Three computational techniques for approximation of counterparty exposure for financial derivatives are presented. The exposure can be used to quantify so-called Credit Valuation Adjustment (CVA) and Potential Future Exposure (PFE), which are of utmost importance for modern risk management in the financial industry, especially since the recent credit crisis. The three techniques all involve a Monte Carlo path discretization and simulation of the underlying entities. Along the generated paths, the corresponding values and distributions are computed during the entire lifetime of the option. Option values are computed by either the finite difference method for the corresponding partial differential equations, or the simulation-based Stochastic Grid Bundling Method (SGBM), or by the COS method, based on Fourier-cosine expansions. In this research, numerical results are presented for early-exercise options. The underlying asset dynamics are given by either the Black–Scholes or the Heston stochastic volatility model.

No abstract

According to Basel III, financial institutions have to charge a credit valuation adjustment (CVA) to account for a possible counterparty default. Calculating this measure and its sensitivities is one of the biggest challenges in risk management. Here, we introduce an efficient method for the estimation of CVA and its sensitivities for a portfolio of financial derivatives. We use the finite difference Monte Carlo (FDMC) method to measure exposure profiles and consider the computationally challenging case of foreign exchange barrier options in the context of the Black-Scholes as well as the Heston stochastic volatility model, with and without stochastic domestic interest rate, for a wide range of parameters. In the case of a fixed domestic interest rate, our results show that FDMC is an accurate method compared with the semi-analytic COS method and, advantageously, can compute multiple options on one grid. In the more general case of a stochastic domestic interest rate, we show that we can accurately compute exposures of discontinuous one-touch options by using a linear interpolation

Insurance companies issue guarantees that need to be valued according to the market expectations. By calibrating option pricing models to the available implied volatility surfaces, one deals with the so-called risk-neutral measure Q , which can be used to generate market consistent values for these guarantees. For asset liability management, insurers also need future values of these guarantees. Next to that, new regulations require insurance companies to value their positions on a one-year horizon. As the option prices at t = 1 are unknown, it is common practice to assume that the parameters of these option pricing models are constant, i.e., the calibrated parameters from time t = 0 are also used to value the guarantees at t = 1 . However, it is well-known that the parameters are not constant and may depend on the state of the market which evolves under the real-world measure P . In this paper, we propose improved regression models that, given a set of market variables such as the VIX index and risk-free interest rates, estimate the calibrated parameters. When the market variables are included in a real-world simulation, one is able to assess the calibrated parameters (and consequently the implied volatility surface) in line with the simulated state of the market. By performing a regression, we are able to predict out-of-sample implied volatility surfaces accurately. Moreover, the impact on the Solvency Capital Requirement has been evaluated for different points in time. The impact depends on the initial state of the market and may vary between −46% and +52%.

The focus of this paper is the efficient computation of counterparty credit risk exposure on portfolio level. Here, the large number of risk factors rules out traditional PDE-based techniques and allows only a relatively small number of paths for nested Monte Carlo simulations, resulting in large variances of estimators in practice. We propose a novel approach based on Kolmogorov forward and backward PDEs, where we counter the high dimensionality by a generalisation of anchored-ANOVA decompositions. By computing only the most significant terms in the decomposition, the dimensionality is reduced effectively, such that a significant computational speed-up arises from the high accuracy of PDE schemes in low dimensions compared to Monte Carlo estimation. Moreover, we show how this truncated decomposition can be used as control variate for the full high-dimensional model, such that any approximation errors can be corrected while a substantial variance reduction is achieved compared to the standard simulation approach. We investigate the accuracy for a realistic portfolio of exchange options, interest rate and cross-currency swaps under a fully calibrated ten-factor model.