Delta Hedging a Short Call Using the Option’s Price Sensitivity
Summary
The document clarifies how a call option’s delta determines the hedge for a written position. Delta approximates how much the option price changes for a small change in the underlying stock price. In the example, the option price rises by about $0.60 when the stock rises by $1, so the short position loses that amount per share covered by the contract. Across the stated 2,000 shares, the option loss totals $1,200, offsetting the gain on the shares held as a hedge.
The key distinction is between the option’s market price and its immediate exercise value. The question’s calculation uses intrinsic value, which does not capture how an option’s price responds to a stock move before maturity. Delta is a local approximation, so the hedge may need adjustment as prices and other market conditions change; the example does not address rebalancing, gamma, or transaction costs.
Key ideas
- Delta estimates the option price change for a small move in the underlying asset.
- A short call loses value when the call price rises, with the loss per share approximated by delta times the stock move.
- The total option loss scales with the number of shares covered by the contracts.
- Intrinsic value alone does not describe an option’s price sensitivity before maturity.
- Delta is a local approximation and does not account for hedge rebalancing or trading costs.
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Full text
# Delta Hedging: Clarification example of the book "Hull, Options, Futures, and Other Derivatives"
# Delta Hedging: Clarification example of the book "Hull, Options, Futures, and Other Derivatives"
> 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:
$$(\$ 100 - \$ 50 ) \times 2,000 = \$ 100.000$$
If the stock price goes up by $1, the investor loss:
$$(\$ 101 - \$ 50 ) \times 2,000 = \$ 102.000$$
so, if the stock price goes up by \$1, the option contract produce a loss of \$ 2.000
Why?
## Answer by Gordon (score 4, accepted)
https://quant.stackexchange.com/a/33892
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 is $2,000 \times \$0.60 = \$1,200$.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.