Exact simulation of financial models with stochastic volatility and jumps, and lattice methods for corporate debt pricing

This thesis is a collection of three essays focused on (1) exact simulation of option prices under financial models that include stochastic volatility and jumps, (2) exact simulation of option Greeks under the same models, (3) pricing risky corporate debt and modeling Chapter 11 proceedings using lattice methods.; The stochastic differential equations for affine jump diffusion models do not yield exact solutions that can be directly simulated. Discretization methods can be used for simulating security prices under these models. However, discretization introduces bias into the simulation results and a large number of time steps may be needed to reduce the discretization bias to an acceptable level. In the second chapter of this thesis, we suggest a method for the exact simulation of the stock price and variance under Heston's stochastic volatility model and other affine jump diffusion processes. The sample stock price and variance from the exact distribution can then be used to generate an unbiased estimator of the price of a derivative security. We compare our method with the more conventional Euler discretization method and demonstrate the faster convergence rate of the error in our method. Specifically, our method achieves an O( s-12 ) convergence rate, where s is the total computational budget. The convergence rate for the Euler discretization method is O( s-13 ) or slower, depending on the model coefficients and option payoff function.; In the third chapter, we derive Monte Carlo simulation estimators to compute option price derivatives, i.e., the 'Greeks,' under the same models. We use pathwise and likelihood ratio approaches together with the exact simulation method to generate unbiased estimates of option price derivatives in these models. By appropriately conditioning on the path generated by the variance and jump processes, the evolution of the stock price can be represented as a series of lognormal random variables. This makes it possible to extend previously known results from the Black-Scholes setting to the computation of Greeks for more complex models. We give simulation estimators and numerical results for some path-dependent and path-independent options.; The fourth chapter of the thesis is on the pricing of risky corporate debt. The pricing of corporate debt is still a challenging and active research area in corporate finance. Starting with Merton [50], many authors proposed a structural approach in which the value of the assets of the firm is modeled by a stochastic process, and all other variables are derived from this basic process. These structural models have become more complex over time in order to capture more realistic aspects of bankruptcy proceedings. The literature in this area emphasizes closed-form solutions that are derived by either PDE methods or analytical pricing techniques. However, it is not always possible to build a comprehensive model with realistic model features and achieve a closed-form solution at the same time. We develop a binomial lattice method that can be used to handle complex structural models such as ones that include Chapter 11 proceedings of the U.S. bankruptcy code. Although lattice methods have been widely used in the option pricing literature, they are relatively new in corporate debt pricing. In particular, the limited liability requirement of the equityholders needs to be handled carefully in this context. Our method can be used to solve the Leland [47] model and its extension to the finite maturity case, the more complex model of Broadie, Chernov and Sundaresan [15], and others.