Better Portfolios with Options

Gerda Cabej, Manfred Gilli and Enrico Schumann

Abstract—In the period following the last financial crisis,equity markets have performed poorly. In consequence, equitylong-only strategies have generally disappointed over this pe-riod. This has motivated the investigation on whether betterperformance can be achieved by including equity options inthe portfolios. We show that simple systematic option strate-gies improve portfolio performance. Results are supported bythorough backtesting and simulations.

I. INTRODUCTION

CLASSICAL portfolio selection generally considers pri-mary asset classes such as equities, bonds and com-

modities. An extension consists in introducing derivatives onsuch assets. Our aim is to investigate whether equity options,written on single assets, enhance the overall performance ofa portfolio.Writing an option, call or put, on an asset is commonly

named option overwriting and corresponds to the sellingof a given option. Some of the questions that arise are:i) what type of option on which asset; ii) what strike; iii)what maturity; iv) how many contracts. We simplify theproblem by considering only maturities of one month andtwo moneyness levels of 100 and 105.1While option pricing is extensively discussed in the litera-

ture, little is published on portfolios with option overwriting,except for the buy-write strategy applied mainly to indexoptions. Also known as covered calls, this is the mostwidespread option strategy and consists in buying a stockand immediately selling a call on this stock. Typical ratiosare one-to-one (fully covered calls) or below one.The literature on covered call portfolios dates back to the

late 1970s following the advent of the Black-Scholes pricingmodel and the expansion of the option exchanges, e.g. [2].Most of this research concludes that writing covered callsis beneficial, in particular if the performance is measuredin risk-adjusted terms. In recent years there seems to be arenewed interest in portfolios with options, due, probably,to the less attractive performance of the equity markets.Most of this work considers index options. Among thesewe can cite [3] and [4] who highlight the favorable riskand return characteristics of the S&P 500 buy-write index(BXM) introduced in April 2002 by the Chicago BoardOptions Exchange (CBOE). In a similar study, [5] arrive at

The first two authors are with the University of Geneva (email:{Gerda.Cabej, Manfred.Gilli}@unige.ch) and Enrico Schumann is withAquila Capital Group (email: es@enricoschumann.net). Manfred Gilli isaffiliated with the Swiss Finance Institute.We thank Chrilly Donninger for helpful comments.1It may be tempting to leave all these decisions to an optimization

algorithm, but it is often preferred to impose structure on a problem soto avoid or at least reduce overfitting [1].

comparable conclusions by analyzing a buy-write strategy onthe Russell 2000 index.Our analysis deals with portfolios built from a set of

15 assets from the DAX. The portfolios with options areconstructed following a covered call and a Delta-hedgingstrategy and are compared with benchmarks consisting ofequity-only and a 1/N portfolio.Section II presents data, backtesting schema and bench-

mark portfolios. Option strategies are introduced in Section 3and Section 4 concludes.

II. DATA, BACKTESTING AND BENCHMARKSData: Our data comprises a set of 15 time series of equity

prices from the DAX2 spanning May 2007 to September2011. The option data consist of series of implied volatilitiesfor the same 15 equities with a maturity of 30 days and twomoneyness levels 100 and 105. Moneyness M is defined asthe ratio of strike X over spot S

Mt = 100×X/St

thus Mt is expressed in percentages. Mt = 100 correspondsto an option at-the-money. Mt = 105 corresponds to a strike5% above current spot. Thus a call would be 5% out-of-the-money and a put 5% in-the-money.Backtesting: To get a realistic view of the performance of

the models we construct the portfolios on a rolling windowof historical data representing the in-sample information. Theportfolios are then held over a fixed horizon and thus theirperformance is evaluated out-of-sample. At the next rebalanc-ing date, which coincides with the business day following theend of the holding period, we shift the historical window andrestart the portfolio construction process. The schema belowillustrates this procedure, where H and F are, respectively,the length of the historical window and the holding horizon.In our application H is set to one year and the portfoliosare held for approximately one month. Rebalancements takeplace on the first business day following the option expirydate (usually the Monday following the third Friday of themonth). Thus, over the entire horizon, the portfolios arereevaluated 55 times.

Period 1

t1−H t1 t1+F

F

H

invest

Period 2

t2−H t2 t2+F

rebalance

2Tickers of assets in dataset: ADS, ALV, BAYN, BAS, BMW, CBK,EOAN, DBK, DPW, LHA, MEO, SAP, SIE, DTE, TKA.

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The procedure just described generates a wealth path. Toquantify the uncertainty of such a wealth path, the process isrepeated a number of times with small changes in the histor-ical data. These changes are obtained by jackknifing, i.e. ran-domly deleting a given small percentage of observations,in our case 10%. Jackknifing directly affects the portfoliosminimizing drawdown or variance as the optimization relieson the historical information alone, whereas the portfoliosminimizing the Omega ratio are built on data simulated bya factor model.3 The latter is estimated from the historicalwindow, thus, in this case, the impact of jackknifing is lessdirect.The randomness of the set of paths can be characterized

in a variety of ways. One could consider a summary of thewhole path using statistics such as the Sortino ratio. Wechoose to focus on overall performance measured in terms ofannualized returns. This allows an easy comparison of port-folios for different time horizons. Figures report the medianpath, with respect to final wealth, of the simulated trajectoriesas well as the empirical distribution of the annualized returns.Benchmarks: The reference portfolios comprise a peri-

odically rebalanced 1/N portfolio and a set of equity-onlyportfolios that minimize variance (MV), drawdawn (DD), theOmega ratio (Om) and a linear combination of the last two(DD-Om). In a later step options are added to these portfolios.For the equity-only portfolios the optimization problem canbe briefly formalized as

minx

Φ(x)∑

j∈J xj p0j = v0

xinfj ≤ xj ≤ xsup

j j ∈ JKinf ≤ #{J } ≤ Ksup

where Φ is the objective function, x is a vector with xj theshares of asset j in the portfolio. The vector of initial pricesis denoted p0, v0 is the initial wealth and J is the set ofassets in the portfolio. The optimization is subject to thebudget constraint, minimum and maximum holding size andupper cardinality constraint. The objective functions Φ to beminimized are then:

• For variance

Φ = Var(∑

j∈Jwj Rti−H:ti,j

)(1)

where w is the weight vector and Rti−H:ti,j is the returnvector of asset j in the ith historical window.

• For drawdown

Φ = max(vmaxt − vt) (2)

where vmaxt is the running maximum of portfolio wealth,

i.e. vmaxt = max{vs|s ∈ [ti − H, ti]}. In this case Φ

measures the largest drop of the portfolio value overthe time horizon.

3For a detailed explanation of the scenario generation procedure see [6].

• For Omega

Φ =Mlo

Mup=−∑

rs 1{rs<0}∑rs 1{rs>0}

(3)

where

Mlo = 1nS

nS∑s=1

(rd − rs)mlo 1{rs<rd}

Mup = 1nS

nS∑s=1

(rs − rd)mup 1{rs>rd}

with rd a return target, set to zero in our applications,and mlo, mup the order of the moments, set to one.

A particular feature inherent to optimization in finance isthe large quantity of noise contained in the observations. Asa consequence only a limited precision is meaningful for thecomputed solutions [7]. Also if the objective function buildson empirical distributions or is subject to realistic constraintsthe problem becomes complex, generally not computablewith a classical method. Altogether this encourages the useof heuristic methods which also presents another importantadvantage, as we do not need to reduce our problem inorder to fit the classical optimization paradigm. Our preferredmethod is Threshold Accepting introduced by [8] and [9].This technique was first applied to portfolio selection by [10].Implementation details as well as applications can be foundin [11] and [12].The performance of the reference portfolios is given in

Figure 1, where, as already mentioned, are shown the medianwealth paths (upper panel) of 100 simulated trajectoriesand the corresponding empirical distributions of annualizedreturns (lower panel). The red triangle in the lower panelof Figure 1 corresponds the annualized return of the 1/Nportfolio, for which no distribution has been computed, asthe strategy does not depend on historical data. All portfoliosyield overall negative returns.

III. PORTFOLIOS WITH OPTIONSIntroducing options in an equity portfolio might be mo-

tivated by a need to hedge or the desire to leverage orenhance performance. Strategies can be based on a bet onthe evolution of the market (directional) or on its volatility.Among the variety of existing strategies we focus on astatic one, buy-write, and a dynamic one, based on volatility,known as Gamma trading.4 For the volatility based strategywe choose at-the-money options as these have the highestGamma. For both strategies the procedure of portfolio con-struction follows the same two consecutive steps: in a firststep the optimization selects an equity-only portfolio which,in a second step, is augmented with positions in single-nameoptions on the portfolio components.

A. A static strategy: covered callsWriting for each equity in the portfolio a number of

options corresponding to the number of shares held is known

4A comprehensive discussion about Gamma trading can be found in [13].

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OmDD−OmDDMV1/N

−20 −15 −10 −5 00

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OmDD−OmDDMV

Fig. 1. Results for equity-only reference portfolios (Om, DD-Om, DD, MVand 1/N). Upper panel: median wealth paths of 100 trajectories obtainedfrom jackknifed observations. Lower panel: corresponding empirical distri-butions of annualized returns.

as a covered call or buy-write strategy.5 Figure 2 shows theresults for portfolios computed with the objective functionspresented in Section II, where, at each rebalancement dateand for each position, out-of-the-money calls are written. Themoneyness of these calls is chosen to beM = 105 restrainingthe upside potential of the portfolio to 5% plus the premium.In case of a downside movement at least the premia areearned. Rebalancement dates are chosen as to match the dayafter the options expire. This simplifies our simulations as wedo not need to worry about liabilities related to outstandingoptions when constructing the new portfolios.

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OmDDMV1/N

Fig. 2. Median wealth paths (upper panel) and empirical distributions ofannualized returns (lower panel) for portfolios of covered calls with strike5% above spot (M = 105).

Comparing Figure 1 with Figure 2 we observe, as expected,

5A thoughtful description of overwriting, from a practitioner’s point ofview, can be found in chapter 17 of [14].

an improved performance of the covered call portfolios,i.e. a shift to the right in the distribution of annualizedreturns. This is true for all objective functions as well asfor the 1/N portfolio. In spite of the achieved enhancement,unfortunately, all returns remain negative.An alternative way to illustrate the improvement due to

the buy-write strategy is given in Figure 3 where the ratio ofcovered call over equity-only portfolios for the median pathsis plotted. We observe that the performance enhancement ismore important for portfolios minimizing drawdown. Further-more, for the first year, overwriting minimum variance (MV)portfolios does not add value (ratio below one).

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1.6MVDD

Fig. 3. Relative performance of covered call vs equity-only portfolios,measured by the ratio of median wealth paths. Covered call portfoliosoutperform, both for minimum variance (MV) and maximum drawdown(DD).

B. A dynamic strategy: volatility tradingThe goal is to construct a volatility position. This can

be achieved by selling or buying options. However, suchportfolios would be exposed to two risk factors, the underlierand volatility, the main drivers of option prices. To reduceunderlier risk we Delta hedge, making the position insen-sitive to small price movements and, as a consequence, ourportfolio becomes mainly a volatility position, hence the termvolatility trading. The strategy is also a two-step procedure.First an optimal equity portfolio is constructed, and then eachequity position is hedged independently with options.Recall that the Delta of an option is defined as its derivative

with respect to the underlying equity. This derivative canalso be computed for a portfolio consisting of an equity andoptions. Such a portfolio is Delta neutral, i.e. insensitive toa small change in price, if its derivative with respect to theunderlier is near zero. As only long positions in equities areconsidered and short selling is not allowed, two types ofcombinations or portfolio strategies are possible to realizethe Delta-hedging:i) long asset and short call, i.e. buying the underlying assetand selling the call;

ii) long asset and long put, i.e. buying the underlying assetand the put.

The hedge strategy is decided at the nper rebalancing dates,based on the forecasts of either realized or implied volatility[15]. In this application realized volatility is considered and adecision has to be made whether to buy or to sell volatility. Iffuture volatility is expected to be lower than the current, we

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sell it by taking a short position on calls. In the oppositesituation we buy volatility by buying puts.6 During thelife of the option, the hedge is adjusted daily, if necessary,by trading the underlying equity, while the option positionremains unchanged until maturity. Again the option expirydate coincides with the last holding date. The implementationis summarized in Algorithm 1.

Algorithm 1 Construction of Delta-hedged portfolios1: for i = 1 : nper do2: Initialize t1 and t2 corresponding to window i

3: Compute PF from historical observations Pt1:t2,1:nA

4: Initialize t3 and t4 corresponding to window i

5: Forecast volatility σFi for period t3 to t4 and generate

signal i, i = 1, 2, . . . , nA

6: At rebalancing date t3 choose hedge strategy according tosignal i, i = 1, 2, . . . , nA

7: for t = t3 + 1 : t4 do8: Adjust hedge for position i and store wealth in Vt,i, i =

1, 2, . . . , nA

9: Compute portfolio value vt =∑nA

i=1 Vt,i

10: end for11: end for

The core of the problem is to forecast the volatility ofthe underlier during the holding period (Statement 5 inAlgorithm 1). For each asset i and at each rebalancing datethe forecast will be translated into a signal according to:

signal i =

⎧⎨⎩

+1 if σFi > σR

i + h/2−1 if σF

i < σRi − h/2

0 otherwise(4)

where σFi and σR

i are the forecasted, respectively realizedvolatility7 and h is a given bandwidth. Thus we distinguishthree situations for the evolution of the underlier, i.e. increas-ing, decreasing and stable volatility.The choice of the hedge strategy (Statement 6 in Algo-

rithm 1) depends on the signal. In periods of decreasingvolatility (signal = −1) the hedge is done by selling at-the-money calls (strategy i) above) such that the Delta ofthe position equals zero and the Gamma8 is negative. In theopposite situation (signal = +1, increasing volatility) thehedge is achieved resorting to strategy ii), i.e. buying puts,thus yielding a long Gamma position. If no significant changeof volatility is forecasted (signal = 0), no option positionis taken for the concerned equity. Adjustments are doneby trading the underlier, if the usual minimum transactionvolume is reached. Options are traded by lots of 100 contractsfor which a fee of e 1 is charged.Summarizing, our strategy consists in selecting option

positions based exclusively on the volatility related signal

6Actually, one should look whether the market has not already priced inthis change in expected volatility. At this point we are not taking this intoaccount.7Realized volatility is estimated by computing the standard deviation on

a sliding window of past returns.8Gamma is the derivative of Delta with respect to the price of the

underlying asset. Note that Gamma takes only positive values for both, callsand puts.

generated by the algorithm. In addition one could also letthe algorithm integrate stock-specific information for thedecision of buying or selling options [16].1) Volatility forecast: Different approaches will be ex-

plored. A first attempt consists in using a GARCH processwith filtered historical simulation. A second trial resorts to anaive estimation of volatility of volatility and finally the twoapproaches are combined.

a) Volatility forecast with GARCH: We need a volatilityforecast for the date of expiry, i.e. approximatively 25 daysahead. To obtain it we partly follow [17] as we calibratea GARCH(1,1) model to the returns and then standardizeresiduals. These enter the model to forecast 1-day aheadvolatility. For the n-day forecast the cycle is repeated ntimes. We slightly deviate in taking arithmetic instead oflog-returns and further we use Differential Evolution for theestimation of the parameters. The use of heuristics for suchan estimation is motivated by the results in e.g. [18]. Asforecasts are less reliable over longer time horizons, we selectour strategy on the basis of the 1-day ahead volatility forecast.Results are shown in Figure 4 and reflect a significantimprovement with respect to the covered call portfolios inFigure 2.

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1.3

1.4

1.5

1.6OmDDMV1/N

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0.1

0.5

0.91

OmDDMV1/N

Fig. 4. Results for the dynamic strategy. At-the-money options and GARCHvolatility forecast. Upper panel: Median wealth paths with respect to finalwealth. Lower panel: Empirical distribution of annualized returns.

b) ‘Naive’ volatility forecast: We compute an approxi-mation of the volatility of volatility ν for a historical windowand compare it with the last observed change of volatilityΔσ = σt − σt−1. Our ‘strategy’ signal is then generated asin (4) where we replace σR with ν and σF with Δσ and h ischosen appropriately. Note that we are only guessing a trendand not trying to forecast a level.As can be observed in Figure 5 using the ‘naive’ volatility

forecast yields good results. The median annualized returnof all strategies is positive and minimum variance (MV)and maximum drawdown (DD) have positive 0-quantiles,i.e. all simulated wealth paths have positive annualizedreturns (lower panel of Figure 5). Comparing the empirical

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distributions in the lower panels of Figures 4 and 5 we wouldconclude that the Delta-hedged portfolios based on the ‘naive’volatility forecast outperform those based on the GARCHforecast. This is true for annualized returns.

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OmDDMV1/N

Fig. 5. Results for the dynamic strategy. At-the-money options and‘naive’ volatility forecast. Upper panel: Median wealth paths. Lower panel:Empirical distributions of annualized returns.

The upper panel in Figure 6 illustrates, for the medianportfolio (out of 100 simulations), premia earned from (orpayed for) the option positions selected by the algorithm.The lower panel shows the corresponding contributions inthe portfolio Gamma. The displayed Gamma are standardizedwith respect to the price of the underlier.

IV. CONCLUSIONS

This study is an investigation of the effect of options onportfolio performance. The market considered is a subsetof 15 components of the DAX. To judge the influence ofthe options, a set of equity-only portfolios is computed asa benchmark. Even though these portfolios outperform theindex, their average annualized returns remain negative, i.e.approximately −10%.Two types of strategies have been tested, one static and the

other dynamic. The static strategy, known as buy-write, con-sists in mechanically writing out-of-the-money (moneyness105) call options on a previously optimized equity portfolioin a one to one ratio. This yields enhanced performance forall portfolios. The empirical distributions of the annualizedreturns shift to the right reaching a median value of around−5%.The dynamic strategy is the best performing one. This is a

simple systematic option strategy, where we first optimize anequity portfolio and then actively choose between long voland short vol positions. In this case we achieve substantialpositive annualized returns despite the transaction costs in-volved.9 The median of annualized returns reaches some 8%

9The transaction costs due to the daily adjustment of the hedge over thebacktesting period amount to approximately 10% of the initial wealth.

Nov06 Jun07 Dec07 Jul08 Jan09 Aug09 Mar10 Sep10 Apr11 Oct11

TKA

SIE

SAP

MEO

EOAN

DPW

LHA

DTE

DBK

CBK

BMW

BAYN

BAS

ALV

ADS

Nov06 Jun07 Dec07 Jul08 Jan09 Aug09 Mar10 Sep10 Apr11 Oct11

TKA

SIE

SAP

MEO

EOAN

DPW

LHA

DTE

DBK

CBK

BMW

BAYN

BAS

ALV

ADS

Fig. 6. Option features of the median portfolio. Dynamic strategywith ‘naive’ volatility forecast. Minimum variance (MV) objective function.Upper panel: Premia for single positions in percentage of portfolio value(space between assets corresponds to 4%). Positive values: premia earnedfrom selling calls. Negative values: premia payed to buy puts. Lower panel:Gamma of the positions standardized with respect to the price of theunderlier. Positive values: long positions in puts. Negative values: shortpositions in calls.

for portfolios minimizing variance when the ‘naive’ volatilityforecast is used.These results are supported by extensive simulations and

backtesting together with a robustness check, presented inthe Appendix, where the models have been tested on thecomplete set of DAX components. Conclusions are invarianteven when performance is measured by risk adjusted returns.One topic for further investigation would be the perfor-

mance comparison of our portfolios with single name optionswith that of the dynamic strategy applied to index options.Another extension to this research would consist in includingthe Gamma of at-the-money options in the objective function.

REFERENCES

[1] S. Geman, E. Bienenstock, and R. Doursat, “Neural networks and thebias/variance dilemma,” Neural Computation, vol. 4, pp. 1–58, 1992.

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[2] R. C. Merton, M. S. Scholes, and M. L. Gladstein, “The returns andrisk of alternative call option portfolio investment strategies,” Journalof Business, vol. 51, no. 2, pp. 183–242, 1978.

[3] R. Whaley, “Return and risk of CBOE buy write monthly index,”Journal of Derivatives, vol. 10, no. 2, pp. 35–42, 2002.

[4] J. M. Hill, V. Balasubramanian, K. B. Gregory, and I. Tierens, “Findingalpha via covered index writing,” Financial Analysts Journal, vol. 62,no. 5, pp. 29–46, 2006.

[5] N. Kapadia and E. Szado, “The risk and return characteristics of thebuy-write strategy on the Russell 2000 index,” Journal of AlternativeInvestments, vol. 9, no. 4, pp. 39–56, 2007.

[6] M. Gilli and E. Schumann, “Risk–reward optimisation for long-runinvestors: an empirical analysis,” European Actuarial Journal, vol. 1,no. 1, pp. 303–327, Supplement 2 2011.

[7] ——, “Optimal enough?” Journal of Heuristics, vol. 17, no. 4, pp.373–387, 2010.

[8] G. Dueck and T. Scheuer, “Threshold Accepting. A General PurposeOptimization Algorithm Superior to Simulated Annealing,” Journal ofComputational Physics, vol. 90, no. 1, pp. 161–175, September 1990.

[9] P. Moscato and J. Fontanari, “Stochastic Versus Deterministic Updatein Simulated Annealing,” Physics Letters A, vol. 146, no. 4, pp. 204–208, 1990.

[10] G. Dueck and P. Winker, “New Concepts and Algorithms for PortfolioChoice,” Applied Stochastic Models and Data Analysis, vol. 8, no. 3,pp. 159–178, 1992.

[11] M. Gilli and P. Winker, “Heuristic optimization methods in economet-rics,” in Handbook of Computational Econometrics, D. A. Belsley andE. Kontoghiorghes, Eds. Wiley, 2009.

[12] M. Gilli, D. Maringer, and E. Schumann, Numerical Methods andOptimization in Finance. Elsevier, 2011.

[13] R. Rebonato, Volatility and Correlation: The Perfect Hedger and theFox. Wiley, 2004.

[14] M. P. O’Connell, The Business of Options: Time-Tested Principles andPractices. Wiley, 2001.

[15] R. Ahmad and P. Wilmott, “Which free lunch would you like to-day, Sir?: Delta hedging, volatility arbitrage and optimal portfolios,”Wilmott Magazine, pp. 64–79, November 2005.

[16] M. Ammann, D. Skovmand, and M. Verhofen, “Implied and realizedvolatility in the cross-section of equity options,” International Journalof Theoretical and Applied Finance, vol. 12, no. 06, pp. 745–765,2009.

[17] G. Barone-Adesi, K. Giannopoulos, and L. Vosper, “Backtestingderivative portfolios with filtered historical simulation (FHS),”European Financial Management, vol. 8, no. 1, pp. 31–58, 2002.[Online]. Available: http://dx.doi.org/10.1111/1468-036X.00175

[18] P. Winker and D. Maringer, “The convergence of estimators based onheuristics: theory and application to a GARCH model,” ComputationalStatistics, vol. 24, no. 3, pp. 533–550, 2009.

APPENDIX

In the following the previous models are tested on adifferent data set composed of all the equities in the DAXindex as well as the corresponding time series of optionimplied volatilities. These data have been downloaded fromBloomberg whereas the original data consisted in a subset ofDAX components with over-the-counter option data. Bothsets of data cover the same time horizon. We proceedas previously by first computing equity-only portfolios forwhich results are shown in Figure 7. When compared withFigure 1 we observe significantly higher annualized returns,most probably due to a selection bias. However this is not anissue as our goal is to investigate whether a given portfoliocan be improved with options.In Figure 7 a spike can be noticed in the wealth paths

around end of October 2008 due to Volkswagen shares in

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Fig. 7. Results for equity-only and 1/N portfolios computed from the fullset of DAX components. Upper panel: median wealth paths. Lower panel:empirical distributions of annualized returns.

the portfolios.10 This pattern is observed in all subsequentresults.The buy-write strategy detailed in Section III-A is now

applied. As explained, in a first step we construct a portfoliominimizing a given criterion and in a second step, we writeout-of-the-money calls (M = 105). The number of writtencalls equals that of shares held. Figure 8 illustrates medianpaths and distributions of annualized returns for portfoliosminimizing maximum drawdown and variance. The empiri-cal distribution of returns is shifted to the right for both (DDand MV) objective functions. In addition wealth paths aresmoother, translating into lower variance and overall higherSharpe ratios. Figures 10 and 11 show the empirical distribu-tions of the Sharpe ratios for equity-only, covered calls andDelta hedged portfolios and both objective functions.Finally results for the dynamic strategy (Delta hedged

portfolios) are shown in Figure 9 where the volatility forecastis done with the ‘naive’ approach. Distributions of annualizedreturns are further shifted to the right, reaching a medianvalue of above 5% for the minimum variance portfolios.Sharpe ratios are also improved as can be seen in the rightpanels of Figures 10 and 11. The enhancement is morepronounced for minimum variance portfolios.

10In 2008 Volkswagen shares went from 194 on October 24 to 870 onOctober 28 and back to 360 on November 3 because of the takeover storyinvolving Porsche.

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Fig. 8. Median wealth paths (upper panel) and empirical distributions ofannualized returns (lower panel) for portfolios of covered calls (M = 105)computed from all DAX components.

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Fig. 9. Median wealth paths (upper panel) and empirical distributions ofannualized returns (lower panel) for portfolios with Delta hedged positionscomputed from all DAX components. Hedge realized with at-the-moneyoptions. Volatility forecasted with the ‘naive’ model.

−0.5 0 0.50

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Δ−hedge

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Fig. 10. Empirical distributions of Sharpe ratios for portfolios constructedfrom the subset (left panel) and the complete set of DAX components (rightpanel) for portfolios minimizing drawdown.

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Δ−hedge

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Fig. 11. Empirical distributions of Sharpe ratios for portfolios constructedfrom the subset (left panel) and the complete set of DAX components (rightpanel) for portfolios minimzing variance.

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