Option greeks as dollar P&L

Question

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 1

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).