Economics - Theses
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ItemEfficient and generic methods for pricing exotic interest rate derivatives including callable exotics( 2010)We extend and improve Monte Carlo techniques used for the pricing and Greek calculations of exotic interest rate derivatives primarily in the displaced-diffusion LIBOR market model. Our new improvements allow for a vast array of increasingly complicated exotic derivatives to be handled practically within this model. A particular focus is the pricing of Bermudan interest rate derivatives; we introduce a number of improvements to the popular least-squares method, policy iteration and the Andersen-Broadie method for upper bounds. Chapter 2 introduces a new arbitrage-free interpolation scheme for the displaced-diffusion LIBOR market model. Not only is this interpolation scheme internally arbitrage-free, but in contrast to internally consistent existing schemes, it also precludes arbitrage with cash in the standard LIBOR market model, and is thus truly arbitrage-free. It is widely believed that when evolving rates in the displaced-diffusion LIBOR market model to step over tenor dates the terminal measure must be used. In Chapter 3, we explain why this is not the case, and show that by very long stepping in the spot measure it is possible to obtain significant accuracy and standard error improvements, leading to substantial improvements in efficiency. In particular, we demonstrate that speed-ups of a factor in the thousands are possible when pricing auto-caps if the same drift approximation is used in both measures. Chapters 4-6 introduce a set of improvements which allow the calculation of very tight lower bounds for Bermudan derivatives using Monte Carlo simulation. These lower bounds can be computed quickly, and with minimal hand-crafting. Our focus is on accelerating policy iteration to the point where it can be used in similar computation times to the basic least-squares approach, but in doing so introduce a number of improvements which can be applied to both the least-squares approach and the calculation of upper bounds using the Andersen-Broadie method. The enhancements to the least-squares method improve both accuracy and efficiency, as do the improvements to the calculation of upper bounds. Chapter 7 studies the simulation of range accrual coupons when valuing callable range accruals in the displaced-diffusion LIBOR market model. We introduce a number of new improvements that lead to significant efficiency improvements, and explain how to apply the adjoint-improved pathwise method to calculate deltas and vegas under the new improvements, which was not previously possible for callable range accruals. One new improvement is based on using a Brownian-bridge-type approach to simulating the range accrual coupons. We apply our new improvements to the pricing of Bermudan derivatives using Monte Carlo and consider a variety of examples, including when the reference rate is a LIBOR rate, when it is a spread between swap rates, and when the multiplier for the range accrual coupon is stochastic. Chapter 8 introduces two new methods to calculate bounds for game options using Monte Carlo simulation. These extend and generalise the duality results of Rogers and Jamshidian to the case where both parties of a contract have Bermudan optionality. It is shown that the Andersen-Broadie method can still be used as a generic way to obtain bounds in the extended framework, and we apply the new results to the pricing of convertible bonds by simulation.