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In this thesis, we study the fi ltering problem for a partially observed jump diffusion (Zₜ)ₜɛ[ₒ,T] = (Xₜ, Yₜ)tɛ[ₒ,T] driven by Wiener processes and Poisson martingale measures, such that the signal and observation noises are correlated. We derive the fi ltering equations, describing the time evolution of the normalised conditional distribution (Pₜ(dx))tɛ[ₒ,T] and the unnormalised conditional distribution of the unobservable signal Xₜ given the observations (Yₛ)ₛɛ[ₒ,T]. We prove that if the coefficients satisfy linear growth and Lipschitz conditions in space, as well as some additional assumptions on the jump coefficients, then, if E|πₒ|ᵖLρ < ∞ for some p ≥ 2, the conditional density π = (πₜ)tɛ[ₒ,T], where πₜ = dPₜ/dx, exists and is a weakly cadlag Lp-valued process. Moreover, for an integer m ≥ 0 and p ≥ 2, we show that if we additionally impose m + 1 continuous and bounded spatial derivatives on the coefficients and if the initial conditional density E|πₒ|ᵖWρᵐ < ∞, then π is weakly cadlag as a Wρᵐ-valued process and strongly cadlag as a Wρˢ - valued process for s ɛ [0;m).

In this thesis, we study some geometrical aspects of metric measure spaces (Rn, psi1/2 , mu)where mu is a locally finite regular Borel measure and a metric on psi1/2 which arises from a continuous negative definite function psi : Rn → R which satisfies psi(xi) ≥ 0 with psi(xi) = 0. This study is motivated by the investigation of a transition density estimate for pure jump processes on a general metric measure space. To gain a better insight into the behaviour of transition functions of symmetric Levy processes in this general setting, it seems desirable to understand geometrical properties of their underlying state spaces. More precisely, we show completeness of the metric spaces (Rn, psi1/2) and study under which circumstances open balls Bpsi(x,r), x ∈ Rn, r > 0, with respect to this metric are convex. Moreover, we focus on conditions of the metric measure spaces (Rn,psi1/2 ,mu) for the balls to satisfy the volume growth property [equation] for mu-almost all x ∈ Rn, 0 < r < R and a constant Cpsi(x,R)≥1. Finally, we show that the homogeneity property of a metric measure space can be applied to our case and provide some results associated with the construction of a Hajlasz-Sobolc space over (Rn,psi1/2, lambda(n)),where lambda(n) denotes the n-dirnensional Lebesgue measure.

This thesis examines the empirical performance of four Affine Jump Diffusion models in pricing and hedging S&P 500 Index options: the Black Scholes (BS) model, Heston s Stochastic Volatility (SV) model, a Stochastic Volatility Price Jump (SVJ) model and a Stochastic Volatility Price-Volatility Jump (SVJJ) model. The SVJJ model structure allows for simultaneous jumps in price and volatility processes, with correlated jump size distributions. To the best of our knowledge this is the first empirical study to test the hedging performance of the SVJJ model. As part of our research we derive the SVJJ model minimum variance hedge ratio. We find the SVJ model displays the best price prediction. The SV model lacks the structural complexity to eliminate Black Scholes pricing biases, whereas our results indicate the SVJJ model suffers from overfitting. Despite significant evidence from in and out-of-sample pricing that the SV and SVJ models were better specified than the BS model, this did not result in an improvement in dynamic hedging performance. Overall the BS delta hedge and SV minimum variance hedge produced the lowest errors, although their performance across moneyness-maturity categories differed greatly. The SVJ model s results were surprisingly poor given its superior performance in out-of-sample pricing. We attribute the inadequate performance of the jump models to the lower hedging ratios these models provided, which may be a result of the negative expected jump sizes.