In this chapter, we provide a detailed survey of simulation methods applied to numerical pricing of European, and, more recently, American options. Since European methods option prices can be calculated as expected values, it is natural to use Monte Carlo for computing them. However, this can often be quite slow, and this chapter reviews and compares different methods used to improve the efficiency of Monte Carlo methods.
Despite advances in finance and risk management, a satisfactory method for measuring the total financial risk faced by a business or bank at any time remains elusive, says Paul Glasserman. Value-at-Risk (VaR) is one attempt, and it has certainly helped people grapple with basic issues in market risk management. Regulatory concern over the use of derivatives has helped popularise it, as have the opportunities for risk capital savings by banks that develop sophisticated VaR measurement systems.
This paper describes,analyzes and evaluates an algorithm for estimating portfolio loss probabilities using Monte Carlo simulation. Obtaining accurate estimates of such loss probabilities is essential to calculating value-at-risk,which is a quantile of the loss distribution. The method employs a quadratic ("delta-gamma") approximation to the change in portfolio value to guide the selection of effective variance reduction techniques; specifically importance sampling and stratified sampling. If the approximation is exact,then the importance sampling is shown to be asymptotically optimal.
I survey and assess the development of continuous-time methods in finance during the last 30 years. The subperiod 1969 to 1980 saw a dizzying pace of development with seminal ideas in derivatives securities pricing, term structure theory, asset pricing, and optimal consumption and portfolio choices. During the period 1981 to 1999 the theory has been extended and modified to better explain empirical regularities in various subfields of finance.
This paper proposes and anlyzes discrete-time approximations to a class of diffusions, with an emphasis on preserving certain important features of the continuous-time processes in the approximations. We start with multivariate diffusions having three features in particular: they are martingales, each of their components evolves within the unit interval, and the components are almost surely ordered.
The valuation of debt and equity, reorganization boundaries, and firm's optimal dividend policies are studied in a framework where we model strategic interactions between debt holders and equity holders in a game-theoretic setting which can accommodate varying bargaining powers to the two claimants. Two formulations of reorganization are presented: debt-equity swaps and strategic debt service resulting from negotiated debt service reductions. We study the effects of bond covenants on payout policies and distinguish liquidity-induced defaults from strategic defaults.
This paper empirically compares a variety of firm-value-based models of contingent claims. We formulate a general model which nests versions of the models introduced by Merton, 1974; Leland, 1994 and Anderson and Sundaresan, 1996, and Mella-Barral and Perraudin (1997). We estimate these using aggregate time series data for the US corporate bond market, monthly, from August 1970 through December 1996. We find that models fit reasonably well, indicating that variations of leverage and asset volatility account for much of the time-series variations of observed corporate yields.
When consumers' willingness-to-pay increases by a uniform amount, the change in the resulting monopoly price is generally indeterminate. Our analysis identifies sufficient conditions on the underlying demand curve which predict both the sign and the magnitude of the resulting price change.
In this paper, we consider American option contracts when the underlying asset has stochastic dividends and stochastic volatility. We provide a full discussion of the theoretical foundations of American option valuation and exercise boundaries. We show how they depend on the various sources of uncertainty which drive dividend rates and volatility, and derive equilibrium asset prices, derivative prices and optimal exercise boundaries in a general equilibrium model.