Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 29K / d

Solution

Write

R_{M, N} = \frac{S _{N}}{S _{M}}

for the product of the last

N - M

stock factors. The forward-start payoff is

(S_{N} - λ S_{M})^{+} = S_{M} (R_{M, N} - λ)^{+} .

Under the risk-neutral measure the binomial factors are independent with the same probability

q

at every node. Hence

S_{M}

and

R_{M, N}

are independent, while

E_{Q} S_{M} = S_{0} (1 + r)^{M} .

The forward-start call option consequently has price

FSC (M, N, λ) = (1 + r)^{- N} E_{Q} [S_{M} (R_{M, N} - λ)^{+}] = S_{0} (1 + r)^{- (N - M)} E_{Q} [(R_{M, N} - λ)^{+}] .

An ordinary

(N - M)

-period call with strike

K

has price

EC (N - M, K) = S_{0} (1 + r)^{- (N - M)} E_{Q} [(R_{M, N} - \frac{K}{S _{0}})^{+}] .

The required equality therefore holds for

K = λ S_{0} .

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