Further P&L analysis for option books

P&L analysis (or P&L attribution) breaks down P&L into components arising from
different sources. The above breakdown removes unwanted components of P&L so
that a clean P&L figure can be calculated for backtesting. Studying these other
components can reveal useful information about the trading operation. For instance,
on a market-making desk, does most of the income come from fees and commissions
and spreads as expected, or is it from positions held from one day to the next? A
change in the balance of P&L from different sources could be used to trigger a further
investigation into the risks of a trading desk.
The further breakdown of interday P&L is now considered. In many cases, the P&L
analysis would be into the same factors as are used for measuring risk. For instance,
P&L from a corporate bond portfolio could be broken down into contributions from
treasury interest rates, movements in the general level of spreads, and the movements
of specific spreads of individual bonds in the portfolio. An equity portfolio could have
P&L broken down into one component from moves in the equity index, and another
from movement of individual stock prices relative to the index. This type of breakdown
allows components of P&L to be compared to general market risk and specific risk
separately. More detailed backtesting can then be done to demonstrate the adequacy
of specific risk measurement methods

P&L for options can be attributed to delta, gamma, vega, rho, theta, and residual
terms. The option price will change from one day to the next, and according to the
change in the price of the underlying and the volatility input to the model, this
change can be broken down. The breakdown for a single option can be written as
follows:
*cóLc
LS
*Sò1
2
L2c
LS2 (*S)2òLc
Lp
*pòLc
Lr
*ròLc
Lt
*tòResidual
This formula can also be applied to a portfolio of options on one underlying. For a
more general option portfolio, the greeks relative to each underlying would be
required. If most of the variation of the price of the portfolio is explained by the
greeks, then a risk measurement approach based on sensitivities is likely to be
effective. If the residual term is large, however, a full repricing approach would be
more appropriate. The breakdown of P&L allows more detailed backtesting to validate
risk measurement methods by risk factor, rather than just at an aggregate level.
When it is possible to see what types of exposure lead to profits and losses,
problems can be identified. For instance, an equity options desk may make profits
on equity movements, but losses on interest rate movements.