte Carlo method for the generation of future scenarios for the risk factors and the finite difference method for solving the option pricing partial differential equation (PDE). Hence this novel method is named the Finite Difference Monte Carlo method (FDMC). An important feature of the method is that it can estimate sensitivities of EE to market risk factors highly efficient.
To compute the EE of an option, we first need scenarios of the underlying that can be generated by a Monte Carlo simulation. At any time point, during the life of the option, an option value for every path is needed. From these prices a distribution can be computed and the EE results as the mean, whereas the PFE can be extracted as a quantile. This calls for an efficient pricing technique in particular when exotic options and nonstandard dynamics such as the Heston stochastic volatility model are in scope. In the FDMC method, the option prices are obtained by solving the PDE by the finite difference method. It fully benefits from the advantages of the individual methods, from the Monte Carlo method, the scenario generation is used, while the finite difference method gives us a grid of option prices. The generated scenarios are interpolated on the solution grid to obtain a distribution of the price.
The validity of the FDMC method is done by a comparative study. This study is presented in a forthcoming paper together with Q. Feng and C.W. Oosterlee where also SGBM is tested. The FDMC method is applied to Bermudan put options under the Heston dynamics and the results are validated by comparison with the semi-analytic COS method.
In modern risk management, CVA needs to be hedged, therefore the sensitivities play an important role. The bump and revalue approach is computationally unattractive. By combining the FDMC method with the pathwise Monte Carlo method, these sensitivities can be obtained fast and accurate.