Past exam of the mathematical models of financial derivatives course of the University of Oxford / 2021 / 2 / c

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- Solution c

Solution

Under the risk-neutral measure, write

S_{T} / s = exp ((r - σ^{2} /2) τ + σ τ Z), Z \sim N (0, 1) .

Complete the square in the normal density after multiplying by the power of

S_{T} / s

. The truncated lognormal moment gives, for

K > 0

,

V_{p} (t, s, K) = M_{p} (τ) Φ (d_{p}), M_{p} (τ) = e^{[p r + p (p - 1) σ^{2} /2] τ}, d_{p} = \frac{lo g ( s / K ) + [ r + ( p - 1/2 ) σ ^{2} ] τ}{σ τ} .

All real powers have finite moments in this lognormal distribution. If

K = 0

, the indicator is identically one and

V_{p} = M_{p}

V_{p}

is an undiscounted conditional moment; the option prices below additionally include the discount factor.

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