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Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 4 / 29K / d / Solution

For the European call option, risk-neutral valuation gives

x = (1 + r)^{- N} E_{Q} [(S_{N} - K)^{+}] = (1 + r)^{- N} E_{Q} [S_{N} 1_{{S_{N} > K}}] - K (1 + r)^{- N} Q (S_{N} > K) .

Define the stock-numeraire measure in a binomial market by

Q (A) = \frac{E _{Q} [ S _{N} 1 _{A} ]}{S _{0} ( 1 + r ) ^{N}} .

It is a probability measure because the risk-neutral martingale property gives

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

. Consequently

x = S_{0} Q (S_{N} > K) - K (1 + r)^{- N} Q (S_{N} > K) .

Tilting one up move by its stock factor changes its probability to

q = \frac{q ( 1 + b )}{1 + r} = \frac{( 1 + b ) ( r - a )}{( 1 + r ) ( b - a )} .

Similarly

1 - q = (1 - q) (1 + a) / (1 + r)

. Thus under

Q

the number of up moves is binomial with parameter

q

, and

Q (S_{N} = S_{0} (1 + b)^{i} (1 + a)^{N - i}) = (i N) q^{i} (1 - q)^{N - i} .

Solved by gpt-5.6-sol high.

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