Diagnosing Irregular Implied Volatility Smiles from American Option Data
Summary
The document describes an attempt to build implied volatility smiles for European options from American option chain data. The author calculates implied volatilities with Black–Scholes, selects out-of-the-money options to reduce early-exercise concerns, and fits a cubic regression across strikes to interpolate missing values. Some sample chains produce smooth, familiar smiles, while others yield a curve that fails to fit the observations despite a high reported R-squared overall.
The example involves a put and call with different implied volatilities at nearby strikes around the underlying share price, with nine months to maturity. The author reports checking the calculations repeatedly and notes that fit quality improves over time, apart from a final-day data error. The document poses possible causes and remedies but supplies no answer. Its setup also leaves open issues such as dividends, negative rates, American exercise effects, data quality, and whether a polynomial in strike is an appropriate surface model.
Key ideas
- The author uses out-of-the-money American options to approximate European option observations for volatility estimation.
- A cubic regression in strike is used to interpolate implied volatility between observed strikes.
- Some chains produce irregular fitted curves even when the overall regression fit is reportedly high.
- The document raises possible data and modeling issues but does not identify the cause or provide a remedy.
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Full text
# What is the cause of a "broken" volatility surface?
# What is the cause of a "broken" volatility surface?
I am currently working on a project for which I need the implied volatility surfaces, to estimate the value of plain-vanilla European options with different strikes (cannot be observed directly in the market). I collected American option chain data from a data source and calculated the implied volatilities with the Black-Scholes formulas. As you know, the price of an European option is not always equal to the price of an American option. Normally, in a positive interest rate environment and for non-dividend paying stocks, the price of an American call is equal to an European call. However, my data set do contain a lot of options which has dividend paying security as underlying and we currently live in a negative interest rate environment.
To solve this problem, I picked only out-of-the-money options to calculate the implied volatilities. I made this choice because out-of-the-money options will never be exercised (in this case the American option converts to an "European" option). When I did a random check for three option chains, I got really nice volatility smiles (typical school examples).
As mentioned, I need to make estimate for plain-vanilla options with different strike prices (most of them cannot be directly observed in the market). So I made the following simple regression, to interpolate between observed strike prices:
$$IV_{i,t,K}=\alpha_{i,t}+\theta_{1,i,t}K_{i,t}+\theta_{2,i,t}K_{i,t}^2+\theta_{3,i,t}K^3_{i,t}+\epsilon_{i,t}$$
Which produces a lot of graphs as follows:
The regression matches the observed implied volatilities, I got a really high r2 score and I have no problem to interpolate the implied volatilities. However, sometimes I get a graph as follows (10-15% of the cases) :
As you can see, the regression is not able to interpolate due to the fact that there is a "broken" volatility surface. The observed implied volatility for a put option with strike 17 is equal to +/- 0.42 and the observed implied volatility for a call option with strike 19 is equal to +/- 0.52 (the share price is around 18.75). The time to maturity is 9 months. I already recalculated a couple of times, so, I am 98% sure I did not make a mistake. Furthermore, one extra detail is the following graph, in which you see that the r2 improves over time (last day is a data error):
My exact question is as follows: Do some has also faced a "broken" volatility surface? And what is it caused by? And how did you solved it?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.