This thesis deals with three problems in financial engineering and Monte Carlo simulation. We first price a financial derivative which is called stock loan under Kou’s double-exponential jump diffusion process. To solve this problem, we first formulate the valuation of a stock loan as pricing of an American call option with a time-dependent strike. We then investigate pricing problems of both infinite- and finite-maturity stock loans. In the infinite-maturity case, we obtain a closed-form pricing formula by deriving the moment generating function of the first passage time for the double-exponential jump diffusion process and solving the related optimal stopping problem. In the finite-maturity case, we provide an analytical approximation to the stock loan price by solving an ordi...[ Read more ]

This thesis deals with three problems in financial engineering and Monte Carlo simulation. We first price a financial derivative which is called stock loan under Kou’s double-exponential jump diffusion process. To solve this problem, we first formulate the valuation of a stock loan as pricing of an American call option with a time-dependent strike. We then investigate pricing problems of both infinite- and finite-maturity stock loans. In the infinite-maturity case, we obtain a closed-form pricing formula by deriving the moment generating function of the first passage time for the double-exponential jump diffusion process and solving the related optimal stopping problem. In the finite-maturity case, we provide an analytical approximation to the stock loan price by solving an ordinary integro-differential equation explicitly. Numerical experiments demonstrate the accuracy of our pricing methods.

The second part of the thesis deals with the importance sampling (IS) estimators of Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR). We first derive asymptotic representations of VaR and CVaR , and then prove the asymptotic properties of the IS estimators of VaR and CVaR, e.g., asymptotic convergence property and asymptotic normality property, under some weak condition of the likelihood ratio function based on the representations. At last, we provide simple conditions for IS procedures to be effective.

The third part of the thesis provides a Gaussian process-based search (GPS) for finding out global optima of discrete black-box functions which can only be estimated via simulation. The most critical issue for this kind of optimization problems is the tradeoff between exploitation and exploration. We first give a general adaptive random search framework and discuss the desired properties of the sampling distribution to balance exploitation and exploration. A Gaussian process is then proposed to facilitate constructing the sampling distribution. This Gaussian process-based algorithm, is fast to construct, can balance exploitation and exploration automatically, and is provable convergent under some weak conditions. The numerical performance of the algorithm is quite good.