Optimal Option Portfolio Strategies

Transcription

1 Optimal Option Portfolio Strategies José Afonso Faias 1 and Pedro Santa-Clara 2 Current version: October 2011 Abstract Options should play an important role in asset allocation. They allow for kernel spanning and provide access to additional (priced) risk factors such as stochastic volatility and negative jumps. Unfortunately, the traditional methods of asset allocation (such as mean-variance optimization) are not adequate for considering options because the distribution of returns is non-normal, and the short sample of option returns available makes it difficult to estimate the distribution. We propose a method to optimize option portfolios that solves these limitations. In an out-of-sample exercise, even when transaction costs are incorporated, the portfolio strategy delivers an annualized Sharpe ratio of 0.50 between January 1996 and October Corresponding author: José Afonso Faias, Universidade Católica Portuguesa - Católica Lisbon School of Business and Economics, Palma de Cima, Lisboa, Portugal. jfaias@clsbe.lisboa.ucp.pt. We thank Joost Driessen, Miguel Ferreira, Mark Grinblatt, José Correia Guedes, Christopher Jones, Ángel León, André Lucas, Pedro Matos, David Moreno, Andreas Rathgeber, Enrique Sentana, Ivan Shaliastovich, and participants at the Informal Research Workshop at Universidade Nova de Lisboa, the QED 2010 Meeting at Alicante, Universidade Católica Portuguesa, the 6th Portuguese Finance Network Conference, the Finance & Economics 2010 Conference, the XVIII Foro de Finanzas, the 2011 AFA Annual Meeting, the 5th Conference on Professional Asset Management, the Oxford Man Institute at Quantitative Finance, the 2011 EFMA Annual Meeting, the EcoMod2011, and the Conference in honor of Richard Roll and Eduardo Schwartz for helpful comments and discussions. This research was funded by grant PTDC/EGE- ECO/119683/2010 of the Portuguese Foundation for Science and Technology-FCT. 1 Universidade Católica Portuguesa - Católica Lisbon School of Business and Economics, Palma de Cima, Lisboa, Portugal. Phone jfaias@clsbe.lisboa.ucp.pt. 2 Millennium Chair in Finance. Universidade Nova de Lisboa - NOVA School of Business and Economics, NBER, and CEPR, Campus de Campolide, Lisboa, Portugal. Phone psc@novasbe.pt.

2 I. Introduction Although options are well known to help span states of nature [Ross (1976)] and to provide exposure to (priced) risk factors like stochastic volatility and jumps, 1 they are seldom used in investment portfolios. 2 Part of the problem is that the portfolio optimization methods we have, like the Markowitz mean-variance model, are ill suited to handle options. There are three main problems in option portfolio optimization. First, the distribution of option returns departs significantly from normality and therefore cannot be described by means and variances alone. Second, the short history of options returns available severely limits the estimation of their complex distribution. For example, we have data for Standard & Poor s 500 options only since 1996, which is not long enough to estimate the moments of their return distribution with sufficient precision. Third, there are high transaction costs in this market. On average, at-the-money (ATM) options have a 5% relative bid-ask spread, while out-of-the-money (OTM) options have relative bid-ask spreads of 10%. We offer a simple portfolio optimization method that solves these problems simultaneously. Instead of a mean-variance objective, we maximize an expected utility function, such as a power utility, which accounts for all the moments of the portfolio return distribution and has the advantage of penalizing negative skewness and high kurtosis. We deal with the limited sample of options returns by relying on data for the underlying asset instead. Our application uses returns of the S&P 500 index since 1950 to simulate returns of the underlying asset going forward and, from the definition of option payoffs, generate simulated option returns. 1 See Bates (1996), Bakshi, Cao, and Chen (1997), Andersen, Benzoni, and Lund (2002), and Liu and Pan (2003), among others. 2 Mutual funds use of derivatives is limited [Koski and Pontiff (1999), Deli and Varma (2002), Almazan, Brown, Carlson, and Chapman (2004)]. Mutual funds generally face legal constraints in terms of shortselling, borrowing and derivatives usage. This does not happen with hedge funds. Most hedge funds use derivatives, but they represent only a small part of their holdings [Chen (2010), Aragon and Martin (2007)]. 1

3 Plugging the simulated option returns into the utility function and averaging across simulations gives us an approximation of the expected utility that can be maximized to obtain optimal portfolio weights. Note that only current options prices are needed in our procedure, as the payoff is determined by the simulations of the underlying asset. 3 We apply our model to portfolio allocation between a risk-free asset and four options on the S&P 500 index with one month to maturity. We define each option by choosing the most liquid contract in a predefined range around the specific moneyness. We consider an ATM call, an ATM put, a 5% OTM call, and a 5% OTM put option. These are liquid options that can be combined to generate a variety of final payoffs. 4 To incorporate transaction costs, we follow Eraker (2007) and Plyakha and Vilkov (2008). For each option, we define two securities: a long option initiated at the ask quote, and a short option initiated at the bid quote. We form a constrained optimization problem for these eight options, and we identify short securities by a negative sign in the optimization algorithm. By not allowing short-selling, only one of two options is ever bought. We study the performance of our Optimal Option Portfolio Strategies, which we denote by OOPS, in an out-of-sample (OOS) exercise. We find the optimal option portfolios one month before option maturity, and examine the return that they would have had at maturity. Investors could have obtained the resulting time series of returns following our method in real time. We can then compute measures of performance such as Sharpe ratios or alphas to assess the interest of the method. For each OOS observation, only one month of option observations is needed. For the entire period of 178 monthly observations, 99% of the sample is OOS. This aspect by itself is significantly different from approaches in other studies. 3 We consider different alternatives for simulating the returns based on parametric distributions fitted to the data or simple bootstrap methods. We can also model a time-varying distribution of returns by simulating their distribution conditional on state variables such as realized volatility. 4 We do not include the S&P 500 in the asset universe because it is spanned by the options. 2

4 OOPS have high Sharpe ratios and positive certainty equivalents in our sample period between January 1996 and October The best strategy yields a Sharpe ratio of This compares well with the Sharpe ratio of the market in the same period of 0.13 (or even in the full sample since 1950 of 0.23). Several strategies also present positive skewness and low excess kurtosis, which is not achievable simply by shorting individual options. We find that our strategies load significantly on all four options, and that the optimal weights vary over time. Finally, our optimal strategies are almost delta-neutral albeit with significant elasticity. Hedging demand and long-term considerations can potentially be very important. Viceira and Campbell (1999) study portfolio choices between equities and the risk-free rate, and show that hedging demand is a substantial part of overall demand, and there are high utility losses for failure to hedge intertemporally. Tan (2009) finds that the benefit of adding options seems to be quite small for long-horizon Constant Relative Risk Aversion (CRRA) investors who can buy put and call options. Liu and Pan (2003) and Driessen and Maenhout (2007) show that improvements by including derivatives are driven mostly by a myopic component. In fact, our results show low predictability of returns of the optimal strategy. There is also little correlation with the market. These two features imply that there would be little hedging demand. A related literature investigates the returns of simple options trading strategies. Coval and Shumway (2001) show that short positions in crash-protected, delta-neutral, straddles present Sharpe ratios of around 1.0 and Saretto and Santa-Clara (2009) find similar values in an extended sample. Driessen and Maenhout (2006) confirm these results for short-term options in US and UK markets. Coval and Shumway (2001) and Bondarenko (2003) also find that selling naked puts offers high returns even after taking into account their considerable risk. These authors, however, do not discuss how to optimally combine options into a portfolio. Interestingly, we find that our portfolios depart significantly from exploiting these simple strategies. For instance, there are extended periods in which the optimal portfolios are net long put options. 3

5 There are five pieces of research closest that also address the optimal portfolio allocation with options. Liu and Pan (2003) model stochastic volatility and jump processes and derive the optimal portfolio policy for a CRRA investor across one stock, a 5% OTM put option, and cash. Although they obtain an analytic solution for the optimal option allocation, they need to specify a particular parametric model and estimate its parameters. They try different parameter sets and obtain ambiguous conclusions in terms of put weights. Also, they require only one option to complete the market as they consider either a pure jump risk or a pure volatility risk setting. In our case, we can use any model (parametric or not) for the distribution of returns of the underlying asset. Our work is empirical in nature, and we impose no restrictions on the number of options that could be used in the optimization problem. Eraker (2007) uses a standard mean-variance framework with a parametric model of stochastic volatility and jumps to choose among three risky assets: ATM straddles, OTM puts, and OTM calls. He provides a closed-form solution for weights and obtains an OOS annualized Sharpe ratio of around 1.6. As in Liu and Pan (2003), estimates are sensitive to the period considered, despeit the long period required for estimation. Our approach is more flexible in term of the distribution of returns and the number of options in the portfolio. Driessen and Maenhout (2007) analyze the importance of derivatives in portfolio allocation by using the Generalized Method of Moments (GMM) to maximize the expected utility of returns for a portfolio of a stock, an option strategy, and cash. They use either an OTM put, an ATM straddle, or corresponding crash-neutral strategies. They conclude that positive put holdings that would implement portfolio insurance are never optimal, given historical option prices. We find, by contrast, that optimal weights are time-varying, and change signs during our sample period. Jones (2006) studies optimal portfolios to exploit the apparent put mispricing. He uses a general nonlinear latent factor model and maximizes a constrained mean-variance objective. He circumvents the short history of data by using daily options 4

6 returns. His model is quite complex, with 57 parameters to estimate even when only one factor is considered. This limits the practical usefulness of his approach. Constantinides, Jackwerth, and Savov (2009) study portfolios made up of either calls or puts with a targeted moneyness; they leverage-adjust their returns using options elasticity. Although they find high Sharpe ratios, mostly for put strategies, these strategies yield negative skewness and high kurtosis. The paper is organized as follows. Section II explains the methodology. Section III describes the data used. Section IV presents results and findings. Robustness checks are performed in Section V. Finally, we present some concluding remarks. II. Portfolio Allocation A. Methodology Let time be represented by the subscript t and simulations indexed by n. Our portfolio allocation is implemented for one risk-free asset and a series of call and put options with one period to maturity. We assume that there are C call options indexed by c where c = c 1,..., c C, and P put options indexed by p where p = p 1,..., p P. We include only options that are not redundant by put-call parity. At time t, the value of the underlying asset is denoted by S t, and each option i has an exercise price of K t,i. The risk-free interest rate from time t to t + 1, known at time t, is denoted by rf t. For each date t, weights are obtained through maximization of the investor s expected utility of end-of-period wealth, which is a linear function of simulated portfolio returns. The latter are derived from option returns, which in turn depend on the underlying asset returns. The steps below describe the algorithm used in detail. The Appendix shows a simple illustration. 1. We simulate N underlying asset log-returns r n t+1 where n = 1,..., N. Several different possible simulation schemes can be used. Our simulation is performed under the empirical density not risk-adjusted measure. We explain the simulation scheme in detail in the next section. 5

7 2. The log-returns from step 1 are used to simulate the next period s underlying asset value, given its current value: S n t+1 t = S t exp(r n t+1) (1) where n = 1,..., N, and S n t+1 t denotes the simulated underlying asset value in period t + 1 conditional on information up to time t. 3. Using known strike prices for call options K t,c and put options K t,p and one-period simulated underlying asset values S t+1 t in equation (1), we simulate option payoffs at their maturity t + 1: C n t+1 t,c = max(s n t+1 t K t,c, 0) and P n t+1 t,p = max(k t,p S n t+1 t, 0) (2) where n = 1,..., N. Using these simulated payoffs in equation (2) and current option prices, returns are then computed by r n t+1 t,c = Cn t+1 t,c C t,c 1 and r n t+1 t,p = P n t+1 t,p P t,p 1 (3) where n = 1,..., N. To compute these returns, we use current option prices C t,c, c = c 1,..., c C and P t,p, p = p 1,..., p P at month t. Notice that only one-period-ahead payoffs are simulated; the denominator of the return is the current observed option price. 4. We construct simulated portfolio returns in the usual way: rp n t+1 t = rf t + c C p P ω t,c (rt+1 t,c n rf t ) + ω t,p (rt+1 t,p n rf t ) (4) c=c 1 p=p 1 where n = 1,..., N. Each simulated portfolio return is a weighted average of the asset returns, and only the risk-free rate is not simulated. 5. We choose weights by maximizing expected utility over simulated portfolio returns Max ω E [ U ( W t [1 + rp t+1 t ] )] Max ω 1 N N U ( W t [1 + rp n t+1 t] ). (5) n=1 The output is given by ω t,c, c = 1,..., C and ω t,p, p = 1,..., P. 6

8 6. One-period OOS performance is evaluated with realized option returns. First, we determine the option realized payoffs: C t+1,c = max(s t+1 K t,c, 0) and P t+1,p = max(k t,p S t+1, 0) (6) Second, we find the corresponding returns: r t+1,c = C t+1,c C t,c 1 and r t+1,p = P t+1,p P t,p 1 (7) Third, we determine the one-period OOS portfolio return: rp t+1 = rf t + c C p P using the weights determined in step 5 above. ω t,c (r t+1,c rf t ) + ω t,p (r t+1,p rf t ) (8) c=c 1 p=p 1 B. Return simulation The first step of the algorithm is to simulate one-period log-returns of the underlying asset. There are many possible approaches to this; see Jackwerth (1999) for a survey of the literature. Aït-Sahalia and Lo (1998) and Jackwerth and Rubinstein (1996) present two examples of potential routes one could follow to recover a density function in a continuous or a discrete time setting, respectively. We follow two approaches, unconditional and conditional simulation. Either is implemented in two ways, historical bootstrap and parametric simulation based on historical estimation of the parameters of the density. In all cases, in each month we use an information set corresponding to an expanding window of data of the underlying asset up to time t so that the results are out of sample. First, we explain the unconditional OOPS which takes into account raw returns of the underlying. Following Efron and Tibshirani (1993), we bootstrap raw returns from the historical empirical distribution of the underlying distribution up to date t. Hence, we resample directly from historically observed returns. Implicitly, this corresponds to drawing returns from their empirical distribution (histogram). We denote this approach as empirical. 7

9 Alternatively, we simulate returns using a parametric distribution f estimated from past data. We use two types of distribution. The first distribution, which is the most standard in the literature, is a normal distribution with density f(r ˆµ t, ˆσ t ) where ˆµ t is the sample mean, and ˆσ t is the sample standard deviation. It is known that the normal distribution does not fit return data well; for one thing, it does not capture the frequency of extreme events. 5 Moreover, the normal distribution does not make sense because it does not take into account skewness and kurtosis of the stock market. Nonetheless, we still present results from this experiment. To extend our analysis to other types of distributions that account for skewness and kurtosis, we consider a family of distributions that is commonly designated by the generalized extreme value distribution (GEV). The GEV distribution is defined by the density f(r ˆλ t, ˆσ t, ˆµ t ) with shape parameter ˆλ t, scale parameter ˆσ t, and location parameter ˆµ t. 6 It provides a flexible framework that generalizes several distributions. Notice that all estimated parameters are time-varying, as we use an expanding window up to time t to estimate them. 7 The unconditional approach does not make sense for two reasons. First, an investor would think options are very expensive when volatility is high. Second, we have not taken into account that returns may be dependent. Both the bootstrap and the parametric density approaches assume that returns are independent and identically distributed (i.i.d.), although it is well known that volatility clusters in time. To capture this issue, we use standardized log-returns, which we denote by x. We select realized volatility as an esti- 5 Jackwerth and Rubinstein (1996) show that using a normal distribution to model returns, the probability of a stock market crash like the ones that we have witnessed in the past is The GEV density function is defined in the following way: f(r λ, σ, µ) = { 1 σ 1 σ ( 1 + λ r µ σ exp( r µ σ ) 1 1/λ exp ( ( 1 + λ r µ ) 1/λ ) σ )exp( exp( r µ σ for λ 0 )) for λ = 0 The distributions depend crucially on the sign of the parameter λ. A positive sign denotes the Fréchet class which include well known fat-tailed distributions such as the Pareto, Cauchy, Student-t and mixture distributions. The zero parameter denotes the Gumbel class and includes the normal, exponential, gamma and lognormal distributions. A negative sign denotes the Weibull class which includes the uniform and beta distributions. 7 Parameters are estimated by maximum likelihood using the built-in functions of MATLAB normfit and gevfit. 8

10 mate of volatility rv t = Dt rt,d 2 where D t is the number of days in month t, and r t,d are d=1 the daily returns on day d in month t. Standardized log-returns are the ratio of log-return to the previous month s realized volatility x t = r t. This is close in spirit to the filtered rv t 1 historical simulation of Barone-Adesi, Giannopolos, and Vosper (1999) in which volatility is estimated by a parametric method such as a generalized autoregressive conditional heterocedastic (GARCH) model. 8 Table I presents summary statistics for raw and standardized returns for the period between 1950 and 2010 and two subperiods, before and after By either measure, returns are lower in the second subperiod. While standardizing returns reduces volatility in the second period, the opposite happens to raw returns. Both processes show negative values for skewness between 0.63 and 0.33, but skewness is lower, in absolute terms, for standardized returns. Raw returns have an excess kurtosis in the first of subperiod around 3 which drops drastically in the second period, and standardization almost eliminates kurtosis. The reason is less frequent extreme standardized returns; note, for example, that the Black Monday extreme negative return is now much smaller. The standardized return is now only 0.36 standard deviation units away from the mean compared to standard deviation units away from the mean for raw returns. For the period between 1950 and 1995, the one- and twelve-month autoregressive coefficients of raw and standardized returns are low and on the order of 0.03, while the autoregressive coefficient of the squared processes is on the order of 0.10 in absolute value. Ljung-Box autocorrelation tests for the residuals of raw and standardized returns show no autocorrelation. According to an ARCH test for the previous one and twelve months, the i.i.d. hypothesis is rejected for raw returns mainly in the period between 1950 and The i.i.d. hypothesis is not rejected for standardized returns at any reasonable significance level. 8 Properties of standardized returns are presented, for instance, in Andersen, Bollerslev, Diebold, and Ebens (2001) for stocks and Andersen, Bollerslev, Diebold, and Labys (2003) for exchange rates. They show that standardized returns are close to i.i.d. normal. 9

11 The conditional approach uses standardized returns rather than raw returns, with a slight modification in the first and second steps of the algorithm of the algorithm in Section A. 1. Simulate standardized returns: x n t+1 = rn t+1 rv t (9) To obtain x n t+1 we consider the same two ways as in the unconditional simulation, bootstrapping and parametric simulation. 2. Scale the bootstrapped standardized returns by the current standardized return: return S n t+1 t = S t exp(x n t+1 rv t ) (10) where x n t+1 are the simulated standardized returns from step 1 and rv t is the realized volatility of the time period between t 1 and t, which is not simulated. In either case, unconditional or conditional, current option prices are used in step 3. This modification takes into account the recurrent change in expectations of the underlying asset conditional distribution (e.g., OTM put options become more expensive if investors think there is a greater probability of a crash). Our concept of conditional variable is the scaling with lagged volatility. We believe that the conditional strategies work better because option prices are evaluated according to current volatility. While we do not think that normal distribution makes sense in this context, we still present those results for comparison reasons. C. Maximizing expected utility In the fourth step, the investor maximizes the conditional expected utility of next period s wealth: max ω i,t R E[u(W t+1 )] subject to the usual budget constraint W t+1 = W t (1 + rp t+1 ). Maximizing expected utility takes into account different return distributions. If returns are normal, rational investors 10

12 care only about the mean and variance of portfolio returns. In practice, this is unlikely to hold, especially for options returns. Investors care also about tail risk (extreme events), so mean and variance do not provide enough information to evaluate asset allocation choices. We use the power utility function [see Brandt (1999)]. This utility function presents constant relative risk aversion (CRRA) and is given by: 1 u(w ) = W 1 γ, if γ 1 1 γ ln(w ), if γ = 1 where γ is the coefficient of relative risk aversion. 9 This utility function is attractive for two reasons. First, because of the homotheticity property, portfolio weights are independent of the initial level of wealth. So maximizing E[u(W t+1 )] is the same as maximizing E[u (1 + rp t )]. Second, investors care about all moments of the distribution, and this particular utility function penalizes skewness and high kurtosis. Brandt, Goyal, Santa-Clara, and Stroud (2005) offer approximations of the optimal portfolio choice. We set the constant relative risk aversion parameter γ equal to 10 for a conservative asset allocation choice. To the extent that the investor has lower risk aversion than this value, this works as a shrinkage mechanism and smoothes the portfolio weights. Rosenberg and Engle (2002) estimate an empirical risk aversion of 7.36 for S&P 500 index options data over the sample period between 1991 and Finally, notice that we could have used any other utility function in applying our methodology For arguments lower than 0.001, we use a first-order approximation of this utility function to avoid extreme negative values, which is standard in the literature. 10 Driessen and Maenhout (2007) discuss several more sophisticated models and implications for disappointment aversion. 11

13 D. Transaction costs There is a large body of literature that documents high transaction costs in the options market that are in part responsible for some pricing anomalies, such as violations of the put-call parity relation. 11 Hence, it is essential to include these frictions in our optimization problem. We discuss only the impact of transaction costs measured by the bid-ask spread. Other types of costs like brokerage fees and market price impacts may be substantial, but are ignored here. Figure 1 shows substantial bid-ask spreads for options between 1996 and 2010 options present. Table II shows average bid-ask spreads of $1.30 for ATM options and $0.70 for OTM options. Dividing this by mid-prices, we measure relative bid-ask spreads for ATM options of around 5%, increasing to 10% on average for OTM options. Relative bid-ask spreads change over time and for OTM options can reach up to 30%. 12 We incorporate transaction costs by decomposing each option into two securities : a bid option, and an ask option. We initiate long positions at the ask quote and short positions at the bid quote, and the latter enters the optimization with a minus sign. This follows the approach approach taken by Eraker (2007) and Plyakha and Vilkov (2008). Then we run the algorithm as a constrained optimization problem by imposing a no shortselling condition. This means that in each month only one of the securities, either the bid or the ask option, is bought. Note that the wider the bid-ask spread, the less likely an allocation to the security, as expected returns will be lower. 11 See, for instance, Phillips and Smith (1980), Baesel, Shows, and Thorp (1983), and Saretto and Santa- Clara (2009). 12 Dennis and Mayhew (2009) shows that the effective spread is about two-thirds of the quoted spread given in OptionMetrics, so quoted spreads may overestimate costs which seems a conservative assumption in terms of OOPS performance. 12

14 III. Data A. Securities We analyze the optimal portfolio allocation from January 1996 through October Choice of the period relates to the availability of options data. We use data that go back to February 1950 for the simulation process, returns of the Standard & Poor s 500 index, from February 1950 through October Figure 2 presents the monthly time-series of Standard & Poor s 500 and VIX indices in the overall period. 13 There are a variety of market conditions over the period as can be seen from the cycles in the index and from the evolution of volatility. Events include the 1997 Asian crisis, the 1998 Russian financial crisis, the 1998 collapse of Long Term Capital Management, the 2001 Nasdaq peak, the September 2001 attack at New York s World Trade Center, the 2002 business corruption scandals (Enron and Worldcom), Gulf War II, and the 2008 subprime mortgage crisis. The empirical analysis relies on monthly holding-period returns, as microstructure effects tend to distort higher-frequency returns. For our empirical analysis, we use S&P 500 index closing prices available from Bloomberg. Based on this, we construct a time-series of monthly log-returns. We use data from the OptionMetrics Ivy DB database for European options on the S&P 500 index traded on the CBOE. 14 The underlying asset is the index level multiplied by 100. Options contracts expire on the third Friday of each month. The options are settled in cash, which amounts to the difference between the settlement value and the strike price of the option multiplied by 100, on the business day following expiration. The dataset includes daily highest closing bid and the lowest ask prices, volume, and open interest for the period between January 1996 and October VIX is calculated and disseminated by Chicago Board Options Exchange. The objective is to estimate the implied volatility of short-term ATM options on the S&P 500 index over the next month. The formula uses a kernel-smoothed estimator that takes as inputs the current market prices for several call and put options over a range of moneyness and maturities. 14 The options trade under the ticker SPX. Average daily volume in 2008 was 707,688 contracts. 13

15 In order to eliminate unreliable data, we apply a series of filters. First, we eliminate all observations for which the bid is lower than $0.125 or higher than the ask price. Second, we eliminate all observations with no volume to mitigate the impact of non-trading. Finally, we exclude all observations that violate usual arbitrage bounds. For the purpose of this study, we assume the risk-free interest rate is represented by one-month US LIBOR. This series comes from Bloomberg for the period between January 1996 and October B. Construction of options returns Our asset allocation uses a risk-free asset and a set of risky securities. We define four options with different levels of moneyness: an ATM call, an ATM put, a 5% OTM call, and a 5% OTM put. This limited number of securities keeps the model simple, but nevertheless generates flexible payoffs as a function of underlying asset price. OTM options are important for kernel spanning [Burachi and Jackwerth (2001) and Vanden (2004)], and a deep OTM put option is much more sensitive to negative jump risks. We do not allow the investor to choose from all available contracts simultaneously, as our investor may then exploit small in-sample differences between highly correlated option returns, leading to extreme portfolio weights (see, e.g., Jorion (2000) for a discussion of this issue). These are also among the most liquid options, using volume as a proxy for liquidity. We choose one-month maturity options. Burachi and Jackwerth (2001) report that most of the trading activity in S&P 500 index options is concentrated in the nearest contracts of less than 30 days to expiration. By choosing the one-month maturity we prevent microstructure problems. This target maturity is also appealing as longer-maturity option contracts may stop trading of the underlying asset moves in such a way that options become very deeply ITM or OTM. Yet another advantage is that holding the options to maturity incurs transaction costs only at inception of the trade. 14

16 We then construct a time-series of options returns, initially using the midpoint of bid and ask prices. We first find all available options contracts with exactly one month to maturity. 15 We then define buckets for option moneyness in terms of the ratio of the underlying price to the strike price less one, S/K 1. We set a range of moneyness of between -2.0% and 2.0% for ATM options and a bound 1.5 p.p. away from 5.0% for OTM options. Basically, we fix target moneyness buckets conditional on one-month maturity options. Following this, at each month and for each bucket we are left with several potential securities, although we want only one option contract in each month. We choose the option with the lowest relative bid-ask spread, defined as the ratio between the bid-ask spread and the mid-price. When more than one contract has the same ratio, we choose the one with the highest open interest. 16 Finally, we construct the synthetic one-month hold-to-expiration option returns: r t,t+1 = Payoff t+1 Price t 1 where Payoff t+1 is the payoff of the option at maturity calculated using the closing price of the underlying asset on the day before settlement, and Price t is the option price observed at the beginning of the period. We obtain a time-series of 178 observations for each option. Figure 3 presents kernel density estimates for each option security. We see that the options return distribution departs significantly from the normal distribution, with considerable negative tail risk for any of the options considered. Mean time-series characteristics for each option by option moneyness are presented in Table II. ATM call and put options have average moneyness of 0.35% and 0.24%, respectively, while OTM call and put options have average moneyness of 4.05% and 4.41%, respectively. These numbers show how close each contract is to the mean value of each bucket. Volume for each contract is around 4,000 and open interest is close to 20, There are other alternatives that we do not follow. For example, Burachi and Jackwerth (2001), Coval and Shumway (2001), and Driessen and Maenhout (2006) select options on the first day of each month and compute returns until first day of the next month. 16 We do not need further criteria, since this already defines a unique contract in each month for each option type and bucket. 15

17 Mean implied volatility varies between 15% and 22% with moneyness, which confirms the known smile effect. Panel B of Figure 1 shows the implied volatility of each option between 1996 and Implied volatility is low between 2003 and the mid-2007 s. We can see four pronounced peaks in these time-series in to the year 1998, over and , and in mid Table III reports summary statistics of returns for the various securities. The primary performance measure in this study is the Sharpe ratio. The S&P 500 index has an average monthly return of 0.5% over the sample period, corresponding to an annualized Sharpe ratio of We also compute the certainty-equivalent of an investor with risk aversion of 10 and incorporate descriptive statistics of the return distribution, in particular, skewness and excess kurtosis. S&P 500 index returns present negative skewness and excess kurtosis. Options present large negative average monthly returns ranging from -0.7% to -39.9%. This suggests that writing options may be a good strategy, as annualized Sharpe ratios range from 0.03 to 0.68 for this period. 18 Writing options, however, has negative tail risk that may be too onerous in some months. The returns to writing options have a maximum of 100%, but the minimum ranges from -463% to -2,496% depending on the option. This leads to large negative skewness and excess kurtosis between 0.71 and The last row of Table III shows a strategy that allocates the same weight to each option. DeMiguel, Garlappi, and Uppal (2009) argue that a naive 1/N uniform rule is generally good. Using this rule for our four risky assets and shorting this strategy, we 17 This is different from the usual stated Sharpe ratios for the US market of on the order of The main reason is the period in question which is not very long. The main problem of a Sharpe ratio is that it takes into account only the first two moments, the mean and the standard deviation. Broadie, Chernov, and Johannes (2009) show that although the Sharpe ratio is not the best measure to evaluate performance in an options framework, other alternative measures as Leland s alpha or the manipulation proof performance metric face the same problems. See Bernardo and Ledoit (2000) and Ingersoll, Spiegel, Goetzmann, and Welch (2007) for problems with the Sharpe ratio. 18 Coval and Shumway (2001) and Eraker (2007) show that writing put options earns Sharpe ratios of 0.68 or above for a different time period. 16

18 obtain smoother skewness and kurtosis, and an annualized Sharpe ratio of This comes with the same problem as with individual options. The construction of bid and ask securities is straightforward. For the chosen contract, we use the bid quote and the ask quote, respectively. The descriptive statistics for these contracts are very similar to the ones using mid-prices and are not presented to save space. IV. Results Table IV presents summary statistics of out-of-sample returns for the optimal option portfolio strategies between January 1996 and October Note first that all strategies display annualized Sharpe ratios of between 0.38 and 0.52, higher than the market ratio of around The unconditional OOPS returns have negative skewness (around -1) and substantial excess kurtosis (around 30). Notice that a -94.3% monthly return may be possible using this strategy. The best strategy is the one that uses a generalized extreme value distribution with an annualized Sharpe ratio of 0.52, although the Sharpe ratios are not very different across simulation methods. 19 Certainty-equivalent values are extremely negative because of the extreme returns. Conditional OOPS returns, however, present positive skewness (around 1) and smoother excess kurtosis (around 7). Figure 4 shows that this intrinsically defines a narrower distribution that limits the downside risk. Note by comparison of this figure with the one for individual options in Figure 3 that these strategies eliminate the substantial left-tail risk, and they yield almost symmetric distributions with returns truncated by -100% and 100% for unconditional OOPS and by -20% and 20% for conditional OOPS. Annualized Sharpe ratios are on the order of 0.50, and annualized certainty-equivalents are 4%. Conditional OOPS returns always have lower standard deviations than unconditional OOPS returns. These strategies result in negative returns for only 40% of the 19 These strategies are i.i.d., as can be shown using an ARCH test or a Ljung-Box tests (results not tabulated for space reasons). Hence, there is no need to correct annualized Sharpe ratios for the potential autocorrelation in strategy returns. 17

19 months. We perform a GEV fit exercise to analyze the type of distribution for these OOS OOPS returns (not presented for reasons of space). The implied GEV distribution is much narrower for the conditional OOPS, with means a little higher than the market, and very low volatility. The ratio of the mean over standard deviation is on the order of 8.0 compared to 1.3 for the unconditional strategies. We then analyze in more detail the OOS returns of the conditional OOPS using the GEV distribution. The negative extreme returns, which are relatively small, occur in September 2007 (-8.7%) and September 2001 (-7.2%), corresponding to two events that were totally unexpected by either the options market or the stock market. On the other hand, positive extreme returns occur in February 1996 (20.0%), July 1996 (7.8%), May 1997 (13.4%), November 1999 (13.2%), March 2007 (9.8%), and October 2008 (14.1%). Figure 5 presents the OOPS cumulative returns starting from one dollar invested in January This shows the excellent increasing stability of conditional strategies compared with the good overall performance of the unconditional strategies with their irregular paths and sudden drops like, for example, the one in September Table V reports average weights of each option in each strategy (panel A) and the proportion of months with positive weights (panel B). This latter measure lets us see if the security is, on average, long or short, as the mean may be affected by outlier values. We complement the analysis using a picture showing the evolution of weights. Figure 6 represents one example of what happens in terms of weights of the four risky assets for the parametric simulation using a GEV distribution for conditional OOPS. All four of the assets have significant weights that offset each other; at some times this offsetting is not always the same. For instance, in November 1998 we write -8.6% of an OTM put option and balance it with an ATM put option weight of 9.4%. In September 1999, call options are more important. We write an ATM call with a weight of -6.1% and hold a long position of 3.9% in an OTM call. In August 2008, we write -0.6% of an OTM put option, and we hold a long position of 2.1% of ATM put options and 1.6% of OTM call options. 18

20 Another important aspect is the no-investment case. This happens 42% of the months for ATM call options; see November 2009, for example. Usually we invest in the other options, except in only 5% of the months. There are also months when we invest only in one option (e.g., November 2008 corresponds to 1.4% of an OTM call option), two options (e.g., April % of an OTM call and 0.1% of an OTM put options), or three options (e.g., July % of an ATM call, 0.9% of an ATM put, and -0.7% of an OTM put option). These weights are quite different over time. The OOPS are quite different from the simple short put strategies described in the literature. Note that Driessen and Maenhout (2007) show that constant relative risk aversion investors always find it optimal to short OTM puts, and only with distorted probability assessments are they able to obtain positive weights for puts using cumulative prospect theory and anticipated utility. Table V also shows that the sign of the position in each option is not really related to the way we choose to simulate returns. On average, we hold long positions of ATM puts and OTM calls and short positions of OTM puts. The holdings of ATM call options change the most. The second point is that simulating the conditional distribution of returns leads to an allocation with lower weights in absolute value. The maximum and the minimum weights of all securities are lower for conditional OOPS, and the mean of the sum of the absolute values of weights of these eight securities is clearly lower than for unconditional OOPS. This makes our portfolio less leveraged. We can check that generally ATM options balance OTM options, which is even more true when extreme weights are set. Correlation figures confirm these results. The strongest correlated pairs are the ATM and OTM calls and the ATM and OTM puts (on the order of -0.70). For the other pairs the correlation is lower than 0.50 in absolute value. Moreover, put options seem to play a more important role in the allocation than call options. This can be checked in terms of individual assets and by comparing the sum of absolute weights of calls and puts. The latter is three times higher for unconditional OOPS and twice as high 19

21 as for conditional OOPS. Moreover, peaks are most pronounced in put options weights. One pronounced weight is attributable to September 11, This seems understandable, as this event was in no way expected by the market. The event was absolutely not priced in the options market, and thus the reason why this affected allocation so much. Following this event, the period surrounding Long-Term Capital Management bailout is also very volatile in terms of weights. September 2008 also sees higher weights, but still not as much as the previous two above. Our results do not match those of Liu and Pan (2003) who tend to buy OTM put options. They do not have a choice between different levels of moneyness. For conditional OOPS we do have a positive net position in put options that agrees with their result. Our results partially confirm the work of Driessen and Maenhout (2007) because we short an OTM put option, but we never write straddles. The bottom right picture in Figure 6 describes the evolution of the risk-free security weight. The mean weight is 98.2%, and 83% of the months the weight is less than 100%. Hence, no borrowing is the most standard case. The maximum weight is 107.8% in June 1996, and the minimum is around 91.5% in February To measure the sensitivity of this portfolio to potential changes in the underlying asset we first analyze the Black-Scholes delta. Panel A of Table VI presents summary statistics of the delta for each of the three strategies for unconditional and conditional OOPS. The main conclusion is that the portfolio delta is very close to zero, ranging from to 0.08 with a mean of around 0. Most months, the portfolio delta is negative. Conditional OOPS have a narrower range of deltas implying less risk. A better measure of risk is the elasticity of the portfolio. The elasticity of an option is described by multiplying the delta by the ratio between the underlying asset value and the option value. This measure has the advantage that it takes into account option leverage. Panel B of Table VI presents summary statistics for elasticity. The mean value is around -9, hence an increase of 1 percentage point in the underlying asset, is expected to 20

22 impact the portfolio by -9 percentage points, on average. This means that we typically hold a large net short position, although our strategies are clearly targeted to prevent extreme bad outcomes. A negative return for the OOPS strategies against a positive return for the market happens only in 28% of the months and with an implied monthly average return of only -1.53%. Note that in only 49% of the months is elasticity higher than 8 in absolute value. In fact OOPS are much less elastic than individual options. 20 Conditional OOPS have much less extreme negative elasticity than unconditional OOPS. Figure 7 shows the evolution of elasticity in the period between January 1996 and October 2010 for a simulation using a GEV distribution of conditional OOPS. This series is quite volatile. There are three main periods below -20, such as around , , and August and September 2008 this series attains its lowest value of around -44. We next investigate which variables explain or predict our results, in particular, the out-of-sample OOPS returns. The variables include r S&P 500, the S&P 500 index monthly return as a proxy for the market; and jump, the variable related to jump risk proxied by a binary variable equal to the absolute value of the S&P 500 index return when the current monthly return is below -5% and zero otherwise. Volatility is measured in two different ways: realized volatility (svol) of the last month using daily data, and the V IX level (vix). We also construct a variable that is the spread between the implied and realized volatility (vixsvol) a standard approach in the literature. Skew represents the probability of lefttail risk, meaning that a higher value is associated with more left-tail risk than a normal distribution. 21 We also use first-differences in the monthly values for some of these variables, and we denoted by the symbol. All these variables are customary in the equities and options literature. 20 Elasticity for individual options is presented in Table II. 21 This time-series is taken from the CBOE website, but we divide by 100 to make the analysis easier. 21

23 Table VII reports the results for both unconditional and conditional OOPS. 22 Many variables show modest explanatory power as demonstrated by a relatively high R 2 corresponding to a maximum of 17%, particularly, market returns, jumps, and volatility for unconditional OOPS. Only skew is irrelevant. Most of the variables have no predictive power. There is some statistical significance of volatility variables for some simulations; the maximum R 2 is 7%. Unconditional OOPS returns are profitable, on average, when the market increases, when a negative jump occurs, and when mean risk declines. 23 Conditional OOPS returns are more difficult to explain. No variable is able to explain all three strategies, but the strategy does well when risk increases. Notice the opposite signs on the conditional compared to the unconditional OOPS for all variables even if they are not significant. Explanatory regressions have a maximum R 2 of 3%. There is almost no predictive power in terms of conditional OOPS, which translates to a maximum R 2 of 2%. We conclude that there is no variable that explains conditional OOPS returns. To summarize the results, conditional OOPS present good stable performance with reduced weights on options and relatively less risk than the unconditional OOPS. V. Robustness Checks We perform some robustness tests only for conditional OOPS, as we have shown these strategies behave the best. The first test examines the impact of choosing fewer assets in each OOPS. A variety of authors use a restricted set of options to develop optimal strategies We tried different explanatory variables and models. There is insignificant exposure to the Carhart factors or Fung and Hsieh hedge fund factors. 23 Jones (2006) confirms that three factors explain the cross-section of option returns: market, volatility, and jump. 24 Liu and Pan (2003) use one stock, a 5% OTM put option, and cash. Eraker (2007) chooses among three risky assets, ATM straddles, OTM puts, and OTM calls. Jones (2006) uses only put options. Driessen and Maenhout (2007) analyze the choices among stock, an option strategy, and cash. They use either an OTM put, an ATM straddle or corresponding crash-neutral strategies. Constantinides, Jackwerth, and Savov (2009) use either call or put options. 22

24 Table VIII presents the most relevant statistics for understanding their performance. Each row describes a different portfolio choice, where each number represents a different option choice considered. The digits 1, 2, 3, and 4 denote the ATM call, the ATM put, the 5% OTM call, and the 5% OTM put options, respectively. When more than one digit appears, that strategy involves a combination of options. The strategy with four option securities yields simultaneously the best Sharpe ratio (around 0.50) and certainty-equivalent values (around 5%) with low kurtosis and positive skewness for the different simulation methods. For portfolios formed by only one option, there is no strategy that yields a positive Sharpe ratio, although using an ATM call delivers a positive annualized certainty-equivalent. A strategy with only a 5% OTM put option yields both the poorest Sharpe ratio and certainty equivalent value. Portfolios formed with two options all present positive Sharpe ratios, mainly around The strategy that combines an ATM call option with a 5% OTM put option achieves a Sharpe ratio of 0.53 for the empirical simulation, but then falls to 0.30 in GEV simulation. The same happens to the certainty-equivalent (from around 4% to close to 0%). Two other strategies seem to work as well: investing in both call options or in both put options. Strategies with three options present moderate or negative Sharpe ratios and certainty equivalents. We conclude that smaller security sets do not work well because they do not allow the investor to control risk by, for example, buying one put and shorting another put. The ones that work better are related to the choice of an even number of securities since there is the need to counterbalance short and long positions in options. Second, we analyze the effect of risk aversion on portfolio choice. Table IX presents the most relevant statistics for understanding performance of the strategies. Each row presents a different risk aversion parameter, γ. We follow two approaches. The first is to understand the effect of changing the risk aversion parameter, γ, from 10 to 5, 3, or 2. In Panel A of Table IX, note as γ declines, the skewness becomes even more positive, but kurtosis also increases. This negatively impacts 23