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

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.