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

Condition on the first risk-neutral move. After a factor

1 + a

, homogeneity of the call payoff gives

((1 + a) S_{N} - K)^{+} = (1 + a) (S_{N} - \frac{K}{1 + a})^{+},

and the analogous identity holds for

1 + b

. Therefore the binomial call-price recursion is

EC (N + 1, K) = u EC (N, \frac{K}{1 + a}) + v EC (N, \frac{K}{1 + b}),

with

u = \frac{( 1 + a ) ( b - r )}{( 1 + r ) ( b - a )}, v = \frac{( 1 + b ) ( r - a )}{( 1 + r ) ( b - a )} .

The inequalities

a < r < b

and

a > - 1

make both coefficients positive.

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