Price a one-year at-the-money Asian call on a $100 stock with 30% volatility, averaging continuously over the year, using the standard effective volatility argument. Compare it to the vanilla and explain the ratio.

Price a one-year at-the-money Asian call on a $100 stock with 30% volatility, averaging continuously over the year, using the standard effective volatility argument. Compare it to the vanilla and explain the ratio.

Approach: Compute the variance of the time average of a Brownian path over the life of the option, convert it to an effective volatility, then apply the at-the-money approximation.

6.93. The time average of a Brownian path has variance T/3 rather than T, since Var((1/T)*integral of W_t dt) = T/3, so the effective volatility of an Asian option is sigma/sqrt(3) = 0.30/1.7321 = 17.32%. Applying the at-the-money approximation gives 0.4*0.1732*1*100 = 6.93, against 12.00 for the vanilla, a ratio of 1/sqrt(3) = 0.577. Averaging suppresses volatility because early prices are already partly known by the end of the window and the average cannot reach the extremes a single terminal print can. That is why Asian structures are the standard hedge for a corporate with a daily physical flow, and why a desk short an Asian carries far less gamma near expiry than the vanilla it might otherwise have quoted.

Follow-up: The averaging window covers only the last month of a one-year option. What effective volatility applies and how does the vega profile change over the year?

Key concepts: asian option, effective volatility, time average, at-the-money approximation.