Interpreting Delta-Hedged P&L Through Gamma and Theta
Summary
The document explains a variance-swap discussion of a short call that is hedged daily, where the trader loses money overall despite realized volatility being below the option’s implied volatility. It clarifies that the charted trading profit and loss is the delta-hedging P&L and addresses why that P&L can initially rise while the underlying price rises, as well as why the chart begins at zero.
The response attributes early gains to positive theta outweighing losses from short gamma when realized volatility is below the implied volatility used to price the option. It relates the gamma and theta contributions through the Black–Scholes pricing equation, simplifying the explanation by assuming a zero interest rate. The initial P&L is zero because an asset purchased at its current price can immediately be sold at that same theoretical price. The explanation is specific to the example and simplified assumptions; it does not detail transaction costs, discrete hedging effects, or other pricing-model differences.
Key ideas
- A short option position can have positive theta alongside negative gamma exposure.
- Delta-hedged P&L depends on the balance between realized variance losses from gamma and theta gains.
- When realized volatility is below the implied volatility, theta can exceed the gamma loss in the example.
- An initial hedge has zero theoretical P&L when marked and traded at the same price.
- The explanation uses a simplified Black–Scholes relation with interest rates set to zero.
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Full text
# Why is this delta-hedging/P&L example on a variance swap call correct? # Why is this delta-hedging/P&L example on a variance swap call correct? I'm looking into this article about var swaps: http://sbossu.com/docs/VarSwaps.pdf and not sure how to correctly interpret Exhibit 2.1.1. > "In this example an option trader sold a 1-year call struck at 110% of the initial price on a notional of 10,000,000 for an implied volatility of 30%, and delta-heged his position daily. The realized volatility was 27.50%, yet his final trading P&L is down $150k. Furthermore, we can see (Figure a) ....." Is the trading p&l meant to be the delta-hedging p&l? It looks like it, but - how come p&l is raising steadily even when stock price is rising? the trader should be losing money on the delta hedging because he is short gamma? - why does it start from zero ? When trader has sold the option, he should have bought the stock and trading p&l at that point was negative? ## Answer by mbison (score 5) https://quant.stackexchange.com/a/19462 To answer your questions: Is the trading p&l meant to be the delta-hedging p&l? Yes, in his example it concerns delta hedged pnl. how come p&l is raising steadily even when stock price is rising? the trader should be losing money on the delta hedging because he is short gamma? He is short gamma but long theta. He is initially making money because as per his chart (the bottom one) initially when the stock is rising the realized vol is below 30%. His gamma pnl would equal his theta pnl if realized would equal 30%. However initially the realized vol is less than 30% therefore he makes more on his theta than he loses on his gamma. This follow from the black scholes PDE. $\theta + 0.5 \sigma^2 S^2 \gamma + r S \delta - rV = 0$ (perhaps a bit sloppy. you can derive this much nicer with r>0) but take r = 0. Then you will see immediately that $\theta = - 0.5 \sigma^2 \gamma$ why does it start from zero ? When trader has sold the option, he should have bought the stock and trading p&l at that point was negative? If at time 0 the value of any asset is say S, and you buy it at S and immediately you look at your pnl then you bought at S and you can sell at S. Therefore your theoretical pnl is 0.
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