Economics - Theses
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ItemNew methods and improvements to Monte-Carlo methods for pricing derivative securities( 2012)This thesis presents new Monte Carlo methods for pricing financial derivative securities. Some of these new methods are entirely original ideas whilst others are improvements upon existing Monte Carlo methods in the literature. In Chapter 4 we present an algorithm for obtaining upper bounds on Bermudan option prices without the need for nested simulations; our new method has a computational order far lower than that of the existing popular methods that require nested simulations. This new algorithm can also be used to reduce the variance of lower bounds estimators. In Chapter 5, we introduce a new generic upper bound for Bermudan prices. We give a special case of this upper bound which has theoretical and practical appealing properties; it can be no greater than the popular Rogers' upper bound estimator and can potentially reduce computation time by terminating simulation paths early. In Chapter 6, we present a new control variate which reduces the variance of Jamshidian's multiplicative upper bound on Bermudan option prices to levels similar to that of the popular Andersen-Broadie upper bound method. The implementation is straightforward and requires almost no additional computations from the original method yet numerical results show that the variance reduction could be in the order of hundreds. In Chapter 7, we introduce new techniques for sampling the payoff of discretely-monitored barrier options; we also apply the likelihood ratio method with our new methods to compute Deltas of the barrier options. When compared to existing Monte Carlo methods, numerical results show that these new methods can significantly improve convergence speeds of the price and Delta estimators; in particular, the variance reduction achieved on the Deltas, compared to finite difference, can be in the order of thousands. In Chapter 8, we present two generic improvements to the popular pathwise method. The first improvement is a new method for finding control variates to the pathwise Greek estimator whilst the second improvement allows the use of the pathwise method on options that have the possibly of knocking out. We also show that these two improvements can be used in conjunction so that the benefits of both improvements are obtained. In Chapter 9, we present a new perspective on the concept of hedging options within a simulation; implementation of this new approach can offer significant variance reductions when compared to the existing view of hedging.
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ItemLimit proxy methods for fast Monte-Carlo Greeks( 2012)This thesis presents new Monte-Carlo techniques for the pricing and Greeks computations of exotic derivatives in the LIBOR market model and the Heston stochastic volatility model. These new techniques allow the rapid computation of Greeks even when pay-offs are discontinuous and underlying densities post-discretization are singular. Chapter 3 introduces a new class of numerical schemes known as quasi mean-shifted proxy simulation schemes for discretizing processes driven by Brownian motions. This is a generalization of the partial proxy simulation scheme developed by Fries and Joshi. Under this class of numerical schemes, Greeks for financial products with discontinuous pay-offs can be evaluated efficiently via finite difference approximations with the main constraint being that the discontinuities at each step must be determined by a one-dimensional function: the proxy constraint function. A specific quasi mean-shifted proxy simulation scheme known as the minimal partial proxy scheme is constructed in this chapter. In Chapters 4 - 5, we show that, for any numerical scheme that belongs to the class of quasi mean-shifted proxy simulation schemes, the pathwise adjoint method can be used to compute price sensitivities even when pay-offs are discontinuous and underlying densities post-discretization are singular. Using this result, new Monte-Carlo techniques known as the pathwise partial proxy method and the pathwise minimal partial proxy method are developed. We also consider linearizing the non-linear proxy constraint functions in order to reduce the computational complexity. In Chapters 6 - 7, we shift our focus to the Heston stochastic volatility model. Specifically, we present three new discretization schemes for the Heston stochastic volatility model - two schemes for simulating the variance process and one scheme for simulating the integrated variance process. These new schemes evolve the Heston process accurately over long steps without the need to sample the intervening values. Hence, prices of financial derivatives can be evaluated rapidly using our new schemes. An efficient approach to computing the first and second order price sensitivities in the Heston model is also presented.