Explaining Derivatives P&L with Risk Factors and Model Attribution
Summary
The document discusses how a derivatives desk can explain changes in position value and compare model-based P&L with movements attributed to risk factors. One response recommends risk-theoretical P&L, using a Taylor-series approximation based on sensitivities and changes in market factors. For positions marked to model, it also suggests cumulative and independent brute-force attribution. Another response describes unexplained P&L as model P&L minus risk-factor-explained P&L, with model P&L adjusted for items such as new positions and financing effects.
The material frames P&L explain as a check on how well the chosen risk factors and model account for realized price changes, and points to regulatory and practitioner references. It mentions first- and second-order sensitivities as tools for estimating explained P&L. The document does not provide a complete implementation guide or establish one universally appropriate presentation. Its stated expectation for the share explained is a contributor’s view, not demonstrated evidence or a general standard.
Key ideas
- Risk-theoretical P&L estimates changes using sensitivities and movements in risk factors.
- Brute-force attribution is suggested for positions marked to model, with cumulative and independent variants.
- Unexplained P&L is framed as the gap between adjusted model P&L and risk-factor-explained P&L.
- P&L explain can be used to assess how well a model’s risk factors account for observed changes.
- The discussion offers references and methods but not a complete desk implementation standard.
Tags
Full text
# Good references on PNL explain?
# Good references on PNL explain?
Can anyone share good references for how PNL explain should be calculated and presented for the best use of a derivatives trading desk?
## Answer by Dimitri Vulis (score 8)
https://quant.stackexchange.com/a/59356
I'm not aware of any great reference. However Peter Nash Effective product control: controlling for trading desks. Wiley (2018) chapter 10 Review of Mark-to-Market P&L is a good start. Andrew Colin Mastering Attribution in Finance: A practitioner's guide to risk-based analysis of investment returns. FT Publishing International (2015) is worth a look too. David Bolder Fixed Income Portfolio Analytics Springer (2015) Part III "Performance" has a decent discussion of risk-theortical P&L attribution for bonds. Carl Bacon Practical Portfolio Performance Measurement and Attribution, 3rd edition Wiley (Wiley 2023) has good literature reviews.
I wrote some notes here that I hope may help.
You should have risk-theoretical P&L (RTPL - Taylor sereis approximation of the P&L) for all positions. For the positions that are marked to model, you should have Brute Force - both "Cumulative" and "Independent". (It is possible to do bruce force P&L explain for positions with observable price, but it's harder and less useful.)
## Answer by Transcending (score 4)
https://quant.stackexchange.com/a/59361
References
- https://www.bis.org/publ/bcbs265.pdf This one is directly used by banks for programs such as FRTB.
- https://assets.kpmg/content/dam/kpmg/xx/pdf/2018/10/frtb-white-paper-july-2018.pdf. This one describes it from a P&L variance ratio point of view.
- https://en.wikipedia.org/wiki/PnL_Explained. Basic summary of P&L attribution.
Summary
The purpose of the P&L explanation (or Volcker P&L attribution) is to test how well your risk factors explain your actual P&L and hence the overall logic and consistency of the model.
$$ P\&L_{unexplained} = P\&L_{model} - P\&L_{risk \ factors}$$
$P\&L_{risk \ factors}$, the "explained" portion of P&L, is estimated using the Greeks/sensitivities of the risk factors (sum of first order sensitivities with respect to individual risk factors multiplied by risk factor shifts). First order sensitivities (i.e. delta) use forward differences while second order sensitivities (i.e. gamma) use central differencing and are typically used for futures options.
$$ P\&L_{model}=P\&L_{comprehensive} - P\&L_{new \ positions} - P\&L_{other} $$
is actual model P&L calculated from the price of a position on two consecutive days, where
$$ P\&L_{comprehensive}=NPV_{T}-NPV_{T-1}-CASH-CVAHedges $$ and $$ P\&L_{new \ positions}=P\&L_{new \ position} + P\&L_{trade \ event} $$ $P\&L_{trade \ event} $ is NPV changes from notional changes in existing positions and $P\&L_{other} $ are finance adjustments.
The $P\&L_{unexplained}$ thus compares the difference between the model P&L and the P&L of the risk factors used to explain the price movements. A modeler would like to expect that $P\&L_{risk \ factors}$ explains more than 90% of P&L. In other words, you would like to minimize the portion of P&L that is unexplained by the risk factors used in the model which are supposed to capture the effects of actual P&L experienced by the position.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.