Limits of Dynamic Hedging and Option Risk Premium

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 envi- ronment, 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 un- hedgeable 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

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