Investors and Central Bank s Uncertainty Embedded in Index Options On-Line Appendix

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1 Investors and Central Bank s Uncertainty Embedded in Index Options On-Line Appendix Alexander David Haskayne School of Business, University of Calgary Pietro Veronesi University of Chicago Booth School of Business, CEPR, and NBER December 2, 213 This online appendix contains additional material that is excluded from the main article. In particular, we cover the following topics: Section 1: Estimation without Options Section 2: Out-of-Sample Fit Section 3: ATM Implied Volatility versus VIX Section 4: 5% P/C Index versus 1% P/C Index Section 5: Alternative Measures of Money Growth Section 6: Moments of SMM Procedure 1. Estimation without Options We believe that options include important information about fundamentals, and thus using them in the estimation of the regime switching model as in the main article help inference for otherwise hard-to-detect regimes. As discussed in David and Veronesi (213) and in Section 3.2 of the paper, Peso Problem situations can effectively be dealt with in our framework as asset prices depend on expectations of regimes that may occur, but might not actually occur in sample on an ex-post basis. For instance, the potential of a deflationary regime may push both bond yields and stock prices down, even if ex post no such regime

2 actually occurred. Our estimation methodology that combines the maximum likelihood estimation using only fundamentals with pricing errors from asset prices (in this sequence, that is, first compute the likelihood using only fundamental shocks, and then, given the extracted beliefs, check pricing errors) ensures that all of the state variables are driven by fundamental shocks, but still elicit information from asset prices in what type of potential composite regimes can actually occur. Nonetheless, it is an interesting question on what we gain from using options, as opposed only stock and bond prices. David and Veronesi (213) only used stocks and bonds to detect six composite economic regimes for inflation and real earnings growth. 1 In this section, we use only stocks and bonds to estimate the eight composite regimes put forward in the main body of the paper. Table 1 contains the parameter estimates. The inflation and earnings regimes are similar to those estimated in Table 2 in the paper. However, some noticeable differences are visible in the composite regimes for Capacity Utilization, Money Growth, and the Taylor Rule. Figures 1 through 5 report the model s fit. We can notice on the right panels of Figure 5 that the model s fit for options, which were not used in this estimation, deteriorated a little bit compared to the estimates in the paper. This empirical result suggests that options contain useful information to detect especially the monetary regimes. Table 2 reports the model s fit over the sample. As it can be seen, all of the variables are strongly significant, on their respective samples, including the stock and bond ATMIV and Put-to-Call ratio, that were not used in this estimation. However, even if stock and bond ATMIVs and Putto-Call ratio s fits are similar to the paper, their average level is a bit off, as it can be seen from the right-hand-panels of Figure 5. In particular, we notice that the OTM Put-to-Call implied volatility ratio is lower in the than in the data. Intuitively, because the shocks driving the model s beliefs are the same in the two estimations, the dynamics of prices is similar across the two models. However, when we estimate using options we can also ensure a better fit of the level of asset prices. Interestingly, we also notice a bit of difference in the beliefs about capacity utilization between model and the Bloomberg probabilities extracted from the dispersion of analyst forecasts (see discussion in paper). Given that capacity utilization is the main driver of monetary policy in our paper, this evidence suggests that indeed using options (as in the paper) leads to a better assessment also monetary policy. 2. Out-of-Sample Fit The second robustness check that we perform is to estimate the model using again options data, but only using the first half of the option s sample. We can then check how the model fits the data in the second half of the sample. Table 3 reports the parameter estimates, which are similar to the estimates that use the full sample. Panel A of Table 4 reports the fit of fundamentals and prices on models counterparts for the full sample up to 211 (including 1 The on-line Appendix of David and Veronesi (213) shows the impact of using a lower number of regimes on the fit of the model. In fact, even with four regimes they show that the fit is so poor that no economic inference can be made, meaning that it is not possible to provide an economic explanation for the main message of their paper. 2

3 also the estimation period). Panel B reports the fit of the model purely out-of-sample, that is, on the sample not used in the estimation. The model performs well, although clearly t-statistics and R 2 are a bit lower. Figures 6 through 9 plot the model fits on the overall sample. 3. ATMIV versus VIX In this subsection we compare two measures at-the-money implied volatility. The first measure is Black-Scholes at-the-money implied volatility (ATMIV) obtained by interpolation of nearly option prices, which we have used in the estimation of our model. The second is a model-free measure of expected forward volatility under the risk-neutral measure from options with about three months to maturity similar to the popularly traded VIX index, which uses 1 month options. We will refer to this latter measure as 3-Month VIX. In this approach the variance contract price estimated from out-of-the-money puts and calls is calculated as: 2(1 log[ K V (t, τ) = ]) S(t) S(t) 2(1 log[ S(t) C(t, τ, K)dK + ]) K P(t, τ, K)dK. (B1) S(t) K 2 K 2 For more details of this calculation see e.g. CBOE Bulletin on VIX, 23, and among other authors by Britten-Jones and Neuberger (2) and Bakshi, Kapadia, and Madan (23). One essential difference is that the ATMIV uses information in essentially close to the being at-the-money options, while the 3-Month VIX uses information in all traded options, albeit with declining weights for options further out-of-the-money. The time series of the two variables in Figure 1 shows that even though the two series are very highly correlated (correlation coefficient of 98%), the gap between the two series has increased in recent years. In the latter part of the sample, there is in increase in the trading of options that are further away from the money, so undoubtedly, in the 3-Month VIX such options have had a larger influence on the index. In particular, in the early part of our sample ( ), obtained from the CBOE, there were very few options traded more that 5% out-of-the-money, and indeed, the two measures were very close. The reason we calibrated our model using ATMIV is that its model counterpart is faster to compute, using only the prices of at-the-money puts and calls at each date, while a model based 3-Month VIX will require computing at each date a different set of options prices to match those in the data (for the latter part of the sample there are 3-5 strikes per quarter). 4. 5% P/C Index versus 1% P/C Index As discussed in the main body of our paper, the 5% OTM put-call ratio is generally pro-cyclical. Here we address the question as to whether the analogous ratio for deeper outof-the-money options has similar cyclical properties, and in particular look at the 1% OTM options. One issue that we face is that in the first part of our sample, with data provided by the CBOE, there were many quarters where deeper OTM strikes were not traded, and extrapolating the traded prices led to a very volatile and likely unreliable time series. We therefore, limited our analysis to the second part of our sample, where the data are provide by Optionmetrics. The time series of the 5% and 1% OTM P/C indices are plotted in Figure 3

4 11, and as evident from the figure, the two series are highly correlated, with a correlation coefficient 76%. In particular, in both recessions in this shorter sample, the P/C ratios have fallen from their recent levels, while the ratios were high in periods of stronger economic conditions (such as 1997 and 21). Overall, considering deeper OTM options does not change our main finding that P/C indices are generally pro-cyclical. 5. Alternative Measures of Money Growth In this subsection we compare some alternative money growth measures. In our analysis of the body of the paper we use the real growth in M1, and we provide some comments for this choice. The top panel of Figure 12 shows the growth in nominal and real M1. While the two series are highly correlated (correlation coefficient of nearly 86 percent), it is evident that the real growth series displays greater variation in the first half of our sample, when inflation was generally higher than in the second half of the sample. Indeed, average nominal M1 growth did not display the sharp declines in the first three recessions of our sample that were evident in real money growth. The bottom panel of Figure 12 shows the growth in real M1 and M2. It is useful to note that M2 is a broader measure of money supply mainly due to the inclusion of savings deposits, which are not part of M1. The two series are highly correlated (correlation coefficient of nearly 55 percent) however, there are some important differences in the 199s. In particular, real M1 growth was strong following the 1991 recession, while real M2 growth was nearly flat. In the late 199s period of very rapid earnings growth, real M1 growth was tight, but real M2 growth was quite stable. Overall, as a signal of real economic activity, it appears real M1 growth is a better measure. On a similar note, Buraschi and Jiltsov (27) use M1 growth to fit the term structure of interest rates. 6. Moments of the SMM Procedure Table 5 reports the moments from the SMM procedure, described ( in) the appendix. Recall that our estimation procedure computes moments as ε(t) = e(t), ˆL, where e(t) collects Ψ the differences between data observed financial quantities and their model s counterpart, which in turn depend on probabilities. The second term ˆL are the scores of the likelihood Ψ function. Details of the procedure are in the appendix of the paper. 4

5 References Bakshi, G., N.Kapadia, and D.Madan, 23, Stock Return Characteristics, Skew Laws, and the Differential Pricing of Individual Equity Options, Review of Financial Studies, 16, Britten-Jones, M., and A.Neuberger, 2, Option Prices, Implied Price Processes, and Stochastic Volatility, Journal of Finance, 55(2), David, A., and P. Veronesi, 213, What Ties Return Volatilities to Price Valuations and Fundamentals?, Journal of Political Economy, 121(4),

6 Table 1: Parameter Estimates: Estimation without Options Composite Regimes Conditional P/E and Rates Infl Earn CapUt Money P/E Conditional Rates (%) Regime Name # β (%) θ (%) ρ (%) w(%) C i r n i y i (1) y i (1) Regular Boom (.3) (3.5) (.9) (.14) Regular Recession (.9) (2.92) (.9) (1.37) Overheating Boom (.9) (3.5) (.12) (.14) Stagflation (.23) (2.92) (.12) (2.75) New Economy Growth (.3) (2.68) (.9) (.53) Deflation (.1) (2.92) (.97) (1.37) Low Capacity Boom (.3) (3.5) (.97) (1.37) Deep Recession (.9) (2.95) (.97) (.53) Diffusion Matrix(%) Jump (%) Taylor Rule ( 1) Inflation κ 5.4 α 1.79 (.2)) (.21) (.7) Earnings µ α β (.48) (.2) (.998) Kernel σ α ρ (2.95) (1.83) (.318) (1.37) Capacity Utilization µ (.1) (.6) (8.593) Money σ (.1) (17.98) Infinitesimal Generator Regime Estimates 1 j λ 1j λ 4 λ 6 λ 2 λ 1 λ 3 λ (.141) 2 λ 4 j λ 2j λ 3 λ 5 λ 1 λ 2 λ 2 λ (.158) 3 λ 6 λ 2 j λ 3j λ 2 λ 1 λ 3 λ 2 λ (.643) 4 λ 4 λ 5 λ 4 j λ 4j λ 2 λ (6.293) 5 λ 2 j λ 5j λ 1 λ (3.9) 6 λ 3 λ 3 λ 2 j λ 6j λ 3 λ (14.18) 7 λ 4 λ 3 j λ 7j λ 3 8 λ 4 λ 3 λ 2 λ 2 λ 3 λ 4 j λ 8j Notes: Simulated Methods of Moments (SMM) estimates of the regime-switching model s parameters. The methodology combines the scores of the (simulated) likelihood function from fundamentals (inflation, real earnings, capacity utilization, and money growth) with pricing errors from financial variables (S&P5 index P/E ratio, 3-months Treasury Rate, Treasury Slope, and the volatilities of stock, the five-year, and ten-year bonds). The last four columns of top panel also report the conditional P/E ratios and conditional yields across the eight composite regimes. The data sample 6 is , except for options, whose sample is Newey-West adjusted standard errors are in parenthesis.

7 Table 2: Fit and Data: Estimation without Options. α β t(α) t(β) R 2 Inflation Real Earnings Capacity Utilization Money Growth P/E Ratio month Treasury Rate Term Structure Slope Stock Implied Volatility Put-to-Call Ratio Bond Implied Volatility Stock Realized Volatility y Bond Realzed Volatility y Bnd Realzed Volatility Notes: Results of the regressions (Fundamentals) t = b + b 1 E t [Fundamentals] + ɛ t (Financial Variable) Data t = b + b 1 (Financial Variable) t + ɛ t where Fundamentals is either inflation, real earnings growth, capacity utilization, or money growth, and financial variables are identified in each row. In these regressions, both expected fundamentals and modelimplied financial variables are conditional on the fitted beliefs. The sample is , except for optionbased quantities, whose sample is All t-statistics are Newey-West adjusted for heteroskedasticity and autocorrelation using four lags. 7

8 Table 3: Parameter Estimates: Sample Composite Regimes Conditional P/E and Rates Infl Earn CapUt Money P/E Conditional Rates (%) Regime Name # β (%) θ (%) ρ (%) w(%) C i r n i y i (1) y i (1) Regular Boom (.63) (.56) (.7) (.2) Regular Recession (.5) (1.78) (.7) (.31) Overheating Boom (.5) (.56) (.6) (.2) Stagflation (.7) (1.78) (.6) (4.42) New Economy Growth (.63) (1.3) (.7) (.8) Deflation (.1) (1.78) (.3) (.31) Low Capacity Boom (.63) (.56) (.3) (.31) Deep Recession (.5) (1.78) (.3) (4.42) Diffusion Matrix(%) Jump (%) Taylor Rule ( 1) Inflation κ 8.29 α 1.47 (.3)) (.1) (.89) Earnings µ α β (1.16) (.3) (2.189) Kernel σ α ρ (1.4) (6.74) (1.67) (.97) Capacity Utilization µ (.5) (.6) (8.593) Money σ (.3) (6.14) Infinitesimal Generator Regime Estimates 1 j λ 1j λ 4 λ 6 λ 2 λ 1 λ 3 λ 1.53 (.13) 2 λ 4 j λ 2j λ 3 λ 5 λ 1 λ 2 λ 2 λ (.19) 3 λ 6 λ 2 j λ 3j λ 2 λ 1 λ 3 λ 2 λ (1.43) 4 λ 4 λ 5 λ 4 j λ 4j λ 2 λ (.34) 5 λ 2 j λ 5j λ 1 λ (1.46) 6 λ 3 λ 3 λ 2 j λ 6j λ 3 λ (12.22) 7 λ 4 λ 3 j λ 7j λ 3 8 λ 4 λ 3 λ 2 λ 2 λ 3 λ 4 j λ 8j Notes: Simulated Methods of Moments (SMM) estimates of the regime-switching model s parameters. The methodology combines the scores of the (simulated) likelihood function from fundamentals (inflation, real earnings, capacity utilization, and money growth) with pricing errors from financial variables (S&P5 index P/E ratio, 3-months Treasury Rate, Treasury Slope, and the volatilities of stock, the five-year, and ten-year bonds). The last four columns of top panel also report the conditional P/E ratios and conditional yields across the eight composite regimes. The data sample 8 is , except for options, whose sample is Newey-West adjusted standard errors are in parenthesis.

9 Table 4: Fit and Data: Estimation Sample Panel A: Fit over the Full Sample α β t(α) t(β) R 2 Inflation Real Earnings Capacity Utilization Money Growth P/E Ratio month Treasury Rate Term Structure Slope Stock Implied Volatility Put-to-Call Ratio Bond Implied Volatility Panel B: Fit over the Out-of-Sample α β t(α) t(β) R 2 Inflation Real Earnings Capacity Utilization Money Growth P/E Ratio month Treasury Rate Term Structure Slope Stock Implied Volatility Put-to-Call Ratio Bond Implied Volatility Notes: Results of the regressions (Fundamentals) t = b + b 1 E t [Fundamentals] + ɛ t (Financial Variable) Data t = b + b 1 (Financial Variable) t + ɛ t where Fundamentals is either inflation, real earnings growth, capacity utilization, or money growth, and financial variables are identified in each row. In these regressions, both expected fundamentals and modelimplied financial variables are conditional on the fitted beliefs. The model s parameter are estimated on the sample In Panel A the sample for the regressions is , except for option-based quantities, whose sample is In Panel B, the sample of the regressions is 2-211, which is then fully out-of-sample. All t-statistics are Newey-West adjusted for heteroskedasticity and autocorrelation using four lags. 9

10 Table 5: Moments and Mean Absolute Errors from SMM A: Pricing Errors Variable Mean Error MAE P/E M Yield (%) Year - 1-Year Yield (%) Sharpe Ratio Stock s AMTIV (%) Stock s P/C.55.1 Bond s ATMIV (%) B: Scores of Likelihood Function Variable Mean Error (Scaled) MAE β 1 2.E-5 5.E-4 β β β 4 6.E-5.3 θ 1 7.4E-6.1 θ 2 8.E-5.7 θ 3 4.7E-6.1 ρ 1 1.E-4.7 ρ 2 6.E-6.2 ρ 3 1.4E-3.19 ω 1 1.3E-6.1 ω 2-3.E-6 5.E-5 ω 3 4.E-4.9 ω 4 1.E-5.2 α -3.E-5 6.E-5 α β -1.E-5 6.E-5 α rho -2.E-5 8.E-5 κ 1.6E-3 1.7E-3 µ 1 5.3E-3 3.5E-2 σ 1 2.7E-3 9.E-3 µ 2-3.E-5 1.E-4 σ 2-1.E-5 1.1E-5 σ M,2-2.E-4 1.E-3 σ M,3 1.E-4 8.E-4 λ 1-3.E-5 1.E-4 λ 2-3.E-5 2.E-4 λ 3 8.E-5 7.E-5 λ 4 3.E-5 6.E-5 λ 5 1.E-6 7.E-5 λ 6-1.E-6 4.E-5 Notes: Panel A reports the pricing errors from the SMM procedure, while Panel B reports the moments from the scores of the likelihood function. The details of the estimation procedure are contained in the Appendix of the paper. 1

11 Figure 1:, Surveys and Fundamentals: Estimation without Options 2 1 A. Inflation 5 Data SPF B. Real Earnings 1 2 Data SPF C. Capacity Utilization 3 2 Data D. Money Growth Data Bloomberg Panel A plots the inflation data, the expected inflation rate from the fitted model, and the Survey of Professional Forecasters (SPF) consensus forecasts for GDP deflator-based inflation. Panel B plots real earnings data, the expected earnings growth from the fitted model, and the SPF consensus forecasts of real GDP growth. Panel C plots capacity utilization, the expected change from the model, and Bloomberg consensus forecasts of capacity utilization one quarter ahead. Finally, Panel D plots money growth and the model expected modeny growth rate. In all panels the solid grey line are the data, the solid black line is the model expectation, and the dashed line is the survey-based forecasts. 11

12 Figure 2: Composite Regime Probabilities: Estimation without Options 1 A. Regular Boom 1 B. Regular Recession C. Over heating Boom D. Stagflation E. New Economy Growth F. Deflation G. Low Capacity Boom H. Deep Recession s fitted beliefs about each of eight composite regimes from 1967 to 211. Shaded areas correspond to NBER-dated recessions. The estimates of the eight composite regimes are in Table??. 12

13 Figure 3: Marginal Probabilities: vs. Surveys. Estimation without Options 1 A. High Inflation 1 B. Medium Inflation 5 corr. = 71% 5 corr. = 57% C. Low Inflation 1 D. Zero Inflation 5 corr. = 84% 5 corr. = 38% E. Recession Probability 1 F. High Capacity Utilization 5 corr. = 47% 5 corr. = 5% G. Medium Capacity Utilization 1 H. Low Capacity Utilization 5 5 corr. = 8% corr. = 39% Survey Panels A to D: s fitted marginal posterior probabilities about the four possible inflation regimes (black lines) and professional forecasters probability assessments of similar levels of next-year inflation (grey lines). Panel E: s fitted marginal probability of a recession (black line) and professional forecasters probability assessment of a GDP decline the following quarter. Panels F to H: s fitted marginal posterior probabilities about high, medium, and low capacity utilization (black lines) and Bloomberg-based probability of the same three high, medium, low level of capacity utilization obtained from the distribution of Bloomberg forecasts. Shaded vertical bars are the NBER-dated recessions. 13

14 Figure 4: Fitted and Data Series: Estimations without Options 3 A. P/E Ratio 5 B. Stock Implied Volatility P/E C. 3 month Treasury Rate 2 D. Put to Call Implied Volatility Ratio P/C E. Term Structure Slope F. 1 year Bond Implied Volatility Data Panels A, C, and E plot the realized price/earnings ratio, 3-month T-Bill rate, and the slope of the term structure (1 year minus 1 year), respectively, and their model-fitted counterparts, over the sample Panels B, D, and F plot the realized stock ATM IV, Put-to-Call implied volatility ratio, and 1 Bond futures option ATM IV, respectively, and their model-fitted counterparts over the option s sample In all panels, the solid grey line is the data and the dashed black line is the model s fitted. Shaded areas correspond to NBER-dated recessions. 14

15 Figure 5: Fitted and Data Series: Realized Volatilities. Estimations without Options 6 A. Stock Realized Volatility B. 5 Year Bond Volatility C. 1 Year Bond Volatility Data Panels A - C plot the realized volatilities of stock returns, the 5-year bond and 1-year bond returns over the option s sample In all panels, the solid grey line is the data and the dashed black line is the model s fitted. Shaded areas correspond to NBER-dated recessions. 15

16 Figure 6:, Surveys and Fudamentals: Sample A. Inflation 5 Data SPF B. Real Earnings 1 2 Data SPF C. Capacity Utilization 3 2 Data D. Money Growth Data Bloomberg Panel A plots the inflation data, the expected inflation rate from the fitted model, and the Survey of Professional Forecasters (SPF) consensus forecasts for GDP deflator-based inflation. Panel B plots real earnings data, the expected earnings growth from the fitted model, and the SPF consensus forecasts of real GDP growth. Panel C plots capacity utilization, the expected change from the model, and Bloomberg consensus forecasts of capacity utilization one quarter ahead. Finally, Panel D plots money growth and the model expected modeny growth rate. In all panels the solid grey line are the data, the solid black line is the model expectation, and the dashed line is the survey-based forecasts. 16

17 Figure 7: Composite Regime Probabilities: Sample A. Regular Boom 1 B. Regular Recession C. Over heating Boom D. Stagflation E. New Economy Growth F. Deflation G. Low Capacity Boom H. Deep Recession s fitted beliefs about each of eight composite regimes from 1967 to 211. Shaded areas correspond to NBER-dated recessions. The estimates of the eight composite regimes are in Table??. 17

18 Figure 8: Marginal Probabilities: vs. Surveys. Sample A. High Inflation 1 B. Medium Inflation 5 corr. = 71% 5 corr. = 55% C. Low Inflation 1 D. Zero Inflation 5 corr. = 81% 5 corr. = 32% E. Recession Probability 1 F. High Capacity Utilization 5 corr. = 6% 5 corr. = 12% G. Medium Capacity Utilization 1 H. Low Capacity Utilization 5 5 corr. = 84% corr. = 83% Survey Panels A to D: s fitted marginal posterior probabilities about the four possible inflation regimes (black lines) and professional forecasters probability assessments of similar levels of next-year inflation (grey lines). Panel E: s fitted marginal probability of a recession (black line) and professional forecasters probability assessment of a GDP decline the following quarter. Panels F to H: s fitted marginal posterior probabilities about high, medium, and low capacity utilization (black lines) and Bloomberg-based probability of the same three high, medium, low level of capacity utilization obtained from the distribution of Bloomberg forecasts. Shaded vertical bars are the NBER-dated recessions. 18

19 Figure 9: Fitted and Data Series: Sample A. P/E Ratio.5 B. Stock Implied Volatility P/E C. 3 month Treasury Rate 2 D. Put to Call Implied Volatility Ratio 15 1 P/C E. Term Structure Slope F. 1 year Bond Implied Volatility Data Panels A, C, and E plot the realized price/earnings ratio, 3-month T-Bill rate, and the slope of the term structure (1 year minus 1 year), respectively, and their model-fitted counterparts, over the sample Panels B, D, and F plot the realized stock ATM IV, Put-to-Call implied volatility ratio, and 1 Bond futures option ATM IV, respectively, and their model-fitted counterparts over the option s sample In all panels, the solid grey line is the data and the dashed black line is the model s fitted. Shaded areas correspond to NBER-dated recessions. 19

20 Figure 1: ATMIV versus VIX. 7 ATMIV VIX ATMIV versus VIX This figure plots the ATMIV used in the paper compared to the VIX index from the CBOE. The VIX index is only available from January

21 Figure 11: 5% OTM P/C Index versus 1% OTM P/C Index % OTM P/C 1% OTM P/C 5% OTM P/C versus 1% OTM P/C P/C This figure plots the 5% OTM put-to-call implied volatility ratio (P/C index) used in the paper compared to the 1% OTM P/C index. We can construct the 1% OTM P/C only on OptionsMetric sample, available from

22 Figure 12: Real and Nominal Money Growth 3 2 Real M1 Nominal M1 Panel A: Real M1 versus Nominal M1 percent M1 M2 Panel B: Real Growth of M1 versus M2 percent Panel A plots the real growth rate of M1, used in the paper, and compares it to its nominal growth. Panel B plots the M1 money growth used in the paper and compares it to the real money growth of M2. 22