Estimating Delta-Hedged Short-Call P&L from Theta and Gamma

Summary

The discussion estimates the one-day profit and loss of a short call when the underlying moves sharply. The answer first assumes the position is delta hedged, so the starting portfolio has no first-order exposure to the spot move. It then treats the stated annualized volatility as implying a typical daily move of roughly one percent, and uses the given daily theta to infer an offsetting gamma loss for a move of that size. Since gamma P&L scales approximately with the square of the move, a two-percent move produces a much larger estimated loss; subtracting it from the theta gain gives the answer’s approximate net loss.

This is only a local approximation, not an exact option valuation. The prompt omits key information such as option gamma, hedge rebalancing, and changes in implied volatility. The estimate also assumes the volatility-based daily move and theta-gamma offset are applicable to this option and horizon. The answer labels the result approximate and its currency wording is inconsistent with the question’s euro inputs.

Key ideas

• A delta hedge removes the first-order spot exposure at the start of the period.
• For a short option, theta income can be offset by losses from negative gamma when the underlying moves.
• Under the stated approximation, gamma P&L scales with the square of the underlying move.
• The proposed calculation is approximate because the prompt lacks enough information for exact option P&L.
• Hedge changes and volatility shifts can materially affect realized P&L.

Tags

# What is the P&L


# What is the P&L












I have a question about the P&L calculation, please.

If we sell a call option on a stock with a volatility of 16%. Theta is worth 100€/day. Let's assume that the spot moves by 2% in one day. What is the P&L ?

Thank you for the help

## Answer by dm63 (score 5)

https://quant.stackexchange.com/a/79078

Can’t be calculated precisely from the information given, but we can make an approximation: -first, assume that you delta hedged your call so we start with a delta neutral portfolio -second, note that 16% vol is equivalent to a daily move of 16%/sqrt(252)= about 1% per day

- we may therefore estimate that a move of 1% would balance the negative gamma versus the theta. Hence , the negative gamma p/l of a 1% move is -100.

- Since gamma is quadratic, the gamma p/l of a 2% move should be -400.

- Hence the overall p/l = theta gain minus gamma loss = 100-400= a loss of 300 dollars.

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