Asymptotic Solutions for Australian Options with Low Volatility

Abstract

In this paper we derive asymptotic expansions for Australian options in the case of low volatility using the method of matched asymptotics. The expansion is performed on a volatility scaled parameter. We obtain a solution that is of up to the third order. In case that there is no drift in the underlying, the solution provided is in closed form, for a non-zero drift, all except one of the components of the solutions are in closed form. Additionally, we show that in some non-zero drift cases, the solution can be further simplified and in fact written in closed form as well. Numerical experiments show that the asymptotic solutions derived here are quite accurate for low volatility.

Key Words:

• Asian options
• Australian options
• stochastic volatility
• asymptotic expansions

Acknowledgement

Both authors acknowledge support from the Australian Research Council Grant DP1095969.

Notes

1 This calculation can be done by integrating the SDE in Equation (2) and then taking conditional expectations.

2 An alternative approach is to change the drift term in Dewynne and Shaw (2008) to replacing by in the individual terms of the corresponding equation in Dewynne and Shaw (2008), expanding these as powers of and then collecting powers of .

3 Test cases 1 and 3 indicate that the Australian call price is increasing with volatility. This fact is established analytically in Ewald et al. (2013) by using the Pontryagin maximum principle. Test cases 3 and 4 indicate that the Australian call price may be decreasing with maturity. This relationship is slightly more subtle. The following relationship between the partial derivatives of the Australian call price function with respect to volatility , time of maturity T and interest rate r can be established:. While

4 is always positive as shown in Ewald et al. (2013), it is not difficult to see that is always negative, causing the ambiguity in monotonicity of the Australian call price as a function of time-of-maturity.