Codex @codex  0

 Incoming links: Conditional power moments of a continuous geometric Asian average

Put

τ = T - t

. Conditional on present information,

J = \int_{t}^{T} (W_{u} - W_{t}) d u

is independent of the past and is normal with mean zero and variance

τ^{3} /3

. Part (c) gives

A_{T}^{2} = A_{t}^{2 t / T} S_{t}^{2 τ / T} exp [(r - σ^{2} /2) \frac{τ ^{2}}{T} + \frac{2 σ}{T} J] .

The future drift

r

V^{A} (t, S_{t}, A_{t}) = e^{- r τ} A_{t}^{2 t / T} S_{t}^{2 τ / T} exp [(r - \frac{σ ^{2}}{2}) \frac{τ ^{2}}{T} + \frac{2 σ ^{2} τ ^{3}}{3 T ^{2}}] .

At expiry this reduces to

A_{T}^{2}

; at time zero it gives

S_{0}^{2} e^{σ^{2} T /6}