Partially exact and bounded approximations for arithmetic Asian options

Abstract

This paper considers the pricing of European Asian options in the Black-Scholes framework. All approaches we consider are readily extendable to the case of an Asian basket option. We consider three methods for evaluating the price of an Asian option, and contribute to all three. Firstly, we show the link between the approaches of Rogers and Shi [1995], Andreasen [1999], Hoogland and Neumann [2000] and Večeř [2001]. For the latter formulation we propose two reductions, which increase the numerical stability and reduce the calculation time. Secondly, we show how a closed-form expression can be derived for Curran's and Rogers and Shi's lower bound for the general case of multiple underlyings. Thirdly, we considerably sharpen Thompson's [1999a,b] upper bound such that it is tighter than all known upper bounds. Finally, we consider analytical approximations and combine the traditional moment matching approximations with Curran's conditioning approach. The resulting class of partially exact and bounded approximations can be proven to lie between a sharp lower and upper bound. In numerical examples we demonstrate that they outperform all current state-of-the-art bounds and approximations.

Bounds for Asian basket options

“…Then, S will not be a comonotonic sum of random variables, making the determination of the lower bound more complicated since it does not follow from the comonotonicity literature. To determine a lower bound, we follow the approach suggested in [15] for basket options. We know that the lower bound can be rewritten as…”

“…Thompson [19] used a first order approximation to the arithmetic sum and derived an upper bound that sharpens those of Rogers and Shi. Lord [15] revised Thompson's method and proposed a shift lognormal approximation to the sums and he included a supplementary parameter which is estimated by an optimization algorithm. In [16], Nielsen and Sandmann applied the Rogers and Shi approach to arithmetic Asian option pricing by using one specific standardized normally distributed conditioning variable and only in a Black and Scholes setting.…”

“…Finally, we generalize the method of Thompson [19] and of Lord [15] to the Asian basket case. In Thompson's approach, we include an additional parameter which is optimized by using an optimization algorithm as in [15].…”

“…In Thompson's approach, we include an additional parameter which is optimized by using an optimization algorithm as in [15]. Numerical results are included and based on several numerical tests, we give a conclusion which should help the reader to choose a precise bound according to the situation of moneyness and time-to-maturity that she is confronted with.…”

“…In Section 3, we derive an analytical closed-form expression for a lower bound in a non-comonotonic situation, which is then used to obtain the upper bound in the Rogers and Shi approach. In Section 4, we generalize the upper bound based on the idea of Thompson [19] and the approach of Lord [15] to discrete arithmetic Asian basket options. In Section 5, we discuss the quality of all these bounds in some numerical experiments and give a guideline of which bound to use in which situation.…”