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This sheet shows, how one can extract a "risk neutral density" from prices of traded options: theoretical one can price options through a probability function (describing the stock's behavior) and if prices would be given 'in a continuous way' one should recover that functions to get information on the stock (differentiating prices twice w.r.t. strike). The first step is a quite brute way: interpolate extracted volatility by a polynomial of low degree to get the observable part of such an density. Now extend to the unobservable left and right tails by selecting a Black-Scholes situation with reasonable parameters. Looking at the statistics given through that one finds plausible numbers - but not all of them are very satisfactory: for example the standard deviation and the volatility from option prices to not agree well. One reason is that tails do not behave according to Black-Scholes. In a second step the statistics is used to fit against a Normal Inverse Gauss (NIG) model, which is known to be theoretical correct over the whole range (note: here only needed for one expiry). This is a stochastic model with an explicitly known density and descriptive statistics. As a result one sees that there is a trade-off between pricing errors versus pdf fitting resulting in smoothing prices to (model dependent) valid prices. Data are settlement data - so only close to real data. The uncertainty of "real, fair" prices and data is reflected by the initial volatility and settlement data, so the problem persists beyond the clinical situation of settlement. BesselK_numeric.mpl is used for faster numerics for Bessel function, load it into the sheet. Download 102_RND and Option Prices.mws or preview as pdf (with hidden input to keep it shorter).
example statistics format

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