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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.

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  • $\begingroup$ What if you had two parametric density functions, the first one (1) from the physical world and the second one (2) from the risk-neutral world? Would you be able to get the Radon-Nikodym derivative? You can get (1) by FHS and then by fitting a log-normal mixture, then you can do the same for (2) by constraining the mean to the forward price. You'll get two parametric densities, which you can even fit with a spline to have a smooth derivative of both. How does that sound? $\endgroup$ – Lisa Ann May 9 at 12:19
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    $\begingroup$ I suppose you have tried / are aware of this approach: papers.ssrn.com/sol3/papers.cfm?abstract_id=170629 $\endgroup$ – ilovevolatility May 9 at 12:30
  • $\begingroup$ @ilovevolatility, just read the paper. Yes, it's like what I am suggesting to do. Nonetheless, forward price constraints can be already imposed when you estimate a log-normal mixture on the underlying (returns) distribution. Maybe there's no need to minimize the relative entropy with such constraint My two cents. $\endgroup$ – Lisa Ann May 9 at 14:01
  • $\begingroup$ @ilovevolatility No, I am not aware of that paper. The issue I have encountered when performing a numerical minimisation of the Lagrangian is that the resulting coefficients mean that the term in the exponential can grow to positive infinity, so the tails of the real world distribution also have to fall off at an exponential rate or the resulting risk-neutral "distribution" will blow up. $\endgroup$ – Confounded May 13 at 10:46
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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.

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  • $\begingroup$ Thank you for your reply. I would like to keep things model-free if possible, that's why I have estimated a pdf of log-returns using a kernel-based estimator. Effectively, I need to calibrate the RD derivative to match the martingale and market prices constraints. $\endgroup$ – Confounded Apr 9 at 12:57
  • $\begingroup$ Apologies I didn't get that you were trying to keep things model free. Do you have the pdf for daily log returns, or for 30 days log returns ? $\endgroup$ – Antoine Conze Apr 9 at 13:37
  • $\begingroup$ I first constructed a kernel-based pdf on 1-day returns and then convoluted this pdf 30 times to get a pdf for 30-day returns (clearly, this assumes that the daily returns are iid). $\endgroup$ – Confounded Apr 9 at 13:54

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