How Delta Hedging Offsets Small Call Option Price Changes
Summary
The document explains why a long stock position can offset gains or losses on a short call option. A call's delta estimates how much its price changes for a small change in the underlying stock price. For the example described, the call delta is positive, so a stock price increase raises the call price; because the investor sold calls, that price rise creates a loss on the option position.
Holding stock in proportion to the calls' delta creates a gain when the stock rises and a loss when it falls, approximately offsetting the short option's price change. The explanation uses a first-order Taylor approximation and scales the per-option hedge to the stated position size. This is a local approximation: delta changes as market conditions change, and it describes small, immediate price moves rather than guaranteeing a complete hedge over time. The document focuses on the direction and intuition of the hedge, not on rebalancing or other risks.
Key ideas
- Call delta approximates the option price change for a small move in the underlying stock.
- A positive call delta means the call price tends to rise when the stock price rises.
- A short call position loses value when the call price rises, while a long stock hedge gains.
- The delta hedge offsets option price changes approximately and locally, so it may need adjustment.
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# Answer by Magic is in the chain (score 3, accepted)
# Do not understand 'The gain (loss) on the stock position would then tend to offset the loss (gain) on the option position'
Currently, I am reading John Hull's Options, Futures and Other Derivatives. On page 401, the author mentions the following:
> Suppose that the delta of a call option on a stock is $0.6$, stock price is $\$100$ and the option price is $\$10$. Imagine an investor who has sold call options to buy $2,000$ shares of a stock. The investor’s position could be hedged by buying $$0.6 \times 2,000 = 1,200 \text{ shares}.$$ The gain (loss) on the stock position would then tend to offset the loss (gain) on the option position.
I do not understand the bold sentence.
My thought: As stock price increases while keeping other factors unchanged, the payoff for a call option increases, as the difference between terminal stock price and strike price increases. Thus the call option value increases. However, it is contradicting the bold sentence.
Furthermore, in this case, delta is a positive number $0.6$. Wouldn't this mean that an increase in underlying asset price leads to an increase in option value?
## Answer by Magic is in the chain (score 3, accepted)
https://quant.stackexchange.com/a/48798
Let $C\left(S,t\right) $ represent the price of the call option when the underlying price is S at time t. Now if S changes by h instantaneously, the call price becomes $C\left(S+h, t\right) $. So the change in the call option price is:
$C\left(S+h, t\right) - C\left(S,t\right) $
Which you can approximate via first order Talyor series:
$C\left(S+h, t\right) - C\left(S,t\right) \approx \frac{\partial C}{\partial S}h$
The derivative on the right hand side is 0.6 in your question.
In summary, if the stock price changes by a small amount h, the price of the call option will change, resulting in a gain or loss (LHS), which will be offset by the position on the right hand side (RHS), which is 0.6 units of the stock per call option, or 1200 shares per 2000 call options .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.