Pricing Models as Tools for Explaining Portfolio P&L and Risk

# How to understand the role of stochastic differential equations models in finance, in particular in managing portfolios?


# How to understand the role of stochastic differential equations models in finance, in particular in managing portfolios?












I just begin to read Stochastic Volatility Modeling by Lorenzo Bergomi. I t is very inspiring to me, but there are some statements which confuse me and I would like to ask for help here.

In the introduction of Chapter 1, it writes "Indeed, a pricing equation is essentially an analytical accounting device:..." and "More sophisticated models enable their users to characterize more precisely their P&L and the conditions under which it vanishes, ... , is not to be able to predict anything, but rather to be able to differentiate risks generated ...".

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Could anyone explain these ideas in a simpler way or example please? Thank you so much!!

## Answer by Ezy (score 1, accepted)

https://quant.stackexchange.com/a/42583

The most important thing to remember in trading is that when a PM/trader gets his pnl at the end of a day of trading, it is important to provide some "pnl explain" which is a decomposition of this pnl in various buckets which can help understand the origin of one's portfolio profit of loss. For example if you trade a collection of stocks you could decompose your pnl, by stock, or by sector for instance. You can see how such decomposition can be seen as a form of accounting.

Now in a slightly more abstract fashion one may want to decompose the pnl not in terms of tradable instruments but rather in terms of risk factors. For instance CAPM is a simplistic risk factor model where each stock return is decomposed into a beta to the overall market factor and some idiosyncratic component. Using such model you could decompose your pnl into a sum of a market pnl (derived from your overall beta exposure) and an idiosyncratic pnl. This particular way of decomposing the pnl is based on a statistical model because the beta is obtained through a statistical procedure, not some fundamental/analytical one.

Now to the meat :) in options trading there's a very natural risk model for explaining the pnl of an option instrument: using the natural variables used in the pricing equation of Black-Scholes: spot, implied-vol, time to maturity, dividends, rates (discount, repo). Even if BS is an idealized/naive model of how financial really work it is nonetheless very good at choosing "degrees of freedom" that "explain" how the value of an option changes as a function of certain market quantities (the "betas" being the greeks which are just the analytical derivatives of the option pricing formula). Those degrees of freedom are the "factors" that one uses to decompose the options portfolio pnl into: delta-pnl, gamma-pnl, vega-pnl, theta-pnl, dividend-pnl, rho-pnl etc...

Intuitively imagine for a second that you knew nothing of Black-Scholes (and were the only one in the market not knowing about it :) ) but nevertheless were just a smart statistical/trader looking at how your portfolio moves from day to day. Surely you would be able to discover that your option instruments have a certain linear (delta) and quadratic (gamma) dependency to the spot. But unfortunately you would soon discover that the closer you get to maturity your regression coefficients to the spot vary pretty often, you would then go on and study how those regresion coefficients change with spot and would realize if you are really smart that you could introduce a new quantity called volatility that would explain both the way your delta and gamma move but also explain quite a bit of the residual of your delta/gamma model. So in essence, the existence of the BS formula allows you to directly identify the main drivers of your pnl.

Sorry for the long post but i hope this clarifies a bit things for your study!