Black Swan(s) – the Fat Tail Issue (Continued) - Gaetan “Guy” Lion May 2010
Preface <ul><li>In December 2009, I studied Taleb’s assertion that the Normal distribution does not capture well the tails of stock returns. And, that instead one should use the Cauchy distribution as advocated by Taleb’s mentor Benoit Mandelbrot. I uncovered the Cauchy distribution overshot with tails that were too fat; and, that the Student’s t distribution provided a better fit. </li></ul>
What to do? <ul><li>I was curious how the market dealt with this issue. My limited understanding told me option market participants did not use either the Cauchy or the Student’s t distributions. And, they still relied on Black Scholes models with Normal distribution. </li></ul>
An hypothesis <ul><li>I figured the easiest way to fatten the tail of a Normal distribution would be to increase its standard deviation in the tail. Let’s say you consider selling a S&P 500 Put with a strike price that stands at 2 full standard deviations below the current S&P 500 level. </li></ul>This is an easy way to fatten the left tail of the Normal distribution and increase your Put Ask price. How about tweaking upward your standard deviation (implied volatility) so instead of your Put being 2 standard deviations away, it is now only 1.33 standard deviation away.
There is much evidence the market does boost implied volatility in the tails. The X axis is a Z value representing the # standard deviations between the S&P 500 level and its Put strike price. Let’s say the S&P 500 index is trading at 1,000 and the Put strike price is 600. Meanwhile, the standard deviation or Implied Volatility is 20% or 200 pts of the index. The Z value = (600 – 1,000)/200 = -2.0. The Put strike price is -2 standard deviations away from the Index level. The Z value is calculated using the Implied Volatility using Black Scholes of a Put with a strike price that is the closest to the current index trading level. The Implied Volatility Multiple shown on the Y axis is equal to the Implied Volatility of a Put (further away from being in the money) divided by the Implied Volatility of the Put closest to being in the money. Let’s say the Implied Volatility for the Put with a strike price of 600 is 30%; meanwhile the one with a strike price of 950 is only 20%. In this case, the Implied Volatility Multiple would be: 30%/20% = 1.5 when a Put is – 2.0 standard deviation away from current price.
Situation on May 19 & 20th The previous slide showed the Implied Volatility multiples on March 17, 2010. The graphs above show the same multiples on May 19 and May 20 th . The relationship between the Z value and the Implied Volatility Multiple is far more stable and linear on May 19 and 20 vs March 17. The picture for May 19 and 20 is nearly identical. It shows the multiple growing smoothly in a near straight line from 1 to 1.5 as the Z value increases from 0 to – 1.5. Data source: OptionsExpress.com. 10 mths Put on S&P 500 (SPY).
Multiple from 3/22 – 5/20/2010 Here the graph shows a single multiple for each trading days over the past two months. And, the multiple is equal to the Implied Volatility of a Put with strike price of 590 vs a Put with a strike price of 1,150 (very close to being in the money throughout this period). As shown, the multiples are mainly above 1.50 associated with Z values of – 1.50 or greater (in absolute term). The Put term over the period varied between 10 to 12 mths as they all had the same expiration date in March 2011.
Scatter Plot combining all the data The scatter plot shows that once the Z value is – 1.5 (as the Put strike price is 1.5 standard deviation below the current level of the S&P 500 index), the Implied Volatility multiple reaches 1.5. The rise of the multiple is very stable and linear until the Z value reaches – 1.75 or so. Above that level (in absolute term) when the Z value reaches – 2.00 and up to -2.50, the Implied Volatility multiple jumps around between 1.6 and up to 2.6. This suggests that the further you go on the left tail, the fatter options trader render that tail by using a higher Implied Volatility multiple. In essence, they do what we hypothesized they would do as expressed on slide 4.
Implied Volatility Multiple Using the same data as the scatter plot, focusing on measures of central tendencies (Average and Median) we see how the Implied Volatility Multiple increases progressively from one Z value bucket to another.
Historical returns observations since 1950 The table shows the Left tail frequencies of S&P 500 9 month returns. The data set includes every trading day since 1950 (15,002 observations). We see how the further out on the Left tail, the more observations the Normal distribution misses out. However, using the Implied Volatility multiple on the previous slide we corrected the Left tail of the distribution. Those results are shown in the right hand column. With the Implied Volatility multiple adjustment, the revised distribution actually overshoots substantially. Thus, Puts way out of the money are no more underpriced. Instead, they are way overpriced. Calculation example: - 15% return corresponded to a Z value of – 1.51. The associated cumulative Normal distribution to the left of (Z) – 1.51 is 6.55%. And, 15,002 observations times 6.55% = 983 observations. At the (Z) – 1.51 level the Implied Volatility multiple is 1.53 (slide 9). The revised Z value is – 1.51/1.53 = - 0.99. The cumulative Normal distribution is now 16.18% resulting in 15,002 times 16.18% = 2,428 observations.
Stumbling upon Jump Diffusion Models <ul><li>What I thought option market participants did manually using an Implied Volatility Multiple to fatten the tails, it appears some of them have actually modeled the process. </li></ul><ul><li>Indeed, Robert Merton has done so by modifying his original Black Scholes model (now called a Jump Diffusion model). Similarly, Cox-Ross-Rubinstein have also modified their Binomial options pricing model accordingly. </li></ul>
And Extreme Value Theory… <ul><li>This branch of statistics deals with extreme deviations from the median of probability distributions. Extreme Value Theory is important for assessing risk for highly unusual events, such as 100-year floods or 100-year market downturns. Much of the work consists in mathematically fitting the tails of the distributions. </li></ul>
But, going manual is still OK… <ul><li>Apparently, many option market participants still use traditional Black Scholes models and simply use an increasingly higher Implied Volatility as they price out options in the tail. This approach is described humorously as using "the wrong number in the wrong formula to get the right price." </li></ul>
Implications for Taleb-like option traders <ul><li>In April 2007, when “The Black Swan” came out, Taleb did harp upon the Normal distribution’s tail-fitting limitation. Also, a fund Taleb advised did well following his strategy of buying way-out-of-the-money Puts in 2008 Q4 (Lehman bankruptcy). </li></ul><ul><li>However, now Taleb’s harping may be outdated. Option participants using various models and methods are valuing Puts in the Left tail with much higher Implied Volatility and resulting prices than the original Black Scholes model would have. </li></ul><ul><li>The market is a learning machine. That makes it challenging to repeatedly remain on the winning side of the exact same trade. That’s especially true when one broadcasts how you have been winning all along (Taleb). Everyone has moved on to Extremistan using Taleb’s metaphors. </li></ul>Mediocristan Extremistan