Relating Delta-Hedged P&L Assumptions to Volatility Models

# How to, from various hypotheses on the P&L, get known models (BS, Heston etc ...)


# How to, from various hypotheses on the P&L, get known models (BS, Heston etc ...)












Usually models in quantitative finance are taught by giving, let's say, stochastic differential equations, initial conditions, and then pricing, under the model, various derivatives written on the underlying. Qualifying a model as good or bad wrt the pricing of a given derivative is often done by saying that the model captures, in a rather pertinent way, "quantities" the derivative is sensitive to. (For instance the Black-Scholes model would be bad to price forward starting options as it fails, least to say, to capture movements of the forward implied volatility.) Then, we can compute the discounted P&L of the porfolio consisting in the sold derivative and underlyings (corresponding to a delta-hedge of the derivative).

Now consider a european option of pay-out function $\varphi$ and maturity $T$ written on a (tradable) underlying $S$. Consider the aforementioned portfolio associated to this option and note $\Delta_t$ the number of shares that we have in the portfolio at time $t$. We can write (and it's quite classic) the discounted P&L of the portfolio over the options's lifetime as $$P\&L_0 = -e^{-\int_0^T r_S ds} \varphi (S_T) + \pi_0 + \int_0^T e^{-\int_0^t r_s ds} \Delta_t \left( dS_t - S_t r_t dt\right)$$ where $\pi_0$ is the price we make a time $t=0$ on the option. Note that I only stress the dependance of $\Delta$ in time, but it can, of course, depend on anything else. Note also that $P\&L_0$ does not depend on any model specification for $S$.

My question is the following : without making any model hypothesis, that is, without specifying $dS_t$ or $r_t$, is there a correspondance

$$\{\textrm{set of hypotheses on $P\&L_0$}\}\to \{\textrm{models on $S$ under a certain measure}\}$$

such that for any given "known" model (BS, Heston, SABR, 3/2, 4/2, Bergomi's P1 etc) there exist a set of hypotheses on $P\&L_0$ (for instance on its expectation, its variance that one would like for instance to minimize, on its others moments etc under some measure or under another, or something else) that lead to the given model and the fondamental theorem of pricing associated to it ?

By lead I mean : wanting to prescrible/minimize some quantities associated to $P\&L_0$ will lead to functional equations (variational, in fact) on the function $\Delta$ that will lead to a PDE that will ultimately lead (through Feynman-Kac) to a model on $S$.

First of all : how to introduce dependance of $\Delta$ in a parameter (in $S$ for instance, to begin with, or later in realized variance) through assements/hypotheses on the $P\&L_0$ ? How to retrieve the Black-Scholes model ? Other models ?

## Answer by q.t.f. (score 3)

https://quant.stackexchange.com/a/38435

You can characterize local volatility (LV) models by the existence of a delta-hedging strategy which reduces the variance of P&L to zero, together with an assumption that P&L is a continuous process.

You can characterize the Black-Scholes (BS) model as the unique LV model with time homogeneity and spacial homogeneity. Under BS we have expected P&L at time $t $ of a payoff $\phi (S_T)$ is a function of $S_t $ and $T-t$ (time homogeneity).