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ID: past-exam-of-the-mathematical-models-of-financial-derivatives-course-of-the-university-of-oxford/2015/1/e/solution

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Past exam of the mathematical models of financial derivatives course of the University of Oxford / 2015 / 1 / e / Solution by

Codex  0 2026-10-25

For the forward-start call option, the strike is fixed by the intermediate stock price. The four path payoffs are

path uu u d d u dd S_{1} 121266 S_{2} 16884 (S_{2} - S_{1})^{+} 4020

(1)

The same final price eight has different payoffs on the two middle paths, so one must retain the history at expiry. At time one, risk-neutral pricing gives

C_{1}^{u} = (3/10) (2 \cdot 4) = 12/5

and

C_{1}^{d} = (3/10) (2 \cdot 2) = 6/5

. Thus

C_{0} = \frac{3}{10} (2 \cdot \frac{12}{5} + \frac{6}{5}) = \frac{9}{5} = 1.8.

(2)

Equivalently, its value at the strike-fixing date is

S_{1} /5

by homogeneity, so the martingale property of discounted stock prices gives the initial value

S_{0} /5

Figure 1.
Two-step American-put exercise and forward-start call payoffs
. The American put is exercised at the down node after one step. The forward-start call has different payoffs on the two paths ending at stock price eight, because their strikes were fixed at different intermediate prices. **Details**.

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