Z. Wahab ENMG 625 Financial Eng g II 04/26/12 Volatility Smiles The Problem with Volatility We cannot see volatility the same way we can see stock prices or interest rates. Since it is a meta-measure (a measure of another measure, the price) the best we can do is evaluate it statistically. Because of this, our quantification of volatility is backward looking while we really want to know what volatility is going to be in the future. Ignoring this means assuming volatility to be constant, something that can clearly be shown to be false through empirical observation. Implied vs. Actual Volatility The implied volatility is the one which when input into the Black-Scholes option pricing formulae gives the market price of the option. It is often described as the market s view of the future actual volatility over the lifetime of the particular option. The actual volatility is very difficult to measure and can be thought of as the amount of randomness in an asset return at any particular time. 1
The differences between these two bring about the phenomena we call volatility smiles and skews. It makes sense to consider volatility a characteristic of the underlying asset. This assumption means that for the same stock, options with different strike prices should have the same implied volatility. If actual volatility were constant, and if the Black-Scholes model were correct and if people prices options correctly then that would hold; however this is rarely the case. Example: Foreign Currency Options If we take options with the same maturity on a certain foreign currency that vary only in strike price we can calculate the implied volatility for each one. Keep in mind that since they share the same underlying asset we expect the volatility to remain constant regardless of the strike price. However, plotting gives the following, 2
The volatility is relatively low for at-the-money options and gets progressively higher as an option moves either into or out of the money. We gain some analytical insight into why this occurs if we compare the implied volatility distribution with the lognormal one with the same mean and standard deviation. Consider a deep out-of-the-money call with strike price above K 2. This derivative will only pay off if the exchange rate closes above K 2, and according to the above figure the probability of this happening is higher for the implied distribution than the lognormal one. A higher probability will generate a higher price, which in turn means a higher implied volatility. The same is true of a deep out-of-the-money put with strike price below K 1. 3
Empirical Results We have just shown that the smile used by traders for foreign currency options implies that they consider that the lognormal distribution understates the probability of extreme movements in exchange rates, but is this accurate? A study of 12 different exchange rates over a 10-year period was conducted. The first step was to calculate the standard deviation of daily percentage change in each exchange rate. Then to note how often the actual percentage of change exceeded one standard deviation, two, three etc. The final step was to calculate how often this would have happened if the percentage changes had been normally distributed. The lognormal model implies that percentage changes are almost exactly normally distributed over a one day time period. The table provides evidence to support the existence of heavy tails and the volatility smile used by traders. 4
Note: In the mid-80 s a few traders knew about the heavy tails of foreign exchange probability distributions. They bought deep out-of-the-money call and put options on a variety of different currencies and proceeded to make lots of money. This is because the lognormal model that everyone else accepted underpriced those options and more of them ended up in-the-money than it expected. By the late-80 s, traders realized that foreign currency should be priced with a volatility smile and the trading opportunity disappeared. Reasons for the Smile in Foreign Currency Options For an asset to have a price with a lognormal distribution two conditions must be met. Its volatility must be constant and the price of the asset should change smoothly with no jumps. Neither of these is satisfied with exchange rates. The percentage impact of jumps on both prices and volatility smiles becomes less pronounced as the maturity of the option is increased. This is because the jumps tend to get averaged out and so the jumps appear to be smoother. This makes the derivative conform to the assumed lognormality more readily. Equity Options Equity (stock) options exhibit a volatility skew 5
The volatility used to price a low-strike-price option (i.e. a deep our-of-the-money put or a deep in-the-money call) is significantly higher than that used to price a high strike-price option. Once again, we turn to a comparison of the probability distribution of the implied volatility and a lognormal one with the same mean and standard deviation for an analytical explanation. 6
To the left (low-strike-price) the implied volatility shows a higher probability than the lognormal one of equal parameters. This means that these options are priced higher than ones with larger strike-prices. This is not the case to the right, thus the skew. Reasons Behind the Skew Interestingly, there was no marked volatility skew before October 1987. It has been suggested that the market crash of 87 instilled fear into traders who therefore trade lower strike-price options more heavily in order to hedge their positions, thus affecting the price of these options with their increased demand. This explanation has some empirical support. A steepening in the skew is observed when the S&P 500 (the market portfolio) declines while a flattening accompanies its rise. This shows that the worse the market is doing, the more people trade low-strike-price options in apprehension of a market crash. Another explanation is that as a company s equity declines, it becomes riskier and therefore more volatile. When the opposite happens a company is viewed as safer and therefore has lower volatility. 7
Equity Options when Expecting a Large Jump In order to better illustrate the behavior of equity options we shall consider the following example. Consider a stock at $50 that is expected, due to the coming determination of a currently unknown event, to change strongly in one month. The risk-free rate is 12% We can calculate the prices of 1-month European puts and calls with different strike prices and from them find the implied volatility. The results are, 8
Plotting the implied volatility against the strike-price reveals a volatility frown. The volatility implied from an option with a strike-price of $50 will overprice an option with a strike-price that is further from being at-the-money. Remember: traders use volatility smiles to allow for non-lognormality. The volatility smile defines the relationship between the implied volatility of an option and its strike price. Volatility Surfaces Implied volatilities are frequently used to quote the prices of options. 9
The implied volatility of a European option on a particular asset as a function of strike price and time to maturity is known as the asset's volatility surface. Every day traders and brokers estimate volatility surfaces for a range of different underlying assets from the market prices of options. Some points on a volatility surface for a particular asset can be estimated directly because they correspond to actively traded options. The rest of the volatility surface is typically determined by interpolating between these points. If the assumptions underlying Black-Scholes held for an asset, its volatility surface would be flat and unchanging. In practice the volatility surfaces for most assets are not and change stochastically. 10
Volatility Modeling Since the Black-Scholes option pricing formulae require volatility as an input we need a way to accurately assess it. Our understanding of volatility is that of meta-data that is not a real characteristic of an asset, but a descriptive one of its price. Given this difficulty, how can we best model it? Simple Models The simplest way to account for volatility is assuming it constant. However, it is clear that this approach is inapplicable to trading when examining real stock prices and their changing volatilities. If we believe that the volatility is close to constant and only varies slowly then we can consider the volatility of recent returns only. Setting a window of N we have, There are obvious major problems associated with this volatility measure; because the returns are equally weighted we will get a plateau-effect associated with a large return. 11
If there is a large one-day return it will increase the volatility instantaneously, but the estimate of volatility will stay raised until N days later when that return drops out of the sample. Exponentially Weighted Moving Average Now let us consider time-varying volatility. We don t just have one σ but must consider σ n, our estimate of the volatility on the nth day, using data available up to that point. Exponentially Weighted Moving Average estimates eliminate the plateau-effect of the simpler moving average volatility estimate. The earlier models give every entry an equal weight whereas this model weighs the most recent ones most and decays the weight exponentially as the returns become further. 12
The parameter λ must be greater than zero and less than one. The more recent the return, the more weight is attached. The sum extends back to the beginning of time. The coefficient of 1 λ ensures that the weights all add to one. This expression can be simplified to, Mean Reversion and GARCH Models If we believe that volatility tends to vary about a long-term mean σ, then we could use, 13
Here there is a weighting assigned to each of the long-run volatility estimate and the current estimate based on the last n returns. This is called an ARCH model (for Autoregressive Conditional Heteroscedasticity). Alpha is the weight that we give to our estimate of mean volatility and the remainder is split between n instances of recent volatility measures. If we combine the ARCH and Exponential Weighting, we arrive at a Generalized ARCH (or GARCH) model, All notations maintain their identity, alpha arbitrating between our mean volatility and recorded data, and lambda distributing the weights of recent measures in an exponential fashion. 14