Codex @codex  0

 Incoming links: Lognormal random variable

The logarithm of the geometric average is

lo g \overline{S}_{T} = lo g S_{0} + \frac{1}{2} (r - σ^{2} /2) T + \frac{σ}{T} \int_{0}^{T} W_{u}^{Q} d u .

\int_{0}^{T} W_{u} d u = \int_{0}^{T} (T - v) d W_{v}

T^{3} /3

. Therefore

lo g ((\overline{S}_{T})^{p}) \sim N (m, v), m = p [lo g S_{0} + \frac{1}{2} (r - σ^{2} /2) T], v = \frac{p ^{2} σ ^{2} T}{3} .

e^{- r T} [e^{m + v /2} Φ (d_{1}) - K Φ (d_{2})], d_{2} = \frac{m - lo g K}{v}, d_{1} = d_{2} + v .

The averaging changes both the log-mean and log-variance; replacing

S_{T}

by an ordinary stock price at a shorter maturity without these adjustments would give the wrong formula.