Essentially, an option is a financial contract between two parties, giving the right, but not the obligation, to the owner of the contract to either buy or sell a specific underlying asset for a given strike price K. If the right to the owner is to buy the asset, it is a call option and if the right is to sell the asset, it is a put option. The Asian option is specific kind of option that do not share the characteristics of plain vanilla options. Mainly, the Asian option is a path dependent option, meaning that its pay-off are affected by how the price at maturity time T was reached and thus depended on the price path of the underlying asset. This differ from the standard European options, whose pay-off is dependent only on the price at which it is exercised [1].
The path dependence of the Asian option is due to its pay-off includes a time average of the price for the underlying asset and typically there is two types of averaging for Asian options, namely arithmetic avergae and geometric average. It is thus also referred to as average-price options [1]. The pay-off for the Asian call option and Asian put option, given the price, S(t), of the underlying asset, is therefore given by
for arithmetic average and
for geometric average. This compared to the pay-off of the European option defined as
where T>0 is the time of maturity of the underlying asset [2].
The financial utility of the Asian option is the reduction of market manipulation at maturity time T, a common problem in European options where the price could be manipulated upwards by speculators before the maturity. Even if prohibited, market manipulation occurs, but with the effect of not being as dependent of the expiration date, Asian options acts as a form of protection from such market manipulation [3]. Also, due to the averaging on the prices, the Asian option has lower volatility than the European options and thus typically making them cheaper than standard European options [4]. Because of this, they are often used by such traders that are being exposed under a period of time by the underlying asset [5]. A possible disadvantage of the Asian option, as previously mentioned is the reduced impact from volatility which implies that the ability to capitalize on the volatility peaks. Therefore, the possible profit from trading Asian options are less than non-averaging option.
In the figures below, the initial price for the Asian call option and Asian put option as well as the standard European call respectively put option for different values of the volatility (top) and initial prices (bottom) is shown.
In terms of the volatility graph, its seen that the standard European call option converges to the initial stock price S0=10 which is expected, whereas the Asian call option converges to a lower value. This is also expected due to the arithmetic average over time as this imply a lower volatility in the option compared to the standard European option, hence implying a lower price. Moreover, the Asian options and the standard European options follows similar behaviour, but with slower convergence for the Asian options. This, also due to the time averaging, as expected. Furthermore, for both the Asian options and the standard European options, the put options have a higher price than the call options. This is due to the choice of initial price S0 and strike price K. As K > S0, the call options are clearly out of the money whereas the put options are in the money. This means that the call options have no intrinsic value and thus there is not profit between the current price S0 and the strike price K. With no intrinsic value, the premium of the call options are cheaper than the put options that have intrinsic value. This thus entails that the put options are more expensive than the call options as observed in the above figure.
For varying values of the initial stock price S0 in the bottom figure, the Asian options and the standard European options follow a similar behaviour, but with some deviations. For the call options, the Asian call option is slightly shifted compared to the standard European call option. This is expected as the price of the Asian option is cheaper than the standard European option as previously discussed. However, due to the observed fluctuations in the Asian options, it is clear that there exist some flaws in the numerical computations. One reason could be the size of the partitions being to small and thus decreasing the accuracy of the numerical approximations. This is also seen in the Asian put option which also follows the behaviour of the standard European put option up until around S0 = 30 where it then deviates to lower values. This is also explained by the same argument as for the call options, that the Asian option is expected to be cheaper than the standard European option. Furthermore, the difference in behaviour between the call options and the put options is due to their intrinsic values. When S0 < K = 50, the call options are out of the money which, as previously discussed, entails lower prices compared to the put options as the figure shows. Furthermore, it is seen that in the inteval for S0 < 30, the call option prices are almost zero. Being so far out of the money the implied volatility is very high and thus, this behaviour is expected. Similarly in the same interval, the put options are far in the money and thus have a very high premium, making them very expensive. The same argument, but other way around holds for the interval at the other extreme, at around S0 > 70 for the Asian put option and around S0 > 90 for the standard European put option.
Furthermore, when the options are at the money, for S0 = K = 50, the expected behaviour is that ΠAC(0) = ΠAP(0) and similarly ΠEC(0) = ΠEP(0). This due to their intrinsic values being equally zero with the same extrinsic values. As the figure shows, this is almost the case for the standard European options as their intersection is around S0 ≈ 48, but not the case for the Asian options whose intersection is around S0 ≈ 45. This result could also be caused by flaws in the numerical computation as in the case of their fluctuations.
Clearly the interval for the values of σ and S0 in the top figure is quite unrealistic, as the volatility is within the interval σ ∈ (0, 1) and the price of the underlying asset (in this case the stock price), is much closer to the strike price K, a more realistic modelling would be to examine the the options in under these conditions.
| Volatility σ | Initial price S0 |
|---|---|
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[2] Steven E. Shreve. Stochastic Calculus for Finance II, chapter 7.
[3] Amanda Harvey. The Asian Option. URL: https://www.stock-options-made-easy.com/asian-option.html. Accessed: 2021-03-14
[4] Qian Zhang Aleksandra Mraovic. Valuation of Asian Options - with Levy Approximation, 2014. URL: https://lup.lub.lu.se/luur/downloadfunc=downloadFile&recordOId=4301159&fileOId=4301160. Accessed: 2021-02-11
[5] Finpricing. EQUITY ASIAN OPTION, 2019. URL: https://finpricing.com/lib/EqAsian.html.Accessed: 2021-03-14.