This is an investigation on how the Option Market deal with the Fat Tail issue. This analysis is focused on Put options on the S&P 500 index with a term around 9 to 12 months.

1. Black Swan(s) – the Fat Tail Issue (Continued) - Gaetan “Guy” Lion May 2010

5. 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.

6. 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).

7. 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.

8. 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.

9. 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.

10. 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.