Past exam of the martingales and financial mathematics course of the University of Oxford / 2008 / B / 6

The terminal put payoff obeys $0\le (K-S_T{)}^{+}\le K$. If the put cost more than the zero-coupon bond paying $K$, sell the put and buy that bond. Invest the strictly positive initial difference in the bank account. At maturity the bond covers every possible put liability and the extra investment remains positive. This arbitrage is excluded, so

A stock share plus a put has terminal value $S_T+(K-S_T{)}^{+}=\max (S_T,K)\ge K$. If $S_t+P(t,S_t)<K{e}^{-r(T-t)}$, buy that package and borrow the price of a bond paying $K$. The package covers the debt at maturity, while the strictly positive initial surplus can be invested. This would be an arbitrage. ConsequentlyTogether with payoff nonnegativity, the lower bound can be strengthened to the maximum of this expression and zero.

The difference between the high-strike and low-strike put payoffs satisfiesIf its price were negative, buying the nonnegative-payoff spread would generate an arbitrage. If its price exceeded the discounted maximum payoff, selling it and buying a bond for that maximum would generate an arbitrage. Thus the sharper inequality isFor the positive-rate convention of the paper this implies the printed upper bound $K_1-K_2$. If rates were allowed to be negative, the undiscounted upper bound need not hold; the discounted bound is the correct rate-independent statement. This distinction is about the stated financial assumptions, not a change to the transcribed inequality.

The call spread in the hint has payoffIt is between zero and one and converges to $1_{\{s>K\}}$ as $ϵ\downarrow 0$. In the Black-Scholes model with positive volatility and $t<T$, the conditional terminal stock distribution has no atom at $K$. The strict and non-strict indicator therefore have the same discounted expectation. The dominated convergence theorem in the risk-neutral pricing formula givesEquivalently this is $e^{-r(T-t)}\Phi (d_2)$. The no-atom condition explains the endpoint subtlety: the displayed spread has value zero at $s=K$, whereas the printed digital pays one there. At maturity the strike derivative can be one-sided or undefined, so the smooth pricing identity is understood before expiry.