Semidefinite Programming Approaches for Bounding Asian Option Prices
Semidefinite programming (SDP) approaches are considered for obtaining bounds for the price of an arithmetic average Asian option. A method for computing the moments of the distribution of prices is developed which enables the method of Bertsimas and Pope
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Department of Industrial and Systems Engineering, University of Florida, Weil Hall, P.O. Box 116595, Gainesville, FL 32611-6595, USA, [email protected] Department of Mechanical & Industrial Engineering, University of Toronto, 5 King’s College Road, Toronto, Ontario, M5S3G8, Canada, [email protected] Department of Industrial and Systems Engineering, University of Florida, Weil Hall, P.O. Box 116595, Gainesville, FL 32611-6595, USA, [email protected]
Summary. Semidefinite programming (SDP) approaches are considered for obtaining bounds for the price of an arithmetic average Asian option. A method for computing the moments of the distribution of prices is developed which enables the method of Bertsimas and Popescu to be extended for the case of the Asian option. In particular, several SDP formulations for upper and lower bounds of the price of an Asian option are given based on different representations of the payoffs of the option. The formulations are amenable to standard SDP computational methods. Key words: Asian options, semidefinite programming.
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1 Introduction Financial derivatives such as options have been a fundamental component of modern finance and risk management for the last few decades (Hull, 2003) . Options are derivative contracts that enable holders (buyers) of the option the right but not obligation to purchase or sell an asset (e.g., common stock or foreign exchange) for a predetermined price at some future point in time. As such, options can be very useful instruments in hedging financial exposures. In the case of a European call or put option, the holder of an option will decide to purchase or sell at the end of a specified time period depending on the price of the underlying asset at the end of the time period (exercise time), i.e., the payoff of the option at exercise time is a function of the price of the underlying
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Georgios V. Dalakouras, Roy H. Kwon, and Panos M. Pardalos
asset. If the payoff is positive, then the holder of the option will exercise the right to purchase or sell else the holder will not exercise this right. A crucial issue is the pricing of an option, i.e., how does one determine a fair price for an option? For European options one can use a variant of the famous Black and Scholes (1973) pricing formula to obtain a fair price. A key assumption that is necessary for the Black-Scholes formula is that the distribution of the price of the underlying asset is lognormal. In this paper, we consider the pricing of arithmetic average European-type Asian options. We will refer to this variant as simply “Asian options” for the rest of the paper. Asian options are financial derivatives whose payoff depends on the average price of an underlying asset over a specified (fixed) time duration where the exercise time is at the end of the time period (Boyle and Emanuel, 1980). Asian options are of great practical and theoretical interest as they are useful instruments for underlying assets such as currencies with low trading v
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