A 2 Dimensional Pde For Discrete Asian Options
This paper presents a new method for the pricing of discrete Asian options when assuming a deterministic volatility as specified in Dupire (1993). Using a homogeneity property, we show how to reduce an n+1 dimensional problem to a 2 dimensional one. Previous research has been intensively focussing on continuous time Asian options using Black Scholes assumptions. However, traded Asian options are based on a discrete time sampling and can exhibit a pronounced volatility smile. Previous works which have tried to find approached closed forms have the major drawback to be not extendable to more complex volatility model, like the Dupire model, as well as to American type options: Vorst (1992), Geman and Yor (1993), Turnbull and Wakeman (1991), Levy (1992), Jacques (1995), Zhang (1997) and Milevsky and Posner (1998). Works which have focussed at numerical methods do not take into account volatility smile and focus at continuous Asian options: Kemma and Vorst (1990), Hull and White (1993), Caverhill and Clewlow (1990), Benhamou (1999), Roger and Shi (1995), He and Takahashi (1996), Alziary et al. (1997) and Zvan et al. (1998). Our paper oñers a solution to the pricing of discrete Asian options with volatility smile. As suggested by Dupire (1993), we assume a volatility function of time and the underlying so as to take the volatility smile into account. Assuming furthermore that our volatility function can be written as a function of the normalized underlying, we prove that the option price is homogeneous in the underlying price and the strike. By means of this property, we show how to reduce the numerical problem of a discrete Asian options with n fixing dates, which normally should be computed as an n +1 dimensional problem (n fixings and the time) to a 2 dimensional problem. This is of considerable interest for the efficient computation of discrete Asian options. This generalizes to discrete Asian options the dimension reduction found for continuous Asian options by Rogers and Shi (1995). We derive a PDE for the computation of the Asian option and solve it with the standard Crank Nicholson method. Because of the dimension reduction, we need to interpolate our conditional price at each fixing dates. Diñerent methods of interpolation are examined. The rest of the article focuses at numerical specification of our finite difference (grid boundaries, time and space steps) as well as an extension to the case of non proportional discrete dividends, using a jump condition. We compare our result with a Quasi Monte Carlo simulation based on Sobol sequences. Our simulations show us interesting results about the delta hedge and its jump over fixing dates.
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