Abstract: American options can be priced by solving linear complementary problems (LCPs) with parabolic partial(-integro) differential operators under stochastic volatility and jump-diffusion models like Heston, Merton, and Bates models. These operators are discretized using finite difference methods leading to a so-called full order model (FOM). Here reduced order models (ROMs) are derived employing proper orthogonal decomposition (POD) and non negative matrix factorization (NNMF) in order to make pricing much faster within a given model parameter variation range. The numerical experiments demonstrate orders of magnitude faster pricing with ROMs.

Abstract: We develop and analyze a second order implicit predictor-corrector scheme based on Exponential time differencing (ETD) method for pricing American put options under Multistate - Regime Switching economy with Jump Diffusion Models. Our approach formulates the problem of American options pricing as a set of coupled partial intgro-diffrential equations (PIDE), which we solve using a primitive tri-diagonal linear system, while we treat the complexity of the dense jump probability generator and the nonlinear regime switching terms explicitly in time. We define both differential and integral terms of the PIDE on the same domain, and discretize the spatial derivatives using a non-uniform mesh. The American option constraint is enforced by using a scaled penalty method approach to establish a conservative bound for the penalty parameter. We also provide a detailed treatment for the consistency, stability, and convergence of the proposed method, and analytically study the impact of the jump intensity, penalty and non-uniform parameters on convergence and solution accuracy. The dynamic properties of the no -uniform mesh and ETD approach are utilized to calibrate suitable values for the penalty and no uniform grid parameters. Superiority of the prosed scheme over recently published methods is demonstrated by numerical examples by discussing the efficiency, accuracy and reliability of the proposed approach

Abstract: Different dividend assumptions consistent with prices of European option can lead to very different prices for American options. In this paper we study the impact of continuous versus discrete and cash versus proportional dividend assumption on the prices of European and American options and discuss the consequences it implies for calibration and pricing of exotic instruments.

Abstract: As per regulations and common risk management practice, the credit risk of a portfolio is managed via its potential future exposures (PFEs), expected exposures (EEs), and related measures, the expected positive exposure (EPE), effective expected exposure (EEE) and the effective expected positive exposure (EEPE). Notably, firms use these exposures to set economic and regulatory capital levels. Their values have a big impact on the capital that firms need to hold to manage their risks. Due to the growth of CVA computations, and the similarity of CVA computations to exposure computations, firms find it expedient to compute these exposures under the risk neutral measure. Here we show that exposures computed under the risk neutral measure are essentially arbitrary. They depend on the choice of numeraire, and can be manipulated by choosing a different numeraire. The numeraire can even be chosen in such a way as to pass backtests. Even when restricting attention to commonly used numeraires, exposures can vary by a factor of two or more. As such, it is critical that these calculations be done under the real world measure, not the risk neutral measure. To help rectify the situation, we show how to exploit measure changes to efficiently compute real world exposures in a risk neutral framework, even when there is no change of measure from the risk neutral measure to the real world measure. We also develop a canonical risk neutral measure that can be used as an alternative approach to risk calculations.

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 is one of the big challenges in risk management. In earlier studies, future distributions of derivative values have been simulated by a combination of finite difference methods for the option valuation and Monte Carlo methods for the state space sampling of the underlying, from which the portfolio exposure and its quantiles can be estimated. By solving a forward Kolmogorov PDE for the future underlying distribution instead of Monte Carlo simulation, we hope to achieve efficiency gains and better accuracy especially in the tails of future exposures. Together with the backward Kolmogorov equation, the expected exposure and quantiles can then directly be obtained without the need for an extra Monte Carlo simulation. We studied the applicability of PCA and ANOVA-based dimension reduction in the context of a portfolio of risk factors. Typically, for these portfolios, a huge number of derivatives are traded on a relatively small number of risk factors. By solving a PDE for one risk factor, it is possible to value all derivatives traded on this single factor over time. However, if we want to solve a PDE for multiple risk factors, one has to deal with the curse of dimensionality. Between these risk factors, the correlation is often high, and therefore PCA and ANOVA are promising techniques for dimension reduction and can enable us to compute the exposure profiles for higher dimensional portfolios. We compute lower dimensional approximations where only one factor is taken stochastic and all other factors follow a deterministic term structure. Next, we correct this low dimensional approximation by two dimensional approximations. We also look into the effect of taking higher (three) dimensional corrections. In our results, our method is able to compute Exposures (EE, EPE and ENE) and Quantiles for a real portfolio driven by 10 different risk factors. This portfolio consists of Cross-Currency Swaps, Interest rate swaps and FX call or put options. The risk factors are: stochastic FX rates, stochastic volatility and stochastic domestic and foreign interest rates. The method is accurate and fast when compared to a full-scale Monte Carlo implementation.