Discrete Bismut formula: Conditional integration by parts and a representation for delta hedging process

The paper gives discrete conditional integration by parts formula using a Malliavin calculus approach in discrete-time setting. Then the discrete Bismut formula is introduced for asymmetric random walk model and asymmetric exponential process. In particular, a new formula for delta hedging process is obtained as an extension of the Malliavin derivative representation of the delta where the conditional integration by parts formula plays a role in the proof.

Pricing and hedging tools for spread option contracts

This thesis examines the problem of pricing and hedging spread options under market models with jumps driven by a Compound Poisson Process. Extending the work of Deng, Li and Zhou we derive the price approximation for Spread options in jump-diffusion framework. We find that the proposed model accurately approximates option prices and exhibits reasonable behavior when tested for sensitivity to the model parameters. Applying the method of Lamberton and Lepeyre, we minimize the squared error between the Spread option price and the hedge portfolio to arrive to an optimal hedging strategy for discontinuous underlying price modes. Additionally, we propose an alternative average Delta-hedging strategy that is derived by conditioning the underlying price processes on the number of jumps and summing over all the possible jump combinations; such an approach allows us to revert to a hedging problem in a Black-Scholes framework. Although the average Delta-hedging strategy offers a significantly simpler approach to hedge Spread options, we conclude that the former strategy performs better by examining the Profit and Loss Probability Density Function of the two competing strategies. Finally, we offer a model parameter calibration algorithm and test its performance using the transitional Probability Density Functions.

Pricing and hedging tools for spread option contracts

This thesis examines the problem of pricing and hedging spread options under market models with jumps driven by a Compound Poisson Process. Extending the work of Deng, Li and Zhou we derive the price approximation for Spread options in jump-diffusion framework. We find that the proposed model accurately approximates option prices and exhibits reasonable behavior when tested for sensitivity to the model parameters. Applying the method of Lamberton and Lepeyre, we minimize the squared error between the Spread option price and the hedge portfolio to arrive to an optimal hedging strategy for discontinuous underlying price modes. Additionally, we propose an alternative average Delta-hedging strategy that is derived by conditioning the underlying price processes on the number of jumps and summing over all the possible jump combinations; such an approach allows us to revert to a hedging problem in a Black-Scholes framework. Although the average Delta-hedging strategy offers a significantly simpler approach to hedge Spread options, we conclude that the former strategy performs better by examining the Profit and Loss Probability Density Function of the two competing strategies. Finally, we offer a model parameter calibration algorithm and test its performance using the transitional Probability Density Functions.

Pricing, Risk and Volatility in Subordinated Market Models

We consider several market models, where time is subordinated to a stochastic process. These models are based on various time changes in the Lévy processes driving asset returns, or on fractional extensions of the diffusion equation; they were introduced to capture complex phenomena such as volatility clustering or long memory. After recalling recent results on option pricing in subordinated market models, we establish several analytical formulas for market sensitivities and portfolio performance in this class of models, and discuss some useful approximations when options are not far from the money. We also provide some tools for volatility modelling and delta hedging, as well as comparisons with numerical Fourier techniques.

Bifractional Black-Scholes Model for Pricing European Options and Compound Options

Recent empirical studies show that an underlying asset price process may have the property of long memory. In this paper, it is introduced the bifractional Brownian motion to capture the underlying asset of European options. Moreover, a bifractional Black-Scholes partial differential equation formulation for valuing European options based on Delta hedging strategy is proposed. Using the final condition and the method of variable substitution, the pricing formulas for the European options are derived. Furthermore, applying to risk-neutral principle, we obtain the pricing formulas for the compound options. Finally, the numerical experiments show that the parameter HK has a significant impact on the option value.