A Variance Gamma model can be efficiently implemented when pricing Asian-style options. Then, using the Bondesson series representation to generate the variance gamma process can increase the computational performance of the Asian option pricer.[12]
Within jump diffusions and stochastic volatility models, the pricing problem for geometric Asian options can still be solved.[13] For the arithmetic Asian option in Lévy models, one can rely on numerical methods[13] or on analytic bounds.[14]
European Asian call and put options with geometric averaging
We are able to derive a closed-form solution for the geometric Asian option; when used in conjunction with control variates in Monte Carlo simulations, the formula is useful for deriving fair values for the arithmetic Asian option.
Define the continuous-time geometric mean as:
where the underlying follows a standard geometric Brownian motion. It is straightforward from here to calculate that:
To derive the stochastic integral, which was originally , note that:
This may be confirmed by Itô's lemma. Integrating this expression and using the fact that , we find that the integrals are equivalent - this will be useful later on in the derivation. Using martingale pricing, the value of the European Asian call with geometric averaging is given by:
In order to find , we must find such that:
After some algebra, we find that:
At this point the stochastic integral is the sticking point for finding a solution to this problem. However, it is easy to verify with Itô isometry that the integral is normally distributed as:
This is equivalent to saying that with . Therefore, we have that:
Now it is possible the calculate the value of the European Asian call with geometric averaging! At this point, it is useful to define:
Going through the same process as is done with the Black-Scholes model, we are able to find that:
In fact, going through the same arguments for the European Asian put with geometric averaging , we find that:
This implies that there exists a version of put-call parity for European Asian options with geometric averaging: