In the beginning of chapter 1.1 "Characterizing a usable model - the Black-Scholes equation of "Stochastic Volatility Model" by Lorenzo Bergomi we read:

Imagine we are sitting on a trading desk and are tasked with pricing and risk- managing a short position in an option – say a European option of maturity $T$ whose payoff at $t = T$ is $f(S,T)$, where S is the underlying.

The bank quants have coded up a pricing function: $P (t,S)$ is the option’s price in the library model. Assume we don’t know anything about what was implemented.

Then few lines below he says:

For the purpose of splitting the total P&L incurred over the option’s lifetime into pieces that can be ascribed to each time interval in between two successive delta rehedges, we can assume that we sell the option at time $t$, buy it back at $t + \delta t$ then start over again. $\delta t$ is typically 1 day.

And gives this formula below for P&L (see this answer for the large excerpt from the book) :

$$ \textit{P & L} = -[ P(t+\delta t, S + \delta S) -P(t,S)] + rP(t,S)\delta t + \Delta (\delta S - rS \delta t + qS \delta t) $$

where $\delta S$ is the amount by which $S$ moves during $\delta t$. $r$ is the interest rate and $q$ the repo rate, inclusive of dividend yield.

From my point of view it does not make any sense to use "the pricing function coded by the bank quants" for P&L calculations and the rest of the paragraph, imho, is then meaningless.

Are there any unstated assumptions which I'm not able to grasp that could make the text plausible?

  • $\begingroup$ As I see it, you can consider the situation of a market maker e.g. a CIB issuing a taylor-made investment product through its retail network. Somehow, the CIB needs to fix the price at which it sells the product at issue date. It asks its quant team to devise a model. For accounting purpose, IFRS requires the product to be priced at "fair value" (see IFRS), so this cannot be "any" model: it should be "fair value" model, hence the discussion in the full excerpt about the impossibility that this model leads to consistently winning/losing money after having hedged. $\endgroup$ – Quantuple Dec 1 '17 at 13:21
  • $\begingroup$ Note that fair value and absence of arbitrage (hence APT) are tied. Actually, Bergomi shows how the BS pricing PDE gets derived from these no arbitrage (or fair value) considerations. $\endgroup$ – Quantuple Dec 1 '17 at 18:13

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