Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 1 / 28J (Stochastic Financial Models)

For positive $S_0,K,T,\sigma$, the Black-Scholes formula usesTo hedge the sold European call option, buy $\Phi (d_1)$ shares and borrow $K{e}^{-rT}\Phi (d_2)$ in cash. This long replicating strategy has initial value $S_0\Phi (d_1)-K{e}^{-rT}\Phi (d_2)$, financed by the option premium. The share holding must subsequently be rebalanced according to the option's option delta in a self-financing strategy.

As $\sigma \to \infty$, $d_1\to +\infty$ and $d_2\to -\infty$, so the European call option price tends to $\boxed{S_0}$. If its traded price $C>S_0$, sell a call, buy one share, and invest the surplus $C-S_0$ at the risk-free rate. The initial net cost is zero, and the maturity payoff isThis is an explicit arbitrage.

As $\sigma \downarrow 0$, the Black-Scholes formula tends to $\boxed{(S_0-K{e}^{-rT}{)}^{+}}$. This also follows by discounting the deterministic zero-volatility payoff; at equality $S_0=K{e}^{-rT}$ the limiting value is zero.

If $S_0>K{e}^{-rT}$ and a call trades at $C<S_0-K{e}^{-rT}$, buy a call, short one share, and buy a bond paying $K$ at maturity. Invest the positive initial surplus $S_0-K{e}^{-rT}-C$. At maturity the call, share and bond together pay $(S_T-K{)}^{+}-S_T+K=(K-S_T{)}^{+}\ge 0$, in addition to the strictly positive invested surplus. If the lower bound is zero, violating it means $C<0$: buy the call and invest the cash received, obtaining a nonnegative option payoff and a strictly positive bank payoff. Both strategies are arbitrages.

Write $\varphi ={\Phi }^{\prime }$ for the standard normal density. Differentiating the Black-Scholes formula and using $S_0\varphi (d_1)=K{e}^{-rT}\varphi (d_2)$ givesThe last equality follows by differentiating $d_1-d_2=\sigma \sqrt{T}$. This derivative is the option's vega.

The Black-Scholes implied volatility of a traded call price $C$ is the volatility $\sigma$ which makes the Black-Scholes formula equal to $C$. Forthe price is a continuous strictly increasing function of $\sigma >0$, by its positive vega, with precisely these endpoint limits. The intermediate value theorem and strict monotonicity therefore give a unique positive implied volatility. The lower endpoint corresponds to the limiting value $\sigma =0$; the upper endpoint is reached only as $\sigma \to \infty$. Prices outside the bounds admit the arbitrages above and have no implied volatility.