Extrapolating Equity Volatility Smiles with Fitted Distributions
Summary
The document considers how to extend an arbitrage-free interpolated equity option price curve beyond the last observed point. The questioner reports that a cited smile extrapolation technique produced prices that could admit arbitrage, and expresses doubt that the method remains arbitrage-free far into the wings.
The response proposes fitting a probability distribution to option data and deriving implied volatilities from option prices calculated under that distribution. A simple example uses a mixture of two Gaussian components to fit FTSE options and construct an implied probability density. The density can be integrated numerically to price options; choosing smooth basis distributions avoids spline-node irregularities in derivatives, and a valid density integrates to one. The answer offers this as an approach rather than a universally validated procedure: it gives no detailed calibration recipe, out-of-sample evidence, or guarantee that every distribution fit will satisfy all relevant arbitrage constraints.
Key ideas
- A fitted probability distribution can be used to extend option prices into smile wings.
- Option prices can be calculated by numerically integrating the fitted density.
- Mixtures of distributions offer flexible shapes and can produce smooth densities.
- The example illustrates a possible approach but does not establish universal arbitrage safety.
Tags
Full text
# Extrapolation of the volatility smile # Extrapolation of the volatility smile Are there any market practices to extrapolate the volatility smile for equities? I already have an arbitrage free interpolated call prices data and I'm looking for a method to extrapolate beyond the last available data. I tried the method described here : S. Benaim, M. Dodgson, and D. Kainth. An arbitrage-free method for smile extrapolation. Technical report, Royal Bank of Scotland, 2008. 1, 2, but I have arbitrageable prices (Plus there is no indication that beyond the vicinity of the last point, we obtain arbitrage free prices. ## Answer by will (score 4) https://quant.stackexchange.com/a/36432 I'm a fan of fitting a distribution, and then implying vols from that in the wings. You'll get an arbitrage free surface, that makes sense. A very simple example is to use gaussians to build a pdf, and then numerically price the options from it. Here's an example fitting FTSE options using just 2 gaussians to create the implied pdf and resulting smile: Where the fit to the options is good: You're of course free to use other distributions as your basis functions to create your pdf, as well as mixtures of different functions - what i liike about doing this is that you get a very nice smooth distribution. No spline nodes causing weird behaviour in the derivatives, and it integrates to 1. You just price the options by numerically integrating your pdf. I welcome anyone to point me at a particularly nasty smile from the past and i will happily fit it and put the results here.
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