# Is there a popular curve fitting formula of options skew vs strike price or vs Delta?

I was trying to build a options trading/optimization system. But it often gets more inaccurate as it scans through the far from ATM options because, you know, options skews.

That is because I did not price in options skews, or jump premium. I am wondering if there is a popular formula that takes "degree of options skew", and either strike price or Delta as inputs, and then give me skews premium in terms of IV as output.

Thank you very much.

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I'd love to know this too. If it helps, I can provide you with data. Also, for a given expiration date, option volatility has a minimum at the forward price (ATM + interest), and increases/decreases linearly with price. However, the slope of the increase and the slope of the decrease can be different. – barrycarter Feb 28 '11 at 14:14

There have been many studies of the shape of volatility skew. There are two simple empirical approaches that have proven most popular:

• Spline (or piecewise polynomial) fits of observable implied vols
• Parabolic parametric fits.

and there is one other approach that is fairly simple yet reliable:

• Edgeworth expansions

Other empirical approaches also exist, of course, but quickly become unwieldy mathematically, computationally, and for calibration.

The overall process is as follows:

1. Choose a function, such as a parabola, that you will use to represent skew and code it into the computer

2. Choose some input volatilities you trust

3. Use them in a fitting scheme to calibrate the parameters (tilt and curvature) of your skew. That is, you find parameters such that your chosen function does the best possible job of reproducing your input volatilities.

4. Use your fitted function to estimate volatilities anywhere on the skew curve.

Sometimes, if data is particularly bad, you can copy tilt and curvature parameters from one fitted curve onto another curve where you have very little input data available. Exercise caution when you do this!

The fits are typically constrained such that the parabola (or other function) cannot dip below zero for any "reasonable" strike. This is hard to achieve with linear fits, which is one major reason why parabolas are more popular. Fits also often have "cutoffs" where volatility is assumed to be constant for strikes above (or below) certain extreme values.

For all such fits, one usually uses existing mathematical libraries and routines (many of which are simple copies of the serviceable code found in Numerical Recipes). You can get spline libraries, and you can fit parabolas using any linear algebra library that has a routine for ordinary least squares. See this site for more information.

Edgeworth expansions are in the family of skew treatments that consider skew as a symptom of non-lognormal returns, and try to represent a non-lognormal distribution in some concise way. A decent place to start learning about them is Rubinstein's paper at this location.

The real trick for any of these fits is choosing a weighting scheme. Typically, you will prefer to weight some input volatilities more than others. For example, most people believe the out-of-the-money put and call prices contain the best information. Others concentrate on weighting the near-the-money options, often by weighting according to the gamma.

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In optimiazation system, you have to weight the price for the different maturities in a way that reflect your confidence in each data point (influenced by liquidity). One way to do so is to weight, each price by its Black-Scholoes Vega (see Tankov (2003)). So when minimazing the squared differences of the sum your weighted option prices, you can use the following approximation (in term of call price).

$\sum_{i=1}^{N} w_i (C^\theta(T_i,K_i)-C(T_i,K_i))^2=\sum_{i=1}^{N} \frac{ (C^\theta(T_i,K_i)-C(T_i,K_i))^2}{ Vega_i^2(T_i,K_i)}$

The advantage of this method is that the option prices scaled by its vega is approximatively equal to its implied volatility and implied volatility are more uniform across maturity and strike than option prices. As a proof, you can apply a taylor approximation.

$C^\theta\approx C+Vega_{BS}(\sigma^\theta-\sigma_{BS}) \Leftrightarrow \frac{C^\theta-C}{Vega}\approx (\sigma^\theta-\sigma)$

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