Good references on PNL explain?

Question

Can anyone share good references for how PNL explain should be calculated and presented for the best use of a derivatives trading desk?

Answer 1

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 2

References

1. https://www.bis.org/publ/bcbs265.pdf This one is directly used by banks for programs such as FRTB.

2. 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.

3. 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.