Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 29L / e

Table of contents
- Solution e

Solution

The average log price is

\frac{1}{T} \int_{0}^{T} lo g S_{t} d t = lo g S_{0} + \frac{1}{2} (r - \frac{1}{2} σ^{2}) T + \frac{σ}{T} I_{T} .

Part (b) gives

var (I_{T}) = T^{3} /3

, so the Gaussian variance of the final term is

v = \frac{1}{3} σ^{2} T .

Rewriting the lognormal factor in the normalized form

e^{- v /2 + v Z}

and discounting gives

S_{0} e^{- (r /2 + σ^{2} /12) T} F (\frac{1}{3} σ^{2} T, \frac{K e ^{- (r /2 - σ^{2} /12) T}}{S _{0}}) .

Thus $α = 1/12$ .

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