# on “recovering probability distributions from option prices” - how to subtract influence of stochastic volatility?

This is based on a 1995 paper by Rubinstein/Jackwerth by the above title where the authors produces a distribution of stock prices inferred from option prices. But their approach only produces a joint distribution of stock prices and whatever other contributing factors, most prominent being volatility.

My question is: is there another way to make out this distribution which also incorporates the expected changes in volatility based on say vix futures? A reference to paper or code will be helpful. Is the difference not sufficient to agonize over? I am interested in computing VaR, that too at time horizons other than the option expiry.

Thanks

• quant_dev, Let me try to rephrase, hoping this will convey better. When one uses the double differentiation you suggested, we get the price distribution assuming that volatility is constant and that's probably the way to go for very short term index options where there are numerous strikes and solid trading volume. But with longer term options, I am wondering aloud if this is a fair approach to recover the price density alone, since we are aware that BSM IV varies quite a bit with moneyness. – Dinesh Feb 10 '11 at 23:02
• No, you don't need to assume that volatility is constant. I agree though that when strikes are far between, double differentiation approach is impossible. – quant_dev Feb 11 '11 at 7:52

You cannot derive the probability distribution you require, because for VaR you need a real-world probability distribution. From the options prices, it is only possible to obtain a risk-neutral distribution.

Now, if you are willing to assume some kind of parametric relationship between the risk-neutral and real-world distributions, then you might find the options prices useful. The resulting mathematics for a stochastic volatility model is somewhat tricky, however. You can find most of it in Jim Gatheral's books. A sloppy treatment would just take the risk-neutral distribution and shift its mean.

Obtaining the approximate risk-neutral distribution is fairly simple. Let p(S) be the time-T risk-neutral probability density. Then we see that (TeX notation alert) \begin{equation} C := Call(T) = B(0,T) \int_0^\infty Max(0,S-K) p(S) dS \end{equation} \begin{equation} \frac{dC}{dK} = B(0,T) \int_0^\infty 1[S>=K] (-1) p(S) dS \qquad\text{[differentiate under integral] } \end{equation} \begin{equation} \frac{dC}{dK} = B(0,T) \int_K^\infty (-1) p(S) dS \end{equation} \begin{equation} \frac{d^2C}{dK^2} = B(0,T) p(K) \qquad \text{ [Fundamental thm of calculus]} \end{equation} Alternatively, you could say that p(S) is the density, and is the derivative of the cumulative distribution function P(S), and write

\begin{equation} C := Call(T) = B(0,T) \int_0^\infty Max(0,S-K) p(S) dS \end{equation} \begin{equation} \frac{dC}{dK} = B(0,T) \int_0^\infty 1[S>=K] (-1) p(S) dS \qquad\text{[differentiate under integral] } \end{equation} \begin{equation} \frac{dC}{dK} = B(0,T) \int_K^\infty (-1) p(S) dS \end{equation} \begin{equation} \frac{dC}{dK} = B(0,T) (-1) ( P(\infty) - P(K)) \end{equation} \begin{equation} \frac{d^2C}{dK^2} = B(0,T) p(K) \end{equation}

Either way you end up finding the density

\begin{equation} p(x) = \frac{1}{B(0,T)} \frac{d^2C(x)}{dx^2} \end{equation} where $x$ is the strike. So an approximate density comes from using the actual option prices available to you. You can spline interpolate, or if you have a regular grid of strikes spaced by dK you can make a histogram of values \begin{equation} \frac{ C(K+dK) - 2C(K) +C(K-dK) }{ dK^2} \end{equation} and divide by the discount factor to find your risk-neutral distribution.

• "The resulting mathematics for a stochastic volatility model is somewhat tricky, however. You can find most of it in Jim Gatheral's books. A sloppy treatment would just take the risk-neutral distribution and shift its mean" speaking of sloppy treatment when not considering stoch vol is misleading at best : the main obstacle when trying to get distribution from option price is that of numerical stability and availability of option prices..... so if you add another layer of complexity, my guess is you wont go far in the real world... – nicolas Jul 9 '11 at 18:37

It would help if you made your question more clear (or included a link to a copy of the paper). If you know call (or put) option prices for a single stock across all strikes $K$ for common expiry $T$, you can then derive the distribution of the stock price at time $T$ in the $T$-forward measure (in which the numeraire is the zero-coupon bond with maturity $T$, and the price of the put option is $V_K(t) = D(T) E[(K-S(T))^+]$, $D(T)$ being the discount factor to expiry $T$). You can do this by differentiating $V_K(T)$ twice over $K$ and dividing by $D(T)$. This will give you simply the distribution of the stock price in the $T$-forward measure, nothing more and nothing else. In other measures (in particular in the real-world measure) this distribution will be different.