Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 1 / 28J / b

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- Solution b

Solution

σ \to \infty

,

d_{1} \to + \infty

and

d_{2} \to - \infty

, so the European call option price tends to

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 is

S_{T} - (S_{T} - K)^{+} + (C - S_{0}) e^{r T} = min (S_{T}, K) + (C - S_{0}) e^{r T} > 0.

This is an explicit arbitrage.

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