Volatility smile is the phrase used to describe how the implied volatilities of options vary with their strikes. A smile means that out-of-the-money puts and out-of-the-money calls both have higher implied volatilities than at-the-money options. Other shapes are possible as well. A slope in the curve is called a skew. So a negative skew would be a download-sloping graph of implied volatility versus strike.

Figure 2.10: The volatility 'smile' for one-month SP500 options, February 2004.

Long answer

Let us begin with how to calculate the implied volatilities. Start with the prices of traded vanilla options, usually the mid price between bid and offer, and all other parameters needed in the Black-Scholes formula, such as strikes, expirations, interest rates, dividends, except for volatilities. Now ask the question: What volatility must be used for each option series so that the theoretical Black-Scholes price and the market price are the same?

Although we have the Black-Scholes formula for option values as a function of volatility, there is no formula for the implied volatility as a function of option value, it must be calculated using some bisection, Newton-Raphson, or other numerical technique for finding zeros of a function. Now plot these implied volatilities against strike, one curve per expiration. That is the implied volatility smile. If you plot implied volatility against both strike and expiration as a three-dimensional plot, then that is the implied volatility surface. Often you will find that the smile is quite flat for long-dated options, but getting steeper for short-dated options.

Since the Black-Scholes formula assume constant volatility (or with a minor change, time-dependent volatility) you might expect a flat implied volatility plot. This does not appear to be the case from real option-price data. How can we explain this? Here are some questions to ask.

• Is volatility constant?

• Are the Black-Scholes formula correct?

• Do option traders use the Black-Scholes formula?

Volatility does not appear to be constant. By this we mean that actual volatility is not constant, actual volatility being the amount of randomness in a stock's return. Actual volatility is something you can try to measure from a stock price time series, and would exist even if options didn't exist. Although it is easy to say with confidence that actual volatility is not constant, it is altogether much harder to estimate the future behaviour of volatility. So that might explain why implied volatility is not constant, and people believe that volatility is not constant.

If volatility is not constant then the Black-Scholes formula are not correct. (Again, there is the small caveat that the Black-Scholes formula can work if volatility is a known deterministic function of time. But I think we can also confidently dismiss this idea as well.)

Despite this, option traders do still use the Black-Scholes formula for vanilla options. Of all the models that have been invented, the Black-Scholes model is still the most popular for vanilla contracts. It is simple and easy to use, it has very few parameters, it is very robust. Its drawbacks are quite well understood. But very often, instead of using models without some of the Black-Scholes' drawbacks, people 'adapt' Black-Scholes to accommodate those problems. For example, when a stock falls dramatically we often see a temporary increase in its volatility. How can that be squeezed into the Black-Scholes framework? Easy, just bump up the implied volatilities for option with lower strikes. A low strike put option will be out of the money until the stock falls, at which point it may be at the money, and at the same time volatility might rise. So, bump up the volatility of all of the out-of-the-money puts. This deviation from the flat-volatility Black-Scholes world tends to get more pronounced closer to expiration.

A more general explanation for the volatility smile is that it incorporates the kurtosis seen in stock returns. Stock returns are not normal, stock prices are not lognormal. Both have fatter tails than you would expect from normally distributed returns. We know that, theoretically, the value of an option is the present value of the expected payoff under a risk-neutral random walk. If that risk-neutral probability density function has fat tails then you would expect option prices to be higher than Black-Scholes for very low and high strikes. Hence higher implied volatilities, and the smile.

Another school of thought is that the volatility smile and skew exist because of supply and demand. Option prices come less from an analysis of probability of tail events than from simple agreement between a buyer and a seller. Out-of-the-money puts are a cheap way of buying protection against a crash. But any form of insurance is expensive; after all, those selling the insurance also want to make a profit. Thus out-of-the-money puts are relatively over-priced. This explains high implied volatility for low strikes. At the other end, many people owning stock will write out-of-the-money call options (so-called covered call writing) to take in some premium, perhaps when markets are moving sideways. There will therefore be an over-supply of out-of-the-money calls, pushing the prices down. Net result, a negative skew. Although the simple supply/demand explanation is popular among traders it does not sit comfortably with quants because it does suggest that options are not correctly priced and that there may be arbitrage opportunities.

While on the topic of arbitrage, it is worth mentioning that there are constraints on the skew and the smile that come from examining simple option portfolios. For example, rather obviously, the higher the strike of a call option, the lower its price. Otherwise you could make money rather easily by buying the low strike call and selling the higher strike call. This imposes a constraint on the skew. Similarly, a butterfly spread has to have a positive value since the payoff can never be negative. This imposes a constraint on the curvature of the smile. Both of these constraints are model independent.

There are many ways to build the volatility-smile effect into an option-pricing model, and still have no arbitrage. The most popular are, in order of complexity, as follows

• Deterministic volatility surface

• Stochastic volatility

• Jump diffusion.

The deterministic volatility surface is the idea that volatility is not constant, or even only a function of time, but a known function of stock price and time, a(S, t). Here the word 'known' is a bit misleading. What we really know are the market prices of vanilla options, a snapshot at one instant in time. We must now figure out the correct function a (S, t) such that the theoretical value of our options matches the market prices. This is mathematically an inverse problem, essentially find the parameter, volatility, knowing some solutions, market prices. This model may capture the volatility surface exactly at an instant in time, but it does a very poor job of capturing the dynamics, that is, how the data change with time.

Stochastic volatility models have two sources of randomness, the stock return and the volatility. One of the parameters in these models is the correlation between the two sources of randomness. This correlation is typically negative so that a fall in the stock price is often accompanied by a rise in volatility. This results in a negative skew for implied volatility. Unfortunately, this negative skew is not usually as pronounced as the real market skew. These models can also explain the smile. As a rule one pays for convexity. We see this in the simple Black-Scholes world where we pay for gamma. In the stochastic volatility world we can look at the second derivative of option value with respect to volatility, and if it is positive we would expect to have to pay for this convexity - that is, option values will be relatively higher wherever this quantity is largest. For a call or put in the world of constant volatility we have

This function is plotted in Figure 2.11 for S = 100, T — t = 1, a = 0.2, r = 0.05 and D = 0. Observe how it is positive away from the money, and small at the money. (Of course, this is a bit of a cheat because on one hand I am talking about random volatility and yet using a formula that is only correct for constant volatility.)

Stochastic volatility models have greater potential for capturing dynamics, but the problem, as always, is knowing which stochastic volatility model to choose and how to find its parameters. When calibrated to market prices you will still usually find that supposed constant parameters in your model keep changing. This is often the case with calibrated

Figure 2.11: d2V/da2 versus strike.

models and suggests that the model is still not correct, even though its complexity seems to be very promising.

Jump-diffusion models allow the stock (and even the volatility) to be discontinuous. Such models contain so many parameters that calibration can be instantaneously more accurate (if not necessarily stable through time).

References and Further Reading

Gatheral, J 2006 The Volatility Surface. John Wiley & Sons Ltd

Javaheri, A 2005 Inside Volatility Arbitrage. John Wiley & Sons Ltd

Taylor, SJ & Xu, X 1994 The magnitude of implied volatility smiles: theory and empirical evidence for exchange rates. The Review of Futures Markets 13

Wilmott, P 2006 Paul Wilmott on Quantitative Finance, second edition. John Wiley & Sons Ltd

Found a mistake? Please highlight the word and press Shift + Enter