# Delta Hedging: Clarification example of the book “Hull, Options, Futures, and Other Derivatives” [closed]

By "Hull, Options, Futures, and Other Derivatives":

Suppose that, in figure,the stock price is \$100 and the option price is \$10. Imagine an investor who has sold 20 call option contracts—that is, options on 2,000 shares. The investor’s position could be hedged by buying $0.6 \times 2,000 =1,200$ shares. The gain (loss) on the stock position would then tend to offset the loss (gain) on the option position.

For example, if the stock price goes up by \$1 (producing a gain of \$1,200 on the shares purchased), the option price will tend to go up by $0.6 \times \$1 = \$0.60$ (producing a loss of \$1,200 on the options written); Why the option price that will tend to go up by \$ 0.60, produce a loss of \$1,200? If strike price$K= \$50$, we have that investor loss:

$$(\ 100 - \ 50 ) \times 2,000 = \ 100.000$$

## 1 Answer

We denote by $C(S_0, K)$ the price for a call option with payoff $(S_T-K)^+$ at the option maturity $T.$ Here $S_0=100$ is the spot stock price. Generally, \begin{align*} C(S_0, K) \ne (S_0-K)^+. \end{align*} Moreover, \begin{align*} C(S_0+\Delta, K)-C(S_0, K) \approx \frac{\partial C}{\partial S_0} \Delta, \end{align*} where $\frac{\partial C}{\partial S_0}=0.6$ is the delta hedge ratio. If the stock price go up by $\Delta = \$1, the shorted option position will loss \begin{align*} \frac{\partial C}{\partial S_0} \Delta = 0.6 \times \1 = \0.60. \end{align*} Then the whole option position loss is2,000 \times \$0.60 = \$1,200$. • My confusion stems from the fact that the value of 1 option contract is \$ 10 written on 100 shares ($2.000 / 20$ option contracts). $$C (S_0, K) = \ 10$$ So $$20 option contracts \times \ 10 = \ 200$$ Could you clarify this aspect? – Mike9 Apr 25 '17 at 22:09
• After the option is sold, that sold option price does not matter any more. Here, we consider the option price change as the underlying stock changes. – Gordon Apr 25 '17 at 22:15
• I'm so sorry but I don't understand why the option position loss is $2,000 \times 0,60 = \$ 1,200\$ ? – Mike9 Apr 25 '17 at 22:24
• The option notional is 2,000: each option has notional of 100, and you sold 20 options; that is, options on 2,000 shares, as you described. – Gordon Apr 25 '17 at 23:26
• OK! I got it. Thank you so much for clarification and for patience – Mike9 Apr 26 '17 at 7:48