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Optimal Option Portfolio Strategies J O S É FA I A S ( C ATÓ L I C A L I S B O N ) P E D RO S A N TA - C L A R A ( N O VA , N B E R , C E P R ) October 2011 THE TRADITIONAL APPROACH 2 Mean-variance optimization (Markowitz) does not work Investors care only about two moments: mean and variance (covariance) Options have non-normal distributions Needs an historical “large” sample to estimate joint distribution of returns Does not work with only 15 years of data We need a new tool! José Faias and Pedro Santa-Clara OOPS - Optimal Option Portfolio Strategies LITERATURE REVIEW 3 Simple option strategies offer high Sharpe ratios Coval and Shumway (2001) show that shorting crash-protected, deltaneutral straddles present Sharpe ratios around 1 Saretto and Santa-Clara (2009) find similar values in an extended sample, although frictions severely limit profitability Driessen and Maenhout (2006) confirm these results for short-term options on US and UK markets Coval and Shumway (2001), Bondarenko (2003), Eraker (2007) also find that selling naked puts has high returns even taking into account their considerable risk. We find that optimal option portfolios are significantly different from just exploiting these effects For instance, there are extended periods in which the optimal portfolios are net long put options. José Faias and Pedro Santa-Clara OOPS - Optimal Option Portfolio Strategies METHOD (1) 4 For each month t run the following algorithm: t 1|t ,c 1. Simulate underlying asset standardized returns t 1|t , p xtn1 rtn1/rvtn , n 1,...,N • • Ct , c Pt , p Historical bootstrap Parametric simulation: Normal distribution and Generalized Extreme Value (GEV) distributions 2. Use standardized returns to construct underlying asset price based on its current level and volatility n t 1|t S St exp x rvt , n 1,...,N n t 1 This is what we call conditional OOPS. Unconditional OOPS is the same without scaling returns by realized volatility in steps 1 and 2. José Faias and Pedro Santa-Clara Max U rptn1|t rt n1|t ,c rt n1|t , p K t 1,c Ctn1|t ,c K t 1,p Pt n1|t , p St Stn1|t t t+1 OOPS - Optimal Option Portfolio Strategies METHOD (2) 5 t 1|t ,c 3. Simulate payoff of options based on exercise prices and simulated underlying asset level: maxK t 1|t , p ,0 , n 1,...,N Max U rptn1|t Ctn1|t ,c max Stn1|t - Kt,c ,0 , n 1,...,N Pt n1|t , p n S t,p t 1|t and corresponding returns for each option based on simulated payoff and initial price rt n1|t ,c Ctn1|t ,c Ct , c - 1 , n 1,..., N rt n1|t , p Pt n1|t , p Pt , p - 1 , n 1,..., N 4. Construct the simulated portfolio return C n t 1|t rp rf t t 1|t ,c r c 1 n t 1|t,c José Faias and Pedro Santa-Clara P rf t t 1|t , p rtn1|t,p rf t p 1 , n 1,...,N Ct , c Pt , p rt n1|t ,c rt n1|t , p K t 1,c Ctn1|t ,c K t 1,p Pt n1|t , p St Stn1|t t t+1 OOPS - Optimal Option Portfolio Strategies METHOD (3) 6 t 1|t ,c 5. Choose weights by maximizing expected utility over simulated returns t 1|t , p N 1 n Maxw E éëU (Wt (1+ rpt+1|t ))ùû » Maxw åU (Wt (1+ rpt+1|t )) N n=1 Ct , c Pt , p Power utility 1 W 1 U (W ) 1 ln(W ) if 1 José Faias and Pedro Santa-Clara rptn1|t rt n1|t ,c rt n1|t , p if 1 which penalizes negative skewness and high kurtosis Output : t 1|t ,c , c 1,...,C Max U K t 1,c Ctn1|t ,c K t 1,p Pt n1|t , p St Stn1|t t t+1 t 1|t , p , p 1,...,P OOPS - Optimal Option Portfolio Strategies METHOD (4) 7 6. Check OOS performance by using realized option returns t 1|t ,c Determine realized payoff Ct 1,c max St 1 - Kt,c ,0 Pt 1, p max Kt,p St 1,0 and corresponding returns rt 1,c Ct 1,c Ct , c -1 rt 1, p Pt 1, p Pt , p P rpt 1 rf t t 1|t ,c rt 1,c rf t t 1|t , p rt 1, p rf t c 1 José Faias and Pedro Santa-Clara p 1 rpt 1 Ct , c Pt , p rt 1,c K t 1,c Ct 1,c K t 1,p Pt 1, p St St 1 t t+1 rt 1, p -1 Determine OOS portfolio return C t 1|t , p OOPS - Optimal Option Portfolio Strategies DATA (1) 8 Bloomberg S&P 500 index: Jan.1950-Oct.2010 1m US LIBOR: Jan.1996-Oct.2010 OptionMetrics S&P 500 Index European options traded at CBOE (SPX): Jan. 1996-Oct.2010 Average daily volume in 2008 of 707,688 contracts (2nd largest: VIX 102,560) Contracts expire in the Saturday following the third Friday of the expiration month Bid and ask quotes, volume, open interest Monthly frequency José Faias and Pedro Santa-Clara OOPS - Optimal Option Portfolio Strategies DATA (2) 9 Jan.1996-Oct.2010: a period that encompasses a variety of market conditions José Faias and Pedro Santa-Clara OOPS - Optimal Option Portfolio Strategies DATA (3) 10 Asset allocation using risk-free and 4 risky assets: ATM Call Option (exposure to volatility) ATM Put Option (exposure to volatility) 5% OTM Call Option (bet on the right tail) 5% OTM Put Option (bet on the left tail) These options combine into flexible payoff functions Left tail risk incorporated José Faias and Pedro Santa-Clara OOPS - Optimal Option Portfolio Strategies DATA (4) 11 Define buckets in terms of Moneyness (S/K‐1) ⇒ ATM bucket: 0% ± 1.5% ⇒ 5% OTM bucket: 5% ± 2% Choose a contract in each bucket Smallest relative Bid‐Ask Spread, and then largest Open Interest José Faias and Pedro Santa-Clara OOPS - Optimal Option Portfolio Strategies DATA (5) 12 José Faias and Pedro Santa-Clara OOPS - Optimal Option Portfolio Strategies TRANSACTION COSTS 13 Options have substantial bid-ask spreads! José Faias and Pedro Santa-Clara OOPS - Optimal Option Portfolio Strategies TRANSACTION COSTS 14 We decompose each option into two securities: a “bid option” and an “ask option” [Eraker (2007), Plyakha and Vilkov (2008)] Long positions initiated at the ask quote Short positions initiated at the bid quote No short-sales allowed “Bid securities” enter with a minus sign in the optimization problem In each month only one bid or ask security is ever bought The larger the bid-ask spread, the less likely will be an allocation to the security Lower transaction costs from holding to expiration Bid-ask spread at initiation only José Faias and Pedro Santa-Clara OOPS - Optimal Option Portfolio Strategies OOPS RETURNS 15 Out-of-sample returns José Faias and Pedro Santa-Clara OOPS - Optimal Option Portfolio Strategies OOPS CUMULATIVE RETURNS 16 José Faias and Pedro Santa-Clara OOPS - Optimal Option Portfolio Strategies OOPS WEIGHTS 18 Proportion of positive weights José Faias and Pedro Santa-Clara OOPS - Optimal Option Portfolio Strategies OOPS ELASTICITY 19 José Faias and Pedro Santa-Clara OOPS - Optimal Option Portfolio Strategies EXPLANATORY REGRESSIONS 20 José Faias and Pedro Santa-Clara OOPS - Optimal Option Portfolio Strategies PREDICTIVE REGRESSIONS 21 José Faias and Pedro Santa-Clara OOPS - Optimal Option Portfolio Strategies CONCLUSIONS 24 We provide a new method to form optimal option portfolios Easy and intuitive to implement Very fast to run Small-sample problem and current conditions of market are taken into account Optimization for 1-month Option characteristics Volatility of the underlying Transaction costs Strategies provide: Large Sharpe Ratio and Certainty Equivalent Positive skewness Small kurtosis José Faias and Pedro Santa-Clara OOPS - Optimal Option Portfolio Strategies