Attributing Option P&L with Greeks and Full Revaluation
Summary
The document explains two ways to attribute an option’s price change to movements in its underlying price, implied volatility, and time. Risk theoretical P&L uses sensitivities measured at the starting point, multiplying them by observed input changes; higher order sensitivities can capture curvature and improve the approximation. Full revaluation reprices the instrument after replacing selected starting inputs with their ending values, helping isolate the effect of each change and their interactions.
It recommends using both approaches to check that the pricing model behaves consistently, including whether changes in first order sensitivities are explained by second order effects. The discussion also notes that attribution becomes more involved when volatility is a surface or the underlying is a curve, where factors such as historical principal components may help describe moves. Real-world events that alter contract terms or timing can also affect P&L. The methods require reliable inputs and a suitable valuation model; no numerical example or comparison of attribution conventions is supplied.
Key ideas
- Greek based attribution approximates P&L by applying starting sensitivities to observed market input changes.
- Full revaluation reprices the option with selected inputs changed to estimate their effects.
- Higher order sensitivities help explain nonlinear effects and changes in first order Greeks.
- Volatility surfaces and interest rate curves may require more structured risk factors.
- Operational or contractual events can create P&L effects beyond standard market Greeks.
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# Attributing change in option prices to greek components # Attributing change in option prices to greek components A noob question. I'm trying to get my head wrapped around this but getting lost, please guide. From the entry and exit prices of an European option, how do I attribute the P&L to various greeks (delta, gamma, theta and vega)? For ex: if my entry price for Strike K is \$100 and if my exit is \$125. I have a P&L = \$25. Is there a way to know how much of this \$25 is due to vega and how much of it is theta (lost/decay) etc? ## Answer by Dimitri Vulis (score 7, accepted) https://quant.stackexchange.com/a/75360 Suppose that the fair value of your option is a function $f$ of 3 inputs: the price of the underlying, the implied volaility, and time. You want to understand why the function value changed from time $T_0$ to time $T_1$. It doesn't matter whether you unwound the investment, or just re-marked to market at time $T_1$. Those who took a better-than-average undergraduate calculus course :), probably recall right away that there are two basic approaches to such problems: - risk-theoretical P&L (RTPL) - at $T_0$, calculate first, second, and sometimes even third order sensitivities of the function to small changes in it inputs. Multiply the observed changes in the inputs by the sensitivities. For example, multiplying the vega at time $T_0$ by the change in implied volatility from $T_0$ to $T_1$ is the first-order approximation of the P&L due to the change in implied volatility. This is, basically, Taylor expansion, although it may be possible to do some attribution a little fancier than Taylor. - Brute Force / Full Reval - recalculate the function value, replacing some of the inputs at time $T_0$ by the inputs at time $T_1$. I.e. compare $f($price at $T_0$, volatility at $T_0$, $T_0)$, $f($price at $T_0$, volatility at $T_0$, $T_1)$, $f($price at $T_0$, volatility at $T_1$, $T_0)$, $f($price at $T_0$, volatility at $T_1$, $T_1)$, $f($price at $T_1$, volatility at $T_0$, $T_0)$, etc. You ideally should do both, with multiple versions of Brute Force, to ensure that your pricing model works correctly. Also verify that the changes in the first order sensitivities are explained by the second order sensitivities. Some of these inputs may have some structure, e.g. the implied volatility may be a surface that changes not in parallel, rather than a single number, and you may want to see the impact of the changes of some parts of the surface. The underlying may have some term structure too. For example, if your option is actually a swaption, and the underlying is an interest rate curve, then in addition to attributing the P&L to just changes in single rates, it may help to calculate historical principal components of the interest rates, and attribute the curve changes and the resulting P&L in terms of historical principal components as discussed in Attribute P&L to PCA vectors (swaps) and Principal Component Analysis for attributing yield curve changes. In a production environment, sometimes other things affect your P&L, and a good P&L explanation will consider those. For example, suppose that a government in India declared, on short notice, that the day when your swap was to mature is going to be a non-working day to hold an election. Now your swap runs for an extra day. How did this affect its value?
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