Introducing the JPMorgan Cross Sectional Volatility Model & Report

Equity Derivatives Introducing the JPMorgan Cross Sectional Volatility Model & Report A multi-factor model for valuing implied volatility For more information, please contact Ben Graves or Wilson Er in Corporate Quantitative Research JPMorgan s European Equity Derivative Strategy Group uses a similar Cross Sectional Model. Please see Equity and Derivatives Markets Weekly Outlook 21 Nov 05 for details on the European model. We introduce our model for valuing implied volatility. It differs from time series models, which value current volatility based on past observations of one or more variables for a single stock, and instead utilizes current values for a group of stocks in an index or sector. The model considers several factors including Beta, market capitalization, realized volatility, and stock returns. It also incorporates CDS spreads, explicitly incorporating the link between these two markets. The model output is a ranking of rich/cheap single stock implied volatility. In practice, the model is not one but many. In total, 33 unique models are maintained for both the broad market (Market Model) and individual sectors (Sector Models), with each being run for 1M, 3M, and 6M implied volatility. The model backtests well. Both Market and Sector Models outperform a simple buy/sell volatility strategy, and produce best results when used together. In addition to buying or selling volatility, the model results are broadly applicable to call overwriting and optimizing single stock baskets for dispersion trades. In the US, JPMorgan has begun publishing the model results daily for a universe of 3 companies via our regular equity derivatives distribution. This note reviews how to read our daily report. Overview JPMorgan s Cross Sectional Model uses a snapshot of the relationship between current implied volatility and other independent variables for a broad group of companies to construct a multi-factor regression. In effect, the model calibrates how implied volatility across the market is impacted by various factors at the present time, and highlights when single stocks deviate from the broader pattern. The factors used in the model are: Beta Market capitalization Realized volatility (1 month & 12 month) CDS spread Stock price returns (1 month & 12 month) Exhibit 1 below shows a comparison between the model and actual implied volatility for the company universe that is liquid in both equity derivatives and CDS. Companies above the line have rich implied volatility vs. the model; companies below the line have cheap implied volatility vs. the model. 15

Exhibit 1: 3M Implied Volatility - model versus actual values* Actual (y-axis) Model (x-axis) 55 45 35 25 15 R2 = 0.76 S BSX YHOO BSC 12 17 22 27 32 37 42 47 52 *Note that several companies that are in the process of a leveraged buy-out have high model volatility relative to actual volatility, as the model does not explicitly consider LBOs. In these cases, the model should be overlaid with a sensibility check to incorporate other factors, such as LBOs, that impact volatility., data as of 3rd October 06 In practice, our Cross Sectional Report is not one model but many models. Implied volatility under each model may depend on some or all of the factors above and have differing sensitivity to each. We discuss the process to select relevant factors for each model in detail later on in this note. Market Model vs. Sector Model It is possible to build a cross sectional regression using any number of company universes. One approach is to build a model for the entire market (e.g. the S&P 0). This approach has the advantage of including many companies and hence a large number of observations across a broad range of company health and volatility. The disadvantage of this approach is that important factors in one sector may not matter in another (e.g., 1M stock returns have been significant for Energy but not for Technology). This generally leads to lower R-squared values for broad Market Models. In contrast, Sector Models are often good at determining rich/cheap volatility within a sector, but they can miss broader market context (e.g., KB Homes currently appears rich vs. its sector, but the sector overall looks cheap to the market). For this reason, we produce both a Market Model for the entire liquid universe and individual Sector Models (one each for the ten GICS sub-sectors). As discussed herein, we find that using both models together produces the best results. Notably, the Market Model is produced with and without CDS spreads as a factor. The Market Model (with CDS) has fewer companies. Due to the limited number of companies where CDS is liquid, we do not include CDS in the Sector Models, as it would too severely limit the number of model observations, in our view. Three different maturities The model is calculated for three different option maturities one month, three month, and six month for each for the Market and Sector Models. This allows for a discussion of relative value at different points along the volatility term structure, as well as the ability to account for differing sensitivities to each factor (e.g., 1M implied volatility may be more impacted by 1M realized, whereas 6M may be more impacted by 6M realized). LEH SEP HET GT TOL AMR FSL KBH AKS 16

Understanding the factors As noted above, we have included six factors in the model: Beta, Market Capitalization, Realized Volatility (1M & 12M), CDS Spread, Stock Return (1M and 12M) 1Y. These are among the strongest factors we have found among the many factors that are traditionally related to implied volatility. We discuss each for the liquid derivatives universe below. Realized volatility: Implied volatility is strongly related to both long term (12M) and short term (1M) realized volatility. Long term realized volatility gives investors a gauge of average volatility, whereas short term realized volatility reflects more recent trends and is generally more relevant for shorter term options. Exhibit 2: 6M Implied Volatility (y-axis) and 1M Realized Vol (x-axis) broad market liquid universe 60 0 0 60 70 80 Market Capitalization: Companies with larger market capitalizations generally have lower volatility expectations (Exhibit 3). We observe, however, that the relationship between market capitalization and implied volatility becomes highly convex for small and distressed companies with low market capitalizations. We find that the LN (market cap) has a more linear fit with implied volatility (Exhibit 4), and use this value in the model. Exhibit 3: 6M Implied Volatility (y-axis) and Market Capitalization in bn (x-axis) broad market liquid universe 60 Exhibit4: 6M Implied Volatility (y-axis) and LN Market Capitalization (x-axis) broad market liquid universe 60 0 0 0 0 0 21 23 25 27 Beta with S&P0: Beta measures stock price sensitivity to movements in the market. As such, higher volatility is often associated with higher betas. 17

Exhibit 5: 6M Implied Volatility (y-axis) and Stock Beta vs. S&P 0 (x-axis) broad market liquid universe 60 0 0 0.5 1 1.5 2 2.5 3 Stock Price Return: Implied volatility is typically inversely related to stock price returns, with negative price returns generally being associated with higher implied volatility, all else equal. One reason for this phenomenon may be the leverage effect. This theory states that as the stock falls, the market value of the equity typically falls faster than the market value of the debt, causing the debt/equity ratio to rise and the risk of the stock to increase. Another reason is a slide down the equity skew, whereby higher implied volatility may be observed at lower prices, even absent a parallel shift in the volatility curve. Additionally, we note that higher implied volatility is also observed for large upside stock returns. This may be due to fears of a reversal, or simply because large past returns are indicative of a more volatile stock. Because large stock price movements both up and down can increase implied volatility, we use the absolute value of stock price returns in the model (Exhibit 6). Exhibit 6: 6M Implied Volatility (y-axis) and stock price return (x-axis) broad market liquid universe 80 70 60 0-35 -25-15 -5 5 15 25 35 CDS Spread: CDS spread is a measure of credit risk. The higher the spread, the more premium a protection buyer is willing to pay for protection against a company's default, presumably due to a higher degree of risk that will also be reflected by implied volatility. We have chosen 5yr CDS as it is the most liquid of the tenures. We expect that increasing attention to cross-asset models that link CDS spreads to implied volatility may reinforce the theoretical correlation between the two markets. Among these models, the most straightforward and transparent include JPMorgan s Cross Asset Relative Performance Report, which publishes univariate regressions between the two markets daily. Other applications include Reduced-form models, which use the insight that both implied volatility for deep out-of-the-money options 18

and CDS spreads can be used to derive a probability of default. Finally, Structural models view corporate credit as the combination of a long risk-free security and short a put struck at the value of the firm s assets. As such, the credit value is contingent on asset volatility, which can be approximated using equity volatility. Exhibit 7: 6M Implied Volatility (y-axis) and 5 year CDS (x-axis) broad market liquid universe 60 0 0 0 0 0 0 0 600 700 800 Principal component analysis and factor selection To determine the key factors, we perform a three step process for each Market and Sector model. The process, commonly known Principle Component Analysis, includes 1) Linear regression and R-Square filters 2) Correlation matrix filter and 3) Multiple regression P-Value filter. For illustrative purpose, we outline the process for the 3M Market Model (with CDS). Step 1: Individual Linear Regression Our goal is to select factors that have a strong correlation with implied volatility, and where the directionality of the impact (i.e. whether it increases or decreases volatility) is stable over time. To accomplish this, we run a cross sectional linear regression for each factor with respect to implied volatility over a period of one year. The rolling 1yr R-squared of these regressions for each factor are shown in Exhibit 8. We can use this process to identify whether each factor has a significant correlation with implied volatility. Our rule is that if the average R-Square over the last year is less than 0.15, the factor will be deemed insignificant and be rejected. For example, although it may seem that Stock Return 1M is decreasing in significance, the average R-Square is still larger than 0.15. Exhibit 8 shows that all our factors were accepted after the first linear regression filter. 19

Exhibit 8: Cross Sectional Linear Regression R-Square for all factors 8A: R-Sq Hist_Vol_1M (Avg:0.58) 8B: R-Sq Hist_Vol_12M (0.78) 8C: R-Sq Beta (0.41) 8D: R-Sq Stock Return 1M (0.18) 8E: R-Sq Stock Return 12M (0.) 8F: R-Sq CDS Spread 5Y (0.44) 8G: R-Sq LN Market Cap (0.45), data as of 3rd October 06 In addition, we wish to determine whether the sign of each factor is the same over time. For example, if stock price returns were strongly correlated with implied volatility, but the direction of the impact changed over time (e.g. if high returns were sometimes associated with an increase in implied volatility and other times were associated with a decrease), this factor would not be especially helpful in predicting implied volatility, despite the high correlation. Exhibit 9 shows the t-statistic of each factor. The t-statistic measures the coefficient of the factor divided by its standard deviation. All of the t-statistics below are consistent with regard to sign, and are well above the threshold for determining statistical significance.

Exhibit 9: Cross Sectional t_stat for all factors 9A: T-Stat Hist_Vol_1M 9B: T-Stat Hist_Vol_12M 9C: T-Stat Beta 9D: T-Stat Stock Return 1M 9E: T-Stat Stock Return 12M 9F: T-Stat CDS Spread 5Y 9G: T-Stat LN Market Cap, data as of 3rd October 06 Step 2: Correlation Test Filter With the remaining factors, we implement a correlation matrix filter to avoid double counting similar or highly correlated data. We have defined a significant level of correlation between factors as having correlation of greater than 0.60. Looking at the pair wise correlation matrix of the factors (Exhibit ), the 12M Hist Vol is eliminated as it is highly correlated with two other factors: 1M Hist Vol and Beta. This process is repeated till there is no high pair-wise correlation between the remaining factors. In this case, no further factors need to be eliminated. Exhibit : Absolute Correlation Matrix for remaining factors after Linear Regression filter Hist_Vol_1M Hist_Vol_12M Beta Market_Cap Stock_Return_1 M Stock_Return_12 M CDS_5Y Hist_Vol_1M 1.00 0.79 0.60 0.54 0.48 0.44 0.46 Hist_Vol_12M 1.00 0.75 0.66 0.44 0.48 0.55 Beta 1.00 0.45 0.31 0.37 0.24 Market_Cap 1.00 0. 0.32 0.54 Stock_Return_1M 1.00 0.24 0.29 Stock_Return_12M 1.00 0.23 CDS_5Y 1.00, data as of 3rd October 06 21

Step 3: Multiple Regression Filter The third and last step is a Multiple Regression filter, which looks at the P-Value of each of the factors. In every regression filter, the factors with largest P-Value (and at least > 0.3) is eliminated. A factor with a high P-Value signifies little significance in the multiple-regression and does not compromise the goodness of fit of the reduced regression, even with its elimination. With six remaining factors, we perform a multiple-regression that, in this example, yields an R-Square of 0.80, and the following statistics for the respective factors (Exhibit 11). The one-month stock return factor has a P-Value of 0.324, and is eliminated. Exhibit 11: Coefficients statistic and P-Value of Multiple-Regression after Correlation filter (1Y Avg) Stdev of R-Sq=0.80 (1Y Avg) Coefficient Coefficient p-stat Constant 46.71 11.880 0.0 Hist_Vol_1M 0. 0.1 0.007 Beta 6.13 2.0 0.008 Market_Cap (1.) 0.4 0.019 Stock_Return_1M 0.04 0.174 0.324 Stock_Return_12M 0.02 0.036 0.227 CDS_5Y 0.02 0.004 0.000, data as of 3rd October 06 Another multiple regression is then performed with the remaining five factors. This time none of the factors are eliminated based on the P-Value threshold of 0.. Resulting factor model for 3M implied volatility, Market Model The three step process ends with the elimination of three factors, with the remaining five factors as shown in Exhibit 12 below. Exhibit 12: Coefficient and P-statistics of multiple regression R-Sq=0.797 Coefficient p-stat Constant 62.76 0.0 Hist_Vol_1M 0.13 0.004 Beta 9.29 0.009 Market_Cap -2.07 0.017 Stock_Return_12M -5.98 0.2 CDS_5Y 0.022 0.000, data as of 3rd October 06 This multi-step process for factor selection and coefficient designation is performed for each model on a quarterly basis. This rebalancing process allows the model to adjust for factors and factor weights that have become more or less important as the market changes. In between the rebalancing periods, the model is updated daily using close of business pricing. Reading the report The report is structured into three main sections Section 1: Summary A one page summary of the top 15 richest and cheapest implied volatility in the liquid equity derivatives universe ranked by the Sector Model z-score. Companies on this list must also have Market Model z-score that agrees with (is the same sign as) the Sector Model. Exhibit 13 shows an excerpt of the summary table for rich 3M implied volatility. 22

Exhibit 13: Summary of the richest 3M Implied volatility If the model volatility is lower than the actual volatility, actual volatility appears rich and the z-score will be positive. The Sector Model is telling us that the actual implied volatility is 2.22 standard deviations too high vs. the Consumer Cyclical model volatility of 34.2%. Additionally, the Market Model also calculates that RAD implied volatility is expensive, at 1.23 standard deviations too high versus the model implied volatility of 33.7%. Section 2: Sector Model and Market Model results This section has the look and feel of Exhibit 2 above, but includes separate pages with the results for all companies in each of the Sector Models and the Market Model. Sector Model results are ranked by the Sector Model z-score, and also show the results of the Market Model (no CDS). Market Model (with CDS) is shown separately and also include the results of the Market Model (no CDS). Section 3: Factors and coefficients A table describing the statistically significant factors and their coefficient for each model is located at the end of the daily report (Exhibit 9). This allows us to understand the sensitivities of implied volatility with respect to the individual variables for each member of the models. These tables will be updated each quarter. 23

Exhibit 14: List of critical factors and respective Coefficient, data as of 3rd October 06 How does the model perform? We performed a series of backtests to assess the performance of the model across the volatility term structure. Specifically, we compare the results across several Cross Sectional Models with the results of a simple model that buys or sells volatility based upon its relationship to historical volatility only. This simple model is used as a base case, and represents a process that is frequently considered by investors in buying and selling volatility, despite its lack of significance in backtests. The five models, and the rules for entering into a trade under each, are outlined below. Market Model (no CDS): Regression of all companies with liquid options. CDS is not used so the universe is larger but the model fit is slightly worse. Buy volatility when z-score > 2.0, sell volatility when z-score < -2.0. Market Model (w/ CDS): Regression of all companies with both liquid options and liquid CDS. The inclusion of CDS reduces the number of companies in the universe. Buy volatility when z-score > 2.0, sell volatility when z-score < -2.0. Sector Model: separate regressions, one for each GICS sector. CDS is not included. Buy volatility when z-score > 2.0, sell volatility when z-score < -2.0. Sector Model + Market Model (no CDS): As above, but enter trade when z- score > 2.0 or < -2.0 for both the Sector Model and Market Model. Simple Model: Regression universe is the same as the Market Model. Buy volatility when implied volatility is less than realized volatility and sell volatility when implied volatility is greater than realized volatility. All illustrated trades are performed assuming interpolated at-the-money options, delta hedged using closing prices with no drift. Trades are entered each day over the last three years ending April 06, are held to maturity, and assume a bid offer 24

spread of one vol. The return percentages listed treat volatility as a security and simply observe its return (i.e. it is not a variance return). Returns are non-annualized. Exhibit 15: Buy and sell volatility: Backtest results for five different models Implied Vol Avg Ret Stdev Ret Info Ratio Ret Total Trades % Good Avg Vega Stdev Vega Info Ratio Vega Model 1M 0.3% 31.8% 0.01 189,472 51.3% 0.0 0.12 0.04 Simple 1M 12.6% 39.9% 0.31 614 64.8% 5.7 22.34 0.25 Market (w/ CDS) 1M 12.2% 35.4% 0.34 3,783 69.7% 8.9 24.47 0.36 Market (no CDS) 1M 11.8% 31.7% 0.37 3,4 68.6% 8.4 22.53 0.37 Sector 1M 15.1% 33.7% 0.45 2,163 73.0% 11.2 26.43 0.42 Sector + Market (no CDS) 3M 0.4% 24.4% 0.02 191,571 43.0% 0.0 0.09 0.04 Simple 3M 6.7% 23.2% 0.29 879 65.1% 3.1 13.12 0.24 Market (w/ CDS) 3M 13.3% 26.8% 0. 3,698 71.7% 9.6 19.82 0.48 Market (no CDS) 3M 11.0% 24.6% 0.45 3,236 68.4% 8.5 19.57 0.44 Sector 3M 14.3% 25.4% 0.56 2,099 72.9% 11.4 22.63 0. Sector + Market (no CDS) 6M 0.5% 22.7% 0.02 191,571 44.6% 0.0 0.09 0.03 Simple 6M 13.4% 21.4% 0.63 1,913 76.3% 7.3 11.09 0.66 Market (w/ CDS) 6M 9.7% 24.0% 0.41 2,283 71.7% 6.6 16.85 0.39 Market (no CDS) 6M 9.4% 21.1% 0.44 1,987 70.7% 6.5 16.06 0. Sector 6M.4% 21.8% 0.48 1,470 73.5% 7.6 17.98 0.42 Sector + Market (no CDS) We can reach several conclusions based on Exhibit 15: Both Market and Sector Models outperform the simple strategy. In terms of information ratio, the simple strategy significantly underperforms for both 1M and 3M trading strategies. The simple strategy does a bit better for 6M strategies, having the second highest information ratio, but also has the lowest average return. The Sector Model + Market Model outperforms the Sector Model alone. For all maturities, requiring a joint "signal" from both the Market Model and Sector Model to enter a trade increases average returns and increases the information ratio. This additional step does, however, reduce the number of trades over the period by about a third. Adding CDS to the Market Model is better for longer dated options. Relative to the Market Model, we find no clear benefit from including CDS for valuing shorter dated options, as average returns and information ratios are lower for both 1M and 3M strategies. 6M strategies, however, outperform in terms of both risk and return when CDS is used in the model. This makes sense to us given that 5yr CDS is used, which has a stronger theoretical link to longer dated options. Average returns for all models is positive. While this is clearly a good thing, it is difficult to draw any conclusions, as the absolute level of returns is driven by the market environment as well as the model output. For example, the backtest period experienced low volatility and selling volatility was generally profitable. We cannot ascertain from Exhibit 15 whether the profits we observe are due to judicious trade selection or a higher tendency to sell volatility. We also can make no claims about the ability of the model to identify buying versus selling opportunities. For this reason, we decompose the results into sell volatility (Exhibit 16) and buy volatility (Exhibit 17) below. 25

The Cross Sectional Models perform better for both selling and buying volatility. All models improve average returns versus the simple strategy for selling and buying volatility, and also improve the risk/return in most cases. In a market environment when selling volatility is generally profitable, we should bring our focus to the results to the relatively stronger performance of the model to the simple strategy, Exhibit 16: Sell volatility only: Backtest results for five different models Implied Vol Avg Ret Stdev Ret Sharpe Ret Total Trades % Good Avg Vega Stdev Vega Sharpe Vega Model 1M 3.61% 32.31% 0.11 126,671 64.6% 0.0 0.12 0.13 Simple 1M 16.0% 34.4% 0.47 4 74.4% 6.6 23.19 0.28 Market (w CDS) 1M 14.5% 34.1% 0.43 3,255 75.2%.1 25.19 0. Market (no CDS) 1M 14.7%.9% 0.48 2,9 74.9% 9.8 23.44 0.42 Sector 1M 16.4% 32.6% 0. 2,044 76.0% 11.9 26.48 0.45 Sector + Market (no CDS) 3M 2.9% 23.8% 0.12 132,846 48.5% 0.0 0.09 0.16 Simple 3M 8.9%.5% 0.43 676 69.8% 3.5 13.69 0.25 Market (w CDS) 3M 14.4% 24.4% 0.59 3,167 75.7%.5.44 0.51 Market (no CDS) 3M 13.3% 23.8% 0.56 2,731 74.0% 9.8.51 0.48 Sector 3M 14.9% 24.3% 0.62 1,956 75.1% 11.9 22.83 0.52 Sector + Market (no CDS) 6M 2.1% 22.7% 0.09 131,3.9% 0.0 0.09 0.11 Simple 6M 16.6% 19.6% 0.85 1,484 83.7% 8.8 11.45 0.76 Market (w CDS) 6M 8.8% 21.7% 0.41 1,988 72.4% 6.3 17.13 0.37 Market (no CDS) 6M 9.8%.3% 0.48 1,721 72.9% 6.8 16.67 0.41 Sector 6M.1%.8% 0.49 1,362 74.2% 7.6 18.17 0.42 Sector + Market (no CDS) Exhibit 17: Buy volatility only: Backtest results for five different models Implied Vol Avg Ret Stdev Ret Sharpe Ret Total Trades % Good Avg Vega Stdev Vega Sharpe Vega Model 1M -11.1% 32.3% (0.34) 62,801 25.4% (0.0) 0.12 (0.29) Simple 1M 4.4% 49.7% 0.09 184 42.4% 3.6.09 0.18 Market (w CDS) 1M -2.2% 39.5% (0.06) 528 35.2% 1.3 17.65 0.07 Market (no CDS) 1M -5.9%.4% (0.19) 490.8% (0.2) 13.12 (0.01) Sector 1M -7.2% 43.4% (0.16) 119.2% (0.8) 22.60 (0.03) Sector + Market (no CDS) 3M -4.6% 24.8% (0.18) 58,725.5% (0.0) 0.09 (0.17) Simple 3M -0.6% 29.3% (0.02) 3 49.3% 2.0.98 0.19 Market (w CDS) 3M 6.8% 37.7% 0.18 531 47.6% 4.3 14.55 0.29 Market (no CDS) 3M -1.1% 25.3% (0.04) 5 38.4% 1.9 11.25 0.16 Sector 3M 6.0% 36.6% 0.17 143 43.4% 4.9 18.63 0.26 Sector + Market (no CDS) 6M -3.12% 22.06% (0.14) 60,268.8% (0.0) 0.08 (0.15) Simple 6M 2.2% 23.5% 0. 429.6% 2.3 7.88 0.29 Market (w CDS) 6M 15.9% 35.1% 0.45 295 67.1% 8.0 14.77 0.54 Market (no CDS) 6M 7.0% 25.8% 0.27 266 56.8% 4.6 11.15 0.41 Sector 6M 14.3% 32.4% 0.44 8 64.8% 8.0 15.51 0.52 Sector + Market (no CDS) 26

Risks to Strategies Option Buyer. Options are a decaying asset, and investors risk losing 0% of the premium paid. Call Option Sale. Investors who sell uncovered call options have exposure on the upside that is theoretically unlimited. Investors who sell calls but also own the underlying stock will have limited losses should the stock rally. Put Sale. Investors who sell put options will own the underlying stock if the stock price falls below the strike price of the put option. Investors, therefore, will be exposed to any decline in the stock price below the strike potentially to zero, and they will not participate in any stock appreciation if the option expires unexercised. Call Overwrite or Buywrite. Investors who sell call options against a long position in the underlying stock give up any appreciation in the stock price above the strike price of the call option, and they remain exposed to the downside of the underlying stock in return for receipt of the option premium. 27