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Presentation for QWAFAFEW, June 8, 2011 The Impact of the Federal Reserve's Interest Rate Target Announcement on Stock Prices: A Closer Look at How the Market Impounds New Information Justin Birru and Stephen Figlewski* New York University Stern School of Business

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The Classic Event Study Market Response Plot Price Time News release Figlewski QWAFAFEW Presentation June 20112

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The Classic Event Study Market Response Plot, with Information Leakage Price Time News release 3Figlewski QWAFAFEW Presentation June 2011

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Figure 1: Intraday Behavior of the Forward Value of the S&P Index on Dec. 11, Figlewski QWAFAFEW Presentation June 2011

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Table 1: Federal Reserve Interest Rate Target Announcements 5Figlewski QWAFAFEW Presentation June 2011

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Hypotheses and Questions about the Impact of the Fed Announcement Is the market's response to the announcement unbiased? Does the information in the announcement all enter the market at the moment of the public announcement? Information leakage beforehand? Sluggish adjustment or overshooting afterward? The mean of the Risk Neutral Density (RND) and the forward value of the spot S&P index are tied together by arbitrage. Do they behave identically on announcement day? Is one a better prediction of the announcement than the other? The RND reveals the market's risk neutral expectation for the future value of the index and also the variance around that expectation, a direct measure of uncertainty. How much uncertainty does a Fed announcement resolve on average? Does it matter if the announcement is viewed as positive or negative by the market? 6Figlewski QWAFAFEW Presentation June 2011

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Hypotheses and Questions about Informational Efficiency Information flow within a time interval can be proxied by the standard deviation of price change over the interval or by volatility within the interval. What is information flow like over an announcement day? How much information does the announcement itself convey? How is the day's price change distributed within the day? Is there a distinct period of re-equilibration after the announcement? How does an announcement day compare to a regular day? Information flow is assumed to be serially independent, so price changes in an efficient market should have zero autocorrelation. Does this hold for the forward index level? Does it hold for the RND mean? 7Figlewski QWAFAFEW Presentation June 2011

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Answers Using the Risk Neutral Density for the S&P 500 We address these questions by exploring the behavior of the risk neutral probability density for the market portfolio on days when the Federal Reserve announces its interest rate target. Data sample: Option Price Reporting Authority (OPRA) National Best Bid and Offer data provides a continuous synchronized record of price movements in the markets for all equity and equity index options and their underlyings S&P 500 index calls and puts, with maturities in March, June, September, and December, 28 announcement days 2005 – 2008 Puts and calls are combined to cover the range of traded exercise prices: Out of the money puts for low X; out of the money calls for high X; a weighted combination of calls and puts for X in the middle range. Options with bid prices less than $0.50 were discarded 8Figlewski QWAFAFEW Presentation June 2011

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The Risk Neutral Probability Distribution for the S&P 500 Index The market price of a risky asset depends on the market's assessment of the probability distribution for its future payoff together with the market's risk aversion. Neither of these can be observed directly, but the two can be combined and expressed as the "Risk Neutral Probability Density" (RND). Breeden and Litzenberger (Journal of Business, 1978) showed how the risk neutral probability distribution for S T, the value of the underlying asset on option expiration day can be extracted from a set of market option prices. Two major problems in constructing a complete risk neutral density from a set of market option prices: 1. How to smooth and interpolate option prices to limit pricing noise and produce a smooth density 2. How to extend the distribution to the tails beyond the range of traded strike prices. But one terrific advantage is that unlike implied volatility, the risk neutral density is model- free. 9Figlewski QWAFAFEW Presentation June 2011

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Extracting the Risk Neutral Density from Options Prices in Theory The value of a call option is the expected value under the risk neutral distribution of its payoff on the expiration date T, discounted back to the present. Taking the partial derivative in (1) with respect to X, 10Figlewski QWAFAFEW Presentation June 2011

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Extracting the Risk Neutral Density from Options Prices in Theory We will use three options with sequential strike prices X n-1, X n, and X n+1 in order to obtain an approximation to F(X) centered on X n. Taking the derivative with respect to X a second time yields the risk neutral density function at X. which is approximated by 11Figlewski QWAFAFEW Presentation June 2011

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Extracting the Risk Neutral Density, a more intuitive approach Consider a call option that allows you to buy a share of some underlying stock for a price of 101 one month from now. If the stock price in one month is above 101, you will exercise the option. The market price for this option is There is a second call option that allows you to buy 1 share of the same stock for a price of 100 in one month. The market price for Option 2 is For every stock price above 101, the second option pays 1 more than the first option. The market values that extra 1 that option 2 pays if the stock price is above 101 as being worth 5.70 – 5.00 = So (roughly speaking) the market is saying the probability the stock price will be above 101 is 70%. 12 Stock price in 1 month Option 1 value Option 2 value Figlewski QWAFAFEW Presentation June 2011

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S&P 500 Index Options Prices Jan. 5, 2005 S&P close Option expiration 3/18/05 At the money 13Figlewski QWAFAFEW Presentation June 2011

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Extracting the Risk Neutral Density from Options Prices in Practice Obtaining a well-behaved risk neutral density from market option prices is a nontrivial exercise. Here are the main steps we follow. 1. Use bid and ask quotes, rather than transactions prices. Eliminate options too far in or out of the money. 2. Construct a smooth curve in strike-implied volatility space 3. Interpolate the IVs using a 4th degree smoothing spline 4. Fit the spline to the bid-ask spread 5. Use out of the money calls, out of the money puts, and a blend of the two at the money 6. Convert the interpolated IV curve back to option prices and extract the middle portion of the risk neutral density 7. Append tails to the Risk Neutral Density from a Generalized Extreme Value distribution 14Figlewski QWAFAFEW Presentation June 2011

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S&P 500 Index Options Prices Jan. 5, 2005 S&P close Option expiration 3/18/05 At the money 15 Not used Blended Figlewski QWAFAFEW Presentation June 2011

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4th Degree Smoothing Spline with Bid-Ask Spread Adjustment 21Figlewski QWAFAFEW Presentation June 2011

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The Generalized Extreme Value (GEV) Distribution We complete the risk neutral density by adding tails from a GEV density. The GEV distribution has three parameters: μ = location σ = scale ξ = tail shape If ξ > 0, the GEV is a Fréchet distribution, that has a heavier tail than the normal distribution; ξ = 0, the GEV is a Gumbel distribution with tails like the normal; ξ < 0, the density is a Weibull, with a finite tail that does not extend to infinity. 23Figlewski QWAFAFEW Presentation June 2011

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95% 98% 24Figlewski QWAFAFEW Presentation June 2011

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95% 98% 25Figlewski QWAFAFEW Presentation June 2011

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The Risk Neutral Density in a Black-Scholes World An Aside: The RND under Black-Scholes Assumptions Risk neutral valuation was first developed in the context of the Black-Scholes model. The returns process is modeled as: The empirical distribution of the date T asset price is lognormal. Risk neutralization simply replaces the empirical drift μ with the riskless rate r: The risk neutral distribution is still lognormal with the same volatility. It is simply shifted to the left, and only the mean changes. Is the curve on the previous slide lognormal? No! It can't be, because it is skewed to the left while the lognormal is skewed to the right. 26Figlewski QWAFAFEW Presentation June 2011

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Figure 2: Initial Impact of the Fed Announcement on the Risk Neutral Density, Dec. 11, Figlewski QWAFAFEW Presentation June 2011

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Figure 3: Re-equilibration of the Stock Market after the Fed Announcement, Dec. 11, Figlewski QWAFAFEW Presentation June 2011

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Information Flow and the Resolution of Uncertainty New information entering the market will produce a change in the market price. The size of the price move measures the flow of new information during a given time interval. We estimate the rate of information flow over different periods of the day by the standard deviation of interval price changes across announcement dates. The RND mean is the market's expectation for the level of the S&P index on option expiration day, and the RND variance is the market's uncertainty about S(T), given S(t). The risk neutral variance measures how much new information the market anticipates will arrive over the time remaining to expiration. If there are T days to expiration, on average 1/T of the uncertainty should be resolved each day and the RND variance is should fall by the fraction 1/T per day. Research shows that this relationship holds very closely in the data. We look at the change in RND variance to measure how much the market's uncertainty is resolved by the information contained in the Fed's announcement 29Figlewski QWAFAFEW Presentation June 2011

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Excerpt from Table 2 Levels and Changes of Key Variables on Fed Announcement Days 30Figlewski QWAFAFEW Presentation June 2011

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Table 6: Volatility of the Forward S&P Index and the RND Mean during the Re-Equilibration Period, by Sub-Intervals 31Figlewski QWAFAFEW Presentation June 2011

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Variance Diminishes Gradually During Re-equilibration To explore the time decay of volatility further, we regress the log of the absolute price change in each minute relative to the volatility within the impact period, as a function of the number of minutes since the end of the impact period. is a minute within date t, with 0 representing 2:24 P.M. Ft,impact is the standard deviation of 1-minute changes in the index forward during the impact period on date t. t-statistics are shown in parentheses. NOBS = 2649 (15) log( | F t,τ | / Ft,impact ) = log( - 0 ) R 2 = (-4.84) (-8.82) 32Figlewski QWAFAFEW Presentation June 2011

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Variance Diminishes Gradually During Re-equilibration Running this regression with the absolute change in the RND mean gives log( | RNDmean t, | / RNDt,impact ) = log( - 0 ) (-7.99) (-7.23) R 2 = The RND variance also shrinks consistently during the re-equilibration period log( RNDvar t, / RNDvar t,impact ) = log( - 0 ) (-2.92) (-5.39) R 2 = Figlewski QWAFAFEW Presentation June 2011

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The Evolution of the RND on December 11, 2007 Movie Time On December 11, 2007 the Federal Reserve announced that it was lowering its interest rate target by 25 basis points. Normally a cut in the Fed funds rate causes the stock market to rise, but this time the market was clearly disappointed that the cut was not larger. Although the market had drifted higher during the day before the announcement, the S&P index fell 27 points (-1.74%) in the next 10 minutes, and a further 18 points by the end of the day. This video shows how the Risk Neutral probability density behaved minute by minute during the course of that day. The density is for the level of the stock market on option expiration day, March 21, The vertical green line shows the current forward level of the S&P index in the market at the same time. 34Figlewski QWAFAFEW Presentation June 2011

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Table 8: Autocorrelation in Intraday Index Option Mid-Quote Changes on Non-Announcement Days 35Figlewski QWAFAFEW Presentation June 2011

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Simulation of Option Quotes when Stock Price Follows a Diffusion and Tick Sizes are Different 36Figlewski QWAFAFEW Presentation June 2011

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Conclusions about the Impact of the Fed Announcement Is the market's response to the announcement unbiased? Does the information in the announcement all enter the market at the moment of the public announcement? Information leakage beforehand? Sluggish adjustment or overshooting afterward? The mean of the Risk Neutral Density (RND) and the forward value of the spot S&P index are tied together by arbitrage. Do they behave identically on announcement day? Is one a better prediction of the announcement than the other? The RND reveals the market's risk neutral expectation for the future value of the index and also the variance around that expectation, a direct measure of uncertainty. How much uncertainty does a Fed announcement resolve on average? Does it matter if the announcement is viewed as positive or negative by the market? YES, IT APPEARS TO BE MAYBE SOME LEAKAGE THERE IS A LOT OF FURTHER ADJUSTMENT DURING RE-EQUILIBRATION AFTER THE ANNOUNCEMENT, BUT NO APPARENT BIAS NOT QUITE IDENTICAL BUT NEITHER SHOWS PREDICTIVE ABILITY ABOUT THE SAME AS ON 5 ORDINARY DAYS YES. MUCH LESS UNCERTAINTY IS RESOLVED WHEN THE MARKET FALLS AFTER THE ANNOUNCEMENT 37Figlewski QWAFAFEW Presentation June 2011

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Conclusions about the Impact of the Fed Announcement, p.2 Information flow within a time interval can be proxied by the standard deviation of price change over the interval or by volatility within the interval. What is information flow like over an announcement day? How much information does the announcement itself convey? STANDARD DEVIATION OVER 10 MINUTES ~ 30-40% OF FULL DAY PRICE CHANGE How much of the day's price change occurs before the market opens? SURPRISINGLY, ABOUT HALF How much during trading hours before the announcement? IN TOTAL ABOUT 1/3 LESS THAN IN THE 10 MINUTES OF IMPACT How much in the re-equilibration period following the initial impact of the announcement? A LOT! 60 – 70% OF FULL DAY MOVE Is there a time pattern of diminishing volatility during re-equilibration? YES, AND VOLATILITY DROPS MORE FOR THE FORWARD THAN THE RND How does an announcement day compare to a regular day? THE TIME PATTERN IS MUCH DIFFERENT 38Figlewski QWAFAFEW Presentation June 2011

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