Taylor-Expansion Attribution of Daily Options P&L

Summary

The document outlines a first-order and second-order method for assigning daily option profit and loss to delta and gamma. Delta contribution is estimated from delta times the underlying price change, while gamma contribution uses one half of gamma times the squared price change. Vega and theta contributions are estimated from their sensitivities multiplied by changes in implied volatility and time, respectively.

These terms correspond to parts of a Taylor expansion, so the attribution is approximate. The answer notes that volatility may depend on the underlying price and time, creating interactions between sensitivities. For example, an underlying move that changes volatility can produce an effect captured by vanna. Such higher-order terms may be small or immaterial for some purposes, but the document gives no numerical example or empirical comparison to establish when they can be ignored.

Key ideas

• Delta P&L is approximated by delta multiplied by the underlying price change.
• Gamma P&L is approximated by one half of gamma multiplied by the squared underlying price change.
• Vega and theta contributions use changes in volatility and time, respectively.
• The Greek-based decomposition is approximate because market variables and sensitivities can interact.
• Vanna can capture an interaction caused when an underlying move changes volatility.

Tags

# How to attribute daily options P&L between Greek sensitivities


# How to attribute daily options P&L between Greek sensitivities












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!

## Answer by D Stanley (score 2)

https://quant.stackexchange.com/a/68267

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.

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.