Consider the European continuously monitored arithmetic average Asian option with terminal payoff max(A T X 1 S T X 2 , 0), where At the current time t 0, the average value A t over the time period 0,t has been realized Let V (S, X 1 ,X 2 ) denote the price function of the Asian option at the start...

To find the value of the Asian option in progress at time t lets denote the current average value as ...

Consider the European continuously monitored arithmetic average Asian option with terminal payoff: max(A T X 1

Question:

Consider the European continuously monitored arithmetic average Asian option with terminal payoff: max(A_T − X₁S_T − X₂, 0), where

AT T 1 =-=-6 T Su du.

At the current time t > 0, the average value A_t over the time period [0,t] has been realized. Let V (S,τ ; X₁,X₂) denote the price function of the Asian option at the start of the averaging period. Show that the value of the in-progress Asian option is given by

T-t XT XT T7V (S.T-t. -V St, T-ti T-t' T-t t; At t - =). T-t