Model Prices and Option P&L in Delta-Hedging Analysis
Summary
The document questions whether a bank’s coded option pricing function can sensibly be used to calculate the profit and loss of a short option position. It presents a one-period P&L expression that combines the change in the option’s model price, interest on the option premium, and the P&L from a delta hedge adjusted for financing and repo costs. The question is whether this model-price change represents the actual trade’s mark-to-market P&L or depends on unstated assumptions about pricing and hedging.
The excerpt provides the formula and describes a hypothetical sale followed by a repurchase at the next rehedge, but it does not include a resolution. The distinction between a model value and a market transaction price is therefore left open. Applying the calculation in practice requires assumptions about how the position is valued, how the hedge is financed, and whether the model price is being used as a proxy for market marks; the document does not specify these details or provide empirical evidence.
Key ideas
- The question distinguishes a bank model’s option value from the price at which an option trades.
- The displayed P&L expression includes option value change, financing, and delta-hedge returns.
- The setup assumes a sale and repurchase around successive delta rehedges.
- The document raises the issue without resolving which valuation assumptions make the calculation appropriate.
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Full text
# Does Bergomi mix up an option model price with option market price?
# Does Bergomi mix up an option model price with option market price?
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?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.