# 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 ?

• It's an interesting question. Just to be sure I understood, you are trying to view this as an "optimal control" problem, where $S_t$ emerges as the optimal policy guaranteeing that $P\&L_0$ verifies certain properties (value function)? And you are now asking, to what value function do each standard models for $S_t$ correspond? For starters, did you happen to read Lorenzo Bergomi's "Stochastic Volatility Modeling"? In the first chapter (media.wix.com/ugd/c4ff5c_ba17141422d44ba99daf19ee2b931544.pdf) he explains some ideas around Black-Scholes that may be of interest to you? – Quantuple Dec 19 '16 at 8:25
• Thank you. This is indeed what I am trying to do. I do possess Bergomi's book, but haven't seen in its BS reminder (nor elsewhere) things pointing towards what I am trying to achieve. (For BS, or for other models, he always uses model equations, that is, he starts with the model given.) That's too bad as, to quote his book : "The argument goes this way and not the other way around - modeling in finance does does not start with the assumption of a stochastic process for $S_t$ and has little to do with Brownian Motion." – ujsgeyrr1f0d0d0r0h1h0j0j_juj Dec 19 '16 at 10:03
• The argument being "specifying break-even conditions for the P&L", which he sadly does using BS dynamics, or Heston dynamics in his "Smile Dynamics 1" paper ... – ujsgeyrr1f0d0d0r0h1h0j0j_juj Dec 19 '16 at 10:09
• I just went through section 1.1 and I fail to see where he imposes any dynamics? IMHO he only postulates that the expectation of future, squared, realised arithmetic returns converges to some value ($\approx$ finite quadratic variation in stochastic calculus). This is similar to what you get when you impose the LHS to be zero under some measure $Q$ in the PnL equation you mention: If discounted asset prices are martingales under $Q$ ($dS_t - S_t r dt = (...)dW_t^Q$), then the premium needs to be defined as $\pi_0 = E^Q[ e^{-\int_0^T r_s ds} \phi(S_T) ]$ to have $E^Q[P\&L_0]=0$. – Quantuple Dec 19 '16 at 10:34
• Now there are many ways, or models, that can be used to specify the $(...)$ part in my previous post (local volatility, stochastic volatility). And 2models that lead to the same expected P&L for a European option $\phi(S_T)$, should have the same expected realised variance over $[0,T]$ (and indeed two local and stochastic volatility models calibrated to the same vanilla implied vol surface will lead to the same variance swap prices). – Quantuple Dec 19 '16 at 10:40

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).