# Suppose we define the flexible geometric average G F (n) of asset prices at n evenly spaced

## Question:

Suppose we define the flexible geometric average G_{F }(n) of asset prices at n evenly spaced time instants by

and S_{i }is the asset price at time t_{i}. Here, ω_{i} is the weighting factor associated with S_{i}. Note that the larger the value of α, the heavier are the weights allocated to the more recent asset price. Under the risk neutral measure, the asset price is assumed to follow the Geometric Brownian process

We consider the fixed strike Asian option with terminal payoff

where X is the strike price, and ∅ is the binary variable which is set to 1 for a call or −1 for a put. Show that the Asian option value is given by (Zhang, 1994)

n is the number of asset prices taken for averaging, Δt is the time interval between successive observational instants, j is the number of observations already passed, B^{f}_{j} can be considered as the weighted average of the returns of those observations that have already passed.

## Step by Step Answer:

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