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

Write $R=e^r$, $u=1.2$ and $d=0.9$. In a one-step binomial options pricing model, absence of arbitrage requires $d<R<u$. If $R\le d$, borrowing to buy the stock gives a nonnegative gain in either state and a positive gain in at least one. If $R\ge u$, shorting the stock and investing the proceeds does the same. Conversely, strict inequalities give the risk-neutral probability $q=(R-d)/(u-d)\in (0,1)$, under which the discounted stock is a martingale; this excludes arbitrage at each node. Hence

Under a risk-neutral measure, the expected stock multiplier is the bank multiplier. With $q=1/3$,Thus $r=0$, and the one-period effective return $e^r-1$ is also zero.

The terminal stock prices and European put option payoffs areLet $q$ be the risk-neutral up probability. Risk-neutral pricing and the supplied discounted-price condition giveso $9q^2-27q+14=0$. Its roots are $2/3$ and $7/3$; only the first is a probability. ThereforeThe 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).

For matching European strike and expiry with no dividends, put-call parity gives $C_0-P_0=S_0-K{e}^{-2r}$. Using the rate and put price from part (c),As a check, only the $uu$ call payoff is positive, at $27$. Its risk-neutral expectation is $27(2/3{)}^2=12$, and discounting by $1.1^2$ gives the same value.

Return to $r=0$ and $q=1/3$ from part (b). An American put option is valued by the larger of immediate exercise and discounted continuation at each node. At time one with stock price $120$, exercise gives zero, while continuation gives $(1/3)0+(2/3)9=6$. With stock price $90$, exercise gives $27$, and continuation gives $(1/3)9+(2/3)36=27$; the holder is indifferent.

At time zero, exercise gives $17$, while continuation gives $(1/3)6+(2/3)27=20$. ConsequentlyIt is optimal to wait initially and at the up node; at the down node either exercise or waiting is optimal. At this zero interest rate, the European put also costs twenty pounds.