When building a P&L attribution system for options, what is the market convention for attributing daily P&L between delta, gamma, vega, and theta Greeks? I'm particularly interested in how the "cross-effects"* between delta and gamma are handled and would love to see a simple numerical example if that's possible. Thanks in advance!
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$\begingroup$ Yes, @Dimitri Vulis, your answer in quant.stackexchange.com/questions/25475/… helps a lot! Is it possible to break out the effects of delta P&L from gamma P&L? If not, then which of these bucket(s) do we put the P&L in? Also, your answer suggests that the order of the shift in risk factors could be important. To that end, In which order should we shift the risk factors when recomputing the T-1 P&L? Thanks. $\endgroup$– equanimityCommented Oct 7, 2021 at 1:07
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$\begingroup$ The order matters only for the cumulatuve brute-force P&L. The order doesn't matter for independent brute-force P&L or for risk-theoretical P&L (Taylor sereis approximation of the P&L using deltas - first order and gammas and cross-gammas - second order risk measures). I think you're asking about RTPL? $\endgroup$– Dimitri VulisCommented Oct 7, 2021 at 2:18
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$\begingroup$ I think I'm getting confused between the "Cumulative" (aka "Waterfall" or "Progressive") approach and the "Independent" (aka "Restore" or "Component Slide") approach. With the "Cumulative" approach, are we changing the significant model inputs one at a time from the T-1 value to the T-0 value and then leaving them in the T-0 state without changing them back before changing the next significant model input? For example, change the spot price from 100 to 101 and leave it at 101, then change volatility from 15% to 16% and leave it at 16% before changing FX rates, etc. $\endgroup$– equanimityCommented Oct 7, 2021 at 2:42
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1$\begingroup$ Cumulative BF example: reprice changing the price from 100 to 101; reprice changing the price from 100 to 101 and the vol from 15 to 16; etc. Here the order matters. In contrast, independent BF example: reprice changing the price from 100 to 101; reprice changing only the vol from 15 to 16 (but not changing the price); etc. here the order doesn't matter. but you seem to be asking about Taylor expansion, no? $\endgroup$– Dimitri VulisCommented Oct 7, 2021 at 2:52
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$\begingroup$ Yes, I was asking about the Taylor expansion. Thank you. $\endgroup$– equanimityCommented Oct 7, 2021 at 2:54
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1 Answer
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I'm not sure what you mean by "cross" effects - the only correlation is that they both are functions of the change in underlying ($\Delta S$)
Delta PnL is $\Delta * (\Delta S)$
Gamma PnL is $(1/2) \Gamma * (\Delta S)^2$
Essentially the first and second terms of a taylor expansion
Vega and Theta are sensetivities to volatility and time, respectively, so their contribution would be:
Vega PnL is $Vega * (\Delta \sigma)$
Theta PnL is $Theta * (\Delta t)$
There are some subtleties to this type of attribution, specifically due to the fact that $\sigma$ is often modeled as a function of $S$ and $t$, so there are cross-effects between the greeks that make it inexact. Meaning if $\sigma$ changes because the underlying changes you could account for that second-order effect with additional sensitivities (vanna specifically), but those effects are generally much smaller and can be insignificant depending on your purpose.
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$\begingroup$ Thank you, @D Stanley. I was (confusingly) referring to the "cross-effects" in the comments in this post: quant.stackexchange.com/questions/59354/…. Following your answer above, what would be the formula to calculate the Vega and Theta P&L? For example, would the Vega P&L be: (1/3*vega) * (change_in_spot^3)? (I'm not clear on Taylor expansion and have more studying to do there) $\endgroup$ Commented Oct 7, 2021 at 3:19
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$\begingroup$ May I humbly suggest Sir Roy George Douglas Allen Mathematical Analysis for Economists Macmillan (1938) Section 12.2 "Partial Derivatives of the Second and Higher Orders", pages 301ff. $\endgroup$ Commented Oct 7, 2021 at 3:30
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$\begingroup$ I read through Section 12.2 of that text. Unfortunately, I last took calculus about 35 years ago and don't remember any of it! I was hoping for a plain-English explanation... :) $\endgroup$ Commented Oct 8, 2021 at 17:52