July 17, 2024

Seminar Series on Quantitative Finance - November 26, 2003


Eliezer Z. Prisman, Schulich School of Business, York University

Arbitrage Violations and Implied Valuations: The Option Market
A relatively recent trend in the option pricing literature is to impute the Risk-Neutral {RN) distribution from market data rather than assuming the distribution of the price of the underlying asset. Given the market prices of European options, one solves for a RN distribution by satisfying the pricing relation. Observed option prices are typically not free of noise and thus usually do not represent arbitrage-free prices. Conducting a study on the imputed RN distribution thus requires eliminating the noise from the prices so they will represent arbitrage-free prices.

Not much attention, however, has been devoted to the process of cleaning the data and no systematic, all-encompassing method has been developed and employed. Methods used in previous literature verify the satisfaction of necessary but not sufficient conditions for the data to be free of arbitrage. This paper develops a necessary and sufficient condition for the satisfaction of the No-Arbitrage {NA) condition in option markets. A consequence of this condition renders the practice of solving for the RN distribution based on discretization as being flawed and allows for arbitrage even though a RN distribution exists for the discretization. The paper also develops a simple and robust methodology that eliminates all displayed arbitrage in option prices. A new procedure of RN distribution estimation is suggested, based on the approximation of convex functions that, until now, have not been utilized for this purpose. (joint work with Ioulia D. Ioffe Carlson School of Management, University of Minnesota)

Ulrich Haussmann, University of British Columbia

Optimizing terminal wealth under partial observation (with J. Sass, RICAM, Linz, Austria)

Typically an investor does not know the instantaneous rates of return of the stocks he holds; he only sees the stock prices. In the literature these rates are frequently modeled as Gaussian processes so that Kalman filtering can be applied to estimate them. We propose an alternate model, a continuous time Markov chain with finitely many states possibly representing the states of the economy. Filtering results are available and allow us to give the optimal trading strategy for the problem of maximizing the expected utility of terminal wealth. Parameters can be estimated using the EM algorithm and the optimal strategy can be determined numerically in some cases.

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