New Tool for Optimal Option Portfolio Strategies by Jos Faias and Pedro Santa-Clara

The traditional mean-variance optimization approach does not work well for options due to their non-normal distribution. Jos Faias and Pedro Santa-Clara propose a new tool called OOPS (Optimal Option Portfolio Strategies) which considers high Sharpe ratios and optimal option portfolios different from simple option strategies. The method involves simulating underlying asset returns and payoffs of options based on exercise prices and asset levels.

• Optimal Option Portfolio
• Strategies
• Finance
• Jos Faias
• Pedro Santa-Clara

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1. Optimal Option Portfolio Strategies JOS FAIAS (CAT LICA LISBON) PEDRO SANTA-CLARA (NOVA, NBER, CEPR) October 2011

2. 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! Jos Faias and Pedro Santa-Clara OOPS - Optimal Option Portfolio Strategies

3. LITERATURE REVIEW 3 Simple option strategies offer high Sharpe ratios Coval and Shumway (2001) show that shorting crash-protected, delta- neutral straddles present Sharpe ratios around 1 Saretto and Santa-Clara (2009) find similar values in an extended sample, although frictionsseverely 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. Jos Faias and Pedro Santa-Clara OOPS - Optimal Option Portfolio Strategies

4. METHOD (1) 4 For each month t run the following algorithm: , | 1 + t t c n t rp Max U | 1 + 1. Simulate underlying asset standardized returns t , | 1 + t t p = = n t n n t /rv n , 1,..., N x r + + 1 1 t n + n + t C, tP, tr Historical bootstrap Parametric simulation: Normal distribution and Generalized Extreme Value (GEV) distributions , | 1 t c c tr p , | 1 t p 2. Use standardized returns to construct underlying asset price based on its current level and volatility ( exp | 1 = + + t t t t x S S n + t C K+ K+ , | 1 t c t 1, c ) n + tP = n n n , 1,..., N rv t 1, p , | 1 t p 1 t This is what we call conditional OOPS. Unconditional OOPS is the same without scaling returns by realized volatility in steps 1 and 2. Jos Faias and Pedro Santa-Clara n + tS tS | 1 t t t+1 OOPS - Optimal Option Portfolio Strategies

5. METHOD (2) 5 3. Simulate payoff of options based on exercise prices and simulated underlying asset level: ( - max | 1 , | 1 = + + t t c t t S C ( K max p t, , | 1 = + p t t S P , | 1 + t t c n t rp Max U | 1 + t , | 1 + t t p ) ) = n n K 0 , c t, n , 1,..., N n + n + t C, tP, tr = n + t n 0 , t n , 1,..., N , | 1 t c c | 1 tr and corresponding returns for each option based on simulated payoff and initial price p , | 1 t p n n t P C , | 1 + P , | 1 + C t t p t c = = = = n n 1 - n , 1,..., N 1 - n , 1,..., N r r n + t C K+ K+ , | 1 + , | 1 + t t c t t p , | 1 t c t 1, c , , t c t p n + tP t 1, p , | 1 t p 4. Construct the simulated portfolio return ( ) ( ) C P = = = + + = n + n + n t n , 1,..., N rp rf r rf r rf n + tS tS | 1 + , | 1 + , | 1 + t t, | 1 c t t, | 1 p t t t t c t t t p t | 1 t c 1 p 1 t t+1 Jos Faias and Pedro Santa-Clara OOPS - Optimal Option Portfolio Strategies

6. METHOD (3) 6 5. Choose weights by maximizing expected utility over simulated returns , | 1 + t t c n t rp Max U | 1 + t , | 1 + t t p N 1 N MaxwE U Wt(1+rpt+1|t) ( ) U Wt(1+rpt+1|t ( ) Maxw n) n=1 n + n + t C, tP, tr Power utility , | 1 t c c tr p 1 , | 1 t p 1 if 1 W W = 1 ( ) U W = ln( ) if 1 n + t C which penalizes negative skewness and high kurtosis Output : K+ K+ , | 1 t c t 1, c n + tP t 1, p , | 1 t p = = , ,..., 1 , ,..., 1 c C p P , | 1 + , | 1 + t t c t t p n + tS tS | 1 t t t+1 Jos Faias and Pedro Santa-Clara OOPS - Optimal Option Portfolio Strategies

7. METHOD (4) 7 6. Check OOS performance by using realized option returns , | 1 + t t c rp + 1 t , | 1 + t t p Determine realized payoff = c t C ( ) ( ) 0 , 1 + t C, tP, tr = max - K 0 , c t, max K S P S , 1 + c c , 1 + + , 1 + 1 t, p t t p t tr , 1 + p p and corresponding returns C r + = P , 1 + , 1 + t p = t c 1 - 1 - r , 1 + , 1 t c t p C P t C K+ K+ , , t c t p , 1 + c t 1, c tP t 1, p , 1 + Determine OOS portfolio return p ( ) ( ) C P = = = + + rp rf r rf r rf + , | 1 + + , | 1 + + 1 t 1, c t 1, p t t t t c t t t p t tS tS + c 1 p 1 1 t t+1 Jos Faias and Pedro Santa-Clara OOPS - Optimal Option Portfolio Strategies

8. 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 (2ndlargest: 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 Jos Faias and Pedro Santa-Clara OOPS - Optimal Option Portfolio Strategies

9. DATA (2) 9 Jan.1996-Oct.2010: a period that encompasses a variety of market conditions Jos Faias and Pedro Santa-Clara OOPS - Optimal Option Portfolio Strategies

10. 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 Jos Faias and Pedro Santa-Clara OOPS - Optimal Option Portfolio Strategies

11. 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 Jos Faias and Pedro Santa-Clara OOPS - Optimal Option Portfolio Strategies

12. DATA (5) 12 Jos Faias and Pedro Santa-Clara OOPS - Optimal Option Portfolio Strategies

13. TRANSACTION COSTS 13 Options have substantial bid-ask spreads! Jos Faias and Pedro Santa-Clara OOPS - Optimal Option Portfolio Strategies

14. 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 Jos Faias and Pedro Santa-Clara OOPS - Optimal Option Portfolio Strategies

15. OOPS RETURNS 15 Out-of-sample returns Jos Faias and Pedro Santa-Clara OOPS - Optimal Option Portfolio Strategies

16. OOPS CUMULATIVE RETURNS 16 Jos Faias and Pedro Santa-Clara OOPS - Optimal Option Portfolio Strategies

17. OOPS WEIGHTS 18 Proportion of positive weights Jos Faias and Pedro Santa-Clara OOPS - Optimal Option Portfolio Strategies

18. OOPS ELASTICITY 19 Jos Faias and Pedro Santa-Clara OOPS - Optimal Option Portfolio Strategies

19. EXPLANATORY REGRESSIONS 20 Jos Faias and Pedro Santa-Clara OOPS - Optimal Option Portfolio Strategies

20. PREDICTIVE REGRESSIONS 21 Jos Faias and Pedro Santa-Clara OOPS - Optimal Option Portfolio Strategies

21. 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 Jos Faias and Pedro Santa-Clara OOPS - Optimal Option Portfolio Strategies