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Efficient lattice methods for pricing contingent claims under stochastic volatility and jumps models
This dissertation develops efficient lattice procedures for pricing American options under stochastic volatility models, and stochastic volatility models extended with jumps in asset returns. It also develops the framework for building lattices for stochastic volatility models extended with jumps in both asset returns and volatility. These lattices allow pricing of American options under stochastic volatility/jump models of Bates , Pan , Duffie, Pan, and Singleton , and others. The first chapter corrects the square root transform of Nelson and Ramaswamy  and develops a more efficient truncated tree for the square root process as well as the entire class of constant elasticity of variance models. As another contribution this chapter proposes jump extensions to the CIR and CEV processes of the short rate. The second chapter develops a new two-dimensional orthogonal transform that allows the construction of two-dimensional lattices for various stochastic volatility models. The transform creates a new process which is conditionally independent of the volatility process. The conditional independence plays a useful role in developing recombining lattices. The results are demonstrated using the examples of Hull and White  and Heston  models. The third chapter shows how the two-dimensional transform developed in the previous chapter could be modified to allow the construction of recombining lattices for the stochastic volatility model extended with jumps in asset returns. This chapter also lays down the foundation for building recombining lattices for the models that allow jumps in both asset returns and volatility. The theory is developed for two cases: (i) jumps between asset returns and volatility process are perfectly correlated, and (ii) jumps between asset returns and volatility process are partially correlated. The contribution of the third chapter is especially significant given no recombining lattice approaches have been developed in the literature to simultaneously account for stochastic volatility and jumps. As a final contribution, the forth chapter estimates the parameters of four models using cross-section of market data on European options on S&P 100 index. The performance of four models is assessed by applying developed lattice procedures to price American options on S&P 100 using parameters estimated from European options.
Beliaeva, Natalia A, "Efficient lattice methods for pricing contingent claims under stochastic volatility and jumps models" (2006). Doctoral Dissertations Available from Proquest. AAI3242380.