Past exam of the martingales and financial mathematics course of the University of Oxford / 2011 / B / 4 / c

Table of contents
- Solution c

Solution

The terminal stock prices and European put option payoffs are

state S_{2} (117 - S_{2})^{+} uu 1440 u d, d u 1089 dd 81 36.

Let

q

be the risk-neutral up probability. Risk-neutral pricing and the supplied discounted-price condition give

8 = e^{2 r} V_{0} = 18 q (1 - q) + 36 (1 - q)^{2},

so

9 q^{2} - 27 q + 14 = 0

. Its roots are

2/3

and

7/3

; only the first is a probability. Therefore

q = \frac{2}{3}, e^{r} = 0.9 + 0.3 q = 1.1, r = lo g 1.1.

The current put price is

V_{0} = 8/1. 1^{2} = 800/121

pounds. This interest rate is specific to part (c), and differs from the zero rate in part (b).

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