Limits of Dynamic Hedging and Option Risk Premium

Meng Tian, Liuren Wu

Zicklin School of Business, Baruch College, The City University of New York

First draft: October 28, 2017; This version: January 2, 2021

Abstract

Classic option pricing theory shows that the risk of underwriting an option can be completely removed via dynamic delta hedging when the underlying security can be traded frictionlessly without cost and the security price moves diffusively with known volatility. In such an idealistic environment, the option contract is redundant and underwriting it does not earn any extra risk premium other than the risk premium embedded in the underlying security. In practice, there are limits to the dynamic hedging strategy that prevent the option underwriters from fully removing their risk exposures. These limits include the non-zero cost associated with trading the underlying security for delta hedging, uncertainty about the security return’s volatility level and its variation, and random security price jumps that cannot be effectively hedged by the delta hedging strategy. To the extent that delta hedging can be costly and cannot fully remove all the risks, investors require a premium for the remaining risk in underwriting an option. This paper documents the effectiveness of delta hedging on U.S. stock options under practical situations, examines the cross-sectional variation of investment returns from underwriting options on different stocks, and attributes the cross-sectional variation to variations in limits to dynamic hedging.

JEL Classiﬁcation: C13; C51; G12; G13

Keywords: Dynamic hedging; Option investment returns; Limits to dynamic hedging; Trading cost; Stochastic volatility; Jumps

We thank Peter Carr, Xi Dong, Yuzhao Zhang, and seminar participants at Baruch College for their comments and suggestions. Liuren Wu gratefully acknowledges the support by a grant from the City University of New York PSC- CUNY Research Award Program. E-mail: [email protected] (Tian) and [email protected] (Wu). 1. Introduction

Black and Scholes (1973) and Merton (1973) (BMS) revolutionized derivative trading not only by deriving an analytically tractable Black-Merton-Scholes (BMS) option pricing formula for Euro- pean options, but also, more importantly, by offering the insight that the risk of underwriting an option contract can be completely removed by dynamic delta hedging with the underlying security, provided that the underlying security can be traded continuously and frictionless without cost and the security price moves diffusively with an ex-ante known volatility level. Underwriting an option contract naked can bring large liabilities to the underwriter. Risk limits prevent investors from underwriting too many option contracts, severely limiting the growth of the derivatives market. The

BMS insight helps underwriters remove a large portion of the underwriting risk from their book via dynamic delta hedging, thus allowing them to greatly expand their underwriting capacity, making the BMS insight one of the biggest contributors to the rapid growth of the derivatives market.

Under the BMS assumption of frictionless market and Brownian movement with constant volatility, the option contract becomes redundant as it can be perfectly replicated via dynamic trading of the underlying security and a riskfree bond. As such, underwriting an option does not earn any extra risk premium, other than the risk premium already embedded in the underlying security. In reality, however, there are limits to the dynamic hedging strategy that prevent the option underwriters from fully removing their risk exposures (Figlewski (1989)). These limits include the non-zero cost associated with trading the underlying security for dynamic delta hedging (Le- land (1985)); uncertainty about the security return’s volatility level upon which the hedging ratio is calculated (Green and Figlewski (1999)); and the presence of other types and sources of risks, such as random jumps and stochastic volatility, which cannot be effectively eliminated by delta hedging. Researchers have proposed reﬁned dynamic and static hedging strategies that can hedge

1 additional risk sources with options as additional hedging instruments,1 but the trading cost of options as hedging instruments can be much higher than the underlying security. To the extent that hedging can be costly and cannot fully remove all the risks, investors can require a premium for the remaining risk in underwriting an option contract. These remaining risks become primary risks in the sense that they cannot be effectively taken with the underlying security alone — The derivative securities in this case play a primary role in risk allocation, a role that cannot be fulﬁlled by the underlying security alone.

In this paper, we document the effectiveness of delta hedging on U.S. stock options under practical situations, and we examine the cross-sectional variation of delta-hedged investment excess returns from underwriting options on different individual stocks. We construct risk factors that capture the delta hedging costs and option investment risks that are difﬁcult or costly to hedge, and we attribute the cross-sectional variation in option investment returns to variations in these risk factors.

We start our analysis with one-month at-the-money options. Each month, we underwrite one- month at-the-money call and put options across the universe of U.S. individual stocks and hold the option positions to expiry. We examine the effectiveness of delta hedging in reducing the risk of the option investment by comparing three strategies: (i) leaving the option investment naked, (ii) performing one-time delta hedge at the initiation but without any delta rebalancing, and

(iii) performing delta rebalancing daily whenever the underlying stock is traded and valid pricing information is available for updating the delta calculation.

1Prominent examples include Renault and Touzi (1996) and Basak and Chabakauri (2012), who consider optimal hedging under a stochastic volatility model; Hutchinson, Lo, and Poggio (1994), who propose to estimate the hedging ratio empirically using a nonparametric approach based on historical data; Branger and Mahayni (2006), who propose robust dynamic hedges in pure diffusion models when the hedger knows only the range of the volatility levels but not the exact volatility dynamics; and Carr and Wu (2004) and Wu and Zhu (2016), who propose static replication strategies with options to better span the risk of jumps of random sizes.

2 When the option positions are left naked, the excess investment returns in percentage of the underlying stock price level have a pooled standard deviation of 8.71% for call options and 7.72% for put options. When we perform one-time delta hedge on the option investment at initiation without update throughout the one-month investment horizon, the pooled standard deviation estimates are reduced to 4.33-4.34% for the put and call investments. Compared to the naked position, one-time delta hedge removes 75% of the variance for the call investment, and 69% of the variance for the put investment. When we further rebalance the delta hedge daily whenever possible to keep the delta exposure small throughout the investment, the standard deviation of the investment return is further reduced to 2.59-2.63%, amounting to 91% variance reduction for the call investment and

88% variance reduction for the put investment.

The large variance reduction of delta hedging highlights the practical signiﬁcance of the BMS insight on dynamic delta hedging. Despite many well-known practical issues in delta hedging implementation, such as unknown true dynamics and accordingly uncertain hedging ratios and additional risk sources such as stochastic volatility and jumps of random size, performing a simple one-time delta hedge at the initiation, although far from perfect, can still remove about 70% of the risk for an 30-day at-the-money option investment. When the underlying stock is liquid and can be traded with low cost, performing daily delta rebalancing can further increase the variance reduction to about 90%. Dynamic delta hedge can therefore drastically reduce the risk of underwriting options, allowing the stock options market to blossom.

Despite the effectiveness of delta hedging, the remaining risks of the delta-hedged option positions are still large and signiﬁcant. For option underwriters to bear such risks, they should demand a risk premium. We construct three risk measures that capture a company’s exposure to the unhedgeable risks: (i) delta hedging cost (HC), as measured by the trading cost of the underlying stock, (ii) stochastic volatility risk (VR), as measured by the volatility of the underlying option

3 implied volatility, and (iii) random jump risk (JR), as measured by the excess kurtosis of the stock return. We hypothesis that the expected excess returns on the option investment increase with these three types of risks. We construct a cross-sectional capital asset pricing model on the excess returns of the option investments, where we regress the excess returns on the three risk factors cross-sectionally, while also controlling for historical average investment returns and the volatility risk premium constructed as the difference between the current option implied volatility and a future return volatility forecast. The coefficient estimate on each risk factor represents the excess return on the corresponding option factor portfolio, which has one unit exposure to the targeted risk factor and zero exposures to other risk factors. The time-series averages of the coefficient estimates reflect the average risk premium charged on each of the risk factors.

Estimation shows that the risk premium estimates on the three risk factors are all positive and statistically significant. In addition, past average returns (historical premium) and volatility risk premium both positively predict future investment returns in the cross section. The average coefficient estimates are similar regardless of whether we perform the regression on excess returns for underwriting the call option, the put option, or the straddle combination, and regardless of whether and how frequent we perform the delta hedge, except that delta hedging increases the statistical significance of the risk premium estimates by removing a large proportion of the return noise due to directional exposures.

In addition to the cross-sectional pricing relation, we also formulate an intertemporal asset pricing model to examine how the excess returns on the different factor portfolios vary intertemporally with the average levels of the risk factors. We ﬁnd that the excess returns on each factor portfolio increase strongly with the level of the corresponding risk factor, suggesting that the market pricings of these risk factors are positive both in the cross section and intertemporally.

4 When we extend the analysis to 30-day investment returns on other option moneyness and maturity buckets, we ﬁnd that daily delta hedging is equally effective across the different buckets.

Compared to at-the-money options, the percentage variance reductions on out-of-the-money options become smaller, averaging around 80% at about 25-delta; nevertheless, the remaining risk level is also smaller, with the lower percentage mainly driven by the smaller risk magnitude to begin with on the naked positions. Across option maturity buckets, as the option maturity increases, the option gamma (and accordingly the option delta variation) becomes smaller, the need for daily delta rebalancing becomes smaller, and the one-time delta hedging at initiation becomes increasingly effective. For at-the-money options with about one-year maturity, just one-time delta hedge at initiation can already remove about 90% of the variance of the 30-day investment return.

Given the increasing effectiveness of one-time delta hedge at initiation for longer-dated options, the delta hedging cost becomes less a concern for underwriting longer-dated options for the same investment horizon. When we estimate capital asset pricing model on 30-day excess returns across different option moneyness and maturity buckets, we ﬁnd that the main pricing variation is across option maturities. While the volatility risk factor and jump risk factor are positively priced across all maturities, the delta hedging cost risk factor is no longer positively priced when the option maturities are three months or longer. In addition, the historical risk premium factor also loses its pricing power on longer-dated options while the pricing of the volatility risk premium factor remains positive and strongly signiﬁcant.

The condition of no dynamic arbitrage has been the starting point for most of the derivative pricing literature; yet this arbitrage-free pricing perspective makes the derivative contracts redundant, leaving little role for its existence in investment analysis. While the limits of arbitrage have long been recognized (Shleifer and Vishny (1997)), it is exactly these limits that allow derivative contracts to play primary roles in spanning the remaining risks. Garleanu,ˆ Pedersen, and Potesh-

5 man (2009) build a theoretical model to show the net demand for these remaining risks interact with the inventory risk faced by options market makers to affect the option valuation. Our analysis and factor model construction provides a systematic framework for analyzing the characteristics of these remaining risks, their marginal contributions to the cross-sectional variation of the option investment returns, and the average risk premiums for bearing each type of the remaining risks.

In related literature, Christoffersen, Goyenko, Jacobs, and Karoui (2018) use intraday option trades and quotes data to construct an option illiquidity measure based on effective bid-ask spreads and show that the average underwriting option returns increase with the option illiquidity. Since option contracts are traded much less frequently than the underlying stock, even though most stock market makers do not hold signiﬁcant stock inventory overnight, option market makers are expected to hold most of their option positions to expiry and manage their inventory risk via delta hedging with the underlying stock while also controlling their net volatility exposures of their option positions. The bid-ask spread the option market makers charge represent largely a compensation for the delta hedging cost they pay and the unhedgeable inventory risk they take. The capital asset pricing model that we construct in this paper traces the illiquidity risk premia to the underlying drivers of the option bid-ask spread.

A long list of studies have examined the behavior of stock option returns and their underlying risk factors.2 Several of the identiﬁed risk factors are directly or indirectly related to the options hedging cost or the unhedgeable risks embedded in our capital asset pricing model. In particular,

Cao and Han (2013) ﬁnd that delta-hedged stock option returns decrease with increasing idiosyncratic volatility of the underlying stock. Idiosyncratic volatility is a key determinant of the trading cost of the underlying stock and accordingly a key determinant of the option’s delta hedging cost.

2See, for example, Coval and Shumway (2001), Bakshi, Kapadia, and Madan (2003), Goyal and Saretto (2009), Cao and Han (2013), and Boyer and Vorkink (2014).

6 Huang, Schlag, Shaliastovich, and Thimme (2019) identify volatility of volatility risk as a signiﬁ- cant risk factor affecting index and volatility index option returns. It is indeed one of the major risk sources that cannot be effectively removed by dynamic delta hedging. Boyer and Vorkink (2014) use return skewness instead of kurtosis as a measure of the jump risk, and identify the unhedgeale jump risk as a key source of expected returns on stock options.

The remainder of the paper is organized as follows. The next section describes the data source, the universe selection criteria, and the option return calculation. Section 3 examines the effectiveness of the delta hedging strategy with different updating frequencies. Section 4 builds a capital asset pricing model on the derivative investments, highlighting the option’s primary risk taking role on the unhedgeable risks. Section 5 extends the analysis from one-month at-the-money options to options at other moneyness and maturity buckets. Section 6 concludes.

2. Data collection and option return calculation

We perform the empirical analysis on the US stock and stock options market using 24 years of data from January 1996 to November 2019.

2.1. Data sources and selection criteria

We obtain the stock options pricing information from Optionmetrics, which also provides the sup- porting information, including the interest rate curve and the underlying stock price and return series.

We sample the data monthly. We determine the sampling date of each month based on the common expiration date for morning-settlement individual stock options that expire the next month

7 so that the options expire exactly 30 calendar days from the chosen sampling date.

On each sampling date, we determine the stock universe of our analysis based on the following

ﬁltering criteria: (i) the underlying security is a common stock, and (ii) the stock price is greater than $5. We also ﬁlter the options data by excluding options with special settlements, with zero open interest, with mid price less than $1/8 or violating arbitrage bounds, with best bid at zero, missing, or higher than the best offer price, with the absolute option delta greater than 1, and with the month containing distributions. For each stock in the chosen universe, we select the strike of the 30-day call and put options closest to the stock’s spot price level for our primary investigation.

We also select options at other moneyness and maturity buckets in a latter section for extended analysis.

The ﬁltered sample includes a total of 6,299 companies over the 24-year period. The size of the universe on each sampling date ranges from 423 companies (in February 1996) to 1,862 companies

(in June 2018), with an average of 1,285 companies per sampling date. Figure 1 plots the number of selected companies per date.

[Figure 1 about here.]

2.2. Underwriting options with and without delta hedging

When an investor underwrites an option, her risk exposure can vary drastically depending on whether and how she hedges the risk exposures of her option position. We compute the underwriting option return based on three different hedging strategies: (i) underwrite the option naked without delta hedging, (ii) underwrite the option and perform one-time delta hedging with the underlying stock at the initiation without further delta rebalancing throughout the holding period,

8 and (iii) underwrite the option and perform delta hedging with the underlying stock with daily

rebalancing at the market close whenever the underlying stock is traded and there is valid pricing

information for updating the delta calculation.

Formally, let t denote the sampling date, T denote the option expiry date 30 days later, K denote the option strike, and (St,ST ) denote the stock price at initiation and expiry, respectively, and

(Ct,Pt) denote the mid prices of the chosen 30-day call and put option at strike K. We use rt(h) to denote the time-t continuously compounding ﬁnancing rate with maturity h, linearly interpolated from the interest rate curve provided by OptionMetrics, and use the interest rate to compute the

ﬁnancing gain or cost from the money market account. The excess proﬁts and losses from underwriting the 30-day call and put options under the three strategies can be computed as follows:

1. The net proﬁts and losses from underwriting the call and put options naked without delta

hedge are,

nh rt τ + πC,t,T = Cte − (ST − K) , (1) nh rt τ + πP,t,T = Pte − (K − ST ) .

To reduce notation cluttering, we henceforth omit the horizon dependence on the interest rate

when no confusion shall occur. At option expiry, we no longer have option price quote and

we replace the option price with its terminal payoff, computed based on the closing stock

price ST at expiry. Although the options are American style and can be exercised early, the

chance is low for one-month at-the-money options with no forthcoming distributions.

2. When the underwriter performs one-time delta hedge at initiation, the net proﬁts and losses

are adjusted as,

ih rt τ + rt τ πC,t,T = Cte − (ST − K) + ∆C,t (ST − Ste ), (2) ih rt τ + rt τ πP,t,T = Pte − (K − ST ) + ∆P,t (ST − Ste ),

9 where (∆C,t ∆P,t) denote the BMS delta of the call and put option, respectively, provided by

OptionMetrics and computed via a lattice method.

3. To perform dynamic rebalancing on the delta hedge, we ﬁrst identify the list of business days

from the date of initiation to the day before expiry when we have valid stock price quote,

valid option price quote, and valid delta computation to perform the delta hedge rebalancing.

We perform delta rebalancing in these days and hold the positions unchanged in between if

there are holidays or days with incomplete pricing information due to, for example, lack of

n trading activities. We use {t j} j=1 to denote the list of rebalancing days with t1 = t denoting the initiation date when we underwrite the option and put on the initial delta hedge position

and tn denoting the last rebalancing date before expiry, with n being the number of total delta

rebalances through the investment horizon. We use tn+1 = T to denote the expiry time and

τ j = t j+1 −t j to denote the time length in between each rebalancing. With these procedural

settings and notations, we can compute the net proﬁts and losses from underwriting the call

and put options with daily delta rebalancing as,

dh + n rt j τ j πC,t,T = Ct − (ST − K) + ∑ j=1 ∆C,t j St j+1 − St j + Ct j − ∆C,t j St j e − 1 , (3) dh + n rt j τ j πP,t,T = Pt − (K − ST ) + ∑ j=1 ∆P,t j St j+1 − St j + Pt j − ∆P,t j St j e − 1 .

With frequent delta rebalancing trades on the underlying stock, we need to separately account

for the capital gains or losses from the stock price movements and the ﬁnancing gains or costs

from the changing money market account at each trading date and sum them together. The

construction in (3) is general and accommodates any rebalancing frequencies, including the

second case of one-time delta hedge at initiation, with n = 1.

To make the investments comparable across different stocks, we scale the net proﬁts and losses

10 on the option investment by the price level of the underlying stock, and label the scaled proﬁts as

the excess returns on the option investments. We compute the excess returns on the call and the

put separately, as well as on the straddle combination. For each of the three strategies, we compute

the option investment returns as,

πC,t,T πP,t,T (πC,t,T + πP,t,T )/2 rC,t,T = , rP,t,T = , rS,t,T = . (4) St St St

Defining returns on derivative investments are not as straightforward as defining returns on buying a primary security because some derivative contracts has low or even negative premiums but can nevertheless have large risk exposures. From the risk management perspective, the returns are usually defined on required investment capital, i.e., the amount of equity capital the investor must put in the margin account for pursuing the investment. The required capital can vary for different types of investors. We use the security price, or the so-called notional amount, as our base for the return calculation. For the straddle investment, we split the investment in the call and put option and use the average net profits and losses to define the excess return.

3. Effectiveness of delta hedge in removing option risks

Before we examine the pricing of unhedgeable risks, we ﬁrst document the effectiveness of delta hedging in removing the risks of underwriting options. For this purpose, we compare the risk behavior of the option underwriting returns under the three hedging strategies. Table 1 reports the summary statistics on the pooled sample of 368,657 company-date observations of investment excess returns from underwriting the 30-day at-the-money call and put options separately as well as in straddle combination, and holding the option positions to expiry.

11 Panel A leaves the option positions naked without delta hedge. On average underwriting call options naked makes only two basis points whereas underlying put options makes 52 basis points.

In both cases, the average excess returns are extremely small compared to the large standard deviation estimates, making naked options investment an extremely risky proposition. The call and put option pricesaverage at slightly higher 5% of the stock price level. The table shows that underwriting call and put options naked can make 4.56-4.59% of the stock price level for 25% of the time, which amounts to collecting the option premium without paying much later; nevertheless, as the underlying stock price goes the other direction, the underwriter can also lose twice as much as the premium for about 10% of the time. Furthermore, underwriting options generate excess returns with negative skewness and high kurtosis, highlighting the inherently adverse risk exposure of options underwriting.

[Table 1 about here.]

By underwriting the call and the put option at the same time as a straddle combination, the investment’s delta exposure becomes much smaller. As a result, the standard deviation of the excess return becomes about half as much as that for the call and put investments separately.

3.1. Variance reduction via one-time delta hedging

Panel B of Table 1 delta hedges the option position with the underlying stock at initiation of the underwriting, but without further dynamic rebalancing. With delta hedging at initiation to remove the directional exposure, the standard deviation estimates are reduced by half for the call and put investments. For the straddle investment, since the initial delta of the straddle is close to zero, the one-time delta hedging position is small and have only a mild effect.

12 In last column of the table, we report the percentage variance reduction through the delta hedge.

The variance reduction is computed as one minus the variance ratio of hedged to unhedged investment returns. Over the pooled sample, performing one-time delta hedging at initiation can reduce the investment return variance by 75% for the call option investments and by 69% for the put option investments.

Combining the two option positions into a straddle drastically reduces the delta exposure and according the delta-hedge need. Nevertheless, due to the discreteness of the option strikes, it is not always practically possible to choose an option strike to make the chosen straddle delta-neutral.

Hedging the remaining small delta exposure of the straddle with the underlying stock can reduce the return variance by another 10%.

Black and Scholes (1973) and Merton (1973) show that, under their specified model environment of frictionless market and a geometric Brownian motion dynamics, an investor can completely remove the option investment risk if she perform dynamic delta hedge with continuous rebalancing. Panel B of Table 1 shows that even though the actual market investment can differ from their specified model environment and investors do not even know the exact dynamics and accordingly the exact risk exposures, investors can still remove about 70% of a 30-day call and put option investment by performing the BMS delta hedge just once at initiation, without any further updating. While the remaining risks are still large, the results nevertheless highlight the practical significance of their revolutionary insights.

3.2. Variance reduction via daily-rebalanced delta hedging

When the underlying stock is heavily traded with small trading cost, investors can afford to perform the dynamic hedging more frequently, moving closer to the BMS suggestion. In Panel C of Table 1,

13 we rebalance the delta of the option position every business day, whenever the underlying stock is traded that day and we have valid pricing information to update the delta calculation.

Through daily delta rebalancing, the standard deviations of the investment returns are further reduced by another 40%. Compared to the unhedged option returns, the average variance reduction is 91% for the call option investment, 88% for the put option investment, leaving only about 10% of risk. While the imperfections of the BMS model are well documented, the 90% variance reduction represents a remarkable achievement.

Even for the straddle investment with little delta exposure at initiation, daily delta rebalancing allows the investor to keep the exposure small across the whole life of the investment and achieve a further 69% variance reduction.

Due to put-call parity for European options, the delta hedged option positions should exhibit similar risk behaviors for calls, puts, and straddles. Indeed, the standard deviation estimates are close to one another from the three contracts in Panels B and C, and the variations in the percentage variance reduction are mainly caused by the variations of the denominator, i.e., the variance of the unhedged returns on the three contracts.

3.3. Stability of variance reduction over time

The variance reduction estimates in Table 1 are computed on the pooled sample. To examine the stability of the variance reduction from the two delta hedging strategies, we estimate the percentage variance reduction with a one-year rolling window. Figure 2 plots the time-series of the rolling- window percentage variance reduction estimates. Panel A plots the rolling estimates on the one- time hedging strategy while Panel B plots the estimates on the daily-rebalanced delta hedging strategy. In each panel, the solid line and the dashed line denote the variance reduction estimates

14 on the call option and put option investment, respectively. The dotted line represents the variance reduction on the straddle.

[Figure 2 about here.]

The rolling estimates vary closely to the pooled average estimates. When investors only perform one-time delta hedge in panel A, the percentage variance reduction estimates stay above 70% for the call option investment. For the put option investment, there are a few incidences when the estimates show sharp drops, with the minimum estimate at 41%. When the market persistently moves in one directions during the horizon of the investment, the delta exposures can become increasingly large after initiation, ultimately leading to large return variations when the market shows large moves.

By contrast, Panel B shows that these incidences of breakdowns do not happen when investors perform daily delta rebalancing whenever they can. Over the whole sample period, the lowest rolling variance reduction estimate for the daily rebalancing strategy is 85% for the call investment and 82% for the put investment.

The large variance reduction of delta hedging highlights the practical signiﬁcance of the BMS dynamic delta-hedging insight. Despite many well-known practical issues in delta hedging implementation, such as unknown true dynamics and accordingly uncertain hedging ratios and additional risk sources such as stochastic volatility and jumps of random size, performing a simple one-time delta hedge at the initiation, although far from perfect, can still remove about 70% of the risk for an 30-day at-the-money option investment. When the underlying stock is liquid and can be traded with low cost, performing daily delta hedge rebalancing can further increase the variance reduction to 90%. The effectiveness of delta hedge can therefore drastically reduces the risk of underwriting options, allowing the stock options market to blossom.

15 Nevertheless, the delta-hedged option positions are far from risk free. Even with daily delta hedging rebalancing, the standard deviation estimates of 2.53% per month for the straddle investment remain very high in absolute terms. It annualizes to about 9% per annum, which is about half of the volatility level of the stock market portfolio. Even with the delta hedging, underwriting options remains highly risky. The 0.21% average monthly return represents a small compensation for bearing the remaining risks.

In practice, option underwriting risks can vary greatly across different underlying stocks. For options on illiquid names, delta hedging can be costly and one-time hedge at initiation is all the underwriters can afford to do. On the other hand, for some highly liquid names, investors can rebalance the delta hedge more frequently to remain delta neutral throughout the investment to reduce the investment risk. For some names with highly actively traded options, investors can potentially also use option positions opportunistically to reduce risk without paying much trading cost. For stocks that are highly correlated with the market, one can also perform beta-hedging with index futures to reduce risk. Finally, stocks that experience lower variations in their return volatility and lower chances of having random price jumps can also experience more variance reduction via delta hedge. Taken together, the actual risk exposure of underwriting options can be quite different for different companies, and investors can ask for different levels of compensation accordingly.

In the following section, we construct company characteristics to capture the cross-sectional variations of the practical limits of dynamic hedging, and accordingly the remaining risks of option investment, and we link these characteristics to the expected returns for underwriting options.

16 4. Primary risk taking in derivative securities

When a derivative contract possesses risks that cannot be fully hedged with the underlying security, the derivative contract becomes a primary security in the sense that it is no longer a redundant security and can play a primary role in risk allocation. Investors can take on these primary risks for a risk premium. In this section, we examine the market pricing of these primary risks in underwriting one-month at-the-money options, from both a cross-sectional perspective and their intertemporal variation.

We have constructed multiple excess returns on one-month at-the-money call and put options as well as on the straddle combination, and with different delta hedging strategies. To streamline the exposition, our main analysis focuses on the excess returns on the one-month at-the-money straddle investment with one-time delta hedge at initiation. Forming a straddle is a cost-effective way of removing much of the delta exposure at initiation. Options exchanges nowadays offer quotes directly on such straddle combinations. Furthermore, because of its reduced delta exposure, the effective bid-ask spread on the straddle combination can even be lower than than the spread on the separate call and put quotes. Hedging the remaining small delta exposure of the straddle at initiation with the underlying stock is readily achievable for most stocks. In a later subsection, we examine how the pricing results vary across the different return constructions with different delta hedging frequencies.

4.1. Constructing factors of hedging costs and unheadgeable risks

We categorize the hedging costs and unhedgeable risks into the following three dimensions:

1. Delta hedging cost (HC): While dynamic delta hedge with the underlying stock can remove

17 a large proportion of the risk in underwriting a stock option, this risk reduction operation is no longer free when trading the stock incurs a signiﬁcant cost. When the trading cost is prohibitively high, it can prevent the option underwriter from performing the delta hedge at all, or at least prevent the underwriter from rebalancing the delta hedge as frequently as they would have wanted, thus leaving the option positions exposed to directional stock price movements.

The cost of buying or selling a certain amount of a stock tends to increase with the return volatility of the stock and the size of the trade as a fraction of the average trading volume on the stock. The ratio of return variance to dollar trading volume has been widely adopted in the industry to capture the cross-sectional variation in trading cost (Barra (1997)). The ratio is similar in spirit to the illiquidity measure of Amihud (2002), and has been justiﬁed from various theoretical perspectives (e.g., Grinold and Kahn (1999), Gabaix, Gopikrishnan,

Plerou, and Stanley (2006), and Gatheral (2010)).

At each date t, and for each stock i, we compute the average daily stock return variance

2 (σt,i), the daily stock return correlation with the market portfolio (ρt,i), and the average daily dollar trading volume (DVt,i, in millions) over the past quarter. We use the ratio of the stock return variance uncorrelated with the market to the average dollar trading volume to proxy the relative magnitude of the delta hedging cost (HCt,i),

2 2 HCt,i = σt,i(1 − ρt,i)/DVt,i. (5)

When the return on an individual stock is highly correlated with the market, one can perform dynamic “beta” hedging with highly liquid index futures to remove a large portion of the directional exposure of the stock, even if the individual stock itself is highly illiquid and

18 costly to trade. The cost of beta hedging can be much smaller than the cost of delta hedging

because of the highly liquid nature of index futures. Therefore, in constructing the hedging

cost measure in (5), we use the idiosyncratic return variance instead of the total variance to

capture the portion of the stock’s return risk that needs delta hedging with the underlying

stock, while excluding the portion that can be hedged with the highly liquid index futures.

2. Stochastic volatility risk (VR): Dynamic delta hedge can remove the directional exposure of

the option position, but the delta-hedged option position can still experience proﬁt and loss

ﬂuctuations when the underlying stock’s return volatility varies randomly over time. The

more volatile the return volatility can vary over time, the larger is this remaining risk.

We measure the volatility risk using the historical standard deviation estimate on the daily

log changes in the three-month at-the-money implied volatility over the past month. Option-

Metrics provides smooth-interpolated implied volatility estimates across a matrix of delta

and time to maturities. We take three month and 50 delta as the central pivot point of the

implied volatility surface and use its historical variation as a proxy for the variability of the

stock return volatility. OptionMetrics provides separate implied volatility estimates for 50-

delta call and 50-delta put. We take the average of the two to create the at-the-money time

series for the volatility of volatility estimation.

3. Random jump risk (JR): Dynamic delta hedging is effective in removing the impacts of small

diffusive directional stock price movements when the return volatility is known ex ante, or

the impacts of large binary jumps with ﬁxed and known jump sizes, but it is not effective in

hedging the risk of jumps of random sizes.

The presence of random price jumps not only reduces the effectiveness of dynamic delta

hedging, but also makes the stock return distribution exhibit fatter tails than normal. We

19 estimate the excess kurtosis of the daily stock return over the past month as a proxy for the

intensity of the jump risk, and we multiply the excess kurtosis with the historical volatility

estimator to create the jump risk factor.

In theory, investors can reduce the volatility risk exposure with dynamic vega hedging using related option contracts (Renault and Touzi (1996) and Branger and Mahayni (2006)) and can reduce exposures to random jump risk via static positioning in options across a whole spectrum of strikes (Carr and Wu (2004) and Wu and Zhu (2016)). In practice, however, the trading cost of options is much higher than the trading cost on the underlying stock, making hedging with options a highly costly practice. When investors underwrite an option contract, they must prepare to hold the contract to expiry, while only making option-related hedging adjustments opportunistically by, for example, taking advantage of occasionally incoming order ﬂows.

To the extent that delta hedging with the underlying stock can be costly and is virtually ineffec- tive in eliminating volatility risk and jump risk, the derivative contracts are no longer redundant, and can play primary roles in one’s investment decisions. Option underwriters can expect to earn positive returns by bearing these unhedgeable, primary risks.

In addition to the above list of hedging costs and unhedgeable risks, we consider two control metrics to capture the average cross-sectional option return variations induced by other structural reasons or risks that our list fails to capture:

1. Historical risk premium (HRP): We compute the average return on the one-month at-the-

money options over the past 12 month excluding the last month, and use this average re-

turn as a proxy for historical cross-sectional variations in the average returns on the option

investments. Heston and Li (2020) consider a similar measure and refer it as the option

momentum factor. If the company characteristic factors that we have constructed to capture

20 the costs and limits of dynamic hedging predict option returns, since these company charac-

teristics tend to persist over time, we expect that the cross-sectional ranking of the average

option returns also persist over time, such that past average option return rankings predict the

cross-sectional variation of future option returns. Thus, using the average past realization as

a control variable can capture persistent cross-sectional option risk premium variations that

our list of risk factors fail to fully capture.

2. Volatility risk premium (VRP): Delta-hedged proﬁt and loss from underwriting options is

approximately proportional to the dollar gamma weighted difference between the option’s

implied variance at initiation and realized return variance over the horizon of the option

maturity (Carr and Madan (2002)). As the dollar gamma is proportional to the stock price

level and the reciprocal of the volatility level, the delta-hedged option return in percentages

of the stock price level is approximately proportional to the difference between implied and

future realized stock return volatility. Therefore, if we can construct a forecast for the future

realized stock return volatility, we can use the difference between the implied volatility and

the volatility forecast, or the volatility risk premium, as a proxy of the future expectation of

the option excess return.

We construct the volatility forecast with a cross-sectional regression of the realized volatility

over the next month against three historical return volatility estimators at one month, one

quarter, and one year horizons, respectively. At each month, we perform the cross-sectional

regression using data from each of the past 12 months and apply the average coefﬁcient es-

timates to the volatility estimators at the current month to generate the volatility forecast for

the next month. Over the sample period, the average weights obtained from the regressions

are 15% on one-month estimator, 21% on three-month estimator, and 56% on the one-year

estimator. On average, the next-month return volatility converges strongly to the longer-term

21 estimator. We construct the volatility risk premium estimated using the current average im-

plied volatility level of the call and put options in straddle and the one-month ahead return

volatility forecast.

The volatility risk premium that we construct can in principle include risk premiums for

bearing the three risk sources that we have identiﬁed that dynamic delta hedging cannot

effectively and costlessly remove. Therefore, we do not regard the volatility risk premium

as a risk factor, but rather a control variable. Once all risk exposures are controlled for, the

remainder of the volatility risk premium can also contain a component of option mispricing,

the reversal of which can also predict future option investment returns independent of any

risk exposures.

4.2. Summary behaviors of one-month option returns and risk factors

We link the above constructed risk characteristics to the excess returns on the option investments.

At each date, we compute the cross-sectional summary statistics on the one-time delta-hedged straddle excess returns and its underlying risk factors. Table 2 reports the time-series averages of these statistics in Panel A, including the cross-sectional mean, standard deviation, the percentile values, the skewness and excess kurtosis, as well as the cross-sectional correlation of each risk factor with the straddle excess return. The straddle excess returns, the volatility of volatility estimates, and the historical and volatility risk premiums are all represented in percentages. The hedging cost is in basis points. The jump risk is in decimals. The particular scaling choices are immaterial for our analysis, but are just to make the numbers in similar, easy-to-represent ranges.

[Table 2 about here.]

22 The straddle excess returns have a sample average of 0.29% per month, slightly higher than the pooled average of 0.27% reported in Table 1. The cross-sectional standard deviation of 4.14% per month is slightly smaller than the pooled standard deviation, and the percentile ranges are similar to the pooled sample, suggesting that the cross-sectional variation dominates the variation of the straddle excess returns.

The three risk factors have different units: The hedging cost is represented as a ratio of daily variance to million dollar volume in basis points, the volatility risk is represented in the volatility of volatility estimate in percentages, and the jump risk is measured by the historical return volatility in decimals multiplied by the return excess kurtosis. All three measures show positive skewness and large excess kurtosis, suggesting that while there are a large number of stocks with low trading costs, small volatility risk, and small jump risk, some stocks can have extremely high trading cost, very large volatility risk, and/or very large jump risk, making the cross-sectional distribution of the three risk factors positively skewed.

All three risk factors show positive cross-correlations with the straddle return on average, largely consistent with our hypothesis that option underwriters demand a higher compensation to invest in options with higher delta hedging costs and unhedgeable risks.

The last two rows in Panel A measure the statistics on the two risk premiums as control variables. Both show positive correlations with the one-month ahead straddle return, more so for the volatility risk premium than for the historical risk premium. The positive correlation with historical risk premium suggests that the cross-sectional variations of the underlying risk factors, and accordingly the risk premiums on the straddle returns, contain a persistent component. The volatility risk premium also captures this persistent component, and it can also contain a misvaluation component: Underwriting the straddle is more likely to make a positive return when the option

23 implied volatility quote is high relative to the volatility forecast.

In Panel B of Table 2, we ﬁrst form an equal-weighted portfolio on the straddle returns and the risk factors at each date. Then, we examine how the portfolio straddle return and the average risk factors vary over time. The standard deviation of the portfolio straddle return at 1.22% is less than one-third of the average cross-sectional standard deviation, suggesting that the cross-sectional variations of the straddle returns tend to be larger than the time-series variation of the returns on the straddle portfolio.

An average monthly excess return of 0.29% translates into an annual risk premium of 3.48%.

The standard deviation of 1.22% translates into an annualized volatility of 4.3%, implying an annualized information ratio of 0.82 for underwriting the straddle portfolio, comparing favorably to holding an equal-weighted stock portfolio.

The three average risk factor series all show signiﬁcant time-series variation, although much smaller than the cross-sectional variation. When we measure the time-series correlation between the straddle portfolio return and the three average risk series, the correlation estimates all remain positive, suggesting that the return compensation for the three types of risks holds both cross- sectionally and over time.

By contrast, the time-series correlation estimates between the straddle portfolio return and the two risk premium control series become highly negative, suggesting that they are, by themselves, not true risk factors that demand compensation, but rather capture cross-sectional variations in some persistent risks and risk compensation. At the aggregate level and over time, the negative correlation suggests that the magnitude of the average risk compensation tends to show reversal behavior.

24 4.3. A cross-sectional capital asset pricing model on derivative securities

We examine the risk-return tradeoff on the straddle investment excess returns by building a cross-

sectional capital asset pricing model of the following form,

rt,T,i = ζt,T + ηt,T,1HCt,i + ηt,T,2VRt,i + ηt,T,3JRt,i + φt,T,1HRPt,T,i + φt,T,2VRPt,i + et,T,i, (6)

where rt,T,i denotes the excess return from underwriting a one-month straddle and performing one- time delta hedge with the underlying stock i at initiation time t and holding the position to expiry

T. By removing the initial directional exposure on the stock via straddle formulation and delta hedge at initiation, we exclude the common stock return risk factors as direct risk exposures and instead focus on the risks that cannot be removed effectively via delta hedge. The coefﬁcients

3 ηt,T = {ηt,T,k}k=1 denote the realized investment excess returns on straddle portfolios constructed 2 to have unit exposures to each of the three risk factors, and the coefﬁcients φt,T = {φt,T,k}k=1 capture the realized excess returns on portfolios formed on the two control variables while holding exposures to the three risk sources to zero. Time-series averages of the realized returns can be regarded as the average risk premium estimates on one unit of risk exposure to each of the risk factors while holding exposures to other risk factors to zero.

To make the coefﬁcient estimates comparable across risk factors, we standardize each of the three risk factors and the two control variables. We winsorize each variable cross-sectionally at

(1-99%) range and compute the mean and standard deviation on the winsorized sample. For the two risk premium measures, we construct the corresponding standard variable by deducting the mean estimate from the measure, and then scaling the demeaned series by the standard deviation estimate. For each of the three hedging cost/risk measures, we regard companies at the bottom one percentile as having minimal hedging cost or unhedgeable risk, and normalize the risk measure for

25 these companies to zero by deducting minimum from the winsorized sample. We then scale the

normalized series by the standard deviation estimate.

With this particular way of standardization, the intercept ζt,T,i represents the average excess return on a straddle portfolio with minimal delta hedging cost, minimal volatility risk, and minimal jump risk, while having the two risk premium control variables at their sample average level. If our hypothesis is right that underwriters of straddles demand a risk premium mainly to compensate for the hedging cost or unhedgeable risk, we would expect that underwriting straddles with minimal hedging cost and minimal unhedgeable risks earn minimal risk premium on average. Accordingly, we would expect the average of the intercept estimates not signiﬁcantly different from zero.

We perform the cross-sectional regression monthly from January 1997 to November 2019. The

ﬁrst year of data in 1996 are used to construct the historical risk premiums and volatility forecasts.

Table 3 reports the summary statistics of the monthly coefﬁcient estimates from the cross-sectional regression in (6). The statistics include the sample average (mean), standard deviation (Stdev),

Newey and West (1987) t-statistics, annualized information ratio (IR), serial monthly autocorrelation (Auto), skewness (Skew), and excess kurtosis (Kurt). The annualized information ratio is constructed as the ratio of the mean to the standard deviation of the portfolio return, annualized √ with 12. In constructing the t-statistics, we adjust for serial dependence with the number of lags

optimally chosen according to Andrews (1991).

[Table 3 about here.]

The straddle portfolios constructed on the three risk factors all generate signiﬁcantly positive

excess returns. The information ratio estimates on the three portfolio excess return series range

from 1.13 on the volatility risk factor, to 1.31 on the delta hedging cost factor, and to 2.54 on the

jump risk factor. The higher risk compensation for jump risk is understandable: Risks induced by

26 small or certain stock price movements can largely be removed with delta hedge. Risks induced by small or certain volatility changes can also largely be removed by vega hedging with an actively traded option contract. Hedging risks from jumps of random size, however, presents particular challenges to dynamic hedging with a few hedging instruments. In principle, each possible jump size demands a position of an option contract at that corresponding strike. Hedging jumps of all possible sizes necessitates positions across a whole spectrum of options, making the hedging cost unbearably large. Furthermore, sudden rare but large jumps, when left unhedged, can induce extremely large negative returns for the straddle underwriters, a major contributor to the negative skew in the straddle return distribution. Bearing such rare but large adverse risks requires a large risk premium compensation. The high information ratio on the jump risk portfolio reﬂects this compensation.

The portfolio constructed on historical risk premium, or the option momentum factor, also generates signiﬁcantly positive average returns, similar to the ﬁndings of Heston and Li (2020), with an information ratio of 0.61. We can think of it as the compensation for the persistent component of the three sources of risks, and potentially some other risks that our constructed risk factors fail to identify.

The investment returns on the volatility risk premium portfolio have the highest t-value at 11.02 and the highest annualized information ratio at 3.03. The results are in line with the ﬁndings of

Goyal and Saretto (2009), who construct a similarly deﬁned volatility risk premium factor by using the historical realized volatility directly as the volatility forecast. With exposures to all three risk factors set to zero, the volatility risk premium portfolio mainly reﬂects the reversal effect of option mispricing component. With all the hedging cost and hedging risks controlled to be minimal, a remaining high option implied volatility level relative to the future volatility forecast suggests that the option is overpriced. Underwriting such overpriced options lead to highly positive investment

27 returns with low risks.

After controlling for the risk factors and risk premiums, the intercept estimates have a small

average that is not signiﬁcantly different from zero, verifying our hypothesis that excess returns

from underwriting straddles are mainly compensation for bearing the three sources of hedging

costs and risks.

4.4. Intertemporal capital asset pricing on derivative securities

In the cross-sectional capital asset pricing model that we propose in (6) for derivative securities,

we standardize the risk factors cross-sectionally at each date by rescaling the risk factors with the

cross-sectional standard deviation estimate. The standardized risk measures represent the relative

ranking of each company in the cross section. The coefﬁcient estimates represent the risk premium

on this cross-sectionally standardized risk. The intertemporal variations of the aggregate risk level

and the cross-sectional dispersion of risks are removed from the cross-sectional pricing model.

In this subsection, we examine the intertemporal variation of the aggregate risk level of each

risk factor and estimate the corresponding average risk premium on each risk factor earned from

the corresponding factor straddle portfolios. For this purpose, we construct an intertemporal capital

asset pricing model of the following form on each of the straddle portfolio return series,

rt,T = α + ζ1HCt + ζ2VRt + ζ3JRt + ψ1HRPt + ψ2VRPt + et,T , (7)

where rt,T denotes the return time series on a particular straddle portfolio and the regressors

(HCt,VRt,JRt,HRPt,VRPt) are the time series on the equal-weighted average levels of the risk

factors and control variables.

28 We estimate the intertemporal relation with a time-series regression on the equal-weighted straddle portfolio, as well as on the five straddle factor portfolios, each with unit exposure to one of the five risk factors and risk premiums. Table 4 reports the slope estimates and the absolute magnitude of the Newey and West (1987) t-statistics (in parentheses) for each portfolio. We use the bold font to highlight the estimates with high statistical significance.

[Table 4 about here.]

Returns on the equal-weighted straddle portfolio increase with the magnitude of the three hedging cost and hedging risk factors, but decrease with the two risk premium control variables. Never- theless, as the identification power of the time-series regression is weak, all but the coefficient on volatility risk premium show low absolute t-values and accordingly weak statistical significance.

Each of the three factor portfolios targets one unit exposure on one of the three risk factors while having minimal exposure on the other two risk factors. Through this targeted portfolio construction, the excess returns on the portfolio show stronger dependence on the risk level of the targeted exposure. The excess returns of the three factor portfolios all increase with the risk magnitude of the targeted exposure, with strong statistical signiﬁcance: Excess returns on the delta hedging cost factor portfolio increase strongly with the hedging cost level; excess returns on the volatility risk factor portfolio increase strongly with the volatility risk level; and excess returns on the jump risk portfolio increase strongly with the jump risk level.

Excess returns on the historical risk premium portfolio are on average small, and do not show any strong dependence on the level of any of the risk factors. By contrast, excess returns on the volatility risk premium portfolio show strongly positive dependence on the delta hedging cost level and the average volatility risk premium level. Furthermore, the average volatility risk premium level also has strongly positive effects on excess returns of the volatility risk and jump risk

29 portfolios.

4.5. Effects of delta hedging on the capital asset pricing relation

We have so far carried out our main asset pricing analysis based on excess returns on underwriting one-month at-the-money straddles with one-time delta hedge at initiation. In this subsection, as a robustness check, we examine how the call, put, or the straddle combination choice and the choices on whether and how frequent to perform the delta hedge affect the capital asset pricing results on derivative securities.

We repeat the cross-sectional capital asset pricing model regression in (6) on excess returns for underwriting the one-month call option, the one-month put option, and the straddle combination, respectively, and managing each option position with (i) no delta hedge, (ii) one-time delta hedge at initiation, and (iii) delta hedge balanced daily whenever the stock is traded and the pricing information is valid for delta update computation, respectively. Table 5 reports the time-series averages of the coefﬁcient estimates and the absolute magnitude of its Newey and West (1987) t- statistics (in parentheses) from the multiple cross-sectional regressions, with each row representing one excess return series, and each panel representing one delta hedging strategy.

[Table 5 about here.]

Qualitatively, regressions on all excess return series lead to the same conclusion: The risk premium of underwriting options increases with the delta hedging cost (HC) and the two types of hard-to-hedge risks: stochastic volatility risk (VR) and random jump risk (JR). Both historical risk premium and volatility risk premium also contribute positively to the average excess return. On the other hand, at minimum hedging cost and unhedgeable risks, the intercept estimates are all small

30 and not signiﬁcantly different from zero.

Quantitatively, the statistical significance of the risk premium estimates increases as the excess return series contain less delta-hedgeable variations. The t-values of the risk premium estimates are the lowest for the unhedged call and put excess return series. The t-values are much higher for the straddle combination due to its much smaller delta exposure. With one-time delta hedge at initiation, the return series on calls, puts, and straddles all generate similar pricing results, all with strong statistical significance. The statistical significance on the risk premium estimates becomes the strongest for the return series with daily-rebalanced delta hedge. By removing a large proportion of the return noise due to directional exposures, the variations of the daily-delta-hedged return series mainly reflect compensations for the unhedgeable risks.

5. Risk and returns across option moneyness and maturity

For each underlying stock, the exchange lists many option contracts across different expiry dates and strike prices. We perform our main analysis by ﬁxing on a 30-day maturity via a deliberate choice of the sampling date per month and we choose one strike per stock that is closest to its stock price level at that date. This particular choice is motivated by several considerations. First, by choosing a uniform time-to-maturity over time, factor portfolio return time series become more comparable in terms of risk exposures. Second, front-month options tend to be where most trading activities occur, and the trading activity tends to center around the spot price level. Therefore, the pricing information on the one-month at-the-money options is likely to be the most accurate and most supported by the trading activities. Third, by choosing the short-term contracts, we can hold the options to expiry and compute the option’s terminal payoffs based on the stock price level at expiry, allowing us to avoid sampling the option prices twice in computing the option return.

31 In this section, we examine how the ﬁndings vary across option contracts at other moneyness and maturities. For this purpose, in addition to the 30-day maturity, we also select option contracts with other maturity buckets around two, three, and 12 months, respectively. At the chosen same sampling date, we have option contracts expiring exactly 30 days later by design, but option contracts are not always available with exactly two, three, or 12-month time to maturity. We hence relax our selection criteria to select an available maturity that is closest to our target maturity bucket. The two-month bucket includes options with time to maturities from 58 to 66 days.

The three-month bucket includes options with a wider maturity range from 86 to 220 days. The one-year bucket has the widest maturity range from 240 to 549 days.

At each maturity, we choose the strike closes to the spot level and obtain both the call and put option at that strike. In addition, we choose an out-of-the-money put option contract with the strike closest to the 25-delta put, and an out-of-money call option with the strike closest to the 25-delta call.

We hold all the options for 30 days and compute their returns over the same holding period based on the three hedging strategies. For one-month options, we compute the terminal payoff based on the stock price at expiry. For longer-dated options, we need to locate the option price 30 days later to compute the option return. The options can become illiquid with unreliable quotes 30- days later, making the return calculation noisier than for the one-month options. We also have 71 months that we fail to identify a maturity falling into the one-year bucket. Accordingly, we expect the return series for the longer-dated options to be noisier and have wider time-series dispersion.

32 5.1. Effectiveness of delta hedging across moneyess and maturity

Table 6 summarizes the pooled mean and standard deviation of the one-month investment excess returns (in percentages) on option contracts across different moneyness and maturity buckets, and with the returns constructed under three hedging strategies: (i) no hedge, (ii) one-time hedge at initiation, and (iii) daily delta hedge whenever the pricing information is available for updating the delta of the option contract. The table also reports the percentage variance reduction from the two hedging strategies relative to the no-hedge benchmark. Each panel represents one maturity bucket.

Within each panel, the four rows denote the four moneyness buckets, which we represent in terms of the approximate option delta and in the order of increasing strike price.

[Table 6 about here.]

At each option maturity bucket, the remaining variance estimates of the option returns become similar across different strikes after delta hedge at initiation, but the variance estimates on the unhedged option returns are lower for the 25-delta options than for the 50-delta options. As a result, the variance reduction percentages are lower for the 25-delta options than for the 50-delta options. Still, even for 25-delta options, dynamic delta hedge with daily rebalancing can reduce

78-83% of the unhedged return variance across all maturities, highlighting the effectiveness of the delta hedging insight.

As the option maturity increases, the one-time delta hedge becomes increasingly more effective in reducing the return variance. For example, the variance reduction on the 50-delta call option increases from 72% for one-month options to 83% for two-month options, 89% to three-month options, and ﬁnally over 92% for 12-month options. The increase in hedging effectiveness is equally impressive for the other option contracts. In particular, at 12-month maturity, the one-time delta hedge at initiation is so effective that daily rebalancing no longer signiﬁcantly increases the

33 variance reduction. Long-dated options have small gamma (i.e., small convexity) so that their delta does not vary as much with the stock price movement. Therefore, once the delta of a long-dated option contract is neutralized at initiation, the delta does not change much over the one-month investment horizon, making daily rebalancing less necessary.

One implication of the increasing hedging effectiveness is that underwriting longer-dated options is not as risky as underwriting short-dated options for the same one-month holding period in the sense that the cost of achieving and maintaining delta neutrality is much lower for longer dated options than for short-dated options. As a result, investors may not ask as much compensation for underwriting longer-dated options than for underwriting shorter-maturity options. Indeed,

Table 6 shows that while the average delta-hedged returns for underwriting one-month options and holding them to maturity are positive, underwriting longer-dated options and holding them for one month no longer generate positive returns on average. Holding longer-dated options for one-month instead of to expiry requires the investors to buy back the option contract one month later. If the option price remains high after one-month, the investment returns are no longer positive.

The analysis suggests that maintaining delta neutrality becomes expensive when the option maturity becomes short and the option gamma becomes large so that the delta of the option becomes unstable and varies a lot with the stock price movement. Accordingly, the option price contains a premium component for this hedging difﬁculty as the option maturity becomes short. This premium is mainly realized as the option approaches expiry. Underwriting long-dated options and holding them for a short period of time do not necessarily earn much of this premium.

34 5.2. Capital asset pricing on options across moneyness maturity

To examine how options at different maturities and moneyness price the unhedgable risks, we repeat the cross-sectional capital asset pricing model regression in (6) on daily-delta-hedged excess returns for underwriting call and put options across different moneyness and maturities buckets.

While we have chosen the sampling date to make the one-month bucket to have chosen option maturity at exactly 30 days, the other maturity buckets tend to have wider dispersion in the maturity choice at different sampling days. To make the cross-sectional regressions results more comparable at different sampling days, we scale the risk factors by the corresponding risk exposures. In particular, we scale the delta hedging cost factor by the absolute value of the delta of each chosen option contract assuming that the hedge need is larger for option contracts with larger absolute delta; we scale the stochastic volatility risk by the vega of the option contract multiplied by the option’s implied volatility level and divided by the stock price level to match the percentage volatility calculation on volatility risk; and we scale the jump risk by the gamma of the option contract, multiplied by the stock price level to remove the scale effect. In addition, the historical risk premium for each regression is estimated on the historical investment returns of each corresponding series, and the volatility risk premium is computed as the difference between the implied volatility of the corresponding contract and the one-month return volatility forecast.

Table 7 reports the time-series averages of the coefﬁcient estimates and the absolute magnitude of its Newey and West (1987) t-statistics (in parentheses) from the cross-sectional regressions.

Each row of the table represents the average regression results on one option bucket, and each panel represents one maturity bucket. The table summarizes results on 12 option buckets spanning four maturity buckets and four option contracts within each maturity bucket.

[Table 7 about here.]

35 When we underwrite one-month options and hold them to maturity while performing daily delta hedging, the return behaviors, as shown in Panel A of Table 7, are similar across options delta. The average risk premiums are strongly positive on all three risk factors, and the magnitudes are similar across option contracts. Both historical risk premium and volatility risk premium also contribute positively to the average excess returnd. On the other hand, at minimum hedging cost and unhedgeable risks, the intercept estimates are all small and not signiﬁcantly different from zero.

When we underwrite longer-dated options and hold them for just one month, the return behaviors, as shown in Panels B to D, become quite different. The average risk premium estimates on the hedging cost risk factor (HC) decline rapidly with increasing option maturity and become virtually zero for one-month returns on options with maturities three months or longer. The average risk premium estimates on the volatility risk (VR) and jump risk (JR) remain signiﬁcantly positive at longer maturities, but the estimates declines with increasing maturity.

The average slope estimates on the historical risk premium factor also decline rapidly with increasing option maturity, and are no longer positive for options with maturities three months and longer. On the other hand, the volatility risk premium factor generates consistently positive slope coefﬁcient estimates across all option maturities.

Due to practical constraints, the option buckets at longer maturities have a wider dispersion in the maturity choice. The wider dispersion in the chosen maturities can increase the variations of the option returns, and reduce the statistical signiﬁcance of the average risk premium estimates.

Nevertheless, the asset pricing results in Table 7 still present us with a general picture on how the pricing of different risks propagates across option moneyness and maturity. As the option maturity increases, the difﬁculty and cost of achieving and maintaining delta neutrality become smaller. As

36 a result, for short-term investments on long-dated options, the pricing on the hedging cost risk factor becomes insigniﬁcant, leaving the volatility risk and jump risk the remaining signiﬁcantly priced risk factor. In addition, volatility risk premium remains an important factor that predicts future option returns, whereas the momentum factor based on historical option investment returns can no longer positively predict future option returns.

Furthermore, at each maturity, the pricing of the hedging cost factor shows an intriguing dependence on the option type and delta, or alternatively the strike. In particular, the pricing on the

25-delta put tends to be the most positive or the least negative; whereas the pricing of the 25-delta call is the least positive or the most negative. In another word, the pricing of the hedging cost declines with the option strike. One conjecture is that underwriting out-of-the-money put options necessitates the underwriter to take short positions in the underlying for hedging, which can be particularly difﬁcult and costly for illiquid names and thus requires a higher risk premium.

6. Concluding remarks

Despite the many well-known market deviations from their model assumptions, the dynamic hedging insight of Black and Scholes (1973) and Merton (1973) revolutionized derivative trading by providing a simple but effective approach in drastically reducing the risks of an option position via dynamic delta hedging. Under their model assumptions, the risk of an option position can be completely removed by continuously updating the delta hedge to keep the position’s directional exposure to zero. We examine the effectiveness of delta hedging in practice on U.S. publicly traded stocks, and ﬁnd that about 70% of the return variance can be removed from underwriting a one-month call or put option and holding the option to expiry if the underwriter neutralizes the delta exposure just one time at the beginning of the underwriting, without any rebalancing. The

37 one-time delta hedge becomes even more effectively for longer-dated options as the option gamma declines with increasing maturity. Furthermore, if the underwriter keeps the delta exposure low throughout the investment horizon via daily rebalancing whenever the stocks are traded and valid pricing information is available, the variance reduction can reach 90%. Although reality is never as perfect as a model’s assumption, the 90% variance reduction highlights the effectiveness of their insight, which has been a key contributor to the booming of the derivatives market.

While the 90% variance reduction is a great achievement, the 10% remaining risk is still sig- niﬁcant for the option underwriters, who should ask for a compensation for bearing such risks.

More important, the presence of the remaining unhedgeable risk renders the derivative securities no longer redundant in real life and allow them to play primary roles in risk allocation. In this paper, we build a capital asset pricing model on the derivative securities to highlight their role in primary risk taking. First, via delta hedging the derivative positions, we focus on the risks that they cannot easily take with the underlying securities and examine how they are priced both cross- sectionally and intertemporally. Second, we identify three main risk factors that contribute to the remaining risk: (i) delta hedging cost, as captured by the trading cost of the underlying security, which prevents investors from fully removing the delta exposure without a cost, (ii) the risk of stochastic volatility, as captured by the volatility of the underlying option implied volatility movements, which cannot be hedged by delta hedging and represents a risk source that can only be effectively allocated via derivatives, and (iii) the risk of random price jumps, as captured by the excess kurtosis of daily return, which reduces the effectiveness of dynamic delta hedging and offers a primary role for the derivative positions. We build a cross-sectional capital asset pricing model that links the cross-sectional variation of the delta-hedged option underwriting excess returns on one-month options to the three risk factors and ﬁnd that all three risk factors ask for signiﬁcantly positive risk premiums on average. We also build factor option portfolios on these risk factors

38 and examine their time-series variation with the aggregate risk level through an intertemporal capital asset pricing model, and show that these risks are also signiﬁcantly priced intertemporally:

The excess returns on the factor portfolio increases signiﬁcantly with the aggregate level of the corresponding risk factors.

Historically, the option pricing literature tends to focus on the no dynamic arbitrage argument by treating the derivative security as a redundant asset and striving to ﬁnd a replication portfolio for the contract. While such an approach can be effective in generating a fair value for the derivative contract that quals the replication cost, it does not offer a rationale for the derivative contract’s existence to begin with, nor does it offer a role for the derivative contract in determining the market pricing of risks. By focusing on the option underwriting risks that delta hedge cannot effectively remove, we offer a rationale for the option contract’s existence, and provide a starting framework for understanding the market pricing of these non-redundant, primary risks that one can only take through derivative securities.

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42 Table 1 Summary statistics of investment returns from underwriting single-name options Entries report summary statistics on the pooled sample of 368,657 company-date observations of investment excess returns from underwriting single-name 30-day at-the-money call and put options separately as well as in straddle combination, and holding the option positions to expiry. The three panels represent three risk-management strategies on the option positions. Panel A leaves the option positions naked. Panel B delta hedges the option position with the underlying stock one time at initiation, without dynamic rebalancing. Panel C delta hedges the option position with the underlying stock, with daily rebalancing on days when the stock is traded and the pricing information is valid. The last column of Panels B and C reports the percentage variance reduction (VR) induced by the one-time delta hedge at initiation, deﬁned as one minus the variance ratio of hedged to unhedged investment returns.

Contract Mean Stdev Percentiles Tails VR % % 10 25 50 75 90 Skew Kurt %

A. Underwriting options without delta hedge Call 0.02 8.71 -9.41 -2.74 2.03 4.59 7.43 -3.71 43.41 — Put 0.52 7.72 -8.71 -1.35 2.32 4.56 7.13 -2.23 8.88 — Straddle 0.27 4.56 -4.73 -1.52 0.90 2.79 4.69 -2.47 27.21 —

B. Underwriting options with one-time delta hedge at initiation Call 0.23 4.34 -4.36 -1.29 0.80 2.54 4.34 -2.36 21.94 75.19 Put 0.32 4.33 -4.25 -1.21 0.86 2.59 4.44 -2.30 21.78 68.59 Straddle 0.28 4.31 -4.30 -1.24 0.85 2.55 4.35 -2.36 22.01 10.45

C. Underwriting options with daily-rebalanced delta hedge Call 0.17 2.59 -2.04 -0.57 0.34 1.26 2.41 -4.24 89.02 91.15 Put 0.24 2.63 -1.96 -0.49 0.40 1.34 2.53 -3.78 67.11 88.41 Straddle 0.21 2.53 -1.94 -0.51 0.38 1.27 2.39 -4.12 77.85 69.27

43 Table 2 Summary statistics of the straddle investment return and its risk factors Panel A reports the time-series averages of the cross-sectional summary statistics of the delta- hedged returns and the underlying risk factors for underwriting one-month at-the-money straddles. Panel B constructs the equal-weighted average of the straddle returns and the risk factors at each date and reports the time-series statistics of each average return and risk series. The statistics includes the mean, standard deviation, percentile values at 10, 25, 50, 75, 90th percentiles, skewness and excess kurtosis, as well as the correlations of each risk factor with the straddle returns.

Mean Stdev Percentile values Tails Corr 10 25 50 75 90 Skew Kurt (%)

A. Time-series averages of cross-sectional statistics Straddle return 0.29 4.14 -4.32 -1.39 0.83 2.67 4.44 -2.04 16.20 —

Delta hedging cost 1.40 2.98 0.03 0.08 0.34 1.25 3.61 4.12 19.47 4.43 Stochastic volatility 3.37 2.41 1.57 2.00 2.68 3.79 5.72 2.73 10.15 2.67 Random jumps 0.75 1.87 -0.27 -0.10 0.15 0.75 2.24 3.84 17.43 3.18 Historical premium 0.32 1.65 -1.56 -0.47 0.40 1.22 2.13 -0.41 2.10 2.14 Volatility premium 3.71 8.97 -5.82 -0.99 3.26 8.02 13.98 0.39 2.58 7.54

B. Time-series statistics of cross-sectional averages Straddle return 0.29 1.21 -0.77 -0.09 0.41 0.87 1.32 -2.04 10.03 —

Delta hedging cost 1.40 1.19 0.57 0.65 0.77 1.89 3.27 1.68 2.36 8.25 Stochastic volatility 3.37 0.94 2.41 2.63 3.23 3.94 4.57 1.16 1.98 1.62 Random jumps 0.75 0.34 0.40 0.51 0.66 0.95 1.25 0.81 0.26 3.80 Historical premium 0.32 0.31 -0.04 0.12 0.32 0.50 0.69 0.11 1.87 -12.43 Volatility premium 3.71 5.72 -2.42 1.06 4.39 7.25 9.88 -1.33 3.87 -17.72

44 Table 3 Risks and risk premiums on one-month straddle portfolios Entries report the summary statistics of the coefﬁcient estimates from the cross-sectional regression of straddle excess returns on three hedging risk factors and two risk premium control variables. The risk factors are standardized such that the intercept represents the return on the straddle portfolio with minimal hedging cost, minimal volatility risk, and minimal jump risk, while having the risk premium controls set to the sample average. The slope coefﬁcient on each risk factor represents the return on a straddle portfolio with one unit of exposure to that risk factor while minimal exposures to the other two risk factors. The statistics include the sample average (mean), standard deviation (Stdev), Newey-West t-statistics, annualized information ratio (IR), serial monthly autocorrelation (Auto), skewness (Skew), and excess kurtosis (Kurt), computed over 274 monthly observations from January 1997 to November 2019.

Mean Stdev t-value IR Auto Skew Kurt

Intercept 0.06 1.27 0.73 0.16 0.07 -2.08 11.27

Delta hedging cost 0.10 0.26 6.27 1.31 0.02 -0.45 0.47 Stochastic volatility 0.07 0.22 5.42 1.13 0.03 1.11 6.69 Random jumps 0.16 0.22 11.84 2.54 0.06 0.18 2.77

Historical premium 0.04 0.25 2.89 0.60 0.01 1.01 8.94 Volatility premium 0.33 0.37 10.98 3.03 0.23 1.58 10.03

45 Table 4 Intertemporal asset pricing on straddle portfolios Entries report the slope estimates and the absolute magnitudes of their Newey-West t-statistics (in parentheses) from regressing one-month ahead delta-hedged excess return series on six straddle portfolios against the average levels of the three risk factors (Hedging cost, stochastic volatility, and jump risk) and two control variables (past risk premium and volatility risk premium). The six portfolios include the equal-weighted straddle portfolio and ﬁve factor portfolios on each of the three risk factors and two control variables.

Portfolios HC VR JR PRP VRP

Equal weight 0.06 ( 0.73 ) 0.04 ( 0.45 ) 0.01 ( 0.13 ) -0.10 ( 1.33 ) -0.18 ( 2.46 )

Delta hedging cost 0.05 ( 2.75 ) 0.00 ( 0.00 ) -0.02 ( 1.03 ) -0.01 ( 0.84 ) -0.02 ( 1.23 ) Stochastic volatility 0.03 ( 1.79 ) 0.03 ( 2.11 ) -0.00 ( 0.30 ) -0.02 ( 1.20 ) 0.06 ( 4.18 ) Random jumps 0.00 ( 0.19 ) -0.02 ( 1.06 ) 0.04 ( 3.18 ) -0.02 ( 1.37 ) 0.05 ( 3.51 )

Historical premium -0.02 ( 0.90 ) 0.00 ( 0.26 ) 0.00 ( 0.31 ) 0.01 ( 0.50 ) -0.00 ( 0.02 ) Volatility premium 0.15 ( 5.64 ) 0.01 ( 0.37 ) 0.04 ( 1.57 ) 0.03 ( 1.48 ) 0.08 ( 3.49 )

46 Table 5 Cross-sectional option pricing on unhedgeable risks Entries report the time-series average and the absolute magnitude of its Newey-West t-statistics (in parentheses) of the coefﬁcient estimates from the cross-sectional regression of excess returns for underwriting one-month options and holding them to expiry against three hedging risk factors and two risk premium control variables: hedging cost (HC), volatility risk (VR), jump risk (JR), historical premium (HRP), and volatility risk premium (VRP). The risk factors are standardized such that the intercept represents the excess return on the option portfolio with minimal hedging cost, minimal volatility risk, and minimal jump risk, while having the risk premium controls set to the sample average. The slope coefﬁcient on each risk factor represents the excess return on an option portfolio with one unit of exposure to that risk factor while minimal exposures to the other two risk factors. Each row represents excess returns on one type of option investment: one-month call option, one-month put option, or the straddle combination. Panel A underwrites the options without delta hedge. Panel A underwrites the options with one-time delta hedge at initiation. Panel C performs daily rebalancing on the delta hedge whenever the underlying security is traded and the pricing information is valid for the delta computation.

Contacts Intercept HC VR JR PRP VRP

A. Underwriting options without delta hedge Call -0.24 ( 1.30 ) 0.10 ( 2.11 ) 0.09 ( 3.16 ) 0.14 ( 4.60 ) 0.12 ( 1.94 ) 0.31 ( 6.38 ) Put 0.34 ( 1.52 ) 0.08 ( 2.39 ) 0.05 ( 1.75 ) 0.18 ( 7.34 ) 0.07 ( 1.50 ) 0.33 ( 6.33 ) Straddle 0.05 ( 0.69 ) 0.10 ( 5.54 ) 0.07 ( 5.27 ) 0.16 ( 10.97 ) 0.06 ( 3.40 ) 0.32 ( 10.34 )

B. Underwriting options with one-time delta hedge at initiation Call 0.02 ( 0.30 ) 0.08 ( 4.98 ) 0.07 ( 4.98 ) 0.16 ( 11.69 ) 0.05 ( 3.19 ) 0.33 ( 10.84 ) Put 0.09 ( 1.17 ) 0.12 ( 7.43 ) 0.07 ( 5.63 ) 0.16 ( 11.89 ) 0.05 ( 3.22 ) 0.33 ( 10.97 ) Straddle 0.06 ( 0.73 ) 0.10 ( 6.27 ) 0.07 ( 5.42 ) 0.16 ( 11.84 ) 0.04 ( 2.89 ) 0.33 ( 10.98 )

C. Underwriting options with daily-rebalanced delta hedge Call -0.05 ( 1.18 ) 0.10 ( 8.15 ) 0.05 ( 6.32 ) 0.18 ( 20.70 ) 0.06 ( 6.68 ) 0.35 ( 12.37 ) Put 0.01 ( 0.12 ) 0.13 ( 11.68 ) 0.06 ( 8.20 ) 0.18 ( 22.26 ) 0.07 ( 7.52 ) 0.36 ( 12.33 ) Straddle -0.02 ( 0.47 ) 0.12 ( 9.99 ) 0.06 ( 7.79 ) 0.18 ( 22.11 ) 0.06 ( 6.47 ) 0.36 ( 12.20 )

47 Table 6 Effectiveness of delta hedging across moneyness and maturity Entries report the mean and standard deviation of the one-month investment excess returns (in percentages) on options across different moneyness and maturity buckets, with different hedging strategies: (i). no hedge, (ii). one-time delta hedge at initiation, and (iii). daily delta rebalancing. We also report the effectiveness of the one-time hedge and daily rebalancing in terms of the variance ratio (VR, %) relative to the investment return with no delta hedge. Each panel contains one maturity bucket, and each maturity bucket contains four option contracts represented in terms of the approximate delta bucket in the order of increasing strike price.

Contracts (i). No hedge (ii). One-time hedge (iii). Daily hedge Mean Stdev Mean Stdev VR Mean Stdev VR

A. One-month options 25-Put 0.39 5.49 0.28 3.84 51.19 0.23 2.34 81.89 50-Put 0.52 7.72 0.32 4.33 68.59 0.24 2.63 88.41 50-Call 0.02 8.71 0.23 4.34 75.19 0.17 2.59 91.15 25-Call 0.12 5.60 0.23 3.95 50.38 0.14 2.38 81.89

B. Two-month options 25-Put 0.07 4.92 -0.00 2.96 63.77 -0.05 2.01 83.37 50-Put 0.18 7.01 0.02 3.33 77.39 -0.04 2.27 89.54 50-Call -0.19 8.23 -0.03 3.38 83.14 -0.07 2.20 92.85 25-Call -0.18 5.10 -0.09 3.10 63.15 -0.13 2.08 83.35

C. Three-month options 25-Put -0.01 4.58 -0.06 2.38 73.08 -0.11 1.85 83.74 50-Put 0.04 6.60 -0.07 2.70 83.32 -0.13 2.09 90.01 50-Call -0.17 7.98 -0.07 2.69 88.67 -0.12 2.00 93.74 25-Call -0.14 4.22 -0.09 2.31 69.93 -0.13 1.73 83.10

A. 12-month options 25-Put 0.08 4.24 -0.05 2.03 77.18 -0.08 1.89 80.04 50-Put 0.20 5.64 -0.05 2.19 84.91 -0.09 2.16 85.29 50-Call -0.31 7.87 -0.02 2.21 92.09 -0.05 1.95 93.89 25-Call -0.16 3.76 -0.03 2.00 71.60 -0.06 1.78 77.57

48 Table 7 Cross-sectional capital asset pricing across moneyness and maturity buckets Entries report the time-series average and the absolute magnitude of its Newey-West t-statistics (in parentheses) of the coefﬁcient estimates from the cross-sectional regression of one-month daily delta-hedged excess returns for underwriting options across different moneyness and maturity buckets against three hedging risk factors and two risk premium control variables: hedging cost (HC), volatility risk (VR), jump risk (JR), historical premium (HRP), and volatility risk premium (VRP).

Contracts Intercept HC VR JR PRP VRP

A. One-month options 25-Put -0.02 ( 0.50 ) 0.11 ( 9.44 ) 0.14 ( 11.03 ) 0.12 ( 18.30 ) 0.04 ( 5.46 ) 0.33 ( 11.03 ) 50-Put -0.02 ( 0.44 ) 0.09 ( 7.96 ) 0.16 ( 12.86 ) 0.13 ( 20.05 ) 0.07 ( 8.08 ) 0.33 ( 9.99 ) 50-Call -0.08 ( 1.84 ) 0.09 ( 7.09 ) 0.12 ( 10.42 ) 0.14 ( 19.84 ) 0.06 ( 6.98 ) 0.33 ( 10.01 ) 25-Call -0.04 ( 1.31 ) 0.04 ( 3.57 ) 0.09 ( 7.67 ) 0.12 ( 17.81 ) 0.03 ( 3.51 ) 0.30 ( 11.23 )

B. Two-month options 25-Put -0.18 ( 3.15 ) 0.05 ( 3.16 ) 0.07 ( 4.98 ) 0.10 ( 10.65 ) 0.01 ( 1.15 ) 0.25 ( 11.43 ) 50-Put -0.18 ( 3.05 ) 0.05 ( 3.36 ) 0.11 ( 6.66 ) 0.10 ( 10.47 ) 0.03 ( 2.50 ) 0.24 ( 9.70 ) 50-Call -0.19 ( 3.75 ) 0.04 ( 2.66 ) 0.07 ( 4.77 ) 0.12 ( 11.29 ) 0.02 ( 2.08 ) 0.24 ( 11.26 ) 25-Call -0.19 ( 5.39 ) -0.02 ( 1.42 ) 0.04 ( 3.75 ) 0.10 ( 10.81 ) 0.01 ( 1.11 ) 0.23 ( 10.44 )

C. Three-month options 25-Put -0.18 ( 2.83 ) 0.00 ( 0.06 ) 0.02 ( 1.74 ) 0.09 ( 11.18 ) -0.01 ( 1.84 ) 0.26 ( 12.88 ) 50-Put -0.20 ( 2.82 ) 0.01 ( 0.56 ) 0.04 ( 2.84 ) 0.09 ( 10.87 ) -0.01 ( 1.15 ) 0.24 ( 10.33 ) 50-Call -0.18 ( 3.32 ) -0.01 ( 0.61 ) 0.03 ( 1.87 ) 0.09 ( 12.82 ) -0.02 ( 2.54 ) 0.25 ( 11.72 ) 25-Call -0.18 ( 4.50 ) -0.04 ( 3.17 ) 0.01 ( 1.04 ) 0.09 ( 13.98 ) -0.02 ( 3.52 ) 0.24 ( 13.12 )

D. 12-month options 25-Put -0.15 ( 1.79 ) -0.01 ( 0.60 ) 0.06 ( 3.15 ) 0.06 ( 5.11 ) -0.02 ( 2.16 ) 0.24 ( 8.81 ) 50-Put -0.11 ( 1.11 ) -0.01 ( 1.00 ) 0.06 ( 3.58 ) 0.04 ( 3.44 ) -0.01 ( 1.03 ) 0.16 ( 6.51 ) 50-Call -0.08 ( 1.23 ) -0.02 ( 1.27 ) 0.04 ( 2.06 ) 0.05 ( 4.93 ) -0.02 ( 1.76 ) 0.20 ( 9.08 ) 25-Call -0.13 ( 2.60 ) -0.05 ( 3.06 ) 0.07 ( 3.97 ) 0.05 ( 5.98 ) -0.01 ( 0.69 ) 0.21 ( 10.08 )

49 2200

2000

1800

1600

1400

1200

1000

800 Number of firms 600

400

200

0 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014 2016 2018 2020

Figure 1 Number of selected ﬁrms per date The bar chart shows the number of selected ﬁrms per date from January 1, 1996 to November, 2019 for a total of 287 months.

50 A. One-time delta hedge at initiation B. Daily-rebalanced delta hedge 100 100

90 90

80 80

70 70

60 60

50 50

40 40

Variance reduction, % 30 Variance reduction, % 30

20 20

10 10

0 0 1997 1999 2001 2003 2005 2007 2009 2011 2013 2015 2017 2019 1997 1999 2001 2003 2005 2007 2009 2011 2013 2015 2017 2019

Figure 2 Stability of variance reduction over time Lines plot the one-year rolling estimates on the percentage variance reduction from the one-time delta hedge at initiation (panel A) and the daily-rebalanced dynamic delta hedge (panel B) on the excess returns from underwriting one-month at-the-money options. The three lines in each panel represent the variance reduction on calls (solid line), puts (dash-dotted line), and straddles (dotted line), respectively.