29K (Stochastic Financial Models)

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},

(4)

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)^{+},

(5)

where

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

(6)

b

 0  0

Solution

 0  0

The terminal payoffs satisfy the pointwise identity

(K - S_{N})^{+} - (S_{N} - K)^{+} = K - S_{N} .

(1)

Pricing both sides by discounted risk-neutral expectation gives

EP (N, K) - EC (N, K) = (1 + r)^{- N} K - S_{0},

(2)

because the discounted stock is a martingale. Thus put-call parity is

EP (N, K) = (1 + r)^{- N} K - S_{0} + EC (N, K) .

(3)

c

 0  0

Solution

 0  0

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})^{+},

(1)

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}),

(2)

with

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

(3)

The inequalities

a < r < b

and

a > - 1

make both coefficients positive.

d

 0  0

Solution

 0  0

Write

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

(1)

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)^{+} .

(2)

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} .

(3)

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)^{+}] .

(4)

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}})^{+}] .

(5)

The required equality therefore holds for

K = λ S_{0} .

(6)

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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 29K (Stochastic Financial Models)

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