Delta-Hedged Short Calls: Gamma Losses and Theta Income
Summary
The document explains why repeatedly adjusting a delta hedge around a short call can lose money when the underlying price moves up and down. A short call position is described as short gamma and vega, and long theta. Delta hedging removes exposure to small directional moves, but it does not remove the effect of curvature in the option’s value, so hedge trades can lose value as the underlying fluctuates.
The response connects those hedge losses to the option’s time decay: theta compensates the seller for bearing gamma risk. It frames the result as a comparison between realized volatility, reflected in hedging costs, and implied volatility, reflected in the option premium. The discussion also relates this trade-off to the Black–Scholes equation, where gamma and theta combine with the financing terms of the hedged portfolio. This is a conceptual explanation rather than a numerical demonstration; the realized-versus-implied comparison is approximate because volatility itself can vary.
Key ideas
- A short call is exposed to negative gamma and positive theta.
- Delta hedging removes first-order price exposure but leaves curvature risk.
- Frequent hedge adjustments can lose value when the underlying moves back and forth.
- Theta income compensates for gamma losses, with the balance related to realized versus implied volatility.
- The Black–Scholes equation links gamma, theta, and the financing of the hedged position.
Tags
Full text
# delta neutral option cost
# delta neutral option cost
I am trying to understand how an delta neutral profile is generated. I sell a call for strike of `50$` and the delat of this call is 0.5. I buy 0.5*100 stocks to remain delta neutral. Now when the market moves up buy `2$` the new delta is 0.6 . So I buy 10 more stocks and the market now moves down by `2$` which makes me sell 10 stocks. Every time I buy at `52$` and sell at `50$` since the total portfolio value seems to decrease when the stock moves up and down. How does this work in the BSM model ?
## Answer by StackG (score 3, accepted)
https://quant.stackexchange.com/a/60161
If you sell a call option, in the language of vol trading you are short gamma and vega, and long theta.
So yes, as the underlying wiggles around, if you keep delta-hedged by trading it you will lose some money. This can alternatively be seen as gamma losses (coming from the second-order move in the underlying, which you have not hedged) or as coming from the hedging strategy.
On the other hand, you are compensated for this by the time decay of the option price, ie. its theta.
By the end of the option lifetime, if realized vol over the period (which is effectively what you pay out to hedge) was higher than the implied vol price you received for the option (which was the premium you received), you probably* lost money, and if it was lower then you made money.
*this isn't exactly true due to vol itself being stochastic... here is a great reference on the topic Wilmott, Ahmad: Which Free Lunch...
EDIT
Responding to comment: The first section of the wikipedia page does a good job answering this... if we re-express BS equation as \begin{align} {\frac {\partial V} {\partial t}} + {\frac 1 2} \sigma^2 S^2 {\frac {\partial^2 V} {\partial S^2}} = rV - rS {\frac {\partial V} {\partial S}} \end{align} then you see that the term on the right (risk-free growth coming from a portfolio of the option and the short delta hedge) exactly cancels the term on the left which is the gamma plus the theta.
So we can hedge away delta, but we are still exposed to these two.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.