# Implied Volatility for Asian option

I am new to the topic of Asian options. Assume I want to price an Asian put (fixed strike, discrete average) in the Black Scholes world. I know implementations to calculate the value but what is the best way to find the implied volatility parameter? Is there a usual way to derive it from the option market of plain vanilla products, e.g. European calls or puts of a certain range of maturities?

• Is this still of interest? I derived an approximation for the Asian implied vol and would like to test it on cases of practical interest. – danp Sep 15 '19 at 11:16
• For mit this is less of interest now. I even don't have access to market data at the moment. – Ric Sep 15 '19 at 14:54
• @demully I don't know what you mean. There is an answer accepted below. In the given case I ask and do not give answers :) – Ric Sep 17 '19 at 6:51
• Your formula K'=S0*(K/S0)^(6/5) seems to be performing well in the case of asian options with almost almost continuous average. Do you happen to know an article mentioning this formula? Or would you know how to get more explanations about this formula? – Hugues Foucher Mar 5 at 9:57
• @HuguesFoucher You are correct, the formula assumes continuous averaging, which should be a good approximation for daily averaging frequency. The formula was derived in this recent paper which just appeared in the April 2021 issue of the Risk magazine risk.net/cutting-edge/banking/7823841/… <br/> I added more details in edits to my older posting below. – danp Apr 20 at 10:40

The easiest way is to use single-expiry volatility that you would get from your volatility surface. It is usually good enough for government work (e.g. to get a sense if you are getting fleeced by a dealer or to understand your vega risk).

A better way is to use local volatility model and the whole volatility surface up to the date of expiry. There is also a bunch of semi-analytical approximations that use weighted volatilities up to there expiry date. Unless you are a dealer and trying to quote these in competition, you don't need to bother with these.

• By the first solution. Do you mean that e.g. if I price an Asian put with 12 months maturity and strike ATM then I take the 12 month maturity ATM volatility of e.g. a European call? Concerning the 2nd point: do you know a good reference (theoretically appealing and pracitcally relevant)? This would be nice ... – Ric Oct 31 '12 at 16:54
• Yes, you would use the volatility for the same strike and same tenor. For the second one, I thin this is what your are looking for: www.dm.unibo.it/~pascucci/web/Ricerca/PDF/39_FPP.pdf – Strange Oct 31 '12 at 21:02
• I have browsed through the paper of your link. I find it theoretically very interesting. I just wonder whether there is some other more applied article...thank you – Ric Nov 2 '12 at 18:28
• Is local vol used in practice for Asian options? I was under the impression most dealers still used analytic approximations. – user3316 Nov 26 '12 at 2:03
• There are some semi-closed form solutions to Asian options. The stupid things trade so tight that it's almost impossible to make out if you are off because of the vol surface differences, business time calculation or modelling differences. I have been to a place where Asians where calculated on local vol model as well as to a place where we used some sort of moment matching model - guess what, both places we some times where off market, except in case of local vol model my junior had to stay till 9 pm to get the P&L and risk for the book. – Strange Nov 26 '12 at 2:09

The approximation I mentioned earlier is that in order to price an Asian option with strike K and maturity T on an asset with spot price S0, one should use the implied volatility at the modified strike K'=S0*(K/S0)^(6/5) and the same maturity T. This assumes an asset with a flat forward curve, like a futures contract.

The derivation of this result is presented in this Risk magazine article What is the volatility of an Asian option?

The approach is based on a short maturity expansion for Asian options in the local volatility model (continuous time averaging) presented in an earlier paper with Lingjiong Zhu.