Converting Option Greeks into Dollar P&L
Summary
The note explains how to translate option sensitivities into approximate dollar gains or losses for specified changes in the underlying price and implied volatility. It defines delta and vega as changes in option value for unit changes in their respective inputs, then scales delta by the underlying’s absolute price move when the move is expressed as a percentage. For gamma, it gives a finite-difference approximation based on option values at nearby underlying prices and applies the second-order Taylor term: one half of gamma times the squared absolute price change.
The explanation clarifies why gamma P&L is nonlinear in the size of the move. It does not provide corresponding calculations for vanna or volga, despite the question asking about them, and it gives no worked numerical example. The formulas are local approximations; actual option P&L can differ for larger moves or when multiple risk factors change together. The note also does not detail contract multipliers or other market conventions that may be needed to convert a quoted option value into total position P&L.
Key ideas
- Delta P&L for a percentage spot move is delta multiplied by the underlying price and the percentage change.
- Gamma P&L is approximated by one half of gamma times the square of the absolute spot change.
- The finite-difference definition of gamma uses option values at nearby underlying prices.
- Greek-based P&L comes from a Taylor approximation and is most directly interpreted for small moves.
Tags
Full text
# Option greeks as dollar P&L
# Option greeks as dollar P&L
If I write the value of an option as O(S, K, T, V), where S is the underlying price, K is the strike, T is the time to expiry and V the implied volatility, how can I compute the dollar amount that I am expected to gain or lose based on specific movements in some of the variables? I know this is what the greeks should tell me and if I use Black and Scholes formula, there are equations for each of the greeks of interest. But I am not sure how to use them to express the move in dollar amount. Let's say that the underlying S moves 1% and that implied volatility moves 1% (in terms of volatility points, i.e. not relative move).
If I want the $ P&L associated to the delta, I can calculate the delta as:
Delta = O(S*1.01, K, T, V) - O(S, K, T, V)
i.e. the difference in option value if the underlying moves up by 1%. For Vega I can do:
Vega = O(S, K, T, V+1%) - O(S, K, T, V)
But what if I want to measure the P&L given by gamma, vanna and volga?
## Answer by Phil-ZXX (score 1, accepted)
https://quant.stackexchange.com/a/41053
The raw Greeks so to speak $-$ e.g. from Wiki - European Option Greeks $-$ usually represent (broadly speaking) the $ amount lost per +1 absolute move in the respective risk-factor.
So for example, if your Spot-Delta is $$\text{Delta} = \Delta \approx O(S+1, K, T, V) - O(S, K, T, V)$$ and spot moves relatively by $x$ (say 1%) then your PnL is $$\text{Delta PnL} = \Delta \cdot S \cdot x = \Delta \cdot S \cdot 0.01$$ because $S \cdot x$ is the absolute move that Spot undergoes.
For completeness, for Spot-Gamma this looks as follows: $$\text{Gamma} = \Gamma \approx O(S+1, K, T, V) - 2\cdot O(S, K, T, V) + O(S-1, K, T, V)$$ $$\text{Gamma PnL} = \frac12\cdot \Gamma \cdot (S \cdot x)^2 = \frac12\cdot\Gamma\cdot (S \cdot 0.01)^2$$
Note that the $\frac12$ and squaring of $(S \cdot x)$ are a result of Taylor expansion (which is what the Greeks ultimately represent).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.