The Pricing Equations. Analysis of Finite Difference Methods. Special Issues. Coordinate Transformations. Numerical Examples. Index.

This thesis demonstrates how to compute Greeks accurately and efficiently for the displaced-diffusion LIBOR market model and the separable LIBOR Markovfunctional model. Chapter 1 introduces the problem and discusses the three main classes of interest rate models. The LIBOR market model is introduced in detail in Chapter 2, where we also provide information for an efficient model implementation. In Chapter 3 we extend the pathwise adjoint method to the displaced-diffusion LIBOR market model and present a 20 percent speed improvement. This improvement is achieved without additional approximations to those of Giles and Glasserman, [34]. We also derive the pathwise adjoint method for the predictor-corrector drift approximation. We compare the deltas and vegas computed using log-Euler and predictor-corrector, showing both methods have the same computational order but the latter is more accurate. In Chapter 4 we introduce the separable LIBOR Markov-functional model. We present a new formulation in the spot measure and find that for a Monte Carlo implementation the spot measure performs better than the terminal measure, which is consistent with well-known results for the LIBOR market model. For a PDE implementation, however, we find that the terminal measure performs better than the spot measure. These results are demonstrated by pricing interest rate caps and double-digital options. Chapter 5 presents the separable LIBOR Markov-functional model as a control variate for the LIBOR market model. This method has been previously suggested in the literature, however, it only works for prices and deltas, not vegas. We present a new technique that is effective for all three. Our approach introduces many innovations, but the essential improvement is an accurate, efficient and stable calibration procedure. It achieves a standard error reduction by a factor of 8 and 10 for the prices of a five-factor, 20-year Bermudan swaption and cancellable inverse floater. For the Bermudan swaption a reduction of 5 is achieved for the vega. In Chapter 6 we introduce the adjoint PDE method for Markov-functional models. This is an accurate and efficient way to compute Greeks, where most of the

background

Pricing Financial Instruments: The Finite Difference Method

Adjoint and PDE methods for pricing and risk management of exotic interest rate derivatives

The price of an option can under some assumptions be determined by the solution of the Black–Scholes partial differential equation. Often options are issued on more than one asset. In this case it ...

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Radial basis function methods for pricing multi-asset options

In this paper we apply the innovative Laplace transformation method introduced by Sheen, Sloan, and Thom\'ee (IMA J. Numer. Anal., 2003) to solve the Black-Scholes equation. The algorithm is of arbitrary high convergence rate and naturally parallelizable. It is shown that the method is very efficient for calculating various options. Existence and uniqueness properties of the Laplace transformed Black-Scholes equation are analyzed. Also a transparent boundary condition associated with the Laplace transformation method is proposed. Several numerical results for various options under various situations confirm the efficiency, convergence and parallelization property of the proposed scheme.

Laplace transformation method for the Black-Scholes equation

In the Basel III accords in 2013, it was stated that financial institutions should charge Credit Value Adjustment (CVA) to their counterparties for (previously under-regulated) Over-The-Counter (OTC) trades. This CVA can be used to hedge a possible default of the counterparty. One important ingredient of CVA is the calculation of the future exposure of the portfolio on which CVA has been charged. This future exposure is also used to determine more recent value adjustments like Debt Value Adjustment (DVA) and Capital Value Adjustments (KVA), and can be calculated from a future distribution of the portfolio value. This distribution can also be used to compute quantiles which are so-called “worst case” scenarios, and are therefore relevant for risk management. Computing the distribution of future portfolio values requires simulating the future states of the risk factors, and then evaluating the portfolio in all these future states. As the number of risk drivers in a typically traded portfolio is high, computing exposure for portfolios is a numerical challenge. In this thesis we look into multiple numerical techniques for the computation of exposures of realistic derivative portfolios. One of the key contributions of this thesis is the Finite Difference Monte Carlo (FDMC) method for an efficient computation of future exposure.

Efficient PDE based numerical estimation of credit and liquidity risk measures for realistic derivative portfolios

Numerical methods are developed for pricing European and American options under Kou’s jump-diffusion model which assumes the price of the underlying asset to behave like a geometrical Brownian motion with a drift and jumps whose size is log-double-exponentially distributed. The price of a European option is given by a partial integro-differential eq uation (PIDE) while American options lead to a linear complementarity problem (LCP) with the same operator. Spatial differential operators are discretized using finite differences on nonun iform grids and time stepping is performed using the implicit Rannacher scheme. For the evaluation of the integral term easy to implement recursion formulas are derived which have optimal computational cost. When pricing European options the resulting dense linear systems are solved using a statio nary iteration. Also for pricing American options similar itera tions can be employed. A numerical experiment demonstrates that the described method is very efficient as accurate option pri ces can be computed in a few milliseconds on a PC. Copyright line will be provided by the publisher

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Pricing Financial Instruments: The Finite Difference Method

Citations

Adjoint and PDE methods for pricing and risk management of exotic interest rate derivatives

Radial basis function methods for pricing multi-asset options

Laplace transformation method for the Black-Scholes equation

Efficient PDE based numerical estimation of credit and liquidity risk measures for realistic derivative portfolios

Numerical valuation of options under Kou's model

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