 Solution

Both probabilities are strictly positive. At every node these are the unique probabilities satisfying the martingale condition, so the fundamental theorem of asset pricing gives a unique no-arbitrage price.

If exactly

n

of the

N

moves use the factor

1 + a

, then

S_{N} = S_{0} (1 + a)^{n} (1 + b)^{N - n},

and there are

(n N)

such paths. Discounted risk-neutral expectation now gives

EC (N, K) = \sum_{n = 0}^{N} w (n, N) (S_{0} (1 + a)^{n} (1 + b)^{N - n} - K)^{+},

where

w (n, N) = \frac{1}{( 1 + r ) ^{N}} (n N) (\frac{b - r}{b - a})^{n} (\frac{r - a}{b - a})^{N - n} .

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