IWORKSHOP ON MATHEMATICAL FINANCE
~ ABSTRACT ~
The value of information: A general stochastic calculus approach to insider trading By Bernt Øksendal
By an insider we mean a person who has access to more information than the information that can be obtained by observing the prices on the market. For example, an insider may have information about the future values of a certain stock. Insider trading is illegal in most countries and it is important to be able to detect it, if it should occur. Central questions are: How much extra can an insider gain compared to an honest trader? How much different is the optimal portfolio of an insider compared to the optimal portfolio of an honest trader? We will give partial answers to these questions by setting up a general stochastic analysis model for insider trading. The model involves the forward integral, the Skorohod integral and Malliavin calculus. The presentation is based on joint work with Francesca Biagini, University of Bologna.
The Stratified Estimators for Value-at-Risk of Portfolios
The standard delta and delta-gamma methods for Value-at-Risk (VaR) estimation are difficult to apply when portfolios have significant exposures to non-linear derivative claims, such as portfolios containing out-of-the-money options. The Monte Carlo simulation is a natural alternative to the delta and delta-gamma methods and the variance reduction is a common technique to improve the precision of Monte Carlo simulation. The purpose of this paper is to develop stratified estimators to assess the value-at-risk of portfolios which may have non-linear response to underlying risk factors , e.g., options. First, we develop a fully stratified VaR estimate for two dimension case, which is suitable to short-term and long-term bonds, where the two risk factors are the factor causing the parallel shifts and the factor causing the change in slope respectively. Second, by combining Latin Hypercube sampling and Monte Carlo sampling, we design VaR estimators for portfolios which may contain a large number of risk factors. Finally, numerical experiments illustrate the potential of those methods for variance reduction.
Pricing American Options using Calibrated Monte Carlo
It is a natural concern that an option pricing model should be able to reproduce the market prices of liquid instruments. In this regards, the weighted Monte Carlo (WMC) method was proposed for calibrating simulation-based models by means of relative entropy minimization. Using the calibrated simulations, it is straightforward and effective to compute the prices of exotic European options on the same underling instruments. The aim of this research work with Lu Chor Sheng is to extend the application of the WMC method to the pricing of respective American-style options by developing a weighted least squares Monte Carlo (WLSM) method, which is an integration of the WMC method and the least squares Monte Carlo (LSM) method. Simulation tests show that the WLSM method is capable of producing more reliable prices of American options under the assumption of stochastic volatility.
Value Creation Through Risk Management: A Corporate
Corporate risk management has the potential to increase shareholder value in the presence of capital market imperfections, such as agency conflicts, costly external financing, direct and indirect costs of bankruptcy, and taxes. In particular, risk management at the firm level can lead to higher firm value by reducing agency conflicts between shareholders, bondholders, and managers; coordinating corporate financing and investment policies; lowering the expected costs of bankruptcy and financial distress; and reducing the corporate tax burden.
Exercise Regions and Efficient Valuation of American Lookback
We presents an efficient method to compute the values and early exercise boundaries of American fixed strike lookback options. The method reduces option valuation to a single optimal stopping problem for standard Brownian motion and an associated path-dependent functional, indexed by one parameter in the absence of dividends and by two parameters in the presence of a dividend rate. Numerical results obtained by this method show that, after a space-time transformation, the stopping boundaries are well approximated by certain piecewise linear functions with a few pieces, leading to fast and accurate approximations for American lookback option values. Such approximations are developed from an explicit decomposition formula for American lookback options, which is also applied to the asymptotic analysis of the early exercise boundary near the expiration date.
Arbitrage Opportunities and Integration Theories
In this talk, we will discuss how the notion of no arbitrage is basically associated with integration theory in fractional Brownian motion model.