Changing from Historical Returns to Risk-Neutral Option Prices

# Estimation of Radon–Nikodym derivative from historical returns and option price data


# Estimation of Radon–Nikodym derivative from historical returns and option price data












Say we have an estimate of empirical density function $f^{\mathbb{P}}_S(s)$ of historical log-returns on a stock $S$ over a 30-day period under the real-world objective measure $\mathbb{P}$. We also have an ATM price of a 30-day European put option on the stock $P(ATM)$. We know that under the risk neutral measure $\mathbb{Q}$ the discounted stock is a martingale, $E^{\mathbb{Q}}[e^{-rt}S(t)|S_0]=S_0$. So, we have two general moments conditions, one for the put and one for the discounted price, which we can use to estimate the change of measure $\frac{d\mathbb{Q}}{d\mathbb{P}}$. The question is how?

It is easy to match one of the two conditions by applying an appropriate shift to the log-returns, but what transformation to apply when we need to match both of the above? And what if we also have, say, two off-strike put prices $P(0.9S_0)$ and $P(1.1S_0)$? In any case, there is not enough option data to estimate the risk-neutral density directly, so I am looking for a way to infer it from the real-world density and a number of risk-neutral moment conditions.

Initially I though about using Maximum Cross Entropy (MCE) method with moments constrains, but this would require solving an optimisation problem to find Lagrangian multipliers, and I would prefer to avoid optimisation. Also, if the Radon-Nikodym derivative is always given by a Doléans-Dade exponential (btw, is this true that the change of measure can only take this form? at least when restricted to a diffusion process driven by the Wiener process?), then maybe we can use this information to calibrate it via, say, OLS?

Maybe there is some literature on this, so I would be grateful for references as well as direct suggestions on how to approach this.

## Answer by Antoine Conze (score 1)

https://quant.stackexchange.com/a/45010

Since Girsanov changes the drift but keeps the volatility unchanged, it would be hard to reconcile say a simple exponential brownian motion under $\mathbb{P}$ with a skew/smile structure under $\mathbb{Q}$. So it seems you first need to choose and estimate a "non EBM" process such as EBM with local vol, EBM with stoch. vol, EBM plus jumps, Levy, ... under $\mathbb{P}$ before moving to $\mathbb{Q}$ by applying the appropriate change of measure for the class of process you have selected. See for instance Cont and Tankov for modelling with jump processes.