Past exam of the martingales and financial mathematics course of the University of Oxford / 2009 / B / 6

If $d_1>d_2$, buy $1/100$ of stock 1 and sell $1/100$ of stock 2. Initial cost is zero; the one-period payoff is $u_1-u_2>0$ in the up state and $d_1-d_2>0$ in the down state. This is an arbitrage. If terminal rather than time-one arbitrage is required, reinvest the positive receipts in either positive-price stock.

For $d_1<d_2$, define the one-period Arrow state pricesThe denominator is positive because $u_1/u_2>1>d_1/d_2$. Both state prices are strictly positive and satisfy ${\pi }_u{u}_i+{\pi }_d{d}_i=1$ for $i=1,2$. Thus every portfolio's present value equals the positive state-price-weighted value of its two future payoffs.

Use the same strictly positive pricing rule at every node of the two-period tree. A nonnegative terminal payoff that is positive in at least one state has strictly positive price by backward induction, so a zero-cost self-financing strategy cannot generate an arbitrage. Therefore the market is arbitrage free when $d_1<d_2$, despite having no initially supplied bond. This is a two-stock binomial market without a supplied bond.

For holdings $h_1,h_2$, a sure payment of 100 requiresSubtracting gives $2h_1+h_2=0$, so $h_1=-1$ and $h_2=2$. Thus sell one share of stock 1 and buy two shares of stock 2. Its initial cost is $100(-1+2)=100$.

This synthetic zero-coupon bond pays exactly its initial cost after one period, and thereforeAt later nodes the holdings can be rescaled to generate the same one-period risk-free return; a physical bond need not be listed separately.

With the rate $r=0$, the risk-neutral probability in a binomial market must satisfyThe first stock gives $0.8+0.6q=1$, hence $q=1/3$. The second gives $0.9+0.3q=1$, confirming the same answer. ThusTake the successive up/down choices independent under this measure, so both stock prices are martingales. The martingale transition condition applies from times zero to one and one to two; the displayed $t+1$ condition in the source should be read for these two transitions.

The exchange option has terminal payoffs determined by the two successive up/down moves:The two stocks span both one-step states, so this finite market is complete. The risk-neutral pricing uses the independent transition probabilities $q=1/3$ and $1-q=2/3$. Since the synthetic risk-free rate is zero,The price is approximately 7.56 in the currency units of the stock prices.