Price vs. Quantity in the Term Structure of Variance Risk Premia Marianne Andries, Thomas Eisenbach, Martin Schmalz and Yichuan Wang⇤ June 2015 Abstract We estimate the term structure of the price of variance risk (PVR), which helps fill a gap in the empirical literature and distinguish between competing asset-pricing theories. First, we measure the PVR as proportional to the Sharpe ratio of shortterm holding returns of delta-neutral index straddles. Second, we estimate the PVR in a Heston (1993) model separately for di↵erent maturities. In both analyses, we find the PVR is negative and decreases in absolute value with maturity. The PVR is more negative and its term structure is steeper when volatility is high. These findings are inconsistent with the calibrations of established asset-pricing models that assume constant risk aversion across maturities, but confirm a key prediction of the model with horizon-dependent risk aversion of Andries et al. (2014).

JEL Classification: G12, G13 Keywords: Volatility Risk, Option Returns, Straddle, Term Structure

⇤

Andries: Toulouse School of Economics, [email protected]; Eisenbach: Federal Reserve Bank of New York, [email protected]; Schmalz: University of Michigan Stephen M. Ross School of Business, [email protected]; Wang: University of Michigan, [email protected]. For helpful comments and discussions, we would like to thank Yakov Amihud, Markus Brunnermeier, Giovanni BaroneAdesi, Ing-Haw Cheng, Max Croce, Ian Dew-Becker, Robert Dittmar, Stefano Giglio, Ralph Koijen, Owen Lamont, Gordon Lawrence, Erik Loualiche, Thomas Mariotti, Stefan Nagel, Christian Schlag, Tyler Shumway, Adrien Verdelhan, and Grigory Vilkov, as well as seminar participants at CMU Tepper, Maastricht University, Toulouse School of Economics, and the University of Michigan. Schmalz is grateful for generous financial support through an NTT Fellowship from the Mitsui Life Financial Center. All errors are our own.

1

Introduction

A fundamental debate in asset pricing has arisen concerning the term structure of risk premia. Well-established theoretical asset-pricing models such as Campbell and Cochrane (1999) and Bansal and Yaron (2004) predict a flat or upward-sloping term structure of excess returns; similarly, the unit price of variance risk is constant across maturities in standard option pricing models such as Heston (1993). However, van Binsbergen et al. (2012) and van Binsbergen and Koijen (2014) find, in the data, one-period returns in equity and equity derivatives markets are actually higher for shorter maturities. Similarly, Giglio et al. (2013) show that very long-run risk premia in housing markets are low compared to observed risk prices for shorter maturities; see van Binsbergen and Koijen (2015) for an overview. At the same time, recent years have witnessed an explosion of new derivative products which allow investors to take long or short positions in shortterm volatility, such as the BXM, VXUP, or VXDN.1 Academic research has not kept up with the speed of this development, and in particular with the question why the short end of the term structure is where the most urgent need for risk sharing appears to arise. A theoretical motivation to study the term structure of risk prices in the variance risk market specifically is that stochastic volatility is a key driver of risk premia in many asset pricing models (Bansal et al., 2013; Campbell et al., 2012) and in the data (Adrian and Rosenberg, 2008; Ang et al., 2006; Menkho↵ et al., 2012; Campbell and Hentschel, 1992). Moreover, the model of Andries et al. (2014) predicts the term structure of risk premia is downward sloping exclusively through the term structure of variance risk prices. Enlarging the body of evidence on the term-structure of returns, and more specifically of risk pricing, has important implications: whereas the valuation of short-to-mediumterm assets can be achieved through arbitrage pricing using assets of similar duration, the pricing of long-term assets and investments crucially depend on having a theoretical asset-pricing model that can match the observed term structure. This paper helps inform this debate by empirically investigating whether the price of variance risk has a non-trivial term structure. We use standard data on S&P 500 index options from February 1996 to April 2011 to estimate the price of variance risk (PVR) separately for di↵erent maturities, ranging from 11 to 252 days. We first measure Sharpe ratios of delta-neutral straddles with di↵erent 1

BuyWrite Index, VixUp and VixDown, traded on the CBOE (Chicago Board Options Exchange)

1

maturities which are a valid qualitative measure of the PVR.2 While we find that Sharpe ratios are negative and large (in absolute value) for short maturities, they are much closer to zero at longer maturities. This indicates a sharply decreasing term structure for the price of variance risk (in absolute value). Because shorting short-maturity straddles generates very high expected returns also when the positions are crash-hedged (Coval and Shumway, 2001), we are confident that one-sided crash risk is not the driver of our results (see also Constantinides et al., 2013). For a more robust estimation, we then adapt the maximum-likelihood approach of Christo↵ersen et al. (2013) to estimate the PVR separately for options of di↵erent maturities and find consistent results. At maturities from 11 to 30 days, investors are willing to pay to protect themselves against index-return variance that is 61 percent higher than what is expected under the physical measure. For maturities of 230 to 250 days, the PVR is only half of that at around 34 percent. Furthermore, higher levels of volatility are associated with more negative prices of variance risk, especially at shorter maturities, leading to a steeper term structure of the PVR. In sum, the finding that the PVR is negative is predominantly driven by short-maturity options and by periods with high market volatility. Our findings contribute to the literature as follows. Possibly guided by existing optionpricing models such as Heston (1993), which predict a constant price of variance risk across maturities, no paper to date in the options literature has investigated if variance risk prices have a non-trivial term structure. For example, work such as Coval and Shumway (2001) or Carr and Wu (2009) measures variance risk premia for options with a single maturity; Christo↵ersen et al. (2013) pool all maturities to estimate the price of variance risk. Choi et al. (2015), using Treasury futures options, find a negative and upward sloping termstructure of variance premia, but do not analyze how prices and quantities contribute. We find estimating a Heston (1993) model separately for di↵erent maturities falsifies its prediction of a flat term structure, indicating the next generation of option-pricing models should allow for risk prices that vary depending on maturity, as observed in the market for variance risk. Our findings are informative not only for option pricing. A conceptually important corollary of the insight that risk prices – and not only quantities – can vary with the horizon is that option prices can be used to distinguish between changes in risk prices and changes in expected risk levels at di↵erent maturities. The distinction between price 2

Under assumptions we discuss below.

2

and quantity is important because it separates the market’s objective belief about the level of future volatility from market beliefs according to the risk-neutral measure. By contrast, the product of price times quantity of variance risk – the variance risk premium – is a somewhat imprecise “fear gauge” of the market that is prone to misinterpretation: an increase in the fear gauge could equally mean the market’s objective expectation has deteriorated or risk aversion has increased. This distinction is important for forecasters and policy makers who use market prices as forward-looking indicators. The implicit assumption that risk prices are flat across horizons would lead them to make biased inferences about market estimates of physical probabilities. In other words, our results emphasize that the conversion between objective and risk-neutral measures depends on maturity. Further, our analysis of both the price per unit and the quantity of variance risk allows us to sharply distinguish between alternative theoretical asset-pricing models. Various frameworks have generated a downward-sloping term structure of equity risk premia with models a↵ecting the expected quantity of risk at various horizons; see van Binsbergen and Koijen (2015) for a comprehensive overview. Because the findings in the present paper also present strict statements about the term structure of the price of variance risk, and not about risk premia (which are the product of price and quantity), they o↵er strong support for also considering a preference-based explanation for the downward-sloping term structure of equity risk premia.3 The long-run-risk model of Bansal et al. (2013) as well as the rare-disaster model of Wachter (2013) correctly predict a negative price per unit of variance risk, but cannot quantitatively match its decline with maturity (in absolute value). Consumption-based asset pricing models with loss aversion, such as Andries (2012) and Curatola (2014), predict a pricing per unit of risk that declines intrinsically (in absolute value) with the quantity of risk, consistent with the evidence on markets where the declines in Sharpe ratios in the term-structure are accompanied by increases in volatility (see van Binsbergen and Koijen (2015) for some such examples). Our results on equity option straddles, however, highlight a decline in both the pricing and quantity of risk in the term-structure, and cannot be simply rationalized by first-order risk aversion. Andries et al. (2014) adopt the long-run risk structure of Bansal and Yaron (2004) but generalize the agents’ Epstein and Zin (1989) preferences to allow for horizon-dependent risk aversion. Their framework 3

Relatedly, the model by Christo↵ersen et al. (2013) is able to generate a downward-sloping variance risk premium driven by a declining quantity of risk by horizon, but cannot explain a downward-sloping price per unit of risk.

3

predicts a negative price of risk as well as a declining term structure (in absolute value), both amplified in times of high volatility, consistent with the results documented in the present paper. Other papers have investigated the term structure of variance risk premia and prices, using di↵erent data sets and di↵erent methodologies than the present paper. Most recently, Dew-Becker et al. (2014) use proprietary data on variance swaps to estimate termstructure models, similar to Amengual (2009) and Ait-Sahalia et al. (2012), but adding realized volatility as a third factor to a standard level-and-slope analysis. They find that only shocks to realized volatility are priced, implying a term structure that is steeply negative at the short end (a one-month horizon) but essentially flat at zero beyond that. Our unconditional results o↵er a more nuanced picture, because we also find a strong concavity in the term structure but with a negative price of variance risk for all maturities. Our conditional results on the relationship between current market volatility and the term structure of risk prices are related to the work of Cheng (2014) who studies the returns of hedging volatility with VIX futures. However, Cheng documents that hedging is cheaper during turbulent times, while we find that the price of variance risk is more negative and that its term structure is steeper when current volatility is high. Barras and Malkhozov (2015) find di↵erences in estimates of variance risk premia in the equity and option markets that are driven by institutional factors. While this finding suggests a potential explanation for the di↵erences between our results and those of Dew-Becker et al. (2014) and Cheng (2014), it also emphasizes the value of using di↵erent methodological approaches on di↵erent data sets. The paper proceeds as follows. Section 2 presents the theoretical derivation of the price of variance risk in the Heston (1993) model as well as its relation to the Sharpe ratios of short-term returns of delta-neutral straddles. Section 3 gives the empirical results. Section 4 concludes.

4

2

Hypotheses Development and Empirical Strategy

2.1

Theoretical Background and Empirical Hypotheses

We use the structure of the option-pricing model of Heston (1993) to isolate the role of variance risk. Specifically, we assume stock price St and variance Vt satisfy: dSt = St µ dt + St dVt =  (✓ dW1t dW2t = ⇢ dt,

p

Vt dW1t , p Vt ) dt + Vt dW2t ,

(1)

where µ denotes the returns drift,  the speed of mean reversion for the returns volatility, ✓ the level to which Vt reverts, and a↵ects the volatility of Vt . Both dW1t and dW2t are Brownian motions, and ⇢ denotes the correlation between shocks to the return and variance processes. In the Heston model, the no-arbitrage price Xt of any option satisfies the partial di↵erential equation: 1 @ 2X @ 2X 1 @ 2X 2 @X 2 V S + ⇢ S V + Vt + rSt t t t t 2 2 2 @S @S@V 2 @V @S @X @X + [ (✓ Vt ) ⇤t ] rXt + = 0, @V @t

(2)

where ⇤t denotes the variance risk premium, and r is the constant risk-free rate. This total risk premium ⇤t can then be decomposed into the unit price of variance risk and the amount of variance Vt : ⇤t = ⇥ Vt , (3) where Vt is expressed in net terms and therefore represents the percentage additional variance under the risk neutral measure. A value of 0.61 means the option is priced with an implied variance that is 61 percent higher than what is expected under the physical measure. In contrast to previous studies that measure the total variance risk premium ⇤t at di↵erent horizons, our interest is in measuring , the unit PVR, at di↵erent horizons. We test three hypotheses: 1. The PVR is negative at all maturities. We hypothesize that investors are willing to pay a positive amount to insure against an increase in volatility, at all maturities. This hypothesis is consistent with predictions of most models that give a role to 5

stochastic volatility, such as Bansal and Yaron (2004) or Andries et al. (2014). The empirical null hypothesis is that the PVR is zero at all maturities. 2. The PVR decreases in absolute value with maturity. This hypothesis corresponds to a central prediction of the horizon-dependent risk-aversion model of Andries et al. (2014). The empirical null hypothesis is that the term structure is flat, that is, that the PVR is constant across horizons. 3. The PVR is more negative and the term structure is steeper when volatility is high. This prediction arises from the model of Andries et al. (2014) as well. The empirical null hypothesis is that the level of volatility does not a↵ect the unit price of risk (although risk premia can increase with volatility because the quantity of risk increases). We now explain the two di↵erent estimation procedures we use to test these hypotheses: a non-parametric estimation using short-horizon Sharpe ratios and a parametric estimation based on Christo↵ersen et al. (2013).

2.2

Non-parametric Estimation: Short-horizon Sharpe Ratios

We show how the short-horizon Sharpe ratios of delta-neutral straddles identify the sign of the PVR and the slope of its term structure. According to equation (2), the option dynamic is given by the following: 

✓ @X dXt = Vt + Xt @V

◆ @X @X @X p @X p St r + µSt dt + St Vt dW1t + Vt dW2. (4) @S @S @S @V

A complication arises with the measurement of because µ is not observable. To address this challenge, we form a measurable portfolio of straddles that are delta neutral so that the portfolio is independent of µ.4 First, note that we can rewrite the option dynamic (4) as ✓ ◆ ✓ ◆ @X @X @X @X p d Xt St = Xt St r + Vt dt + Vt dW2t . (5) @S @S @V @V 4 Delta-neutral straddles are not necessarily at the money. While at-the-money straddles are approximately delta neutral for short maturities, the delta-neutral moneyness increases with maturity; see the decreasing ratios of St /K in the delta neutral straddles described in Table 1.

6

Then we discretize the dynamic in (5) and rearrange to arrive at p

Vt t

+"=

Xt

@X S @S t @X @V

Xt p Vt t

@X S @S t

r t

,

(6)

p where " = W2t / t. Using equation (6), note that the PVR di↵ers from the Sharpe ratio of an deltap neutral straddle only by a factor of Vt t/ . Note further that the denominator of the right hand side of equation 6 is just the standard deviation of the process in equation 5. Hence when Xt is a delta-neutral straddle, we have p

Vt t

⇡ =

Xt Xt

@X S @S t @X @V @X S @S t

q ⇥ Var

Xt Xt r t ⇡ p Var[ Xt ] = SR(Xt )

p

Xt Vt t Xt

Xt

@X S @S t

r t

@X S @S t

r t

@X S @S t

⇤

(7)

Equation (7) shows that the Sharpe ratios of delta-neutral straddles are a qualitatively valid measure of both the sign and slope of the price of variance risk across maturities, even though they are not quantitatively comparable to the results from the parametric p estimation in section 2.3. Because the factor Vt t/ is guaranteed to be positive, the p Sharpe ratio is a robust test of the sign of PVR. Moreover, the extra factor Vt t/ does not change with maturity, so it does not a↵ect the sign of the slope of the term structure of the PVR. In contrast to our approach, Coval and Shumway (2001) look at returns from holding one-month delta-neutral straddles to maturity. The long time period means they cannot use the discretization necessary for equation (7) to hold. The straddles analyzed by van Binsbergen and Koijen (2015) have deltas that increase with maturity, and thus depart from the delta neutrality required by equation (7). The instantaneous Sharpe ratio of investing in delta-neutral straddles can be estimated by E[ Xt /Xt r t] SR = p , Var[ Xt /Xt r t] 7

where

Xt Xt+ t Xt = . Xt Xt

We estimate the Sharpe ratios of options with di↵erent maturities ranging from 11 days to 252 days, using daily returns. To estimate the Sharpe ratio SR⌧ for options with maturity ⌧ , we use returns from options with maturities in the range [⌧, ⌧ + 20) and compute the average divided by the standard deviation of such returns. Figure 1 shows that these returns are little auto-correlated over time, and as such asymptotic standard errors for the Sharpe ratios can be computed by the bootstrap, treating each return as an independent observation. The results of our analysis are described and analyzed in Section 3.

2.3

Parametric Estimation Procedure

p In Equation (7), Vt t/ , while constant in the term-structure, varies in the time series. These time series variations can be correlated (and we show in Section 3 that they are) with variations in the slope of the term-structure of PVR. Such covariations may introduce bias into the magnitude of the estimated slope in the Sharpe ratios analysis just described. This concern encourages us to also estimate the parameter directly, using a discretetime method based on Christo↵ersen et al. (2013).5 These authors estimate the price of variance risk using a pooled sample of options of di↵erent maturities and strike prices. We adapt the procedure to subsets of at-the-money options, and run the estimation of separately for options of di↵erent maturities and volatility levels.6 We first describe the economic intuition, and then explain the formal estimation procedure. In Christo↵ersen et al. (2013), the price of variance of risk is identified by measuring the wedge between expected variance on the basis of a GARCH process for the underlying stock price process, and the expected variance based on option prices. In their model, the index follows a GARCH-in-means process. The excess return over a given period has variance ht . The variance process is ARMA(1, 1) with autoregressive coefficient , a moving average coefficient ↵, and an intercept !. A coefficient governs the correlation between the return process and the stochastic volatility process. An equity risk-aversion 5

Heston and Nandi (2000) show that the discrete time model described below has the continuous time model of Heston (1993) nested as a special case when the number of trading periods per unit of physical time goes to infinity. Therefore the GARCH approach described below is precisely the discrete time analogue of the continuous time Heston (1993) model. 6 In principle, the MLE can be applied to options that are not at the money. However, we restrict the MLE to at-the-money options to limit the e↵ect of jumps on the estimation since jumps have a much larger e↵ect on the price of out-of-the-money options.

8

parameter ⇣ determines the first moment, such that the expected excess return ⇣ 12 ht is linear in the variance. Let Rt be the net stock return on day t, let rt be the daily risk-free rate on day t, and let ht be the value of the variance on day t. Then (Rt rt ) | ht is normally distributed with mean ⇣ 12 ht and variance ht , and the variables evolve according to Rt

1 2

rt = ⇣

ht +

p

ht z t ⇣ ht = ! + h t 1 + ↵ zt 1

p

zt ⇠ i.i.d. N (0, 1) .

ht

1

⌘2

(8)

In addition to the equity risk-aversion parameter ⇣, Christo↵ersen et al. (2013) assume a variance risk-aversion parameter ⇠. Whereas ⇣ describes aversion to states of the world with a low stock price, ⇠ describes aversion to states of the world with high volatility state variable ht . The two risk-aversion parameters ⇣ and ⇠ imbue the risk-neutral process for ht with a higher level and a higher persistence. Formally, denote ht under the risk-neutral measure as h⇤t . It follows the process h⇤t = ! ⇤ + h⇤t

⇣ ⇤ + ↵ zt⇤ 1

zt⇤ ⇠ i.i.d. N (0, 1) ,

1

⇤

p

h⇤t

1

⌘2

, (9)

where7 ! 1 2↵⇠ ↵ ↵⇤ = (1 2↵⇠)2

!⇤ =

= ⇤

=

1 2

⇣

+

(1

2↵⇠) +

1 2

.

(10)

Conditional on the GARCH parameters ⇥ = {!, , ↵, ⇣, }, a value of the parameter ⇠ generates risk-neutral volatilities h⇤t that are used to price options. The details of the option-pricing model come from Heston and Nandi (2000) and are replicated in the appendix. 7

The risk-neutral shocks are given by zt⇤ =

p

1

⇣ ⇣ 2↵⇠ zt + µ +

9

↵⇠ 1 2↵⇠

⌘p ⌘ ht .

We perform the estimation in two stages: In the first stage, we pool the data across maturities to estimate the parameters ⇥ governing the common GARCH process in index returns. In the second stage, we use this set of common GARCH parameters to estimate the price of variance risk separately for subsets of options by maturity and volatility state. These subsets are the same ones used for the analysis of straddle Sharpe ratios above.8 For the first stage, we estimate the GARCH parameters through maximum likelihood. Given a daily return series {Rt , rt }Tt=1 , we solve ˆ = ⇥

( ✓ T 1X 1 log ht + Rt 2 t=1 ht

argmin ⇥={!, ,↵, ,⇣}

rt

✓

⇣

1 2

◆

ht

◆2 )

,

where ht = ! + h t zt = h1 =

Rt

rt

1

p

!+↵ 1 ↵

⇣ + ↵ zt 1 2

⇣ ht 2

1

ht

⌘2 p ht 1

.

ˆ from the first stage and a particular subset of For the second state, given a value of ⇥ option prices {Pi }N i=1 , we estimate ⇠ through maximum-likelihood estimation: ⇠ˆ = argmin ⇠

N ⇢ 1X "2 log sˆ2" + i2 2 i=1 sˆ"

We treat the Black-Scholes Vega (BSV) weighted pricing errors as Gaussian random variables, following the method of Christo↵ersen et al. (2013): sˆ2"

N 1 X 2 = " N i=1 i

"i =

PiMkt PiMod (⇠) BSVMkt i

8

Christo↵ersen et al. (2013) show that a joint maximum-likelihood procedure with both options and returns gives estimates of comparable to those of a procedure that estimates the models sequentially with returns first and options second. The sequential procedure is particularly important in our case because the options all derive value from the same underlying time series for stock returns, so it makes sense for them to share the same time-series parameters.

10

We then derive the continuous-time parameter by calibrating it to obtain the same unconditional variance of stock returns and the same ratio between physical and riskneutral unconditional variances as in the discrete-time model. To test the di↵erent hypotheses, we perform the second stage on several subsets of the data: 1. We estimate by considering the prices of options in maturity buckets ranging from 11 to 250 to see if the PVR changes across the term structure. 2. We split the options into two regimes for instantaneous volatility ht . Doing so enables a first look into how the term structure of the price of variance risk changes in crisis periods (high ht ) and calm periods (low ht ). In our approach, we do not smooth the inputs by computing a volatility surface. Instead, we smooth the outputs from the estimation procedure. This ensures that we are basing our estimates on actual observed prices and that we do not inflate our dataset with interpolated values. The results of our analysis are described and analyzed in Section 3.

3 3.1

Data and Empirical Results Data Sources and Summary Statistics

We use daily closing data from February 1996 to April 2013 of European SPX index options and SPX index levels from OptionMetrics. Value-weighted S&P 500 returns, excluding dividends, from January 1990 to December 2014 come from CRSP. The three-month risk-free rate data are taken from FRED. The risk-free rate for a given daily observation is defined as log (1 + rm ) /252, where rm is the risk-free rate recorded for the last week of the previous month. We clean the data by removing duplicate observations of calls or puts on the same day that have the same expiration date, strike price, and midprice. Next, we keep only options that have a maturity between 11 and 252 trading days on the day of observation.9 We exclude shorter-maturity options to avoid microstructure noise close to expiration a↵ects our results, and we exclude longer-maturity options because they are thinly traded. 9 Using trading days to measure maturity is essential. The GARCH estimation treats the index return series as a continuous series without weekends. Thus, to be consistent, the option maturities should also be expressed in trading days.

11

For the non-parametric estimation, for each maturity and strike on a given day, we estimate the Black-Scholes implied volatility by the average of the call and put BlackScholes implied volatilities. We then use this implied volatility to estimate the BlackScholes delta of the call and the put at that strike and maturity observation on that day. We then pick the strike and maturity such that the straddle delta, which is the sum of the put and call deltas, is closest to zero. We drop observations that have straddle deltas greater than 0.10 in absolute value and whose bid ask spread is more than 10 percent of the midprice.10 As such the options under consideration are highly liquid and close to delta neutral. We also follow Bakshi et al. (1997) in excluding any options that do not obey the futures arbitrage constraints.11 We further restrict our sample to options that satisfy the delta constraint and have a maturity between 11 and 252 days during the entire [ 1, +1] day window relative to the observation date. If a given option contract violates an arbitrage bound or goes out of the money in the [ 1, +1] window, then its return is not used in the calculations.12 Hence, the Sharpe-ratio analysis excludes options in periods when the index changed dramatically in the span of 3 days, thereby excluding crisis periods. We thus ensure that abnormal events do not drive our results. We keep only calls and puts that can be paired into a straddle. Lastly, for any given day, we keep only the option with the smallest di↵erence between the strike price and the index level. For the parametric estimation, for each year and each maturity bin of 10 days starting at every 10th day, we drop the observations corresponding to the top and bottom 1 percent of residuals in a third-order polynomial regression of the option price against the GARCH volatility variable ht .13 The results are not quantitatively sensitive to the exact level of truncation. We present summary statistics for the sample of 47,416 option-day observations in Table 1. We note in Table 2 that the dollar value of the bid-ask spread increases along the maturity structure, but decreases as a percentage of the option price. We view this observation as an indication of good liquidity across the entire term structure. 10

We find that the results are not sensitive to changing the straddle delta threshold to a lower value of 0.05. Sample sizes decrease substantially, however, and as such we do not focus on those results. 11 For a call with maturity ⌧ , C(⌧ ) max{0, St Xt e r⌧ }, and for a put, P (⌧ ) max{0, Xt e r⌧ St }. 12 We do not need to make such corrections for the parametric estimation because the parametric estimation fits prices, not returns. 13 Put prices are converted to equivalent call prices by put-call parity.

12

3.2

Non-parametric Estimation Results

We provide a model-free estimation of the sign of the intercept and slope of the term structure in Figure 3. We present the point estimates for maturity buckets of length 20 days in Table 3.14 The term structure of Sharpe ratios is concave and trends upwards at almost all maturities. Between the first two maturity buckets, the 11–30 day maturity bucket and the 30–50 day maturity bucket, the Sharpe ratio increases from 1.15 to 0.71. This sharp increase represents 40 percent of the overall range in Sharpe ratios over the entire term structure, showing that most of the variation happens in the short-term. The Sharpe ratio continues to drift upwards, albeit more slowly, for maturities beyond 50 days. It is 0.54 for intermediate maturities 50–70 days, more than three times as negative as for the 230–250 day straddles, 0.16. It steadily approach zero for longer maturities. Our findings indicate that measures of the negative price of variance risk, if obtained from a pooled sample, are mainly driven by short maturities. These results are qualitatively consistent with those reported for a coarser sample of straddle returns in van Binsbergen and Koijen (2015).

3.3

Parametric Estimation Results

We present the parametric estimation of the term structure of the price of variance risk (PVR) in Figure 4, and a selection of data points grouped by maturity bucket in Table 4.15 Our results show that the PVR decreases in absolute value with maturity, with a point estimate 0.61 for maturities 11–30, and 0.34 for maturities 230–250.16 We observe a dip in the unconditional term structure, going against the trend, at the 70–90 day maturity bucket. We attribute this anomaly to a change in the distribution of option maturities before and after 2007. As seen in Figure 2, prior to 2007 most traded p As noted in Section 2.2, the factor Vt t/ is not constant in the time series so interaction with the sample size may introduce bias into the magnitude of the estimated slope. As this issue is not present in the parametric estimation, it can account for di↵erent slope estimates across the analyses. Further, di↵erences in liquidity between the delta-neutral straddles and the at-the-money options used in the non-parametric and parametric analysis, respectively, may introduce di↵erences. 15 We only show second-stage results. Our first-stage GARCH estimation yielded ! = 0, = 0.835, ↵ = 3.54 ⇥ 10 6 , ⇣ = 3.48, and = 191.03. These results are in line with estimates from Christo↵ersen et al. (2013). 16 ˆ in the asymptotic MLE standard errors: variations in ⇥ ˆ might We do not correct for variations in ⇥ ˆ increase the variance of the estimate ⇠ on any given subset, but do not imply we are overstating di↵erences in ⇠ between subsets, which is the measure we are interested in. For a more detailed discussion on how we correct the asymptotic MLE standard errors, see the appendix. 14

13

options had maturities between 0 and 60 days. After 2007, however, the maturity range for most traded options increases to 0–90 days. Our results for the pricing of variance risk in the 70–90 day maturity bucket are thus artificially driven by the post 2007 period. We split our sample between pre and post 2007 and find, in Figure 5 and Table 5, that the anomalous dip in the term-structure for the 70–90 day maturity bucket disappears.17 In line with our Sharpe ratio results, the term structures split by time period are smoothly concave and upward trending at all maturities. In both time periods, most of the change in the PVR occurs in the 11–50 day maturity range, with only 30 percent of the overall term-structure variation occurring at the intermediate maturities, between 50 and 250 days. Next, we explore how volatility levels can a↵ect the term-structure of the PVR. We divide days into two categories of expected future volatility, based on whether ht+1 from the GARCH estimation is above or below the sample median of ht , and we provide our results in Figure 6 and Table 6. Compared to the low volatility state, the PVR in the high volatility state is more negative and the term structure is steeper. For the shortest maturities, it is valued at 1.15 in the high volatility states, versus 0.31 in the low volatility states. For the longest maturities, the respective values are 0.58 and 0.13. We find similar results when we split the data into two time periods, pre and post 2007, in Figure 7 and Table 7.

4

Conclusion

We provide estimates of the price of variance risk at various horizons, first, by measuring model-free Sharpe ratios of straddle returns with varying maturities and, second, by estimating the unit price of variance risk in a Heston (1993) model, based on the empirical approach developed by Christo↵ersen et al. (2013). We find the price of insurance against increases in volatilities varies with the horizon of the risk insured: short-term insurance is more expensive than long-term insurance, and this e↵ect is more pronounced in times of higher volatility. These results extend the accumulating evidence for non-trivial termstructure features to the market for variance risk. They provide strong support for the key prediction of the general equilibrium model with horizon-dependent risk aversion of Andries et al. (2014). 17

Prices of variance risk are more negative in the post 2007 data. As such, the dip at the 70–90 day maturity in Figure 4 and Table 4 is evidence for a time varying level of the term structure, and is not noise from estimation.

14

References Adrian, T. and J. Rosenberg (2008). Stock returns and volatility: Pricing the short-run and long-run components of market risk. Journal of Finance 63 (6), 2997–3030. Ait-Sahalia, Y., M. Karaman, and L. Mancini (2012). The term structure of variance swaps, risk premia, and the expectation hypothesis. Princeton University Working Paper . Amengual, D. (2009). The term structure of variance risk premia. Princeton University PhD Thesis. Andries, M. (2012). Consumption-based asset pricing with loss aversion. Working Paper . Andries, M., T. M. Eisenbach, and M. C. Schmalz (2014). Asset pricing with horizondependent risk aversion. Technical report, University of Toulouse, Federal Reserve Bank of New York, University of Michigan. Ang, A., R. J. Hodrick, Y. Xing, and X. Zhang (2006). The cross-section of volatility and expected returns. Journal of Finance 61 (1), 259–299. Bakshi, G., C. Cao, and Z. Chen (1997, December). Empirical performance of alternative option pricing models. Journal of Finance 52 (5), 2003–2049. Bansal, R., D. Kiku, I. Shaliastovich, and A. Yaron (2013). Volatility, the macroeconomy, and asset prices. Journal of Finance. Bansal, R. and A. Yaron (2004). Risks for the long run: A potential resolution of asset pricing puzzles. Journal of Finance 59 (4), 1481–1509. Barras, L. and A. Malkhozov (2015). Does variance risk have two prices? Evidence from the equity and option markets. Working Paper. van Binsbergen, J., M. Brandt, and R. Koijen (2012). On the timing and pricing of dividends. American Economic Review 102 (4), 1596–1618. van Binsbergen, J. and R. Koijen (2014). Real excess volatility. Technical report, Stanford University and London Business School. van Binsbergen, J. and R. Koijen (2015). The term structure of returns: Facts and theory. Working Paper . 15

Campbell, J. Y. and J. H. Cochrane (1999). By force of habit: A consumption-based explanation of aggregate stock market behavior. Journal of Political Economy 107 (2), 205–251. Campbell, J. Y., S. Giglio, C. Polk, and R. Turley (2012). An intertemporal CAPM with stochastic volatility. Working Paper. Campbell, J. Y. and L. Hentschel (1992). No news is good news: An asymmetric model of changing volatility in stock returns. Journal of Financial Economics 31 (3), 281–318. Carr, P. and L. Wu (2009). Variance risk premiums. Review of Financial Studies 22 (3), 1311–1341. Cheng, I.-H. (2014). The returns to fear. Working Paper . Choi, H., P. Mueller, and A. Vedolin (2015). Bond variance risk premiums. Working Paper . Christo↵ersen, P., S. Heston, and K. Jacobs (2013). Capturing option anomalies with a variance-dependent pricing kernel. Review of Financial Studies 26 (8), 1962–2006. Constantinides, G. M., J. C. Jackwerth, and A. Savov (2013). The puzzle of index option returns. Review of Asset Pricing Studies 3 (2), 229–257. Coval, J. D. and T. Shumway (2001). Expected option returns. Journal of Finance 56 (3), 983–1009. Curatola, G. (2014). Loss aversion, habit formation and the term structures of equity and interest rates. Working Paper . Dew-Becker, I., S. Giglio, A. Le, and M. Rodriguez (2014). The price of variance risk. Working Paper. Epstein, L. G. and S. E. Zin (1989). Substitution, risk aversion, and the temporal behavior of consumption and asset returns: A theoretical framework. Econometrica, 937–969. Giglio, S., M. Maggiori, and J. Stroebel (2013). Very long-run discount rates. Working Paper. Heston, S. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies 6 (2), 327–343. 16

Heston, S. and S. Nandi (2000). A closed-form GARCH option valuation model. Review of Financial Studies 13 (3), 585–625. Menkho↵, L., L. Sarno, M. Schmeling, and A. Schrimpf (2012). Carry trades and global foreign exchange volatility. Journal of Finance 67 (2), 681–718. Wachter, J. A. (2013, June). Can time-varying risk of rare disasters explain aggregate stock market volatility? Journal of Finance 52 (3), 987–1035.

17

Figures Daily Straddle Returns by Maturity Facet and by Day in Sample 11 − 30

30 − 50

50 − 70

70 − 90

90 − 110

110 − 130

130 − 150

150 − 170

170 − 190

190 − 210

210 − 230

230 − 250

0.10 0.05 0.00 −0.05 −0.10

0.10

Return

0.05 0.00 −0.05 −0.10

0.10 0.05 0.00 −0.05 −0.10 2000

2005

2010

2000

2005

2010

2000

2005

2010

2000

2005

2010

Date

Figure 1: Daily straddle returns used in Sharpe ratio analysis. Daily option and index price data from February 1996 to April 2011 come from OptionMetrics. Each dot represents the net arithmetic return of a straddle-day observation. Each facet of the plot contains options of di↵erent maturities, labeled at the top of each facet.

18

Maturity Calendar by Year, No Window Restriction 1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

2008

2009

2010

2011

250 200 150 100 50 0 250

Maturity of Option Observation

200 150 100 50 0 250 200 150 100 50 0 250 200 150 100 50 0 0

100

200

300

0

100

200

300

0

100

200

300

0

100

200

300

Day of Year

Figure 2: Calendar of options by maturity and date. Daily option and index price data from February 1996 to April 2011 come from OptionMetrics. Each dot represents an option-day observation. The vertical axis is the maturity, and the horizontal axis has the days of the year. Diagonal lines reflect the fact that not every maturity is traded every day, and that certain maturities are only observed on certain calendar days.

19

Daily Sharpe Ratios of Delta Neutral Straddles 0.0

Sharpe Ratio

−0.5

−1.0

0

50

100

150

200

Maturity in Days

Figure 3: Estimates of the Sharpe ratio of delta-neutral straddle returnsSR⌧ . Daily option and index price data from February 1996 to April 2011 come from OptionMetrics. The Sharpe ratio SR⌧ for options with maturity ⌧ is computed by collecting all returns from options with a maturity in the interval [⌧, ⌧ + 20), and then dividing the sample mean by the sample standard deviation. Dotted lines mark 95 percent confidence intervals formed by the 2.5 and 97.5 percentiles of 10,000 bootstrap estimates, and the solid line is the mean of such estimates.

20

Parametric Estimation of PVR

Unit Price of Volatility Risk

0.0

−0.2

−0.4

−0.6

0

50

100

150

200

Maturity in Days

Figure 4: Parametric estimation of the term structure of the price of variance risk, ⌧ . Daily option and index price data from February 1996 to April 2011 come from OptionMetrics. Return data from January 1990 to December 2014 come from CRSP. The first maturity bucket includes options with maturities of 11 through 29 trading days. Each subsequent point at maturity ⌧ represents the estimation results on options with maturity [⌧, ⌧ + 20). The last bucket contains options with maturities 230 through 250. Dotted lines mark asymptotic 95 percent confidence intervals conditional on the given realization of GARCH parameters from the second-stage maximum-likelihood estimation.

21

Parametric Estimation of PVR by Time Period 1996 − 2007

2008 − 2011

Unit Price of Volatility Risk

0.0

−0.5

−1.0

−1.5

0

50

100

150

200

0

50

100

150

200

Maturity in Days

Figure 5: Parametric estimation of the term structure of the price of variance risk, ⌧ , by time period. Daily option and index price data from February 1996 to April 2011 come from OptionMetrics. Return data from January 1990 to December 2014 come from CRSP. The first maturity bucket includes options with maturities of 11 through 29 trading days. Each subsequent point at maturity ⌧ represents the estimation results on options with maturity [⌧, ⌧ + 20). The last bucket contains options with maturities 230 through 250. Dotted lines mark asymptotic 95 percent confidence intervals conditional on the given realization of GARCH parameters from the second-stage maximum-likelihood estimation.

22

Estimates of PVR by Volatility State Low Volatility

High Volatility

Unit Price of Volatility Risk

0.0

−0.5

−1.0

0

50

100

150

200

0

50

100

150

200

Maturity in Days

Figure 6: Parametric estimates of ⌧ at di↵erent states of forecasted GARCH volatility ht+1 . Daily option and index price data from February 1996 to April 2011 come from OptionMetrics. Error bars mark asymptotic 95 percent confidence intervals conditional on the given realization of GARCH parameters from the second-stage maximum-likelihood estimation.

23

Estimates of PVR by Volatility State and by Time Period 1996 − 2007

2008 − 2011

0.0 −0.5 Low Volatility

−1.5 −2.0 −2.5 0.0 −0.5

High Volatility

Unit Price of Volatility Risk

−1.0

−1.0 −1.5 −2.0 −2.5 0

50

100

150

200

0

50

100

150

200

Maturity in Days

Figure 7: Parametric estimates of ⌧ at di↵erent states of forecasted GARCH volatility ht+1 and by di↵erent time periods. Daily option and index price data from February 1996 to April 2011 come from OptionMetrics. Error bars mark asymptotic 95 percent confidence intervals conditional on the given realization of GARCH parameters from the second-stage maximum-likelihood estimation.

24

25

Maturity 11–30 30–50 50–70 70–90 90–110 110–130 130–150 150–170 170–190 190–210 210–230 230–252 Total

N 3,042 3,547 3,180 1,876 1,500 1,479 1,494 1,510 1,488 1,485 1,469 1,638 47,416

St /K 99.9 99.7 99.4 99.1 98.6 98.4 98.1 97.8 97.5 97.2 96.9 96.6

Midprice ($) 24.45 34.15 41.98 50.26 52.94 57.63 62.66 66.31 69.65 73.19 76.32 78.36 52.07

Calls Bid-Ask Spread ($) 1.51 1.91 2.12 2.33 2.34 2.39 2.36 2.44 2.56 2.60 2.64 2.78 2.2 Bid-Ask Ratio (%) 6.30 5.63 4.87 4.19 3.81 3.53 3.21 3.06 3.00 2.87 2.78 2.75 4.68

Midprice ($) 24.08 35.00 44.73 56.95 62.28 68.66 75.10 81.37 86.20 91.88 96.84 101.64 98.58

Puts Bid-Ask Spread ($) 1.50 1.88 2.11 2.29 2.29 2.33 2.34 2.39 2.54 2.59 2.58 2.73 2.23

Bid-Ask Ratio (%) 6.17 5.7 5.19 4.71 4.46 4.17 3.91 3.75 3.77 3.67 3.54 3.62 4.22

Table 1: Summary statistics of options used in parametric estimation by maturity. Daily option and index price data from February 1996 to April 2011 come from OptionMetrics. A price on a day is defined as the midprice between the closing best bid and best ask. Maturity is defined as the number of days from the observation date to expiration. All maturity ranges are inclusive on the left and exclusive on the right. Midprice is the average between the best closing bid price and the best closing ask price on a particular day. St /K refers to the average ratio of the underlying stock price to the strike price of the option taken within that maturity category in percentage points. Bid-Ask Spread is the di↵erence between the best bid and best o↵er on a given day. Spread Bid-Ask Ratio is a percentage computed as Bid-Ask ⇥ 100. N refers to the total number of calls and puts Midprice at each maturity. All statistics except the observation count are computed as arithmetic means over option-day observations.

Tables

Table 2: Liquidity (bid-ask ratio) of options of various maturities and moneyness. All intervals are inclusive on the left and exclusive on the right. Bid-Ask Spread Ratio is computed as Bid-Ask ⇥ 100. Data come from the full universe Current Price of OptionMetrics data from February 1996 to April 2011, after cleaning for duplications.

Maturity 0–11 11–30 30–50 50–70 70–90 90–110 110–130 130–150 150–170 170–190 190–210 210–230 230–252

0–1% 159.78 53.46 7.14 5.25 4.57 3.95 3.92 3.54 3.36 3.34 3.27 3.26 3.23

Black-Scholes Delta 1–5% 5–10% 77.50 26.64 33.78 12.03 10.56 8.57 5.47 6.05 4.53 4.63 4.19 4.31 3.83 3.88 3.55 3.54 3.40 3.41 3.34 3.37 3.24 3.32 3.15 3.18 3.25 3.25

26

> 10% 35.90 17.21 12.40 8.51 6.54 5.47 4.86 4.41 4.16 4.04 3.87 3.70 3.64

27

N 749 1729 1865 1235 1039 1159 1215 1262 1282 1322 1333 1359

Maturity ⌧

11–30 30–50 50–70 70–90 90–110 110–130 130–150 150–170 170–190 190–210 210–230 230–250

-3.23 -1.63 -1.09 -0.78 -0.70 -0.56 -0.40 -0.41 -0.38 -0.33 -0.24 -0.25

E[r] 2.80 2.32 2.02 1.83 1.82 1.66 1.67 1.69 1.49 1.55 1.54 1.53

(r) -1.15 -0.71 -0.54 -0.43 -0.39 -0.34 -0.24 -0.24 -0.25 -0.21 -0.15 -0.16

SR⌧

95% CI Lower Bound -1.27 -0.77 -0.59 -0.49 -0.45 -0.4 -0.3 -0.3 -0.31 -0.27 -0.21 -0.22

95% CI Upper Bound -1.05 -0.65 -0.49 -0.36 -0.32 -0.27 -0.18 -0.18 -0.19 -0.16 -0.10 -0.11

Table 3: Point estimates of daily expected returns, standard deviation of returns, and Sharpe ratios of straddles of various maturities. The expected value and standard deviation are computed on log arithmetic returns multiplied by 5. Confidence intervals are also included for Sharpe ratios. Daily option and index price data from February 1996 to April 2011 come from OptionMetrics. N refers to the number of straddle-day return observations used to compute the Sharpe ratio. All maturity buckets are inclusive on the left and exclusive on the right. 95 percent confidence intervals are formed by the 2.5 and 97.5 percentiles of 10,000 bootstrap estimates, and the final estimate reported is the mean of such bootstrap trials.

Table 4: Point estimates and confidence intervals of the unconditional price of variance risk from the parametric estimation. Daily option and index price data from February 1996 to April 2011 come from OptionMetrics. An estimate of ⌧ is estimated by maximum likelihood in order to best price the subset of options in a given maturity bucket. N refers to the number of options observations used to compute ⌧ . All maturity buckets are inclusive on the left and exclusive on the right. Standard errors are derived from the delta method applied to the mapping from ⇠ ⌧ and ⌧ . Maturity ⌧

N

11–30 30–50 50–70 70–90 90–110 110–130 130–150 150–170 170–190 190–210 210–230 230–250

5932 6912 6200 3658 2920 2878 2908 2940 2896 2890 2858 2920

⌧

-0.61 -0.47 -0.46 -0.55 -0.40 -0.38 -0.39 -0.37 -0.37 -0.37 -0.36 -0.34

28

Standard Error 0.03 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02

29

11–30 30–50 50–70 70–90 90–110 110–130 130–150 150–170 170–190 190–210 210–230 230–250

Maturity ⌧

4554 5232 4332 1900 1970 1904 1956 1978 1886 1908 1900 1898

N -0.41 -0.30 -0.27 -0.25 -0.21 -0.20 -0.19 -0.19 -0.18 -0.17 -0.17 -0.16

⌧

1996–2007 Standard Error 0.03 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 1378 1680 1868 1758 950 974 952 962 1010 982 958 1022

N -1.41 -1.09 -0.94 -0.91 -0.82 -0.77 -0.84 -0.77 -0.73 -0.77 -0.78 -0.70

⌧

2008–2011 Standard Error 0.11 0.06 0.04 0.04 0.04 0.04 0.05 0.04 0.04 0.04 0.04 0.03

Table 5: Point estimates and confidence intervals of the unconditional price of variance risk from the parametric estimation by time period. Daily option and index price data from February 1996 to April 2011 come from OptionMetrics. An estimate of ⌧ is estimated by maximum likelihood in order to best price the subset of options in a given maturity bucket. N refers to the number of options observations used to compute ⌧ . All maturity buckets are inclusive on the left and exclusive on the right. Standard errors are derived from the delta method applied to the mapping from ⇠⌧ and ⌧ .

30

11–30 30–50 50–70 70–90 90–110 110–130 130–150 150–170 170–190 190–210 210–230 230–250

Low Volatility (Below Median) ⌧ N Standard Error 2991 -0.31 0.03 3590 -0.19 0.02 3180 -0.19 0.02 1681 -0.24 0.02 1379 -0.13 0.03 1537 -0.15 0.02 1438 -0.14 0.02 1399 -0.11 0.02 1500 -0.15 0.02 1448 -0.14 0.02 1368 -0.10 0.02 1448 -0.13 0.02

High Volatility (Above Median) ⌧ N Standard Error 2941 0.07 -1.15 3322 0.04 -0.93 3020 0.03 -0.87 1977 0.04 -0.89 1541 0.03 -0.69 1341 0.04 -0.71 1470 0.03 -0.67 1541 0.03 -0.65 1396 0.03 -0.64 1442 0.03 -0.62 1490 0.03 -0.62 1472 0.03 -0.58

Table 6: Point estimates and confidence intervals of the conditional price of variance risk from the parametric estimation. Daily option and index price data from February 1996 to April 2011 come from OptionMetrics. Volatility quartiles are formed based on the first-stage GARCH estimation parameter h. An estimate of ⌧ is estimated by maximum likelihood in order to best price the subset of options in a given maturity bucket. N refers to the number of options observations used to compute ⌧ . All maturity buckets are inclusive on the left and exclusive on the right. Standard errors are derived from the delta method applied to the mapping from ⇠ ⌧ and ⌧ .

31

11–30 30–50 50–70 70–90 90–110 110–130 130–150 150–170 170–190 190–210 210–230 230–250

Maturity ⌧ -0.20 -0.09 -0.06 -0.03 0.02 -0.02 -0.01 0.03 -0.01 0.00 0.03 0.02

0.03 0.02 0.02 0.03 0.03 0.02 0.02 0.03 0.02 0.02 0.02 0.02

-0.84 -0.70 -0.62 -0.56 -0.50 -0.48 -0.44 -0.46 -0.42 -0.41 -0.42 -0.37

0.05 0.03 0.03 0.03 0.03 0.04 0.03 0.02 0.03 0.03 0.02 0.03

1996–2007 Low Volatility High Volatility ⌧ ⌧ Std Error Std Error -0.83 -0.65 -0.54 -0.56 -0.54 -0.46 -0.54 -0.50 -0.46 -0.52 -0.51 -0.44

0.06 0.04 0.03 0.03 0.04 0.03 0.03 0.04 0.03 0.03 0.03 0.03

-2.06 -1.51 -1.34 -1.21 -1.05 -1.08 -1.06 -0.95 -0.97 -0.96 -0.94 -0.91

0.19 0.09 0.07 0.06 0.07 0.06 0.07 0.06 0.05 0.06 0.05 0.05

2008–2011 Low Volatility High Volatility ⌧ ⌧ Std Error Std Error

Table 7: Point estimates and confidence intervals of the conditional price of variance risk from the parametric estimation by time period. Daily option and index price data from February 1996 to April 2011 come from OptionMetrics. Volatility categorites are formed based on whether the first-stage GARCH estimation parameter ht is above or below its median over the full sample from 1996 to 2011.An estimate of ⌧ is estimated by maximum likelihood in order to best price the subset of options in a given maturity bucket. N refers to the number of options observations used to compute ⌧ . All maturity buckets are inclusive on the left and exclusive on the right. Standard errors are derived from the delta method applied to the mapping from ⇠ ⌧ and ⌧ .

Appendix Option Pricing in Discrete Time under a GARCH Model of the Underlying Appendix B of Christo↵ersen et al. (2013) gives a closed-form solution for the call price. Let t denote the current trading day, and let T denote the future trading day on which the option expires. Let St be the current stock price. Observe that the option price depends on both the particular current and future days, and not just the maturity, because there is dependence on the current state of volatility h⇤t+1 : CiMod = Ct St , h⇤t+1 , K, T = St P1 (t)

K exp ( r (T t)) P2 (t)  ˆ ⇤ K i' gt,T (i' + 1) 1 exp ( r (T t)) 1 P1 (t) = + < d' 2 ⇡ i'S (t) 0  ˆ ⇤ K i' gt,T (i') 1 1 1 P2 (t) = + < d' 2 ⇡ 0 i'

⇤ gi,T (') = exp ' log St + At,T (') + Bt,T (') h⇤t+1 1 At,T = At+1,T (') + 'r + Bt+1,T (') ! ⇤ log (1 2 AT,T = 0 1 Bt,T = ' + Bt+1,T (') + Bt+1,T (') ↵⇤ · ( ⇤ )2 2 1 2 ' + 2Bt+1,T (') ↵⇤ ⇤ (Bt+1,T (') ↵⇤ 2 + 1 2Bt+1,T (') ↵⇤

BT,T = 0 h⇤t+1

⇤

=! + =

h⇤t

ht . 1 2↵⇠

+↵

⇤

⇣

⇤

z (t)

⇤

2Bt+1,T (') ↵⇤ )

⇤

')

p ⌘2 h⇤t

The put price is computed using put-call parity. Observe that delta is just P1 (t). In the parametric estimation, we use Black-Scholes vegas, which come from inverting

32

the Black-Scholes formula: ˆ = arg min St (d1 ) log SKt + r + p ⌧ p d2 = d1 ⌧ p BSVMkt = ⌧ St 0 (d1 ) t d1 =

1 2

2

Ker(T

t)

(d2 )

C Mkt

2

⌧

= CDF of Standard Normal. It remains to link the discrete-time parameters above to the price of variance risk in equation (3). We follow Christo↵ersen et al. (2013) by calibrating , in order to replicate the same unconditional variance of ht and the same ratio of unconditional variance under the risk-neutral measure and under the physical measure: = 1

↵

2

⇥ 252

!+↵ 1 ↵ 2 ⇤ ! + ↵⇤ E ⇤ (h⇤t ) = 1 ↵⇤ ⇤2 E ⇤ (h⇤t ) E(ht ) =  . E ⇤ (h⇤t ) E(ht ) =

(11)

There are two potential reasons the standard errors computed from the parametric estimation are too narrow: 1. There is uncertainty in the estimation of GARCH parameters, and the sequential estimation procedure does not propogate this uncertainty into the standard errors for . 2. The parametric procedure treats the pricing errors as independent over time, which may not be the case if there are unobserved shocks that cause the prices of all options of a certain expiration date to move together. There are two reasons why the uncertainty in the GARCH parameters is unlikely to play a quantitatively large role on our results. First, the time period for the GARCH estimation is very long – it covers over 6,301 trading days. Any uncertainty would be very small. Second, recall that the key mechanism that determines ⇠ˆ is trying to match the wedge 33

between the implied physical volatility from the GARCH process and the implied riskneutral volatility from the options price. Although variation in the GARCH parameters may change the size of the wedge and therefore increase the variance of the estimate ⇠ˆ on any given subset, that variation does not imply we are overstating di↵erences in ⇠ˆ between subsets. To address the problems due to correlated pricing errors in the time dimension, observe that in a given maturity window of 20 trading days there are up to 20 repeated observation of contracts with a given expiration date. If we adopt the standard convention in the option pricing literature that pricing errors are independent at the weekly horizon18 , then it is as if we only observe a fourth as many observations and therefore the standard error when computed under an assumption of iid pricing errors should be multiplied by 2. These are the standard errors reported in the plots and figures for the parametric estimation. Note that no similar correction needs to be made for the Sharpe ratios as those look at returns, which would still be independent if there were a persistent shock that raised prices.

18

Christo↵ersen et al. (2013) for example consider weekly options in their estimation of ⇠, and interpret the resulting log likelihood as the result of iid pricing errors.

34