$($c$)$ Once a trader has sold such an option, she will try to hedge it by trading in the underlying

share. This results in a portfolio II which is

Short a call option;

Long $n$ shares

where $n$ is chosen to make the portfolio value relatively insensitive to small changes in the

share price. In theory, this hedging must be done continuously: As soon as the share price

changes a little, the portfolio must be rebalanced, and a new value of $n$ is calculated. $($In

practice, rebalancing is typically done daily.$)$

As the share price changes, the change in the value of the portfolio is

$=nS-C$

Dividing through by $S$ and taking the limit $S0,$ we obtain

$\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{S}}=n-\frac{\text{d}\text{e}\text{l}\text{C}}{\text{d}\text{e}\text{l}\text{S}}$

Given that $S_0=100,K=100,T=1,r=0\mathrm{.}1,$ and $=0\mathrm{.}4,$ estimate how many shares

should the trader hold to make the portfolio value insensitive to small changes in the value

of the underlying share.

This is called delta hedging $($because the quantity $$:$=\frac{\text{d}\text{e}\text{l}\text{C}}{\text{d}\text{e}\text{l}\text{S}}$ is called the delta of the option$)$: As

the underlying share changes in value, so will the call price. The trader will try to ensure that

the portfolio is immunized against changes in the underlying, by holding just the right amount

of shares. This amount will change from day to day, and the trader must frequently rebalance

the portfolio to prevent the hedge from slipping.