### Talk

```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
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
xtn1  rtn1/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
rptn1|t
rt n1|t ,c
rt n1|t , p
K t 1,c
Ctn1|t ,c
K t 1,p
Pt n1|t , p
St
Stn1|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:

 maxK
t 1|t , p

,0 , n  1,...,N
Max U
rptn1|t
Ctn1|t ,c  max Stn1|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 n1|t ,c 
Ctn1|t ,c
Ct , c
- 1 , n  1,..., N
rt n1|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 rtn1|t,p  rf t
p 1

, n  1,...,N
Ct , c
Pt , p
rt n1|t ,c
rt n1|t , p
K t 1,c
Ctn1|t ,c
K t 1,p
Pt n1|t , p
St
Stn1|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
rptn1|t
rt n1|t ,c
rt n1|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
Ctn1|t ,c
K t 1,p
Pt n1|t , p
St
Stn1|t
t
t+1
t 1|t , p , p  1,...,P
OOPS - Optimal Option Portfolio Strategies
METHOD (4)
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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)
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 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)
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 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)
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 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

José Faias and Pedro Santa-Clara
OOPS - Optimal Option Portfolio Strategies
DATA (5)
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José Faias and Pedro Santa-Clara
OOPS - Optimal Option Portfolio Strategies
TRANSACTION COSTS
13
José Faias and Pedro Santa-Clara
OOPS - Optimal Option Portfolio Strategies
TRANSACTION COSTS
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 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


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

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
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 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
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