Financial Engineering

Unlike European-type derivative securities, there are no simple analytic valuation formulas for finite-lived American options, even when the underlying asset price has constant volatility. The early exercise feature considerably complicates the valuation of American contracts. The strategy taken in this paper is to rely on nonparametric statistical methods using market data to estimate the call prices and the exercise boundaries. A comparison is made with parametric constant volatility model-based prices and exercise boundaries.

This chapter develops a simulation method for pricing path-dependent American options, and American options on a large number of underlying assets, such as basket options. Standard numerical procedures (lattice methods and nite difference methods) are generally inapplicable to such high-dimensional problems, and this has motivated research into simulation-based methods. The optimal stopping problem embedded in the pricing of American options makes this a nonstandard problem for simulation. This chapter extends the stochastic mesh introduced in Broadie and Glasserman.

An important recent development in the pricing of interest rate derivatives is the emergence of models that incorporate lognormal volatilities for forward Libor or forward swap rates while keeping interest rates stable. These market models have three attractive features: they preclude arbitrage among bonds, they keep rates positive, and, most distinctively, they price caps or swaptions according to Black's formula, thus allowing automatic calibration to market data. But these features of continuous-time formulations are easily lost when the models are discretized for simulation.

This paper introduces a class of state dependent jump (SDJ) models in which the arrival intensity and jump sizes depend on a given set of state variables, including lagged jumps. With this model, we investigate the structure of jumps to U.S. equity indices, concentrating on the predictability of jumps times if found for all of the indices considered: Standard and Poor's 500 and Mid-Cap, the Russell 1000, 2000, and 3000 indices, the Wilshire 5000 and the Nasdaq 100 (NDX).

This paper develops a variance reduction technique for Monte Carlo simulations of path-dependent options driven by high-dimensional Gaussian vectors. The method combines importance sampling based on a change of drift with stratified sampling along a small number of key dimensions. The change of drift is selected through a large deviations analysis and is shown to be optimal in an asymptotic sense. The drift selected has an interpretation as the path of the underlying state variables which maximizes the product of probability and payoff—the most important path.

We provide a simple binomial framework to value American-style derivatives subject to trading restrictions. The optimal investment of liquid wealth is solved simultaneously with the early exercise decision of the nontraded derivative. No-short-sales constraints on the underlying asset manifest themselves in the form of an implicit dividend yield in the risk-neutralized process for the underlying asset. One consequence is that American call options may be optimally exercised prior to maturity even when the underlying asset pays no dividends.

This paper considers efficient estimation of value-at-risk, which is an important problem in risk management. The value-at-risk is an extreme quantile of the distribution of the loss in portfolio value during a holding period. An effective importance sampling technique is described for this problem. The importance sampling can be further improved by combining it with stratified sampling. In this setting, an effective stratification variable is the likelihood ratio itself.

The author argues that trade liberalization must be balanced in agenda, process and outcomes, including not only sectors in which developed countries have a comparative advantage, like financial services, but also those in which developing countries have a special interest, like agriculture and construction services. Account must be taken of the marked disadvantage that developing countries have in participating meaningfully in negotiations.