Portfolio Protection Strategies a Study on the Protective Put and Its Extensions

DEGREE PROJECT IN MATHEMATICS, SECOND CYCLE, 30 CREDITS STOCKHOLM, SWEDEN 2018

Portfolio Protection Strategies A study on the protective put and its extensions

GUSTAV ALPSTEN

SERCAN SAMANCI

KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES

Portfolio Protection Strategies

A study on the protective put and its extensions

GUSTAV ALPSTGEN

SERCAN SAMANCI

Degree Projects in Financial Mathematics (30 ECTS credits) Degree Programme in Applied and Computational Mathematics KTH Royal Institute of Technology year 2018 Supervisor at KTH: Anja Janssen Examiner at KTH: Anja Janssen

TRITA-SCI-GRU 2018:303 MAT-E 2018:70

Royal Institute of Technology School of Engineering Sciences KTH SCI SE-100 44 Stockholm, Sweden URL: www.kth.se/sci

Acknowledgements

We would like to express our gratitude to Claes Wachtmeister for introducing us to the subject and his supervising of this thesis. We would also like to thank Anja Janssen for her support and valuable remarks towards our report.

Abstract

e need among investors to manage volatility has made itself painfully clear over the past century, particularly during sudden crashes and prolonged drawdowns in the global equity markets. is has given rise to a liquid portfolio insurance market in the form of options, as well as aracted the aention of many researchers. Previous literature has, in particular, studied the eectiveness of the widely known protective put strategy, which serially buys a put option to protect a long position in the underlying asset. e results are oen uninspiring, pointing towards few, if any, protective benets with high option premiums as a main concern. is raises the question if there are ways to improve the protective put strategy or if there are any cost-ecient alternatives that provide a relatively beer protection. is study extends the previous literature by investigating potential improvements and alternatives to the protective put strategy. In particular, three alternative put spread strategies and one collar strategy are constructed. In addition, a modied protective put is introduced to mitigate the path dependency in a rolling protection strategy.

e results show that no option-based protection strategy can dominate the other in all market situations. Although reducing the equity position is generally more eective than buying options, we report that a collar strategy that buys 5% OTM put options and sells 5% OTM call options has an aractive risk-reward prole and protection against drawdowns. We also show that the protective put becomes more eective, both in terms of risk-adjusted return and tail protection, for longer maturities.

Abstrakt

Hantering av volatilitet i nansiella marknader har under de senaste decennierna visat sig vara nodv¨ andigt¨ for¨ investerare, framfor¨ allt i samband med krascher och langdragna˚ nedgangar˚ i de globala aktiemarknaderna. Dea har ge upphov till en likvid derivatmarknad i form av optioner samt vackt¨ e intresse for¨ forskning i omradet.˚ Tidigare studier har i synnerhet undersokt¨ eektiviteten i den valk¨ anda¨ protective put-strategin som kombinerar en lang˚ position i underliggande aktie med en put-option. Resultaten ar¨ oa inte tilltalande och visar fa˚ fordelar¨ med strategin, dar¨ dess hoga¨ kostnader lys upp som e stort problem. Saledes˚ vacks¨ fragan˚ om protective put-strategin kan forb¨ aras¨ eller om det mojligtvis¨ nns nagra˚ kostnad- seektiva alternativ med relativt bare¨ sakerhet¨ mot eventuella nedgangar˚ i underliggande. Denna studie utvidgar tidigare forskning i omradet˚ genom a undersoka¨ forb¨ aringsm¨ ojligheter¨ for¨ och alternativ till protective put-strategin. Sarskilt¨ stud- eras tre olika put spread-strategier och en collar-strategi, samt en modierad version av protective put som amnar¨ a minska pa˚ vagberoendet¨ i en lopande¨ option- sstrategi.

Resultatet fran˚ denna studie pekar pa˚ a ingen optionsbaserad strategi ar¨ universellt bast.¨ Generellt se ger en avyring av delar av aktieinnehavet e mer eektivt skydd, men vi visar a det nns situationer da˚ en collar-strategi som koper¨ 5 % OTM put-optioner och saljer¨ 5% OTM call-optioner har en araktiv risk-justerad prol och sakerhet¨ mot nedgangar.˚ Vi visar vidare a protective put-strategin blir mer eektiv, bade˚ i termer av en risk-justerad avkastning och som sakerhet¨ mot svansrisker, for¨ langre¨ forfallodatum¨ pa˚ optionerna.

Contents

Introduction7

eoretical Background 12

2.1 Option contracts...... 12

2.1.1 Mathematical denition of European options...... 13

2.1.2 ATM, ITM and OTM...... 15

2.2 Black-Scholes-Merton Model...... 15

2.2.1 Black-Scholes-Merton model with dividends...... 17

2.2.2 Black-Scholes-Merton model for a stock index...... 18

2.2.3 Merton’s Jump Diusion Model...... 19

2.2.4 Convexity of options...... 21

2.3 Volatility...... 22

2.3.1 Realized versus implied volatility...... 22

2.3.2 Volatility risk premium...... 23

2.3.3 Volatility surface...... 25

2.4 Protection strategies...... 27

2.4.1 Protective Put Strategy...... 27

2.4.2 Bear Put Spread Strategy...... 28

2.4.3 Collar Strategy...... 29

Methodology 31

3.1 Overview...... 31

3.2 Assumptions and delimitations...... 32

3.3 Option-based protection strategies and portfolios...... 33

3.3.1 Protective Put...... 34

3.3.2 Divested Equity Portfolio...... 35

4 3.3.3 Low-cost Portfolios...... 35

3.3.4 Fractional Protective Put...... 36

3.3.5 Fractional Put Spread...... 37

3.4 Evaluation methods...... 37

3.5 Monte Carlo Simulation...... 40

3.6 Backtesting...... 41

3.6.1 Data...... 41

3.6.2 Outline...... 42

Results 44

4.1 Monte Carlo Simulation...... 44

4.2 Backtesting...... 54

4.2.1 Whole period: December 2002 - Mars 2018...... 54

4.2.2 Financial crisis...... 62

Discussion 66

5.1 Monte Carlo Simulation...... 66

5.1.1 Risk-adjusted performance...... 66

5.1.2 Peak-to-trough drawdown characteristics...... 71

5.2 Backtesting...... 73

5.2.1 Risk-adjusted performance...... 73

5.2.2 Peak-to-trough drawdown characteristics...... 76

5.2.3 Financial crisis...... 80

5.3 Conclusions...... 82

Appendix 86

6.1 Modelling examples...... 86

6.1.1 Protective Put...... 86

6.1.2 Divested Equity...... 87

5 6.2 Tables...... 88

6.2.1 Monte Carlo Simulation...... 88

6.2.2 Backtesting...... 93

6.3 Figures...... 98

6.3.1 Monte Carlo Simulation...... 98

6.3.2 Backtesting...... 101

6.3.3 Total cost of strategies...... 104

Introduction

e watchword among traders of nancial instruments has been, and is today, volatility. It is a measure that captures the degree of uncertainty in the price movements of nancial assets. Following a sequence of unprecedented global economic events, such as the nancial crisis in 2008, and the rise of derivatives, it has become increasingly important to understand the causes and implications of volatility. More importantly, it has become crucial for traders to manage the presence of volatility in their assets, either by risk-reducing measures or by making a bet on its direction. e need among investors to address the volatility has made itself painfully clear over the past century, in particular during sudden crashes and prolonged drawdowns in the global equity markets.

ere are today a whole variety of measures and instruments that allow market participants to protect their equity investments against market drawdowns. Two widely used hedging methods are short-selling and the use of derivatives. Unlike long positions, short-selling equity can become very costly due to margin and collateral requirements, and there are major risks that can lead to unexpected losses given its speculative character. However, the short-selling strategy is rather limited, since not all stocks can be sold short. More commonly, investors use derivatives to hedge their equity investments. Derivatives are contracts that can be used to reduce the portfolio’s equity exposure by providing downside protection during market drawdowns. e most common instruments used for hedging are futures and options. Both are traded on liquid and transparent markets. e distinctive dierence between them, from a hedging point of view, resides in the investment exibility and liability aributed to the contract parties. Sophisti- cated hedgers oen seek exibility to construct suitable pay-o proles, which are best achieved through option contracts. Futures provide lile exibility in this maer and is generally inferior to options when it comes to hedging equity investments. Also, the nancial liability associated with futures are larger than those for options making them less aractive. e popularity of options as hedging instruments has led to a multitude of protection strategies (Fodor, Doran, Carson and Kirch, 2010).

7 e concept of hedging equity investments against market drawdowns, while allowing upside participation, is oen referred to as portfolio insurance (PI). PI strategies have, over the past decades, become an important tool for investors as well as a popular subject for research. Leland and Rubinstein (1976) introduced the Option Based Portfolio Insurance (OBPI), which consists of a long position in a risky asset (usually equity) and long a position in a put option with a strike price equal to the insured amount. Another strategy, called Constant Proportion Portfolio Insurance (CPPI), was introduced by Perold (1986) and Black and Jones (1987) as an alternative to the OBPI. CPPI ensures a predened oor, or a cushion, by dynamically rebalancing allocations between the risky and risk-less asset (Bertrand and Prigent, 2005).

e eectiveness of the OBPI and CPPI strategies have been thoroughly studied and compared in the literature. e aggregated results from previous studies, Zhu and Kavee (1988), El Karoui et al. (2005), Bertrand and Prigent (2005), Zagst and Kraus (2011) and Bertrand and Prigent (2011), conclude that there is no dominant strategy. Perold and Sharpe (1988) and Black and Rouhani (1989) show that the relative performance of the strategies depend on the market trend and the level of the volatility. Annaert et al. (2009), similarly concludes that no strategy can dominate the other in all market situations. e appropriate strategy must be chosen based on the investor’s expectations on the market situation.

In more recent studies, Figlewski et al. (2013) and Israelov (2017) examined the performance of the protective put strategy, which is a development of the OBPI method. Figlewski et al. (2013) conducted a simulation study to examine the performance of the protective put strategy using three dierent types of strike methods: a xed strike, a xed percentage strike and a combination of both. Figlewski et al. (2013) argue that the xed percentage strike method, which resets the strike price at a xed percentage of the stock’s current price at the time of a rollover, is a more accurate description of the protective put strategy used by actual investors. It is found that the xed percentage strike is much less protective than the xed strike method in a prolonged bear market and provides very limited protection when out-of-the-money (OTM) puts are used. e xed strike method is more costly during low volatility periods and resembles more a long stock position over longer investment horizons, as the stock price tends to dri away from the xed strike price. e results from the study, in terms of combined limited downside risk and aractive mean return, favors a xed percentage strategy using in-the-money (ITM) or at-the-

8 money (ATM) puts. Figlewski et al. (2013) conclude that the protective put is more approriate when the true expected return on the stock is higher than the risk-free rate, but the stock is expected to underperform.

Israelov (2017) measured the eectiveness of the protective put strategy by comparing its peak- to-trough drawdown characteristics to those of a static risk-reducing strategy. e risk-reducing strategy, referred to as the divested equity strategy, manages downside risk by reducing the exposure to the risky asset and does not involve any use of options. Israelov (2017) split his study into two parts, a real-world implementation using the CBOE S&P 500 5% Put Protection Index and an idealized environment through a Monte Carlo simulation. Israelov (2017) nds that unless the option purchases and their maturities are timed just right around market drawdowns, the protective put strategy may oer lile downside protection. e culprits are the high cost of put options and the path dependency of a rolling option strategy. Buying put options reduces the portfolio’s beta relative to S&P 500 index and provides negative alpha due to the existence of volatility risk premium. e combined eect of reduced beta and negative alpha impacts the return more negatively than not having put options in the portfolio. Israelov (2017) continues to show that even in an idealized environment, where there is no volatility risk premium (i.e. no extra costs incurred for the options), portfolios that are protected with put options have worse peak-to-trough drawdown characteristics per unit of expected return than portfolios that have instead statically reduced their equity exposure in order to reduce risk. In a real-world scenario, in which there is non-zero volatility risk premium, the situation was shown to be worse.

Israelov (2017) and Figlewski et al. (2013) both extended the theoretical research done by Zhu and Kavee (1988), El Karoui et al. (2005), Bertrand and Prigent (2005), Annaert et al. (2009), Zagst and Kraus (2011) and Bertrand and Prigent (2011) to option-based strategies over longer investment horizons. However, none of the studies examined alternative protection strategies and their performance relative to the protective put. It would be interesting to shed light on potential extensions of, or substitutes to, the protective put strategy in order to nd out if there are alternatives that have more aractive protective characteristics.

A common way to reduce the cost of the protective put is to nance the strategy by selling an OTM put or an OTM call. e former creates a payo prole referred to as a bear put spread and sacrices some protection in exchange for lower net premium paid. e laer is known as

9 a collar and caps the upside of the protective put. As both the downside and upside are limited, the collar strategy usually has a lower beta than the bear put spread. A sold put or call generates positive alpha that partially osets the negative alpha from the purchased put (Benne, 2014).

Israelov and Klein (2016) compares the protectiveness of the collar to the divested equity strategy. ey nd, in particular, that investors would be beer o simply reducing their equity exposure rather than investing in a collar strategy. eir study shows that investing in a collar has provided lower returns and a lower Sharpe ratio than investing directly in the S&P 500 index. ese ndings point to yet another inferior strategy compared to the divested equity strategy.

e previous literature on the protective put, as well as the article on collar, do not shed a positive light on their risk-adjusted performance and protectiveness. is raises the question if there are other portfolio insurance constructions that provide more aractive risk-adjusted return and tail protection relative to the previous ndings. It does indeed raise the question if the divested equity strategy can be outperformed in both risk-adjusted return and tail protection. Although the divested equity strategy is simple by construction, it requires static risk-reduction, and it is not an easy task to manually time each market crash. Options, on the other hand, are convex instruments that automatically reduce equity exposure as markets crash, which is oen a more convenient construction for the investor who may not have the time to continuously reset the equity exposure. is creates incentives to search for protection strategies that are beer alternatives than the divested equity strategy.

is study aims to draw upon the previous ndings and add several more protection strategies to the analysis. e basic setup of the study is similar to the ones carried out by Israelov (2017) and Figlewski et al. (2013). It consists of a Monte Carlo simulation and a backtesting on the S&P 500 index in the period 20 December 2002 - 23 March 2018. In contrast to previous studies, we will employ a stochastic jump model in our Monte Carlo simulation to create occasional crashes similar to the nancial crisis and other preceding comparable events. is study will also include more advanced options strategies, including a version of the Bear Put Spread and the Collar. In addition, we present a modied version of the protective put and the put spread strategies in an aempt to reduce the negative impact of path dependence, which was particularly emphasized as detrimental to returns by Israelov (2017). e strategies are benchmarked against each other and a divested equity strategy which does not include any use of options.

10 We formulate our research questions as follows:

• Is the protective put strategy really an eective way to hedge an equity portfolio?

• Are there any other protection strategies that provide relatively better tail protection?

• Is it, in particular, possible to improve risk-adjusted returns and tail protection by:

– Making a protection strategy more cost-ecient?

– Reducing the path dependency of a rolling option strategy?

e performance of the strategies are based on their risk-adjusted return measured by the Sharpe ratio in periods of positive excess return. When excess return is negative, which happens during crashes and prolonged bear markets, the risk-adjusted return is instead measured by the adjusted Sharpe ratio as proposed by Israelsen (2009). ere are various metrics for a strategy’s tail protection, where one of the most widely used is the Value at Risk (VaR). An alternative, which is employed in this study, is to measure the portfolio’s peak-to-trough drawdowns over predetermined time windows. Peak-to-trough drawdowns provide a quantity on the protectiveness of the dierent strategies against periods of falling asset prices that persist over dierent time windows.

e study will be structured as follows: We begin by presenting, in Chapter 2, the preliminaries of option theory. is is followed by a presentation, in Chapter 3, of the examined protection strategies, the evaluation methods employed and the setup of the Monte Carlo simulation and the backtesting. e results of their performance are presented in tables and graphs in Chapter 4 and are discussed in detail in Chapter 5.

11 eoretical Background

2.1 Option contracts

An option contract (also called an option) is a nancial derivative that gives the holder of the option the right to take action on an underlying asset, but with no obligation to execute this right. ere are two basic types of options. A call option gives the holder of the option the right to buy an underlying asset by a predetermined date for a predetermined price. A put option, on the other hand, gives the holder of the option the right to sell an underlying asset by a predetermined date for a predetermined price. e predetermined date and the predetermined price are commonly referred to as the maturity date and the strike price, respectively (Hull, 2012).

ere are various styles of options, each with dierent terms specifying under what conditions they are allowed to be executed. ese are broadly categorized into vanilla options and exotic options. Vanilla options are further classied as either European or American.

• European options can be exercised only on the maturity date

• American options can be exercised at any time up to the maturity date

Exotic options have more complex features and may have several triggers relating to their exe- cution. Options derive their value from the underlying asset, which can be anything from a stock to a foreign currency, a bond, a commodity, an index or a futures contract (Hull, 2012).

e remaining parts of this report will focus only on European options on stocks. European options are generally easier to analyze and are one of the most actively traded option types on the markets. Hence, any mention of stock options hereaer implicitly refers to European stock options.

12 2.1.1 Mathematical denition of European options

e holder of a European option will claim the payo Φ(ST ) at maturity time T , where Φ is the contract function and ST is the stock price at maturity.

e claim, or value, of a European call option on a non-dividend-paying stock at maturity is mathematically represented as:

ΦC(S ) = max(S K,0) = (S K)+ (2.1.1) T T − T − where K is the strike price. e call option has a non-zero payo at maturity when the stock price ST exceeds the strike price K, otherwise it expires worthless. Figure 2.1 shows the payo- diagram for a European call option (Bjork,¨ 2009).

With the same notations, the value of a European put option on a non-dividend-paying stock at maturity is given by the formula:

ΦP (S ) = max(K S ,0) = (K S )+ (2.1.2) T − T − T where the strike price K must exceed the stock price ST at maturity to result in a non-zero payo (Bjork,¨ 2009). Figure 2.2 shows the payo-diagram for a European put option.

e value calculated in equations 2.1.1 and 2.1.2 is the intrinsic value of the option. For any time t < T , the intrinsic value of an option is calculated as:

+ (St K) (for a call option) − (2.1.3) (K S )+ (for a put option) − t

(Benne, 2014).

13 50

25 Payoff 20

0 0 10 20 30 40 50 60 70 80 90 100 Stock price, S T

Figure 2.1: Payo-diagram for a European call option with strike price K = 50.

25 Payoff 20

0 0 10 20 30 40 50 60 70 80 90 100 Stock price, S T

Figure 2.2: Payo-diagram for a European put option with strike price K = 50.

For any time t < T it is, however, not obvious how to calculate a ”fair” option price. e price is determined by the market, inuenced by the participants’ aitude to risk and expectations about the future stock prices. ese elements are captured by another value component, namely the time value. At maturity, the time value of the option decays to zero. In general, the value of an option can be decomposed into two components:

Value of option = Intrinsic value + Time value where the time value usually is non-zero for 0 t < T , i.e. all times t prior to maturity T ≤ (Benne, 2014).

14 2.1.2 ATM, ITM and OTM

When the current stock price equals the strike price, St = K, the option is said to be at-the-money (ATM). An ATM option has zero intrinsic value and non-zero time value. e intrinsic value is also zero when the current stock price is less than the strike price, St < K, for a call option, and when the current stock price is greater than the strike price, St > K, for a put option. In both cases, the option is said to be out-of-the-money (OTM). On the contrary, when the intrinsic value is non-zero, an option is said to be in-the-money (ITM), (Benne, 2014).

In general, the time value is greatest for ATM options. OTM options tend to trade cheapest, whereas ITM options are relatively expensive and hence tend to trade in lesser volumes than their cheaper OTM counterparts. Whether an option is ATM, ITM or OTM thus has an impact on its market price (Benne, 2014).

2.2 Black-Scholes-Merton Model

ere are various models to calculate the ”fair” price of a stock option. One of the most common pricing models used by market participants is the Black-Scholes-Merton model. e Black- Scholes-Merton model is widely used by option market participants and is perhaps the world’s most well-known option pricing model. e model is both arbitrage free and complete, making the prices it produces unique. Furthermore, the model is developed based on the following assumptions (Hull, 2012):

1. e stock price S(t) dynamics is described by the geometric Brownian motion:

dSt = µStdt + σStdWt (2.2.1)

where µ is the stock’s annual expected rate of return, σ is the annual volatility of the stock

price and Wt is a Wiener process. e stock price is hence assumed to have a lognormal distribution: σ2 (µ )t+σWt St = S0e − 2

15 where S0 is the initial stock price.

2. Volatility is constant.

3. Short selling is permied.

4. ere are no transaction costs or taxes.

5. ere are no dividends during the life of the option.

6. ere are no arbitrage opportunities.

7. Trading is continuous.

8. e risk-free rate r is deterministic and the same for all maturities.

Given these assumptions, the price formulas for European call and put options are solutions to the Black-Scholes-Merton dierential equation problem:

∂F 1 ∂2F (t,S ) + S2σ 2 (t,S ) rF(t,S ) = 0 ∂t t 2 t 2 t t ∂St − (2.2.2)

F(T,ST ) = Φ(ST )

F(t,St) is the price of an option as a function of time and underlying stock price, where time t extends from the day the contract was wrien to maturity, 0 t T . e boundary condition ≤ ≤ F(T,ST ) = Φ(ST ) ensures that the price of the option at maturity is equal to the payo of the contract at maturity (Hull, 2012).

Solving equation 2.2.2 for the price of a call option and a put option, denoted by c(t,St) and p(t,St), respectively, with strike price K and time of maturity T yields:

h i r(T t) h i c(t,St) = StN d1(t,St) e− − KN d2(t,St) − (2.2.3) r(T t) h i h i p(t,S ) = Ke− − N d (t,S ) S N d (t,S ) t − 2 t − t − 1 t where N is the cumulative distribution function for the N[0,1] distribution and

  1  S 1  d (t,S ) = ln t + r + σ 2 (T t) 1 t   σ√T t K 2 − (2.2.4) − d (t,S ) = d (t,S ) σ√T t 2 t 1 t − −

16 (Hull, 2012).

e presence of an expected rate of return, µ, in equation 2.2.1 introduces a risk preference, which may vary among investors. e Black-Scholes-Merton dierential equation 2.2.2 is independent of investors’ risk preferences. In deriving the Black-Scholes-Merton formula, one can therefore assume a risk-neutral world, where the expected rate of return on all stocks is the risk-free rate, r. Under a risk-neutral measure, the stock price process then becomes:

˜ dSt = rStdt + σStdWt (2.2.5)

˜ where the expected rate of return is equal to the risk-free rate r and Wt is a Wiener process under the risk-neutral measure.

2.2.1 Black-Scholes-Merton model with dividends

e Black-Scholes-Merton formulas in 2.2.3 are derived based on non-dividend paying stocks. In reality, stocks oen pay dividends and to take this into account the Black-Scholes-Merton formula must be modied. Dividends are paid out to the holder of the stock on the ex-dividend dates. On this dates the stock price declines by the amount of the dividend. Stock prices can be decomposed into two components:

• A riskless component, D, that represents the present value of all the dividends during the life of the option discounted from the ex-dividend dates to the present at the risk-free rate.

• A risky component, S, that corresponds to the stochastic part of the stock price following the volatility process σ.

e current stock price, t, including the present value of all the dividends to be paid out from today until maturity, is the sum of both these components:

St∗ = St + Dt

17 Given that the Black-Scholes-Merton formula is derived based only on the risky component, the equations in 2.2.3 can be used if the stock price, including dividends, is reduced by the present value of all the dividends during the life of the option:

S = S∗ D t t − t

In principle, the stock price is adjusted for the anticipated dividends and then the option is valued as though the stock pays no dividend (Hull, 2012).

In the case of a known dividend yield (continuous dividend), q, the adjusted stock price becomes:

q(T t) St∗e− − where t is the current time and T is the time of maturity. e risk-neutral price process of a stock with dividend yield q can be wrien as

dS = (r q)S dt + σS dW˜ (2.2.6) t − t t t where the expected rate of return is reduced by the dividend yield (Hull, 2012).

2.2.2 Black-Scholes-Merton model for a stock index

In valuing stock index options, the underlying index can be treated as a stock paying a known dividend yield, q. e theory presented in the previous section 2.2.1 can thus be used to derive the call and put formulas for stock index options. With the assumptions that the underlying q(T t) stock index has the current price, St∗e− − and pays no dividends, the Black-Scholes-Merton equations for call and put options on a stock index are:

18 q(T t) q(T t) h q(T t) i c(t,St∗e− − ) = St∗e− − N d1(t,St∗e− − ) r(T t) h q(T t) i e− − KN d2(t,St∗e− − ) − (2.2.7) q(T t) r(T t) h q(T t) i p(t,S∗e− − ) = Ke− − N d (t,S∗e− − ) t − 2 t q(T t) h q(T t) i S∗e− − N d (t,S∗e− − ) − t − 1 t

  1  S∗ 1  d (t,S e q(T t)) = ln t + r q + σ 2 (T t) 1 t∗ − −   σ√T t K − 2 − (2.2.8) − q(T t) q(T t) d (t,S∗e− − ) = d (t,S∗e− − ) σ√T t 2 t 1 t − − where q is the anticipated annual dividend yield during the life of the stock index option, St∗ is the current value of the stock index and σ is the volatility of the stock index (Hull, 2012).

2.2.3 Merton’s Jump Diusion Model

e Black-Scholes-Merton model assumes that stock returns have a lognormal distribution described by the geometric Brownian motion presented in section 2.2. Empirically, stock returns tend to have fat tails, i.e. a distribution that assigns higher probability to extreme returns compared to the lognormal distribution. Merton’s Jump Diusion model was introduced as an alternative to the Black-Scholes-Merton model to address the issue of fat tails. By modifying the geometric Brownian motion to include an independent Poisson process, dpt, the modelled stock prices will occasionally experience a jump, more similar to the empirically encountered behaviour. e suggested model is dependent on two additional parameters: the average number of jumps per year, λ, and the average jump size measured as a percentage of the stock price, k (Hull, 2012).

e jumps contribute to the growth rate of the stock price with a value equal to λk. e modied − risk-neutral process for the stock price is:

dS = (r q λk)S dt + σS dW˜ + dp (2.2.9) t − − t t t t

19 ˜ where dWt and dpt are assumed to be independent.

e percentage jump size, X, is assumed to be drawn from a particular probability distribution. Commonly, the distribution is chosen such that ln(1+X) N(γ,δ2) with mean percentage jump ∼ 2 size k = E(X) = eγ+δ /2 1. It is assumed that the two sources of randomness, the Poisson process − for when a jump occurs and the lognormal distribution of the jump size, are independent of each other.

With these modications to the geometric Brownian motion and assumptions for the jumps, one can show that the price of the European option can be wrien as:

X∞ e λ0T (λ T )n − 0 f (2.2.10) n! n n=0 where λ0 = λ(1+k) and fn is the Black-Scholes-Merton option price obtained from equation 2.2.7 for an underlying asset with dividend yield q, variance equal to:

nδ2 σ 2 + T and risk-free rate equal to: nln(1 + k) r λk + − T

(Hull, 2012).

Merton’s Jump Diusion model creates the fat tail distribution that is more in line with reality, but leads to an incomplete market due to the addition of another random source, the Poisson process. e price obtained in 2.2.10 is not unique, since there is no unique risk-neutral measure in an incomplete market. However, by a change of probability measure, and eectively by changing the dri according to equation 2.2.9, Merton constructed an arbitrage-free model. Yet, due to the incompleteness, it is not possible to construct a replicating portfolio and no perfect hedge.

20 2.2.4 Convexity of options

e aractiveness of options for hedging revolves around their non-linear (”convex”) payo structure. For example, put options provide downside protection while preserving upside potential. e value of put options rises as the price of the underlying stock falls. is is captured by the measure delta, ∆. It is given by the rst derivative of the price equation for the put option in 2.2.3 with respect to the underlying stock price:

∂p ∆ = = N(d ) 1 (2.2.11) put ∂S 1 − and its value ranges between -1 and 0 (Hull, 2012).

More importantly, a long position in an option has a positive exposure to the second derivative of the price equations in 2.2.3 with respect to the underlying stock price - or the rst derivative of ∆ with respect to the underlying stock price. e rate of the increase in the price of a put option increases as the price of the underlying stock falls. Conversely, the rate of the decrease in the price of a put option decreases as the price of the underlying stock rises. is measure is known as the gamma, Γ , and quanties the sensitivity of ∆ to changes in the stock price:

∂∆ ∂2p N (d ) Γ 0 1 put = = 2 = (2.2.12) ∂S ∂S S0σ√T which varies with respect to the underlying stock price, S, as a bell-shaped curve. e described relationship between gamma and the underlying stock price is typical for a long position in a put option. is asymmetric behavior, which makes a put option more valuable as the price of the underlying stock falls while preserving upside potential, is what makes it a popular instrument for portfolio protection (Hull, 2012).

21 2.3 Volatility

2.3.1 Realized versus implied volatility

e volatility, σ, measures the amount of variability in the stock returns. ere are two volatility measures that are of interest to participants in the options market. e rst one, realized volatility, also called historical volatility, is an ex-post estimate of stock return variation. It is dened as the annualized standard deviation of daily stock returns:

uv t N 252 X σ = (r r¯)2 (2.3.1) realized N 1 t − − t=1 where N + 1 is the number of observations and

N St 1 X rt = ln , r¯ = rt (2.3.2) St 1 N − t=1

(Hull, 2012).

e realized volatility implies nothing about the future. To estimate future volatility, option traders look at a second volatility, the implied volatility, which is derived from the Black-Scholes- Merton formula by inserting the current market price of the option. It is the volatility parameter in the geometric Brownian motion process:

dSt = µdt + σimplieddWt (2.3.3) St

(Hull, 2012).

Understanding the dierence between realized and implied volatility is fundamental to any investor engaging in options trading. As can be seen in equations 2.2.3 and 2.2.4, an increase in the implied volatility, σ, all else being equal, increases the value of a long position in an option, while the opposite is true for a short position. e time value of an option is, in practice, heavily dependent on the volatility in the underlying stock. A long position in an option is a long volatility exposure, i.e. the realized volatility, at the time of maturity, is expected to exceed the

22 implied volatility at which the option was bought (Benne, 2014).

2.3.2 Volatility risk premium

In a perfect market, there exists only one volatility parameter, σrealized = σimplied = σ. However, in reality option writers (sellers) tweak the Black-Scholes-Merton formulas in equation 2.2.3 to compensate for the risk of losses during periods when realized volatility suddenly increases, also called crash risk. Note in Figure 2.3 how realized volatility spiked higher than the implied volatility during the nancial crisis 2007 - 2008. e implied volatility, however, tends to exceed the realized volatility on average. e discrepancy between implied volatility and the subsequent realized volatility is known as the volatility risk premium. e volatility risk premium indicates how expensive an option is and reduces realized returns for the buyer (Israelov, 2017).

e volatility risk premium over a certain period can, for example, be measured by the dierence between the S&P 500 annualized 1-month ATM implied volatility and the S&P 500 annualized subsequent 1-month realized volatility. e volatility risk premium at time t is thus the dierence between the S&P 500 annualized 1-month ATM implied volatility at time t and the annualized standard deviation of the S&P 500 measured from time t and one month forward. In Figure 2.3, the volatility risk premium is ploed over the period December 2002 - March 2018.

23 1 Implied Volatility Realized Volatility Volatility Risk Premium

0.8

0.6

0.4

0.2

Annualized volatility 0

-0.2

-0.4

-0.6 2002 2005 2007 2010 2012 2015 2017 2020

Figure 2.3: e volatility risk premium measured as the dierence between the S&P 500 annualized 1-month ATM implied volatility and the S&P 500 annualized subsequent 1-month realized volatility.

In the period December 2002 - March 2018, the volatility risk premium amounted to 0.8% on average, with a median of 1.7%. See table 2.1 for more statistics.

24 Table 2.1: Statistics of the volatility in the S&P 500 index in the period December 2002 - March 2018. Percentiles 99th 95th 90th 75th Median Mean Implied volatility 53.3% 31.7% 25.8% 18.9% 14.0% 16.4% Realized volatility 74.2% 32.2% 25.5% 17.7% 13.0% 15.6% Volatility risk premium 11.3% 8.2% 6.4% 4.1% 1.7% 0.8%

2.3.3 Volatility surface

e implied volatility that is used by traders to price an option depends on its strike price and time to maturity. A plot of these three dimensions gives a volatility surface, which shows how implied volatility depends on each of the two parameters. For simplicity, the dependencies are oen ploed as two separate two-dimensional graphs.

e plot of implied volatility and strike price is known as the volatility skew, because the curve is generally skewed. For example, the implied volatility for the S&P 500 on 16 January 2018 is decreasing with increasing strike price and reaches a minimum before it slightly increases again (see Figure 2.4). Hence, the volatility used to price an option with low strike price (i.e. a deep OTM put or a deep ITM call) is signicantly higher than that used to price options with higher strike price (i.e. a deep ITM put or a deep OTM call). By xing the time to maturity, the volatility skew can be depicted in a two-dimensional graph (see Figure 2.5), (Hull, 2012).

Ploing implied volatility against the time to maturity shows the term structure, which provides information on how implied volatility varies with increasing time to maturity. e implied volatility for the S&P 500 as of 16 January 2018 exhibits a high volatility for the shortest time to maturity, but approaches quickly a minimum, from where it increases with increasing time to maturity (see Figure 2.5). is reects an expectation that the volatility will increase over time, leading to more richly priced long-dated options (Hull, 2012).

25 1

0.8

0.6

0.4

Implied volatility 0.2

0 Q2-20 Q1-20 Q4-19 Q3-19 Q2-19 2 Q1-19 Q4-18 1.5 Q3-18 Q2-18 Q1-18 1 Q4-17 Q3-17 0.5 Q2-17 Time of maturity Q1-17 0 Strike percentage

Figure 2.4: A volatility surface for options on the S&P 500 Index. e time of maturity spans from 2 April 2018 to 23 March 2021, where the laer is the longest dated option available in the dataset. e strike price is given as a percentage, where 1 represents an ATM option.

0.8 0.19

0.185 0.7

0.18 0.6 0.175

0.5 0.17

0.4 0.165

0.16 0.3 Implied volatility Implied volatility

0.155 0.2 0.15

0.1 0.145

0 0.14 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Q1-17 Q2-17 Q3-17 Q4-17 Q1-18 Q2-18 Q3-18 Q4-18 Q1-19 Q2-19 Q3-19 Q4-19 Q1-20 Q2-20 Strike percentage Time of maturity

Figure 2.5: Le: Volatility skew for a xed time to maturity. Right: Term structure for a xed strike price.

e volatility surface informs traders about how OTM, ATM and ITM puts and calls are priced in terms of volatility, which in turn relates to the time value of the option. is information is useful for building suitable option strategies. For example, a at (or aer) volatility skew curve - OTM and ITM levels are equal to the ATM level - presents excellent opportunities to put on portfolio hedges at a cheaper cost than the observed skew in Figure 2.5. us, the observed volatility skew in Figure 2.5 informs the trader that the OTM puts are more expensively priced relative to the ATM level. On the other hand, the skew presents opportunities for option writers who would like to take advantage of the relatively more expensive OTM puts.

26 2.4 Protection strategies

A myriad of portfolio insurance strategies have been developed to manage volatility and tail risk exposure. is section presents some of the most widely employed option-based protection strategies.

2.4.1 Protective Put Strategy

e core idea of the protective put strategy is to limit the downside risk of a portfolio consisting of a stock or a stock index (e.g. S&P 500) held over a certain investment horizon by purchasing a sequence of shorter-term put options. e strategy protects the portfolio from losses below a certain strike level, while allowing unlimited prots as long as the underlying asset’s price rises. e total return on the portfolio is reduced by the cost of the put options and also depends on the path that the underlying asset’s price follows. e path dependency arises from the fact that the put options must be rolled over as they mature into new puts at prices that will depend heavily on the underlying asset’s current price and implied volatility. Figure 2.6 shows the payo diagram for a protective put strategy.

0 Payoff

-10

-20

-30

-40

0 10 20 30 40 50 60 70 80 90 100 Stock price, S T

Figure 2.6: Protective put strategy: A long position in a put option with strike price K = 50 and a long position in an underlying asset.

e investor primarily needs to consider two parameters in this strategy - the time to maturity and the strike price of the put options. e time to maturity is usually shorter than the investment horizon, e.g. 1 month, 3 months, or 12 months. e strike price can be one of the following:

27 • A xed value, K = C, where C is a constant

• A xed percentage of the underlying asset’s current price, K = p S , where p is given in · t decimal form (e.g. p = 0.95) and St is the underlying asset’s price at the time of purchase

• A combination of both.

Furthermore, it can be chosen such that the option is either ITM (p > 1), ATM (p = 1) or OTM (p < 1), where the laer is more common since its the cheapest alternative and is expected to reduce the portfolio return the least. e choice of parameters and strike method should therefore depend on the investor’s expectations on the market.

2.4.2 Bear Put Spread Strategy

A bear put spread involves a long position in a put option on a particular underlying asset, while simultaneously taking a short position in a put option on the same underlying asset with the same maturity date, but with a lower strike price. e strategy takes advantage of the observed volatility skew in Figure 2.5 by selling a richly priced deeper OTM put and buying a relatively cheaper OTM put with a higher strike price, which eectively reduces the net premium paid. e le graph in Figure 2.7 illustrates a payo diagram for a bear put spread. e right graph shows the payo diagram when it is used as a hedge on a long position in an underlying asset, e.g. a stock (Benne, 2014).

50 50

40 40

30 30

20 20

10 10

0 0 Payoff Payoff

-10 -10

-20 -20

-30 -30

-40 -40

0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 Stock price, S Stock price, S T T

Figure 2.7: Le: A bear put spread consisting of a long position in a put option with strike price K = 60 and a short position in a put option with strike price K = 30. Right: A bear put spread and a long position in a stock.

28 A bear put spread as a hedging strategy is more complex compared to its protective put counter- part. e maximum protection provided by the bear put spread is equal to the size of the spread less the net premium paid. e strategy does not cover any losses beyond the size of the spread, which requires that the bearish investor is able to proportionately match the spread to the size of the expected fall in the stock price. However, it lies in the investor’s interest to keep the spread as small as possible, since the premium gained from the short put is higher the closer it is chosen to the ATM level. Hence, the strategy becomes a trade-o between keeping the net premium paid as low as possible while maximizing the coverage of potential losses. e right graph in Figure 2.7 illustrates the extent of the hedge obtained by the bear put spread. e stock is hedged only down to ST = 30, below which any additional losses are not covered (Benne, 2014).

2.4.3 Collar Strategy

e Collar strategy consists of a long position in a put option to cover the downside risk of an underlying asset and a short position in a call option to reduce (or to completely oset) the premium paid for the put. Typically, both the put and the call options are OTM with the same maturity on the same number of underlying assets. e net cost of a collar is oen less than the net cost of a put spread (see section 2.4.2) and is therefore considered as a low-cost method for protection. However, to obtain this signicant reduction in cost, the strategy gives up some upside. e short call position results in a cap on performance, which makes the collar strategy less exposed to volatility. When the premium received from the short call option completely osets the premium paid for the long put option, the strategy is called a zero cost collar (Benne, 2014).

e le graph in Figure 2.8 illustrates a payo diagram for the collar strategy. e right graph shows the payo diagram when it is used as a hedge on a long position in an underlying asset, e.g. a stock (Benne, 2014).

29 40 50

40 30 30

20 20

10 10

0 Payoff Payoff 0 -10

-10 -20

-30 -20 -40

-30 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 Stock price, S Stock price, S T T

Figure 2.8: Le: A collar consisting of a long position in a put option with strike price K = 40 and a short position in a call option with strike price K = 60. Right: A collar and a long position in a stock.

e collar is more exposed to skew compared to the bear put spread because of the parabola-like shape around the ATM level (see the le graph in Figure 2.5).

30 Methodology

3.1 Overview

is study aims to evaluate the protective performance of dierent protection strategies involv- ing options. e performance will be evaluated based on their risk-adjusted return and peak- to-trough drawdowns. Moreover, each option strategy will be benchmarked against a divested equity strategy to analyze how it fares against an alternative in which there is no use of options.

e study is split into two parts. First, a Monte Carlo simulation is performed to evaluate the dierent strategies under ideal conditions. e simulation aggregates the performance of the dif- ferent strategies over varying stock price developments, including rising markets, bear markets and extreme crashes. Section 3.5 covers this part in more detail.

e results from the Monte Carlo simulation cannot be generalized beyond the ideal conditions it was performed under. erefore, the strategies will also be backtested against historical data of the S&P 500 index to assess their performance in a real-world implementation. In reality, there are signicant variations in the implied volatility (as displayed in Figure 2.3) driven by prevailing market conditions. Implied volatilities tend to move inversely with equity returns, which may provide positive pressure on a put option’s price during equity losses, improving its downside hedging properties. At the same time, rolling over into new put options become much more expensive. It is important to note that the results of a backtesting provide information only on how the portfolio performed in the specic market environment that prevailed during the backtesting.

In short, the Monte Carlo simulation enables a more extensive analysis, while the backtesting serves as a reality check.

31 3.2 Assumptions and delimitations

Certain assumptions and delimitations have been made in order to simplify the analysis and to isolate the protective properties of the dierent option strategies. In particular, the construction of the option strategies is based on the following:

1. e portfolio is fully invested in the underlying asset (S&P 500 index)

2. Rolling purchases and selling of options:

(a) e rst option is nanced by borrowing at the risk-free interest rate and is repaid at its maturity date, i.e. the cost of the purchased option (including interest expense) is deducted from the value of the portfolio at the maturity date

(b) e following options are nanced by borrowing at the risk-free interest rate if, at each rollover date, the accumulated payos from previous expired options are not sucient to cover the cost of buying a new option

(d) No reinvestments of option payos

3. Other cash positions also grow with the risk-free interest rate

4. No dividends are collected from the underlying asset (S&P 500 index)

5. No transaction costs

In contrast to a realistic scenario, it is implicitly assumed that the portfolio can never run short of funds to purchase an option. e options are either funded by accumulated payos or by borrowing at the risk-free interest rate, where the laer would reduce returns with an extra amount equal to the interest expense.

32 3.3 Option-based protection strategies and portfolios

e core objective of the option-based strategies that are covered in this study is to provide protection against a long position in the underlying S&P 500 index. e strategies are intended to be alternatives to the regular protective put strategy and should therefore be consistent with its objective. Essentially, the hypothetical investor is bullish on the underlying S&P 500 index, but seeks protection against potential losses should the price of the S&P 500 index fall below a tolerated level. Although the hypothetical investor seeks to maximize the risk-adjusted returns, she does not intend to realize prots from the options. e protection obtained from the option- based strategies should therefore be seen as an insurance policy, rather than an investment.

As documented by previous studies, the protective put strategy suers from high put option premiums and path dependency. Besides providing protection, the selected option-based strategies are constructed to address these two problems. In particular, the selected option-based protection strategies aim to:

1. Reduce the high cost related to put options.

2. Minimize exposure to the path dependency in a rolling put option strategy.

Table 3.1 provides an overview of the selected option-based protection strategies. e strategies presented in the table will not only be compared to each other, but also benchmarked against a risk-reducing strategy that does not involve any options. A divested equity strategy will be implemented to serve this purpose, which will be further explained in Section 3.3.2.

33 Table 3.1: Overview of the selected option strategies.

Strategy Description Protection Maximum lossa,b Rationale Full protection below Protective Put Long 5% OTM put 5% + premium Simple strike Long 5% OTM put, Put Spread 95/80 Protection up to 20% dip 5% + 80% + net premium Low cost short 20% OTM put Long 5% OTM put, Put Spread 95/85 Protection up to 15% dip 5% + 85% + net premium Low cost short 15% OTM put Long ATM put, Put Spread 100/90 Protection up to 10% dip 90% + net premium Low cost short 10% OTM put Long 5% OTM put, Full protection below Collar 95/105 5% + net premium Low cost short 5% OTM call strike, capped upside Long overlapping f Full protection aer Reduced path Fractional Protective Put 5% + (1 f ) 95% + f premium of 5% OTM put a certain time − · · dependency Long/Short overlapping Full protection aer 5% + 85% + (1 f ) 10% + Low cost, Reduced Fractional Put Spread 95/85 − · f of 5%/15% OTM put a certain time f net premium path dependency · a f represents a fraction of one option. For example, the Fractional Protective Put strategy based on 3m options involves buying 1/3 of a 3m option every month. Consequently, the portfolio is fully protected aer 3 months. b e maximum loss of the Fractional Protective Put strategy occurs in the rst month, when the portfolio is only hedged by a fraction f of a put option, e.g. for 3m options: 5% +(2/3) 95% + (1/3) premium · ·

e option strategies in Table 3.1 will be implemented in four dierent portfolios. e portfolios are presented in Table 3.2 and are designed to show how the dierent option-based strategies perform across dierent maturities of the constituent options.

Table 3.2: Description of the four dierent portfolios. Portfolio Descriptiona 1m portfolio Rolling purchases/selling of 1m options every month. 3m portfolio Rolling purchases/selling of 3m options every 3 months. 6m portfolio Rolling purchases/selling of 6m options every 6 months. 12m portfolio Rolling purchases/selling of 12m options every 12 months. ”1m”, ”3m”, ”6m” and ”12m” are abbreviations for 1-month, 3-month, 6-month and 12-month. a e fractional strategies are exceptions. ey buy fractional amounts of an option every week in the 1m portfolio and every month in the 3m, 6m and 12m portfolios.

3.3.1 Protective Put

e protective put strategy buys a 5% OTM put at every rollover date. Documenting its performance across the four portfolios (1m, 3m, 6m and 12m) will be important for the analysis, as the other protection strategies are built to improve its deciencies. For example, the low-cost portfolios are designed to address the negative impact of high premiums on the returns of the protective put strategy. See Appendix 6.1.1 for a modelling example.

34 3.3.2 Divested Equity Portfolio

Option-based protection strategies are one approach to protect against rising volatility (or downside risk). However, volatility can also be dampened via asset allocation. One such passive approach is the static reduction of equity exposure - in this context, called the divested equity strategy. It will serve as a benchmark for each option strategy to shed light on how option-based protection strategies stand against a hedging method that does not use options. e divested equity portfolio allocates a portion of its net asset value (NAV) to a long position in the underlying asset, i.e. S&P 500 index, and the remainder to cash. It weighs the investor’s willingness to assume risk against expected returns through the exposure to the risky asset. See Section 6.1.2 in Appendix for a modelling example.

For the purpose of this study, the allocation given to the S&P 500 is chosen such that the divested equity portfolio yields the same geometric return (see Section 3.4 for the denition) as the option-based strategy it is compared to. is enables a fair comparison between their drawdown characteristics. Although no transaction costs are considered, the portfolio will be rebalanced on a monthly basis to maintain a more realistic approach.

3.3.3 Low-cost Portfolios

e put spread 95/80, put spread 95/85, put spread 100/90 and collar 95/105 in Table 3.1 are all low-cost strategies compared to the protective put strategy. e put spreads are of special interest because they will shed light on the benets, if there are any, of forgoing some protection for lower option premiums. e collar, on the other hand, caps the upside in exchange for lower premiums, which gives an equity exposure that is dierent from the protective put, but still interesting for comparative purposes. Ultimately, the comparison provides a perspective on what is worth sacricing for improved risk-adjusted return and reduced tail risk.

e size of the spreads were chosen based on S&P 500 index drawdown percentiles in the period 2 January 1990 - 20 December 2002. e 10-year period was intentionally chosen to precede the period that this study covers. It represents the information available to the hypothetical investor at the time the backtesting commences. e 99th and 95th percentiles of the 20-Day drawdown

35 window are 14.0% and 8.4%, respectively. Over a 63-Day drawdown window, the 99th and 95th percentiles are 22.6% and 15.8%, respectively. With these numbers in mind, the put spread 95/85 strategy is based on a rounded assumption that the market should fall at maximum 15.0% from the ATM level during the holding period. e put spread 95/80 strategy assumes the more extreme case of a 20% dip, while the put spread 100/90 protects against a moderate 10% dip in the market from the ATM level.

Indeed, the 99th and 95th percentiles of drawdowns over longer windows (e.g. 250 days) are larger. If the strategies were designed to match these, then the spreads would be too large and barely reduce the premiums paid, and therefore fail their purpose.

All four low-cost option-based protection strategies, except for one, buys 5% OTM put options. e reason is to keep the costs low, while not giving up too much protection from the ATM level. In order to capture the eect of expensive put options and for comparative purposes, one of the strategies, namely the put spread 100/90, buys ATM puts.

3.3.4 Fractional Protective Put

As explained in Section 2.4.1, the protective put strategy is exposed to path dependency risk. When a put option with a xed percentage strike expires and is rolled over into a new one, the cost will mainly depend on the implied volatility of the S&P 500 index at that particular time. If the rollover of the put option is timed with a sudden spike in implied volatility, then this will result in a higher cost than it would if volatility remained constant. Figure 2.3 shows how implied volatility spiked during the nancial crisis, which at the time led to very high premiums for the put options.

As a measure to address the path dependency risk, a modied protective put strategy has been constructed. e fractional protective put strategy aims to reduce the exposure to path dependency risk by diversifying the purchases of put options over overlapping maturity cycles. e put options are purchased in fractional amounts to keep the costs on par with the regular protective put. For example, the 3m portfolio would buy 1/3 of a 3m option every month. Consequently, full protection is not obtained before the third month.

36 Based on the same rationale, the 6m portfolio buys 1/6 of a 6m option every month and reaches full protection by the sixth month. Similarly, the 12m portfolio buys 1/12 of a 12m option every month and reaches full protection by the twelh month. e 1m portfolio is slightly dierent, but follows the same logic. It buys 1/4 of a 1m option every week and reaches full protection by the rst month.

In the rst months of the strategy, the portfolio is only partially hedged and therefore more exposed to market downturns. e longer the maturity of the strategy, the longer it takes for the portfolio to become fully protected and is thus more exposed to market risk.

3.3.5 Fractional Put Spread

e fractional put spread 95/85 is designed to combine the benets of lower net cost and reduced path dependency. Similar to the regular put spread 95/85, it buys 5% OTM puts and sells 15% OTM puts. Hence, the strategy protects, at maximum, against 15% market dips.

3.4 Evaluation methods

Annualized portfolio return

Both the annualized arithmetic and geometric returns on the portfolio are calculated.

h i 1 Arithmetic return = 252 R1 + R2 + ...RN 1 N 1 · − · − 252 (3.4.1) h i N 1 Geometric return = (1 + R1) (1 + R2) ...(1 + RN 1) − 1 · · − − where Ri is the daily return and N is the number of days in the investment horizon.

37 Risk-adjusted performance

e risk-adjusted performance of a portfolio will be assessed on its excess return per unit of volatility. is measure is captured by the Sharpe ratio. It is dened as the excess return over the risk-free rate per unit of volatility: r r S = p− rf (3.4.2) p σp where rp is the portfolio arithmetic return, rrf is the risk-free interest rate and σp is the portfolio volatility.

When the market is in a downward trend, the Sharpe ratio can be a misleading tool (Scholz, 2007). In this case, the risk-free rate oen outperforms the portfolio returns. A negative excess return gives a negative Sharpe ratio, which makes the relative ranking between the dierent portfolios misleading. In order to address the ranking issue, Israelsen (2009) proposed an adjusted Sharpe ratio:

ep Sp,adj = ep/abs(ep) (3.4.3) σp where e = r r is the excess return on the portfolio. is modication solves the ranking p p − rf issue. e less negative the adjusted Sharpe ratio is, the beer is the risk-adjusted performance, whereas larger negative values indicate the opposite.

Peak-to-trough drawdowns

In order to measure the eectiveness of a strategy’s tail protection, we will examine the returns on the portfolio during peak-to-trough drawdowns over rolling overlapping time windows of sizes 5, 10, 20, 63, 125, and 250 days. e peak-to-trough drawdowns are measured as the largest decline in the portfolio value over a given time window. Graphically, it would be seen as a decline in the portfolio value from its highest peak to its lowest trough over that specic time window.

Let VP (ti) and VL(ti) represent the value of the portfolio at its highest peak and lowest trough, respectively, in the time interval (ti k,ti), where k is the length of the drawdown window (e.g. −

38 5, 10, etc.). en the peak-to-trough drawdown, D, over overlapping windows is calculated by:

VP (ti) VL(ti) D(ti k) = − (3.4.4) − VP (ti) for i = k,k + 1,...,n, where t0,t1,...,tN 1 is the set of days within the investment horizon. { − }

e eectiveness of the protection is measured at the 99th, 95th and 50th (median) percentiles of the peak-to-trough drawdowns over the dierent windows. e 99th and 95th percentiles, in particular, quantify the eectiveness of the tail protection of the strategy.

Capital Asset Pricing Model: Beta and Alpha

e capital asset pricing model (CAPM) describes the relationship between systematic (undiver- siable) risk and expected return on an asset, e.g. a stock. e excess return on the asset over the risk-free rate is regressed against the excess return on the market over the risk-free rate. e formula is given by: R = R + β(R R ) (3.4.5) a f M − f where Ra is the expected return on the asset, RM is the return on the market (e.g. an index such as the S&P 500), Rf is the return on a risk-free investment (e.g. the risk-free rate) and β is the value of the slope obtained from the linear regression. β captures the sensitivity of returns from the asset to returns from the market. If the asset is a stock, then the excess return on the market R R is sometimes referred to as the equity risk premium. Oen the linear relationship M − f in equation 3.4.5 has a non-zero intercept, α, which captures abnormal returns that cannot be explained by the correlation with the market returns (Hull, 2012).

Investors use β and α as portfolio performance metrics, in which case Ra would be the expected return on the portfolio. For the purposes of our analysis, a β below 1 means that the portfolio is less sensitive to changes in the market returns, while a β above 1 implies the opposite. Hence, a β equal to 1 means that the portfolio excess returns perfectly follow the excess return on the market. e β and α will help us to explain the volatility exposure of the dierent strategies and the eect of volatility risk premium on the returns.

39 3.5 Monte Carlo Simulation

A Monte Carlo simulation is performed to produce dierent sets of future stock prices following a distribution based on the Merton’s Jump Diusion model presented in section 2.2.3. e simulation enables an idealized environment and captures many dierent possible outcomes for the underlying stock. e dierent outcomes allow for more generality in the results, but their ap- plicability to the reality is restricted to the ideal conditions, which oen are very dierent from the actual ones. It is, however, worthwhile to examine how the dierent protection strategies compare against each other and how they fare against the divested equity strategy in an ideal seing.

We simulate 1,000 stock paths with a 5 year investment horizon, i.e. T = 5 252 = 1,260 days. · e stock prices are drawn from a lognormal distribution with the addition of a Poisson-driven jump process in accordance with the risk-neutral process 2.2.9 in the Merton’s Jump Diusion model. e jump size is drawn from a lognormal distribution, N( 0.5,0.052), and with a jump − frequency of 0.1 per year. ese numbers are based on an assumption that the stock is expected to fall ca 50% every tenth year similar to a major crash experienced in the real-world equity markets. e jump component contributes with an amount of λk = 0.0393 to the dri of the − stock price.

Furthermore, without loss of generality, it is assumed that the risk-free interest rate and the dividend yield are both equal to zero. e volatility of the stock price is assumed to be 20% based on an annualized historical volatility of ca 18.8% in the S&P 500 index in the period December 2002 - March 2018. e volatility risk premium is set to a rounded number of 2.0 percentage points based on the 1.7% median of the volatility risk premiums in the same period (see Table 2.1). us, we add 2.0 percentage points on top of the stock price volatility in the pricing formula for options. Option prices are modeled according to the pricing formula in the Merton’s Jump Diusion model, equation 2.2.10.

To summarize, the input parameters are:

40 r = 0% (risk-free interest rate)

S0 = 100 (e price of the stock at the initial investment) q = 0% (dividend yield)

σs = 20% (stock volatility)

σp = 22% (stock volatility + volatility risk premium) (3.5.1) 1 dt = 252 (size time steps) λ = 0.1 (jump frequency per year) γ = 0.5 (expected value of the jump size) − δ = 0.05 (standard deviation of the jump size)

e assumption of constant volatility in the Monte Carlo simulation implies no volatility skew with respect to strike prices and no term structure with respect to time to maturity. is is an idealized environment, since OTM put options are cheaper relative to the ATM level compared to the more realistic scenario where the volatility curve is skewed (see Figure 2.5). Also, the cost of options is constant across the dierent maturities. Moreover, the assumption of constant volatility leads to another unrealistic consequence, namely, the absence of spikes in the volatility during crashes. We have modelled crashes, but kept the volatility deterministic, which is a super- idealized environment. us, as crashes occur in the simulated stock paths, the time value of the dierent protection strategies will not be aected. In reality, the cost of put options would rise signicantly, making, for example, the protective put more expensive. is scenario is covered in the backtesting part of the study.

3.6 Backtesting

3.6.1 Data

e backtesting of the option-based protection strategies will be performed on the S&P 500 index, which is a market capitalization weighted index of the 500 largest listed companies in the U.S. by market value. Options on the S&P 500 index give exposure to the broad-based market with relatively lile capital and are actively traded on the Chicago Board Options Exchange (CBOE).

41 eir popularity, as well as the high data availability, motivates their use in this study.

Daily data is extracted from Bloomberg spanning the period from 20 December 2002 to 23 March 2018 and consists of the constituent parameters of the Black-Scholes-Merton model. Table 3.3 presents the data set in more detail.

Table 3.3: Daily data points for the Black-Scholes-Merton input parameters spanning the period 20 December 2002 - 23 March 2018. Parameter Daily data Underlying asset S&P 500 index Strike pricea Fixed percentages of the S&P 500 index spot level (30%, 35%, …, 200%) Time to maturitya 5 days (1w), 21 days (1m), 63 days (3m), 126 days (6m) and 252 days (12m) Volatility Implied volatility of S&P 500 index across strike price and time to maturity Risk-free interest rate Zero coupon rates for the dierent time to maturities Dividend yield Expected S&P 500 dividend yield for the dierent time to maturities a ese are xed values and do not vary by days. Source: Bloomberg

3.6.2 Outline

e option-based protection strategies and portfolios presented in Table 3.1 and Table 3.2 are modelled based on the assumptions presented in Section 3.2 and are performed on the data set in Table 3.3. All four portfolios (1m, 3m, 6m and 12m) implement each of the dierent strategies and their performance is measured in terms of risk-adjusted return and peak-to-trough drawdowns.

e backtesting consists of two parts:

e whole period: 20 December 2002 - 23 March 2018 As depicted in Figure 2.3, the volatility in the S&P 500 index varied signicantly during the whole period. is part examines the performance of the protection strategies over a long-term investment horizon and how they fare during shiing market environments.

e nancial crisis: 14 December 2007 - 5 March 2009 Backtesting the protection strategies on the data from the nancial crisis show how they fare during a period of very high volatility. As they are expected to perform during such events, it is especially interesting to assess how well they live up to these expectations as well as their

42 relative performance.

43 Results

4.1 Monte Carlo Simulation

In some cases, the geometric return on an option-based protection strategy could not be matched with the geometric return on the divested equity strategy. ose cases (or stock paths) are omied to enable a fair comparison between them.

In addition to the results presented in this section, Tables 6.1- 6.4 in Appendix report the drawdown characteristics in more detail. Figure 6.13 in Appendix visualizes the change in the total cost as a percentage of the initial investment with respect to option maturity.

44 Table 4.1: Mean value of annualized returns and performance measures for the simulated strategies across the four portfolios. e adjusted Sharpe ratio is used on stock paths with negative annualized arithmetic return. Each strategy is benchmarked against the divested equity. Arithmetic return Geometric return Volatility Sharpe ratio 1m 3m 6m 12m 1m 3m 6m 12m 1m 3m 6m 12m 1m 3m 6m 12m Protective Put -0.2% -0.7% -1.0% -1.4% -1.2% -1.5% -1.8% -2.0% 16.5% 14.7% 13.9% 13.4% 0.17 0.16 0.15 0.15 Divested Equity -0.5% -0.8% -1.0% -1.2% -1.2% -1.5% -1.8% -2.0% 12.8% 12.2% 12.6% 12.7% 0.27 0.26 0.25 0.25 Di +0.3% +0.1% 0.0% -0.1% 0.0% 0.0% 0.0% 0.0% +3.8% +2.5% +1.2% +0.7% -0.10 -0.10 -0.09 -0.10 Put Spread 95/80 -0.8% -1.1% -1.5% -1.4% -2.2% -2.3% -2.6% -2.6% 19.1% 17.5% 17.3% 17.8% 0.19 0.18 0.18 0.19 Divested Equity -1.0% -1.1% -1.4% -1.3% -2.2% -2.3% -2.6% -2.6% 16.2% 15.5% 15.8% 16.5% 0.27 0.25 0.23 0.24 Di +0.2% +0.1% 0.0% -0.1% 0.0% 0.0% 0.0% 0.0% +2.9% +2.0% +1.4% +1.3% -0.07 -0.07 -0.06 -0.05 Put Spread 95/85 -0.2% -0.8% -1.0% -1.0% -1.7% -2.2% -2.4% -2.4% 19.6% 18.4% 18.4% 19.1% 0.21 0.19 0.19 0.20

45 Divested Equity -0.5% -1.0% -1.1% -1.0% -1.7% -2.2% -2.4% -2.4% 15.9% 16.0% 16.5% 17.6% 0.28 0.25 0.23 0.24 Di +0.4% +0.2% +0.1% 0.0% 0.0% 0.0% 0.0% 0.0% +3.8% +2.4% +1.9% +1.6% -0.07 -0.05 -0.04 -0.04 Put Spread 100/90 -1.8% -1.3% -1.1% -1.0% -2.7% -2.3% -2.4% -2.4% 15.2% 16.4% 17.6% 18.6% 0.17 0.18 0.19 0.20 Divested Equity -1.8% -1.3% -1.2% -1.0% -2.7% -2.3% -2.4% -2.4% 13.6% 14.7% 16.1% 17.0% 0.26 0.24 0.24 0.25 Di 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% +1.6% +1.6% +1.4% +1.6% -0.09 -0.05 -0.04 -0.04 Collar 95/105 0.3% 0.2% 0.3% 0.3% -0.2% 0.0% 0.1% 0.3% 12.1% 7.9% 5.8% 3.9% 0.18 0.19 0.20 0.25 Divested Equity 0.3% 0.3% 0.4% 0.5% -0.2% 0.0% 0.1% 0.3% 10.5% 8.0% 6.6% 5.5% 0.23 0.23 0.22 0.22 Di +0.1% -0.1% -0.2% -0.2% 0.0% 0.0% 0.0% 0.0% +1.6% -0.1% -0.8% -1.6% -0.05 -0.04 -0.02 +0.02 Fractional Protective Put -0.2% -0.5% -0.5% -0.3% -1.3% -1.4% -1.4% -1.2% 16.5% 15.0% 14.6% 14.8% 0.17 0.17 0.16 0.17 Divested Equity -0.6% -0.7% -0.6% -0.4% -1.3% -1.4% -1.4% -1.2% 12.6% 12.6% 12.5% 12.6% 0.27 0.25 0.24 0.25 Di +0.4% +0.1% +0.1% +0.1% 0.0% 0.0% 0.0% 0.0% +3.9% +2.4% +2.1% +2.2% -0.10 -0.09 -0.08 -0.08 Fractional Put Spread 95/85 -0.7% -0.8% -0.6% -0.6% -2.2% -2.2% -2.2% -2.3% 19.8% 18.7% 19.1% +20.0% 0.21 0.19 0.19 0.19 Divested Equity -0.9% -0.9% -0.8% -0.7% -2.2% -2.2% -2.2% -2.3% 16.7% 16.7% 17.2% 18.3% 0.27 0.24 0.23 0.21 Di +0.2% +0.1% +0.2% +0.2% 0.0% 0.0% 0.0% 0.0% +3.0% +2.0% +1.9% +1.6% -0.06 -0.04 -0.03 -0.02 Table 4.2: Mean value of the annualized alpha, beta and total cost (as % of initial investment) for the simulated strategies across the four portfolios. e total cost is measured as the accumulated cost over the whole investment horizon divided by the initial investment. e size of the initial investment is equal for all strategies. Annualized alpha Beta Total cost as % of initial investment 1m 3m 6m 12m 1m 3m 6m 12m 1m 3m 6m 12m Protective Put -3.1% -2.9% -3.1% -3.5% 0.63 0.56 0.52 0.48 56.7% 55.1% 47.9% 39.3% Divested Equity 0.0% 0.0% 0.0% 0.0% 0.52 0.50 0.51 0.51 0.0% 0.0% 0.0% 0.0% Di -3.1% -2.9% -3.1% -3.5% +0.12 +0.06 0.0 -0.03 +56.7% +55.1% +47.9% +39.3% Put Spread 95/80 -2.3% -2.0% -2.0% -2.0% 0.78 0.71 0.71 0.73 47.2% 44.5% 35.2% 24.8% Divested Equity 0.0% 0.0% 0.0% 0.0% 0.65 0.63 0.65 0.68 0.0% 0.0% 0.0% 0.0% Di -2.3% -2.0% -2.0% -2.0% +0.13 +0.08 +0.06 +0.05 +47.2% +44.5% +35.2% +24.8% Put Spread 95/85 -2.0% -1.6% -1.6% -1.6% 0.82 0.76 0.77 0.80 44.3% 38.3% 28.0% 18.5%

46 Divested Equity 0.0% 0.0% 0.0% 0.0% 0.65 0.66 0.69 0.73 0.0% 0.0% 0.0% 0.0% Di -2.0% -1.6% -1.5% -1.5% +0.17 +0.10 +0.09 +0.08 +44.3% +38.3% +28.0% +18.5% Put Spread 100/90 -2.2% -1.6% -1.6% -1.6% 0.60 0.67 0.73 0.79 139.8% 66.6% 39.3% 22.8% Divested Equity 0.0% 0.0% 0.0% 0.0% 0.56 0.62 0.68 0.71 0.0% 0.0% 0.0% 0.0% Di -2.2% -1.6% -1.6% -1.6% +0.03 +0.06 +0.06 +0.07 +139.8% +66.6% +39.3% +22.8% Collar 95/105 -1.0% -0.7% -0.4% -0.1% 0.49 0.31 0.22 0.14 3.0% 0.1% -1.0% -2.1% Divested Equity 0.0% 0.0% 0.0% 0.0% 0.47 0.36 0.30 0.25 0.0% 0.0% 0.0% 0.0% Di -1.0% -0.7% -0.4% -0.1% +0.02 -0.05 -0.08 -0.11 +3.0% +0.1% -1.0% -2.1% Fractional Protective Put -3.0% -2.7% -2.5% -2.4% 0.64 0.59 0.58 0.59 56.7% 55.1% 47.9% 39.3% Divested Equity 0.0% 0.0% 0.0% 0.0% 0.51 0.52 0.52 0.52 0.0% 0.0% 0.0% 0.0% Di -3.0% -2.7% -2.5% -2.3% +0.13 +0.07 +0.06 +0.06 56.7% 55.1% 47.9% 39.3% Fractional Put Spread 95/85 -1.9% -1.4% -0.01 -0.7% 0.82 0.78 0.81 0.85 44.3% 38.3% 28.0% 18.5% Divested Equity 0.0% 0.0% 0.0% 0.0% 0.68 0.69 0.72 0.77 0.0% 0.0% 0.0% 0.0% Di -1.9% -1.4% -1.0% -0.7% +0.14 +0.09 +0.09 +0.08 44.3% 38.3% 28.0% 18.5% 1m Portfolios 3m Portfolios 6 6 Protective Put, (-0.1, 0.14) Protective Put, (-0.1, 0.15) Put Spread 95/80, (-0.07, 0.1) Put Spread 95/80, (-0.07, 0.1) Put Spread 95/85, (-0.07, 0.09) Put Spread 95/85, (-0.05, 0.08) Put Spread 100/90, (-0.09, 0.14) Put Spread 100/90, (-0.05, 0.1) 5 Collar 95/105, (-0.05, 0.11) 5 Collar 95/105, (-0.04, 0.14) Frac. Protective Put, (-0.1, 0.14) Frac. Protective Put, (-0.09, 0.13) Frac. Put Spread 95/85, (-0.06, 0.08) Frac. Put Spread 95/85, (-0.04, 0.07)

4 4

3 3

2 2 Probability Density Function Probability Density Function

1 1

0 0 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Diff. in Sharpe ratio versus Divesting Diff. in Sharpe ratio versus Divesting

6m Portfolios 12m Portfolios 8 12 Protective Put, (-0.09, 0.15) Protective Put, (-0.1, 0.14) Put Spread 95/80, (-0.06, 0.09) Put Spread 95/80, (-0.05, 0.08) Put Spread 95/85, (-0.04, 0.07) Put Spread 95/85, (-0.04, 0.06) 7 Put Spread 100/90, (-0.04, 0.07) Put Spread 100/90, (-0.04, 0.06) Collar 95/105, (-0.02, 0.15) 10 Collar 95/105, (0.02, 0.19) Frac. Protective Put, (-0.08, 0.12) Frac. Protective Put, (-0.08, 0.11) Frac. Put Spread 95/85, (-0.03, 0.05) 6 Frac. Put Spread 95/85, (-0.02, 0.03)

8 5

4 6

3 4 Probability Density Function Probability Density Function

2 1

0 0 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 Diff. in Sharpe ratio versus Divesting Diff. in Sharpe ratio versus Divesting

Figure 4.1: Distribution of the dierence in Sharpe ratio between the option-based protection strategies and the divested equity strategy.

47 1m Portfolios 3m Portfolios 1.8 1.8 Protective Put, (0.31, 0.25) Protective Put, (0.31, 0.24) 1.6 Put Spread 95/80, (0.33, 0.27) 1.6 Put Spread 95/80, (0.34, 0.27) Put Spread 95/85, (0.34, 0.28) Put Spread 95/85, (0.35, 0.28) Put Spread 100/90, (0.33, 0.26) Put Spread 100/90, (0.34, 0.28) 1.4 Collar 95/105, (0.36, 0.27) 1.4 Collar 95/105, (0.37, 0.28) Frac. Protective Put, (0.31, 0.25) Frac. Protective Put, (0.32, 0.25) 1.2 Frac. Put Spread 95/85, (0.34, 0.28) 1.2 Frac. Put Spread 95/85, (0.36, 0.28)

1 1

0.8 0.8

0.6 0.6

Probability Density Function 0.4 Probability Density Function 0.4

0.2 0.2

0 0 0 0.5 1 1.5 0 0.5 1 1.5 Sharpe ratio Sharpe ratio

6m Portfolios 12m Portfolios 1.6 1.6 Protective Put, (0.32, 0.25) Protective Put, (0.33, 0.26) Put Spread 95/80, (0.35, 0.28) Put Spread 95/80, (0.34, 0.28) 1.4 Put Spread 95/85, (0.35, 0.28) 1.4 Put Spread 95/85, (0.36, 0.29) Put Spread 100/90, (0.36, 0.28) Put Spread 100/90, (0.36, 0.29) 1.2 Collar 95/105, (0.38, 0.31) 1.2 Collar 95/105, (0.47, 0.37) Frac. Protective Put, (0.33, 0.25) Frac. Protective Put, (0.34, 0.26) Frac. Put Spread 95/85, (0.36, 0.28) Frac. Put Spread 95/85, (0.37, 0.29) 1 1

0.8 0.8

0.6 0.6

0.4 0.4 Probability Density Function Probability Density Function

0.2 0.2

0 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 0.5 1 1.5 2 2.5 Sharpe ratio Sharpe ratio

Figure 4.2: Probability density function of the Sharpe ratios of the dierent strategies during periods of positive return on the portfolios.

48 1m Portfolios 3m Portfolios 100 100 Protective Put, (-0.01, 0.01) Protective Put, (-0.01, 0.01) 90 Put Spread 95/80, (-0.02, 0.01) 90 Put Spread 95/80, (-0.01, 0.01) Put Spread 95/85, (-0.02, 0.01) Put Spread 95/85, (-0.01, 0.01) 80 Put Spread 100/90, (-0.01, 0.02) 80 Put Spread 100/90, (-0.01, 0.01) Collar 95/105, (0, 0) Collar 95/105, (0, 0) Frac. Protective Put, (-0.01, 0.01) Frac. Protective Put, (-0.01, 0.01) 70 Frac. Put Spread 95/85, (-0.02, 0.01) 70 Frac. Put Spread 95/85, (-0.01, 0.01)

60 60

50 50

40 40

30 30 Probability Density Function Probability Density Function 20 20

10 10

0 0 -0.14 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 -0.1 -0.08 -0.06 -0.04 -0.02 0 Sharpe ratio Sharpe ratio

6m Portfolios 12m Portfolios 100 100 Protective Put, (-0.01, 0.01) Protective Put, (-0.01, 0) 90 Put Spread 95/80, (-0.01, 0.01) 90 Put Spread 95/80, (-0.01, 0.01) Put Spread 95/85, (-0.02, 0.01) Put Spread 95/85, (-0.02, 0.01) 80 Put Spread 100/90, (-0.01, 0.01) 80 Put Spread 100/90, (-0.02, 0.02) Collar 95/105, (0, 0) Collar 95/105, (0, 0) Frac. Protective Put, (-0.01, 0) Frac. Protective Put, (-0.01, 0.01) 70 Frac. Put Spread 95/85, (-0.01, 0.01) 70 Frac. Put Spread 95/85, (-0.02, 0.02)

60 60

50 50

40 40

30 30 Probability Density Function Probability Density Function 20 20

10 10

0 0 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 -0.1 -0.08 -0.06 -0.04 -0.02 0 Sharpe ratio Sharpe ratio

Figure 4.3: Probability density function of the adjusted Sharpe ratios of the dierent strategies during periods of negative return on the portfolios.

49 5-Day 10-Day 0% 0.2% Protective Put Put Spread 95/80 Put Spread 95/85 -0.2% Put Spread 100/90 Collar 95/105 2.2204e-16% Fractional Protective Put Fractional Put Spread 95/85 -0.4%

-0.6% -0.2%

-0.8%

-0.4%

-1%

-1.2% -0.6%

-1.4% Diff. in 99th percentile drawdown versus Divesting Diff. in 99th percentile drawdown versus Divesting Protective Put Put Spread 95/80 -0.8% Put Spread 95/85 -1.6% Put Spread 100/90 Collar 95/105 Fractional Protective Put Fractional Put Spread 95/85 -1.8% -1% 0 2 4 6 8 10 12 0 2 4 6 8 10 12 Maturity (months) Maturity (months)

20-Day 63-Day 6% 3% Protective Put Put Spread 95/80 Put Spread 95/85 5% Put Spread 100/90 Collar 95/105 2% Fractional Protective Put Fractional Put Spread 95/85 4%

0% 2%

1% -1%

0% -2%

-1% Diff. in 99th percentile drawdown versus Divesting Diff. in 99th percentile drawdown versus Divesting Protective Put Put Spread 95/80 -3% Put Spread 95/85 -2% Put Spread 100/90 Collar 95/105 Fractional Protective Put Fractional Put Spread 95/85 -3% -4% 0 2 4 6 8 10 12 0 2 4 6 8 10 12 Maturity (months) Maturity (months)

125-Day 250-Day 3% 2%

2% 1%

1% 0%

0% -1%

-1% -2%

-2% -3%

-3% -4% Diff. in 99th percentile drawdown versus Divesting Diff. in 99th percentile drawdown versus Divesting Protective Put Protective Put Put Spread 95/80 Put Spread 95/80 Put Spread 95/85 Put Spread 95/85 -4% -5% Put Spread 100/90 Put Spread 100/90 Collar 95/105 Collar 95/105 Fractional Protective Put Fractional Protective Put Fractional Put Spread 95/85 Fractional Put Spread 95/85 -5% -6% 0 2 4 6 8 10 12 0 2 4 6 8 10 12 Maturity (months) Maturity (months)

Figure 4.4: Dierence in the mean 99th percentile peak-to-trough drawdown between the option- based protection strategies and the divested equity strategy versus maturity. e dierence is measured in percentage points.

50 5-Day 10-Day 0.2% 0.4%

0.1% 0.2%

0% 1.1102e-16%

-0.1%

-0.2%

-0.4% -0.3%

Protective Put Protective Put Diff. in 50th percentile drawdown versus Divesting Diff. in 50th percentile drawdown versus Divesting Put Spread 95/80 -0.6% Put Spread 95/80 -0.4% Put Spread 95/85 Put Spread 95/85 Put Spread 100/90 Put Spread 100/90 Collar 95/105 Collar 95/105 Fractional Protective Put Fractional Protective Put Fractional Put Spread 95/85 Fractional Put Spread 95/85 -0.5% -0.8% 0 2 4 6 8 10 12 0 2 4 6 8 10 12 Maturity (months) Maturity (months)

20-Day 63-Day 0.6% 1%

0.4% 0.5%

0.2% 0% 0%

-0.5% -0.2%

-0.4% -1%

-0.6% -1.5%

-0.8% -2% -1% Protective Put Protective Put Diff. in 50th percentile drawdown versus Divesting Put Spread 95/80 Diff. in 50th percentile drawdown versus Divesting Put Spread 95/80 Put Spread 95/85 Put Spread 95/85 Put Spread 100/90 -2.5% Put Spread 100/90 -1.2% Collar 95/105 Collar 95/105 Fractional Protective Put Fractional Protective Put Fractional Put Spread 95/85 Fractional Put Spread 95/85 -1.4% -3% 0 2 4 6 8 10 12 0 2 4 6 8 10 12 Maturity (months) Maturity (months)

125-Day 250-Day 1% 1%

0.5%

0% 0%

-0.5% -1%

-1%

-1.5% -2%

-2%

-3% -2.5%

-3% Protective Put Protective Put Diff. in 50th percentile drawdown versus Divesting Diff. in 50th percentile drawdown versus Divesting Put Spread 95/80 Put Spread 95/80 -4% Put Spread 95/85 Put Spread 95/85 Put Spread 100/90 Put Spread 100/90 -3.5% Collar 95/105 Collar 95/105 Fractional Protective Put Fractional Protective Put Fractional Put Spread 95/85 Fractional Put Spread 95/85 -4% -5% 0 2 4 6 8 10 12 0 2 4 6 8 10 12 Maturity (months) Maturity (months)

Figure 4.5: Dierence in the mean 50th percentile peak-to-trough drawdown between the option- based protection strategies and the divested equity strategy versus maturity. e dierence is measured in percentage points.

51 5-Day 10-Day -1.5% -2%

-2% -3%

-2.5% -4% Protective Put Put Spread 95/80 Put Spread 95/85 Protective Put -3% -5% Put Spread 100/90 Put Spread 95/80 Collar 95/105 Put Spread 95/85 Fractional Protective Put Put Spread 100/90 Fractional Put Spread 95/85 -3.5% Collar 95/105 -6% Fractional Protective Put Fractional Put Spread 95/85 -4% -7%

-4.5% -8% Mean 99th percentile drawdown Mean 99th percentile drawdown -5% -9%

-5.5% -10%

-6% -11%

-6.5% -12% 0 2 4 6 8 10 12 0 2 4 6 8 10 12 Maturity (months) Maturity (months)

20-Day 63-Day -4% -5%

-6% Protective Put Put Spread 95/80 Protective Put Put Spread 95/85 -10% Put Spread 95/80 Put Spread 100/90 -8% Put Spread 95/85 Collar 95/105 Put Spread 100/90 Fractional Protective Put Collar 95/105 Fractional Put Spread 95/85 Fractional Protective Put -10% Fractional Put Spread 95/85 -15% -12%

-14% -20%

-16% Mean 99th percentile drawdown Mean 99th percentile drawdown

-18% -25%

-20%

-22% -30% 0 2 4 6 8 10 12 0 2 4 6 8 10 12 Maturity (months) Maturity (months)

125-Day 250-Day -5% -5%

-10% Protective Put -10% Put Spread 95/80 Put Spread 95/85 Put Spread 100/90 Protective Put Collar 95/105 Put Spread 95/80 Fractional Protective Put Put Spread 95/85 -15% Fractional Put Spread 95/85 -15% Put Spread 100/90 Collar 95/105 Fractional Protective Put Fractional Put Spread 95/85

-20% -20%

-25% -25% Mean 99th percentile drawdown Mean 99th percentile drawdown

-30% -30%

-35% -35% 0 2 4 6 8 10 12 0 2 4 6 8 10 12 Maturity (months) Maturity (months)

Figure 4.6: Mean 99th percentile peak-to-trough drawdown for the option-based protection strategies versus maturity.

52 5-Day 10-Day 0% 0%

-0.2%

-0.2% Protective Put Put Spread 95/80 -0.4% Protective Put Put Spread 95/85 Put Spread 95/80 Put Spread 100/90 Put Spread 95/85 Collar 95/105 -0.6% Put Spread 100/90 Fractional Protective Put Collar 95/105 -0.4% Fractional Put Spread 95/85 Fractional Protective Put Fractional Put Spread 95/85 -0.8%

-0.6% -1%

-1.2%

-0.8% Mean 50th percentile drawdown Mean 50th percentile drawdown -1.4%

-1.6% -1%

-1.8%

-1.2% -2% 0 2 4 6 8 10 12 0 2 4 6 8 10 12 Maturity (months) Maturity (months)

20-Day 63-Day 0% 0%

-0.5% -1%

Protective Put Put Spread 95/80 Protective Put Put Spread 95/85 Put Spread 95/80 -1% Put Spread 100/90 -2% Put Spread 95/85 Collar 95/105 Put Spread 100/90 Fractional Protective Put Collar 95/105 Fractional Put Spread 95/85 Fractional Protective Put Fractional Put Spread 95/85

-1.5% -3%

-2% -4% Mean 50th percentile drawdown Mean 50th percentile drawdown

-2.5% -5%

-3% -6% 0 2 4 6 8 10 12 0 2 4 6 8 10 12 Maturity (months) Maturity (months)

125-Day 250-Day -1% -2%

-2% -4%

Protective Put Protective Put -3% Put Spread 95/80 Put Spread 95/80 Put Spread 95/85 Put Spread 95/85 Put Spread 100/90 Put Spread 100/90 Collar 95/105 -6% Collar 95/105 -4% Fractional Protective Put Fractional Protective Put Fractional Put Spread 95/85 Fractional Put Spread 95/85

-5% -8%

-6% -10% Mean 50th percentile drawdown Mean 50th percentile drawdown

-7%

-12% -8%

-9% -14% 0 2 4 6 8 10 12 0 2 4 6 8 10 12 Maturity (months) Maturity (months)

Figure 4.7: Mean 50th percentile peak-to-trough drawdown for the option-based protection strategies versus maturity.

53 4.2 Backtesting

4.2.1 Whole period: December 2002 - Mars 2018

In addition to the results presented in this section, Tables 6.5- 6.8 in Appendix report the drawdown characteristics in more detail. Figures 6.5- 6.8 in Appendix visualize the value development of the dierent option-based protection strategies over the whole investment horizon. Figure 6.14 in Appendix visualizes the change in the total cost as a percentage of the initial investment with respect to option maturity.

54 Table 4.3: Summary of annualized returns and performance measures for the dierent strategies across the four portfolios. Each strategy is benchmarked against the divested equity. Arithmetic return Geometric return Volatility Sharpe ratio 1m 3m 6m 12m 1m 3m 6m 12m 1m 3m 6m 12m 1m 3m 6m 12m Protective Put 5.5% 5.1% 5.5% 5.8% 4.0% 4.3% 4.9% 5.4% 17.7% 13.3% 11.8% 10.9% 0.22 0.27 0.33 0.39 Divested Equity 4.2% 4.5% 5.2% 5.8% 4.0% 4.3% 4.9% 5.4% 7.4% 8.2% 9.8% 11.1% 0.36 0.36 0.38 0.38 Di +1.3% +0.6% +0.2% -0.0% 0.0% 0.0% 0.0% 0.0% +10.3% +5.1% +2.0% -0.3% -0.14 -0.10 -0.04 +0.01 Put Spread 95/80 6.1% 6.0% 6.8% 7.3% 4.6% 5.0% 5.8% 6.4% 17.6% 15.1% 15.1% 15.0% 0.26 0.30 0.35 0.38 Divested Equity 5.0% 5.4% 6.4% 7.2% 4.6% 5.0% 5.8% 6.4% 9.1% 10.1% 12.5% 14.2% 0.37 0.38 0.39 0.40 Di +1.1% +0.6% +0.4% +0.1% 0.0% 0.0% 0.0% 0.0% +8.5% +4.9% +2.6% +0.8% -0.11 -0.08 -0.04 -0.01 Put Spread 95/85 6.6% 6.7% 7.4% 7.8% 5.1% 5.6% 6.3% 6.7% 18.2% 15.9% 16.0% 16.2% 0.28 0.32 0.37 0.39 Divested Equity 5.5% 6.1% 7.1% 7.7% 5.1% 5.6% 6.3% 6.7% 10.3% 11.9% 14.1% 15.5% 0.38 0.39 0.39 0.40 55 Di +1.1% +0.6% +0.3% +0.1% 0.0% 0.0% 0.0% 0.0% +7.9% +4.0% +1.9% +0.7% -0.10 -0.06 -0.03 -0.01 Put Spread 100/90 4.2% 5.5% 6.8% 7.8% 2.9% 4.5% 5.8% 6.8% 16.6% 15.0% 15.4% 15.6% 0.16 0.26 0.34 0.40 Divested Equity 2.9% 4.7% 6.4% 7.8% 2.9% 4.5% 5.8% 6.8% 4.4% 8.6% 12.4% 15.6% 0.31 0.37 0.39 0.40 Di +1.3% +0.8% +0.4% +0.0% 0.0% 0.0% 0.0% 0.0% +12.1% +6.4% +3.1% 0.0% -0.15 -0.11 -0.05 -0.00 Collar 95/105 5.9% 5.9% 4.6% 2.1% 5.3% 5.7% 4.4% 2.0% 12.5% 8.4% 6.7% 5.5% 0.35 0.52 0.45 0.10 Divested Equity 5.8% 6.3% 4.7% 2.0% 5.3% 5.7% 4.4% 2.0% 11.0% 12.2% 8.6% 2.2% 0.38 0.39 0.37 0.19 Di +0.2% -0.4% -0.1% +0.1% 0.0% 0.0% 0.0% 0.0% +1.5% -3.8% -1.9% +3.3% -0.03 +0.13 +0.08 -0.09 Fractional Protective Put 5.6% 5.3% 5.8% 5.9% 4.2% 4.5% 5.2% 5.5% 17.4% 13.2% 11.7% 10.6% 0.23 0.28 0.36 0.41 Divested Equity 4.4% 4.8% 5.7% 6.0% 4.2% 4.5% 5.2% 5.5% 7.8% 8.7% 10.7% 11.6% 0.36 0.37 0.38 0.39 Di +1.2% +0.5% +0.1% -0.1% 0.0% 0.0% 0.0% 0.0% +9.5% +4.5% +1.0% -1.0% -0.13 -0.09 -0.02 +0.03 Fractional Put Spread 95/85 6.7% 6.8% 7.6% 7.9% 5.2% 5.7% 6.5% 6.9% 17.9% 15.9% 16.1% 16.1% 0.29 0.33 0.37 0.40 Divested Equity 5.7% 6.3% 7.3% 7.9% 5.2% 5.7% 6.5% 6.9% 10.7% 12.3% 14.6% 15.9% 0.38 0.39 0.40 0.40 Di +1.0% +0.5% +0.2% 0.0% 0.0% 0.0% 0.0% 0.0% +7.1% +3.6% +1.5% +0.2% -0.10 -0.06 -0.02 -0.00 Table 4.4: Summary of annualized alpha, beta and total cost (as % of initial investment) for the protection strategies across the four portfolios. e total cost is measured as the accumulated cost over the whole investment horizon divided by the initial investment. e size of the initial investment is equal for all strategies. Annualized alpha Beta Total cost as % of initial investment 1m 3m 6m 12m 1m 3m 6m 12m 1m 3m 6m 12m Protective Put -2.3% -0.8% 0.3% 1.0% 0.83 0.59 0.51 0.46 +182.0% 177.1% 162.4% 148.2% Divested Equity -0.1% -0.1% 0.0% 0.0% 0.39 0.43 0.52 0.59 0.0% 0.0% 0.0% 0.0% Di -2.3% -0.8% +0.3% +1.1% +0.44 +0.16 -0.01 -0.13 +182.0% 177.1% 162.4% 148.2% Put Spread 95/80 -2.1% -1.3% -0.7% -0.2% 0.88 0.77 0.78 0.79 162.6% 141.2% 114.6% 85.6% Divested Equity -0.1% 0.0% 0.0% 0.0% 0.48 0.54 0.66 0.76 0.0% 0.0% 0.0% 0.0% Di -0.02 -1.2% -0.6% -0.1% +0.39 +0.23 +0.12 +0.03 +162.6% +141.2% +114.6% +85.6% Put Spread 95/85 -2.1% -1.1% -0.5% -0.2% 0.94 0.82 0.84 0.85 143.9% 116.6% 89.8% 63.3%

56 Divested Equity 0.0% 0.0% 0.0% 0.0% 0.55 0.63 0.75 0.82 0.0% 0.0% 0.0% 0.0% Di -0.02 -0.01 -0.4% -0.2% +0.39 +0.19 +0.09 +0.03 +143.9% +116.6% +89.8% +63.3% Put Spread 100/90 -3.5% -1.9% -0.8% 0.0% 0.82 0.77 0.81 0.82 +442.0% 222.6% 139.2% 83.8% Divested Equity -0.1% -0.1% 0.0% 0.0% 0.24 0.46 0.66 0.83 0.0% 0.0% 0.0% 0.0% Di -3.4% -1.8% -0.8% +0.1% +0.58 +0.31 +0.15 -0.01 +442.0% +222.6% +139.2% +83.8% Collar 95/105 0.3% 1.9% 1.3% -0.7% 0.56 0.36 0.28 0.22 106.8% 70.9% 45.1% 25.8% Divested Equity 0.0% 0.0% -0.1% -0.1% 0.58 0.65 0.45 0.12 0.0% 0.0% 0.0% 0.0% Di +0.4% +2.0% +1.3% -0.7% -0.02 -0.29 -0.18 +0.10 +106.8% +70.9% +45.1% +25.8% Fractional Protective Put -2.1% -0.6% 0.5% 1.2% 0.81 0.59 0.52 0.45 181.6% 175.8% 162.4% 140.1% Divested Equity -0.1% -0.1% 0.0% 0.0% 0.42 0.46 0.57 0.62 0.0% 0.0% 0.0% 0.0% Di -2.0% -0.6% +0.6% +1.2% +0.40 +0.13 -0.06 -0.16 +181.6% +175.8% +162.4% +140.1% Fractional Put Spread 95/85 -1.8% -0.9% -0.4% 0.0% 0.92 0.82 0.85 0.85 143.6% 116.1% 89.3% 60.5% Divested Equity 0.0% 0.0% 0.0% 0.0% 0.57 0.65 0.77 0.85 0.0% 0.0% 0.0% 0.0% Di -1.8% -0.9% -0.4% 0.0% +0.35 +0.17 +0.07 0.00 +143.6% +116.1% +89.3% +60.5% 0.55 0.15 Protective Put Protective Put Put Spread 95/80 Put Spread 95/80 0.5 Put Spread 95/85 Put Spread 95/85 Put Spread 100/90 Put Spread 100/90 Collar 95/105 0.1 Collar 95/105 0.45 Fractional Protective Put Fractional Protective Put Fractional Put Spread 95/85 Fractional Put Spread 95/85 0.4 0.05

0.35

0.3 0

Sharpe ratio 0.25

-0.05 0.2

0.15 -0.1

0.1 Diff. in Sharpe ratio versus Divested Equity

0.05 -0.15 0 2 4 6 8 10 12 0 2 4 6 8 10 12 Maturity (months) Maturity (months)

Figure 4.8: Le: Sharpe ratio of the option-based protection strategies versus maturity. Right: Dierence in Sharpe ratio between the option-based protection strategies and the divested equity strategy versus maturity.

1m Portfolios 3m Portfolios 95% 100%

90% 90%

85% 80%

Protective Put Put Spread 95/80 80% 70% Put Spread 95/85 Put Spread 100/90 Collar 95/105 Fractional Protective Put 75% 60% Fractional Put Spread 95/85

70% 50% Protective Put Put Spread 95/80 Put Spread 95/85 Probability Divesting has better Drawdown Probability Divesting has better Drawdown 65% Put Spread 100/90 40% Collar 95/105 Fractional Protective Put Fractional Put Spread 95/85 60% 30% 0 50 100 150 200 250 0 50 100 150 200 250 Windows (days) Windows (days)

6m Portfolios 12m Portfolios 100% 90%

90% 80%

80% 70% Protective Put 70% Put Spread 95/80 Put Spread 95/85 Put Spread 100/90 60% Collar 95/105 60% Fractional Protective Put Fractional Put Spread 95/85 50% 50% Protective Put Put Spread 95/80 Put Spread 95/85 Probability Divesting has better Drawdown Probability Divesting has better Drawdown 40% 40% Put Spread 100/90 Collar 95/105 Fractional Protective Put Fractional Put Spread 95/85 30% 30% 0 50 100 150 200 250 0 50 100 150 200 250 Windows (days) Windows (days)

Figure 4.9: e probability that the divested equity strategy has beer peak-to-trough drawdowns than the option-based protection strategies.

57 5-Day 10-Day 2% 4%

1% 2%

0% 0%

-1%

-2%

-4% -3% Diff. in 99th percentile drawdown versus Divesting Protective Put Diff. in 99th percentile drawdown versus Divesting Protective Put Put Spread 95/80 -6% Put Spread 95/80 -4% Put Spread 95/85 Put Spread 95/85 Put Spread 100/90 Put Spread 100/90 Collar 95/105 Collar 95/105 Fractional Protective Put Fractional Protective Put Fractional Put Spread 95/85 Fractional Put Spread 95/85 -5% -8% 0 2 4 6 8 10 12 0 2 4 6 8 10 12 Maturity (months) Maturity (months)

20-Day 63-Day 5% 15%

10%

0% 0%

-5%

-10% -5%

-15% Diff. in 99th percentile drawdown versus Divesting Diff. in 99th percentile drawdown versus Divesting Protective Put Protective Put Put Spread 95/80 Put Spread 95/80 Put Spread 95/85 Put Spread 95/85 Put Spread 100/90 -20% Put Spread 100/90 Collar 95/105 Collar 95/105 Fractional Protective Put Fractional Protective Put Fractional Put Spread 95/85 Fractional Put Spread 95/85 -10% -25% 0 2 4 6 8 10 12 0 2 4 6 8 10 12 Maturity (months) Maturity (months)

125-Day 250-Day 20% 30%

20% 10%

10%

0% 0%

-10% -10%

-20% -20%

-30% Diff. in 99th percentile drawdown versus Divesting Protective Put Diff. in 99th percentile drawdown versus Divesting Protective Put -30% Put Spread 95/80 Put Spread 95/80 Put Spread 95/85 Put Spread 95/85 Put Spread 100/90 -40% Put Spread 100/90 Collar 95/105 Collar 95/105 Fractional Protective Put Fractional Protective Put Fractional Put Spread 95/85 Fractional Put Spread 95/85 -40% -50% 0 2 4 6 8 10 12 0 2 4 6 8 10 12 Maturity (months) Maturity (months)

Figure 4.10: Dierence in the 99th percentile peak-to-trough drawdown between the divested equity strategy and the option-based protection strategies versus maturity. e dierence is measured in percentage points.

58 5-Day 10-Day 0.1% 0.1%

1.1102e-16%

-0.1%

-0.1% -0.2%

-0.3%

-0.2%

-0.4%

-0.3% -0.5%

-0.6% Diff. in 50th percentile drawdown versus Divesting Diff. in 50th percentile drawdown versus Divesting Protective Put Protective Put -0.4% Put Spread 95/80 Put Spread 95/80 Put Spread 95/85 Put Spread 95/85 Put Spread 100/90 -0.7% Put Spread 100/90 Collar 95/105 Collar 95/105 Fractional Protective Put Fractional Protective Put Fractional Put Spread 95/85 Fractional Put Spread 95/85 -0.5% -0.8% 0 2 4 6 8 10 12 0 2 4 6 8 10 12 Maturity (months) Maturity (months)

20-Day 63-Day 0.2% 0.5%

0% 0%

-0.2% -0.5%

-0.4%

-1%

-0.6%

-1.5% -0.8% Diff. in 50th percentile drawdown versus Divesting Protective Put Diff. in 50th percentile drawdown versus Divesting Protective Put Put Spread 95/80 -2% Put Spread 95/80 -1% Put Spread 95/85 Put Spread 95/85 Put Spread 100/90 Put Spread 100/90 Collar 95/105 Collar 95/105 Fractional Protective Put Fractional Protective Put Fractional Put Spread 95/85 Fractional Put Spread 95/85 -1.2% -2.5% 0 2 4 6 8 10 12 0 2 4 6 8 10 12 Maturity (months) Maturity (months)

125-Day 250-Day 0.5% 0%

-0.5% 0%

-1% -0.5%

-1.5%

-1%

-2%

-1.5%

-2.5%

-2% -3% Diff. in 50th percentile drawdown versus Divesting Diff. in 50th percentile drawdown versus Divesting Protective Put Protective Put Put Spread 95/80 Put Spread 95/80 -2.5% Put Spread 95/85 Put Spread 95/85 Put Spread 100/90 -3.5% Put Spread 100/90 Collar 95/105 Collar 95/105 Fractional Protective Put Fractional Protective Put Fractional Put Spread 95/85 Fractional Put Spread 95/85 -3% -4% 0 2 4 6 8 10 12 0 2 4 6 8 10 12 Maturity (months) Maturity (months)

Figure 4.11: Dierence in the 50th percentile peak-to-trough drawdown between the divested equity strategy and the option-based protection strategies versus maturity. e dierence is measured in percentage points.

59 5-Day 10-Day -2% -3%

-4%

-3%

-5%

-4% -6%

Protective Put -7% Put Spread 95/80 -5% Put Spread 95/85 Put Spread 100/90 Collar 95/105 -8% Fractional Protective Put Fractional Put Spread 95/85 99th percentile drawdown 99th percentile drawdown -6% -9%

-10% Protective Put -7% Put Spread 95/80 Put Spread 95/85 -11% Put Spread 100/90 Collar 95/105 Fractional Protective Put Fractional Put Spread 95/85 -8% -12% 0 2 4 6 8 10 12 0 2 4 6 8 10 12 Maturity (months) Maturity (months)

20-Day 63-Day -4% -5%

-6% -10%

-8% -15%

-10% -20% 99th percentile drawdown 99th percentile drawdown -12% -25%

Protective Put Protective Put -14% Put Spread 95/80 -30% Put Spread 95/80 Put Spread 95/85 Put Spread 95/85 Put Spread 100/90 Put Spread 100/90 Collar 95/105 Collar 95/105 Fractional Protective Put Fractional Protective Put Fractional Put Spread 95/85 Fractional Put Spread 95/85 -16% -35% 0 2 4 6 8 10 12 0 2 4 6 8 10 12 Maturity (months) Maturity (months)

125-Day 250-Day -10% -10%

-15% -15%

-20%

-20% -25%

-25% -30%

-30% -35%

-40% -35% 99th percentile drawdown 99th percentile drawdown -45%

-40%

Protective Put -50% Protective Put Put Spread 95/80 Put Spread 95/80 Put Spread 95/85 Put Spread 95/85 -45% Put Spread 100/90 Put Spread 100/90 Collar 95/105 -55% Collar 95/105 Fractional Protective Put Fractional Protective Put Fractional Put Spread 95/85 Fractional Put Spread 95/85 -50% -60% 0 2 4 6 8 10 12 0 2 4 6 8 10 12 Maturity (months) Maturity (months)

Figure 4.12: 99th percentile peak-to-trough drawdown for the option-based protection strategies versus maturity.

60 5-Day 10-Day 0% -0.1%

-0.2% -0.1%

-0.3%

-0.2% -0.4%

-0.5% -0.3%

-0.6%

-0.4% -0.7% 50th percentile drawdown 50th percentile drawdown -0.8% -0.5%

Protective Put -0.9% Protective Put Put Spread 95/80 Put Spread 95/80 -0.6% Put Spread 95/85 Put Spread 95/85 Put Spread 100/90 Put Spread 100/90 Collar 95/105 -1% Collar 95/105 Fractional Protective Put Fractional Protective Put Fractional Put Spread 95/85 Fractional Put Spread 95/85 -0.7% -1.1% 0 2 4 6 8 10 12 0 2 4 6 8 10 12 Maturity (months) Maturity (months)

20-Day 63-Day -0.2% 0%

-0.4% -0.5%

-0.6%

-1%

-0.8%

-1.5%

-1% 50th percentile drawdown 50th percentile drawdown -2% -1.2%

Protective Put Protective Put Put Spread 95/80 -2.5% Put Spread 95/80 -1.4% Put Spread 95/85 Put Spread 95/85 Put Spread 100/90 Put Spread 100/90 Collar 95/105 Collar 95/105 Fractional Protective Put Fractional Protective Put Fractional Put Spread 95/85 Fractional Put Spread 95/85 -1.6% -3% 0 2 4 6 8 10 12 0 2 4 6 8 10 12 Maturity (months) Maturity (months)

125-Day 250-Day -0.5% -0.5%

-1% -1%

-1.5%

-1.5% -2%

-2% -2.5%

-3% 50th percentile drawdown -2.5% 50th percentile drawdown

-3.5% Protective Put Protective Put -3% Put Spread 95/80 Put Spread 95/80 Put Spread 95/85 Put Spread 95/85 Put Spread 100/90 -4% Put Spread 100/90 Collar 95/105 Collar 95/105 Fractional Protective Put Fractional Protective Put Fractional Put Spread 95/85 Fractional Put Spread 95/85 -3.5% -4.5% 0 2 4 6 8 10 12 0 2 4 6 8 10 12 Maturity (months) Maturity (months)

Figure 4.13: 50th percentile peak-to-trough drawdown for the option-based protection strategies versus maturity.

61 4.2.2 Financial crisis

In addition to the results presented in this section, Figures 6.9- 6.12 in Appendix visualize the value development of the dierent option-based protection strategies over the whole investment horizon. Figure 6.15 in Appendix visualizes the change in the total cost as a percentage of the initial investment with respect to option maturity.

62 Table 4.5: Summary of annualized returns and performance measures for the dierent strategies across the four portfolios. Each strategy is benchmarked against the divested equity strategy.

Arithmetic return Geometric return Volatility Adj. Sharpe Ratio 20-Day Drawdown (Median) 1m 3m 6m 12m 1m 3m 6m 12m 1m 3m 6m 12m 1m 3m 6m 12m 1m 3m 6m 12m Protective Put -46.6% -28.6% -28.7% -22.7% -38.8% -25.6% -25.4% -20.9% 21.9% 13.0% 11.4% 12.1% -0.10 -0.04 -0.03 -0.03 -4.9% -2.3% -1.9% -1.3% Divested Equity -44.4% -27.8% -27.7% -22.3% -38.8% -25.6% -25.4% -20.9% 30.4% 18.4% 18.3% 14.7% -0.13 -0.05 -0.05 -0.03 -3.7% -2.3% -2.3% -1.8% Di -2.2% -0.8% -1.0% -0.3% 0.0% 0.0% 0.0% 0.0% -8.5% -5.3% -6.9% -2.6% +0.03 +0.01 +0.02 +0.01 -1.2% -0.1% +0.3% +0.5% Put Spread 95/80 -43.5% -30.1% -40.9% -42.6% -37.4% -28.2% -36.1% -37.5% 25.9% 24.4% 28.2% 29.7% -0.11 -0.07 -0.12 -0.13 -4.7% -2.7% -3.2% -3.2% Divested Equity -42.6% -30.9% -41.0% -42.8% -37.4% -28.2% -36.1% -37.5% 29.1% 20.6% 27.8% 29.2% -0.12 -0.06 -0.11 -0.12 -3.5% -2.5% -3.4% -3.5% Di -0.9% +0.9% +0.1% +0.2% 0.0% 0.0% 0.0% 0.0% -3.2% +3.8% +0.4% +0.5% +0.01 -0.01 -0.0 -0.0 -1.1% -0.2% +0.2% +0.3% Put Spread 95/85 -47.8% -34.6% -45.9% -47.8% -40.7% -32.0% -40.0% -41.4% 29.7% 27.8% 31.9% 33.3% -0.14 -0.10 -0.15 -0.16 -5.0% -2.8% -3.5% -3.7% 63 Divested Equity -47.0% -35.7% -46.0% -47.9% -40.7% -32.0% -40.0% -41.4% 32.4% 23.9% 31.6% 33.1% -0.15 -0.09 -0.15 -0.16 -3.9% -2.9% -3.8% -4.0% Di -0.8% +1.0% +0.1% +0.1% 0.0% 0.0% 0.0% 0.0% -2.7% +3.9% +0.3% +0.2% +0.01 -0.01 -0.0 -0.0 -1.1% +0.1% +0.3% +0.3% Put Spread 100/90 -26.0% -31.8% -41.6% -49.0% -25.0% -29.9% -37.1% -42.2% 23.3% 27.3% 30.9% 34.0% -0.06 -0.09 -0.13 -0.17 -2.5% -2.5% -3.0% -3.6% Divested Equity -27.1% -33.1% -42.2% -49.0% -25.0% -29.9% -37.1% -42.2% 17.9% 22.1% 28.7% 34.0% -0.05 -0.07 -0.12 -0.17 -2.2% -2.7% -3.5% -4.1% Di +1.1% +1.3% +0.6% 0.0% 0.0% 0.0% 0.0% 0.0% +5.4% +5.2% +2.1% 0.0% -0.01 -0.01 -0.01 -0.0 -0.2% +0.2% +0.5% +0.5% Collar 95/105 -34.2% -13.3% -10.8% -3.3% -29.7% -12.6% -10.4% -3.3% 13.9% 6.5% 5.3% 3.8% -0.05 -0.01 -0.01 -0.0 -3.9% -1.3% -1.0% -0.6% Divested Equity -32.8% -13.1% -10.7% -3.4% -29.7% -12.6% -10.4% -3.3% 21.9% 8.7% 7.1% 2.6% -0.07 -0.01 -0.01 -0.0 -2.7% -1.1% -0.9% -0.3% Di -1.4% -0.2% -0.1% 0.0% 0.0% 0.0% 0.0% 0.0% -8.0% -2.2% -1.9% +1.3% +0.2 0.0 0.0 0.0 -1.2% -0.2% -0.2% -0.3% Fractional Protective Put -45.7% -31.1% -24.5% -24.6% -38.0% -27.5% -22.5% -23.0% 20.2% 13.9% 13.8% 17.6% -0.09 -0.04 -0.03 -0.04 -4.8% -2.6% -2.2% -2.0% Divested Equity -43.3% -30.1% -24.2% -24.8% -38.0% -27.5% -22.5% -23.0% 29.6% 20.0% 15.9% 16.3% -0.13 -0.06 -0.04 -0.04 -3.6% -2.5% -2.0% -2.0% Di -2.4% -1.0% -0.3% +0.2% 0.0% 0.0% 0.0% 0.0% -9.4% -6.1% -2.1% +1.3% +0.04 +0.02 0.0 0.0 -1.2% -0.1% -0.2% +0.0% Fractional Put Spread 95/85 -43.2% -39.9% -46.0% -50.2% -37.6% -35.8% -40.1% -43.1% 27.9% 29.5% 32.3% 35.2% -0.12 -0.12 -0.15 -0.18 -4.7% -3.4% -3.8% -4.2% Divested Equity -42.8% -40.5% -46.1% -50.3% -37.6% -35.8% -40.1% -43.1% 29.2% 27.4% 31.7% 35.0% -0.13 -0.11 -0.15 -0.18 -3.6% -3.4% -3.8% -4.2% Di -0.4% +0.6% +0.2% +0.1% 0.0% 0.0% 0.0% 0.0% -1.3% +2.0% +0.6% +0.3% 0.0 -0.01 -0.0 -0.0 -1.1% -0.0% +0.1% -0.0% Table 4.6: Summary of annualized alpha, beta, total cost (as % of initial investment) and maximum drawdown for the protection strategies across the four portfolios. e total cost is measured as the accumulated cost over the whole investment horizon divided by the initial investment. e size of the initial investment is equal for all strategies.

Annualized alpha Beta Total cost as % of initial investment Maximum drawdown 1m 3m 6m 12m 1m 3m 6m 12m 1m 3m 6m 12m 1m 3m 6m 12m Protective Put -21.5% -17.0% -20.2% -15.7% 0.43 0.20 0.15 0.12 22.2% 15.2% 14.1% 14.6% -55.6% -33.3% -33.3% -26.5% Divested Equity -1.8% -2.0% -2.0% -1.8% 0.74 0.44 0.44 0.36 0.0% 0.0% 0.0% 0.0% -55.8% -33.6% -33.4% -26.6% Di -19.7% -14.9% -18.1% -13.9% -0.30 -0.24 -0.30 -0.23 +22.2% +15.2% +14.1% +14.6% +0.2% +0.2% +0.1% +0.1% Put Spread 95/80 -9.0% 2.8% -2.2% -1.8% 0.60 0.57 0.67 0.70 17.9% 10.7% 8.4% 6.8% -53.2% -38.55% -51.0% -53.5% Divested Equity -1.9% -2.1% -2.0% -1.9% 0.70 0.50 0.67 0.71 0.0% 0.0% 0.0% 0.0% -53.3% -37.6% -51.0% -53.5% Di -7.1% +4.9% -0.3% +0.1% -0.11 +0.07 -0.01 0.00 +17.9% +10.7% +8.4% +6.8% +0.2% -1.0% 0.0% 0.0% Put Spread 95/85 -7.3% 3.4% -1.7% -1.5% 0.70 0.66 0.76 0.80 14.9% 8.4% 6.3% 4.9% -59.4% -45.6% -58.1% -60.8% 64 Divested Equity -1.6% -2.1% -1.6% -1.5% 0.78 0.58 0.77 0.80 0.0% 0.0% 0.0% 0.0% -59.5% -43.8% -58.1% -60.8% Di -5.8% +5.6% 0.0% 0.0% -0.09 +0.08 0.00 0.00 +14.9% +8.4% +6.3% +4.9% +0.1% -1.8% 0.0% 0.0% Put Spread 100/90 4.9% 5.8% 1.1% -1.7% 0.53 0.65 0.74 0.82 28.4% 12.4% 8.2% 5.8% -32.5% -43.01% -52.8% -62.4% Divested Equity -2.0% -2.1% -1.9% -1.4% 0.43 0.53 0.7 0.82 0.0% 0.0% 0.0% 0.0% -32.7% -40.43% -52.8% -62.3% Di +6.9% +8.0% +3.0% -0.4% +0.10 +0.12 +0.04 -0.01 +28.4% +12.4% +8.2% +5.8% +0.1% -2.6% 0.0% -0.1% Collar 95/105 -19.4% -6.9% -6.2% -0.6% 0.26 0.11 0.08 0.05 6.1% 1.9% 0.3% -0.8% -39.9% -15.2% -12.4% -4.7% Divested Equity -2.1% -1% -0.7% 0.2% 0.53 0.21 0.17 0.06 0.0% 0.0% 0.0% 0.0% -40.0% -15.3% -12.5% -4.1% Di -17.2% -5.9% -5.4% -0.8% -0.27 -0.10 -0.09 -0.01 +6.1% +1.9% +0.3% -0.8% +0.1% +0.1% +0.1% -0.6% Fractional Protective Put -21.7% -17.5% -12.4% -6.7% 0.41 0.24 0.21 0.31 21.2% 15.2% 11.4% 8.3% -54.1% -36.2% -28.4% -29.2% Divested Equity -1.8% -2.1% -1.9% -1.9% 0.72 0.48 0.38 0.39 0.0% 0.0% 0.0% 0.0% -54.2% -36.5% -28.9% -29.7% Di -19.9% -15.4% -10.6% -4.8% -0.30 -0.25 -0.18 -0.09 +21.2% +15.2% +11.4% +8.3% +0.2% +0.4% +0.5% +0.5% Fractional Put Spread 95/85 -5.0% 0.9% -1.1% -1.1% 0.66 0.70 0.77 0.85 14.2% +8.0% 5.1% 3.1% -53.4% -50.1% -58.1% -64.1% Divested Equity -1.9% -2.0% -1.6% -1.2% 0.71 0.66 0.77 0.85 0.0% 0.0% 0.0% 0.0% -53.6% -50.3% -58.2% -64.2% Di -3.1% +2.9% +0.6% +0.1% -0.05 +0.04 +0.01 0.00 +14.2% +8.0% +5.1% +3.1% +0.2% +0.2% +0.2% +0.1% 1m Portfolios 3m Portfolios 80% 90% Protective Put Put Spread 95/80 Put Spread 95/85 Put Spread 100/90 70% 80% Collar 95/105 Fractional Protective Put Fractional Put Spread 95/85 70% 60%

60% 50%

50%

40%

30% 30% Probability Divesting has better Drawdown 20% Probability Divesting has better Drawdown 20% Protective Put Put Spread 95/80 Put Spread 95/85 10% 10% Put Spread 100/90 Collar 95/105 Fractional Protective Put Fractional Put Spread 95/85 0% 0% 0 50 100 150 200 250 0 50 100 150 200 250 Windows (days) Windows (days)

6m Portfolios 12m Portfolios 60% 80% Protective Put Put Spread 95/80 Put Spread 95/85 Put Spread 100/90 70% Collar 95/105 50% Fractional Protective Put Fractional Put Spread 95/85 Protective Put Put Spread 95/80 60% Put Spread 95/85 Put Spread 100/90 40% Collar 95/105 Fractional Protective Put 50% Fractional Put Spread 95/85

30% 40%

30% 20% Probability Divesting has better Drawdown Probability Divesting has better Drawdown 20%

10% 10%

0% 0% 0 50 100 150 200 250 0 50 100 150 200 250 Windows (days) Windows (days)

Figure 4.14: e probability that the divested equity strategy has beer peak-to-trough drawdowns than the option-based protection strategies.

65 Discussion

5.1 Monte Carlo Simulation

5.1.1 Risk-adjusted performance

e mean of the annualized arithmetic and geometric returns, in Table 4.1, are all negative except for the collar 95/105 strategy. is is because the chosen jump process makes the distribution of stock paths more weighted towards negative paths. In total, 58.7% of the simulated stock paths yielded a negative annualized geometric return. us, the strategies were run more oen on bear markets. We begin by analyzing the overall performance of the protection strategies.

As presented in Table 4.2, all protection strategies have beta values lower than 1, indicating that they collect less equity risk premium. Beta values below 1 mean that the strategies earn less during rising markets and lose less during falling markets. Buying put options and selling call options yield beta values lower than 1 as they reduce the portfolio’s equity exposure. Table 4.2 also presents negative alphas. When portfolios are net long volatility, i.e. net long positions in options, the additional cost of volatility risk premium adds negative alpha to the portfolio return, meaning that the return is reduced by the same amount. e put spread and collar strategies are constructed to reduce the negative alpha by adding positive alpha through a short position in a put and a call, respectively. Obviously, the divested equity strategy has zero alpha, since it only has equity exposure.

Table 4.1 displays that, on average, the divested equity strategy outperformed the option-based protection strategies in terms of risk-adjusted return. e mean Sharpe ratios for the divested equity strategy are superior to the other strategies in almost all portfolios, except in the 12m portfolio where the collar 95/105 is slightly beer. is is also seen in Figure 4.1, which shows that all strategies are more likely to underperform against divesting. Although the collar 95/105 has a positive dierence of 0.02 to divesting, it exhibits a larger variation of Sharpe ratios and

66 is very likely to underperform. Oseing the negative alphas in the put spreads and the collar 95/105 by selling puts and calls was, on average, not sucient to outperform a zero-alpha strategy on a risk-adjusted basis.

Table 4.1 further shows that, as maturity was increased, the Sharpe ratio of the collar 95/105 and put spread 100/90 improved. is is a result of a large decline in volatility for the collar 95/105, due to a sharp decrease in its beta values. and a slight improvement in the return for the put spread 100/90, despite an increase in volatility. As evidenced in Table 4.2, the put spread 100/90 is a very expensive strategy for short-dated options. Its cost in the 1m portfolio is almost 2.5 times more higher than the second most expensive strategy. As such, it also realized the worst return on the 1m portfolio. Yet, its risk-adjusted return is comparable to the protective put and the fractional protective put. e payos obtained through protecting from the ATM level during bear markets, evidently, neutralized the impact from the expensive premiums. e total cost drops signicantly for longer maturities and thus has a less drag on returns, which explains its stronger risk-adjusted performance in the portfolios with longer-dated options. In fact, the total cost of all strategies decreased with increasing maturity, as visualized in Figure 6.13 in Appendix.

Longer maturities led to a lower equity exposure in the protection strategies that provide full downside protection, namely the collar 95/105, the protective put and the fractional protective put. is behaviour was less evident in the put spreads given their partial protection, which is sensible as longer maturities mean that there is more time for the market to fall beyond the spreads. However, despite a lower equity exposure, the mean risk-adjusted return on the protective put worsened as maturity was increased. Table 4.2 reports that its alpha values became more negative, eectively implying that there is no escape from the expensive volatility risk premium. Table 4.1, or for beer visualization Figure 6.13 in Appendix, shows that it is the second most expensive strategy. e protective put underperformed against all strategies, in accordance with the criticism earned in previous studies.

Interestingly, the fractional protective put saw an improvement in absolute returns and alpha values over the regular protective put. Its beta values did, however, not reduce as much with increasing maturity, because of its partial protection in the rst months. For example, it takes twelve months for the fractional protective put to be fully protected in the 12m portfolio. Un- til then it is vulnerable to potential declines in the market. Consequently, it realized a higher

67 volatility for the longer maturities and therefore saw only a slight improvement in risk-adjusted return. e same behaviour was documented for the fractional put spread 95/85. ese ndings suggest that buying overlapping fractional amounts of put options, on average, has a marginal impact on the risk-adjusted return when implied volatility is assumed to be constant.

In order to understand the dierence between full downside protection and partial protection beer, we split the risk-adjusted performance into two cases. Figures 4.2 and 4.3 depict the distribution of the Sharpe ratio when the long-term excess return is positive and negative, respectively.

Positive long-term excess return

Figure 4.2 shows that when the long-term excess return is positive, the protective put was the poorest performer among the dierent protection strategies. e high cost of put options have a noticeable impact on the returns. e mean Sharpe ratio, however, improved modestly from 0.31 in the 1m portfolio to 0.33 in the 12m portfolio. Yet, not enough to make the protective put a more aractive protection strategy than just simply divesting equity or pursuing another option-based strategy.

e fractional protective put shows a marginal improvement by 0.01 in mean Sharpe ratio over the regular protective put for the longer-dated options (3m, 6m and 12m portfolios). e put spread strategies, on average, exhibit beer risk-adjusted return than both the protective put and the fractional protective put when excess return is positive. e put spread 95/85 is the cheapest alternative and the slightly beer performer in the 1m and 3m portfolios. Its comparably tighter spread is apparently beer suited for short-dated options, as longer maturities would give the market more time to fall below its protection level. On the other hand, it is the strategy with the highest equity exposure with a beta value ranging between 0.76 - 0.82, only second to the fractional put spread 95/85, and therefore earns relatively more equity risk premium when the stock price rises. Nonetheless, it is the strategy with the highest return volatility on average. In the 6m and 12m portfolios, the put spread 100/90 demonstrated comparable or marginally beer mean Sharpe ratio, because of its comparable beta values and relatively lower return volatility.

68 e put spread 100/90 is the most expensive strategy, but generated alphas that are more or less on par with the put spread 95/85. Buying ATM puts reduced equity exposure more than buying 5% OTM puts in the 1m and 3m portfolios. is is sensible because in the shorter term, i.e. one to three months, the market has less time to fall more than 10%, which is fully within the protection capacity of the put spread 100/90, while the other put spreads would only cover 5% because of the 5% OTM puts. In the longer term, from six to twelve months, the market is more likely to dip beyond the 10% spread, which is also why we see an increase in the beta value for the put spread 100/90 from 0.60 in the 1m portfolio to 0.79 in the 12m portfolio. is increase allows the put spread 100/90 to collect more equity risk premium during periods of positive excess return. Consequently, its mean Sharpe ratio in Figure 4.2 marginally improved with increasing maturity, despite an increase in mean volatility.

e put spread 95/80, on the other hand, sells a 20% OTM put, which adds less positive alpha and makes it more burdened by the volatility risk premium. erefore, the alphas for the put spread 95/80 is, on average, slightly worse. Its mean beta values, however, are between the other two put spreads and its mean total cost is comparable to the put spread 95/85. As such its performance was not very dierent from the others, perhaps slightly less aractive. Based on the aggregated results, the investor would be indierent between the three put spreads on a risk-adjusted basis in this idealized environment. However, if the investor is bullish on the underlying stock and if achieving high absolute returns is the objective, then she should refrain from employing a put spread 100/90 strategy based on 1m puts.

Similar to the fractional protective put, the risk-adjusted return on the fractional put spread 95/85 displayed a modest improvement of 0.01 over the regular put spread 95/85 as maturity was increased. As seen in Table 4.1, it yielded slightly beer returns, but higher return volatilites on average.

It is not odd that the collar 95/105 strategy generated relatively beer alpha and lower beta values compared to the other strategies. e strategy is designed to cost lile to nothing and to limit its returns on both the downside and the upside. As seen in Table 4.2 and Figure 6.13, both its mean beta values and mean total cost are signicantly lower than those for the other strategies. Its outperformance on a risk-adjusted basis is visualized in Figure 4.2. e collar 95/105 displays superior mean Sharpe ratio across all four portfolios, with signicant improvement as

69 the maturity increases, despite its comparably lower beta. It is not because the collar yield higher returns, but because its return volatility, on average, is much lower than the other strategies. Put dierently, the collar earns its returns with lower risk compared to the other strategies.

Overall, the dierence in risk-adjusted performance between the dierent protection strategies is not huge.

Negative long-term excess return

When the long-term excess return is negative, Figure 4.3, the collar 95/105 is by far the best performing strategy due to its more conservative prole and signicantly lower beta. e protective put does a relatively good job and improves signicantly as maturity increases. Evidently, the protective put demonstrates a more aractive risk-adjusted performance when the long-term excess return is negative. Likewise, the fractional protective put exhibits a strong risk-adjusted prole when the excess return is negative - not very dierent from the regular protective put. All three benet from their full downside protection.

As would be expected, the put spread strategies fare worse due to their limited protection capacity and higher equity exposure. e put spread 95/80 stands out with its beer distribution, as displayed in Figure 4.3, given its wider spread and thus more extensive protection. e put spread 100/90 exhibits a beer mean adjusted Sharpe ratio than the put spread 95/85 in the 1m portfolio, despite its signicantly higher total cost. Hence, we see a marginal gain in protecting from the ATM level for short-dated options. e risk-adjusted performance of the put spread 95/85, put spread 100/90 and fractional put spread 95/85 converges with increasing maturity and is more or less the same in the 12m portfolio. e small dierences in the size of their spreads do not maer over longer horizons, as the market is more likely to fall far beyond them.

Not surprisingly, when the long-term excess return is negative, the most important factor appears to be the extent of the protection and not the cost of the options. is is, in particular, conrmed by the superior risk-adjusted performance of the protective put and the fractional protective put. However, combining full downside protection with lower cost, as in a collar, is even beer.

Based on the aggregated results in Table 4.1 and Figures 4.2- 4.3, the collar 95/105 strategy

70 displayed the most aractive risk-reward prole. As it caps the upside performance, it is mainly suitable when a rally is not expected or when a downturn is more likely. e overweight on negative returns in the simulated stocks paths therefore favored the collar strategy.

5.1.2 Peak-to-trough drawdown characteristics

Tables 6.1- 6.4 in Appendix show that the mean peak-to-trough drawdowns for the dierent protection strategies at the 99th, 95th and 50th percentiles are all lower than those for the simulated stock paths. However, the prevailing question is how eective the provided protection is and how it compares to the divested equity strategy.

Tail protection

A rolling protection strategy that buys 1m options is expected to protect the portfolio, at least, against peak-to-trough drawdowns that persist over a 20-day window. Similarly, buying 3m, 6m and 12m options are expected to provide eective protection, at least, over the 63-day, 125-day and 250-day windows, respectively. Figures 4.4 and 4.6 show that this, indeed, is the case for the protective put, the fractional protective put and the collar 95/105 strategy. Both the protective put and the collar 95/105 outperforms the divested equity strategy at the mean 99th percentile over windows that align with the maturity of the purchased options. For example, over the 20- day window, the protective put’s and collar’s mean 99th percentile peak-to-trough drawdowns in the 1m portfolio are -10.4% and -8.1%, respectively. ese are 5.0 and 2.0 percentage points beer than divesting, respectively. e fractional protective put exhibits similar strong tail protection characteristics in the 1m and 3m portfolios, but becomes less eective against tail risk in the 6m and 12m portfolios, both in terms of absolute value and versus divesting. As discussed earlier, the fractional protective put is more vulnerable to declines in the market for longer maturities.

It must be emphasized, though, that the fractional protective put has slightly beer tail protection than the regular protective put in the 1m and 3m portfolios (see Figures 4.4 and 4.6). It gains full protection faster in these portfolios and therefore sees the benet of reduced path dependency. A fair comparison between them would require that we cut o the period where it is partially

71 protected. However, the results indicate that there indeed is an improvement over the regular protective put, given that the fractional protective put has gained full protection shortly aer the initial investment.

Figure 4.6 shows that the mean 99th peak-to-trough drawdowns for the collar 95/105 consistently improves with increasing maturity, while the protective put peaks over the windows that align with the maturity length. e quality of the protective put’s protection improves when the option maturity is most closely aligned with the length of the peak-to-trough drawdown cycle. Overall, as displayed in Figure 4.4, the outperformance in terms of percentage points of the collar 95/105 and the protective put against the divested equity strategy grows with increasing maturity, suggesting that these strategies are more suitable for tail protection than divesting when long-dated options are used.

e put spreads, overall, provide very poor tail protection. As seen in Figure 4.6, their mean 99th peak-to-trough drawdowns are generally the worst across all windows. Due to their partial protection the drawdowns become worse with increasing maturity. Figure 4.4 further shows that they consistently underperformed against the divested equity strategy. According to these results, the put spreads are unarguably not suitable as tail protection strategies. It should, however, be noted that the put spread 100/90 had beer mean 99th percentile drawdowns in the 1m and 3m portfolios compared to the other put spreads. e put spread 95/80 displays a superior performance for longer maturities. Evidently, protecting from the ATM level with a 10% spread appeared to be beer for short-dated options, while wider spreads are more suitable for longer maturities. e fractional put spread 95/85 did not see an improvement over the other put spread strategies. In fact, it performed worse than those on average.

An observation that is true for all strategies is that divesting provides a beer protection against 5- to 10-day mean 99th percentile drawdowns.

e discussed ndings are also visualized in Figures 6.1- 6.4 in Appendix. As evidenced by the gures, the collar 95/105, the protective put and the fractional protective put are signicantly more likely to yield beer tail protection than the put spread strategies. e put spreads consistently show a large variation in its 99th percentile drawdown, fortifying the great uncertainty in their protection capacity.

72 Modest market declines

Figure 4.5 reports that the protective put, overall, falls short at the mean 50th percentile versus the divested equity strategy, indicating that its protection is most oen eective against tail risk drawdowns and not during modest market dips, in agreement with the conclusions drawn by Israelov (2017). As evidenced in Figure 4.7, the collar’s mean 50th percentile is the lowest across all windows and portfolios. In particular, in the 12m portfolio, it consistently outperformed the divested equity strategy across all three percentiles. is is explained by its far lower mean beta value of 0.14 (compare to the 0.25 for the divested equity strategy). Likewise, it exhibits a lower mean beta value in the 3m and 6m portfolios, which makes it very conservative and less exposed to market downturns.

e put spread strategies’ mean 50th percentile drawdowns fared beer against the divested equity strategy, compared to the protective put and the fractional protective put. e partial protection is a good idea when the market does not fall beyond the extent of the spread, which would correspond to a modest volatility environment in the real-world equity markets. Still, the mean 50th peak-to-trough drawdowns for the put spreads, as displayed in Figure 4.7, are fairly unaractive.

Unlike the other strategies, the put spreads show no dependency on the choice of maturity. e mean 50th percentile for the protective put, the fractional protective put and the collar improves with increasing maturity, both in terms of absolute value and versus divesting, i.e. their protection against modest dips in the market improves.

5.2 Backtesting

5.2.1 Risk-adjusted performance

Table 4.3 and Figure 4.8 display that all protection strategies underperformed against the divested equity strategy in the 1m portfolio in terms of mean Sharpe ratio. All but the collar 95/105 consistently improved for longer maturities, but not suciently to outperform divesting. e

73 poor performance across all strategies can be mainly aributed to their much higher volatility - because of their signicantly higher beta values - compared to the divested equity strategy. us, an investor that seeks monthly protection would have achieved a beer risk-adjusted return by statically reducing the portfolio’s exposure to the S&P 500 index.

Among the 1m portfolios, the collar 95/105 had the best Sharpe ratio at 0.35 and just 0.03 below the divested equity strategy. Its superior risk-adjusted return to the other protection strategies can also be seen through its less volatile value development over time in Figure 6.5 in Appendix. As evidenced by Table 4.4, its beta value is lower than the divested equity strategy and all the other protection strategies. Moreover, it has the lowest total cost, resulting from the short 5% OTM call which by theory should be closely priced with the long 5% OTM put due to the parabola- like volatility skew near the ATM level. We see, though, that, in practice, the puts are more expensively priced, leading to a total net cost of 106.8% of the initial investment. e collar 95/105 was 41% cheaper than the protective put, but as much as 95% cheaper, on average, in the Monte Carlo simulation in which both the call and the put were priced based on the same volatility. is underscores the signicance of variations in volatility for option-based protection strategies. Furthermore, despite its lower equity exposure, the collar 95/105 realized a higher volatility than divesting. Consequently, its risk-adjusted return was only inferior to the divested equity strategy.

e collar 95/105 was most closely followed by the fractional and the regular put spread 95/85, which realized a Sharpe ratio of 0.29 and 0.28, respectively. eir underperformance against the divested equity strategy was more severe, though, given their negative alphas and signicantly higher equity exposure - their beta values are ca 60-70% higher than those for the divested portfolio - which eectively made them more risky. e put spread 95/80 was not far behind with a Sharpe ratio of 0.26, but with comparably lower volatility than both the fractional and regular put spread 95/85. It was, however, ca 13.0% more expensive than the 95/85 spreads, due to its wider spread. Evidently, the drag on returns because of higher cost had a bigger impact on the risk-adjusted return than its comparably lower volatility.

It is striking how much of an impact the total cost of the strategies have on the risk-adjusted return. It is particularly evident in the 1m portfolios. e put spread 100/90, the protective put and the fractional protective put were the most expensive strategies among the 1m portfolios,

74 where, in particular, buying ATM puts led to a total cost of 442.0% over the initial investment. is is ca 143.0% more than the cost for the protective put, which is the second most expensive strategy among the 1m portfolios. Obviously, selling 10% OTM puts as a measure to reduce the cost did not make much of a dierence. e most expensive strategies were also those who yielded the worst Sharpe ratios and with most severe underperformance against divesting. In Figure 6.5 in Appendix, the drop in the portfolio value due to higher cost is clearly visible. We found in the Monte Carlo simulation that the total cost is less important than the extent of protection when the long-term return is negative. We nd now that this is not the case when the long-term return is positive - the S&P 500 have a positive annualized return in the examined period.

Figure 4.8 shows that all strategies, except for the collar 95/105, consistently improved their risk- adjusted return with increasing maturity. e highest Sharpe ratios were achieved by the collar 95/105, which outperformed the divested equity strategy in both the 3m and 6m portfolios at Sharpe ratios of 0.52 and 0.45, respectively. Its risk-adjusted return dramatically worsened in the 12m portfolio, sinking down to only 0.10 and making it the worst performing strategy for 12m options. e results are sensible given that the S&P 500 index enjoyed a huge upswing over the long-term, which meant that the sold long-dated calls in the collar 95/105 were more likely to expire ITM, eectively reducing the return on the portfolio. is cap on the upside reduced beta to 0.22 in 12m portfolio from the 0.56 in the 1m portfolio. Conversely, the higher equity exposure in the 1m, 3m and 6m portfolios allowed the collar 95/105 to participate more in the upturn. See Figures 6.6- 6.8 in Appendix.

e protective put and the fractional protective put were the only ones that outperformed the divested equity strategy in the 12m portfolio. eir beta values consistently dropped with increasing maturity, making them less exposed to the volatility in the S&P 500 index, while their alpha values improved and turned positive. Overall, the fractional protective put showed a slight improvement in risk-adjusted performance over the regular protective put. Hence, while there are some merits of diversifying the puts over overlapping cycles, they are not very convincing on a risk-adjusted basis. Figures 6.6- 6.8 in Appendix show that the value of the fractional protective put portfolio is consistently, slightly above the protective put.

e Sharpe ratios for the put spreads also improved with increasing maturity, but none of the strategies outperformed the divested equity strategy in any of the portfolios. e investor would

75 have achieved a beer risk-adjusted return by divesting rather than sacricing some protection for lower premiums. On the other hand, the risk-adjusted returns for the put spread 95/80 and put spread 95/85 are higher than those for the protective put, which can be aributed to their higher beta values and the long-lasting upswing in the market aer the nancial crisis. Figures 6.5- 6.8 in Appendix show that the higher equity exposure allows them to grow faster with a rallying market despite their greater loss during the nancial crisis. In particular, based on the Sharpe ratios, the investor would choose the put spread 95/85 over the protective put in the 1m, 3m and 6m portfolios. e put spread 100/90 only became competitive in the 6m and 12m portfolios, and was particularly the best alternative among the put spreads in the 12m portfolio. Less frequent purchases of ATM puts reduced the total cost from 442.0% in the 1m portfolio to 83.8% in the 12m portfolio. Similarly, the negative alpha improved from -3.5% to 0.0%, implying that the expensive volatility paid for the shorter-dated options diminished with increasing maturity. is led to a Sharpe ratio of 0.40, which is on par with the fractional put spread 95/85 and only exceeded by the fractional protective put with a Sharpe ratio of 0.41.

e fractional put spread 95/85, like the fractional protective put, only led to a marginal improvement over the regular put spread 95/85. e Sharpe ratio improved at most by 0.01 across all portfolios. e eect on risk-adjusted return, as stated earlier, is modest.

By taking a look at the Black-Scholes-Merton formulas in 2.2.3, we conclude that, all else being equal, the price of an option rises with increasing time to maturity. e option writer demands a higher cost for the additional risk that comes with longer time to maturity. Yet Figure 6.14 shows that the total cost for all strategies decreased with increasing maturity. is is because rolling purchases of long-dated options reduce the time value cost, which is higher for shorter-dated options since the time value decays faster near expiry. us, we see that the choice of maturity has a material impact on the returns through the total cost incurred.

5.2.2 Peak-to-trough drawdown characteristics

As seen in Figure 4.9, the probability that divesting has a beer drawdown than the option-based protection strategies is generally high across all portfolios, with some exceptions. e collar 95/105 stands out in the 3m and 6m portfolios, where it is more likely to yield beer drawdowns

76 than divesting. In the 12m portfolio, the put spread 100/90 is more likely to yield beer drawdown than divesting over the shorter windows (5, 10 and 20 days). Indeed, protecting from the ATM level over shorter windows with reduced total cost should cover more of the losses.

Tail protection

Figure 4.10 shows that the 99th percentile peak-to-trough drawdown provided by the option- based protection strategies versus divesting was overall poor, except for the collar 95/105, which outperformed the divested equity strategy in the 1m, 3m and 6m portfolios, beneting from the full downside protection and its comparably lower beta value. Table 6.5 in Appendix, in particular, reports that the collar 95/105 is beer than divesting at the 99th percentile peak-to-trough drawdown over the 20-day, 63-day and 125-day windows in the 1m portfolio and over all windows in the 6m portfolio. Despite its underperformance against divesting in the 12m portfolio, we see that the peak-to-trough drawdowns in Figure 4.12 look aractive in comparison to the other strategies. Looking at these numbers blindly gives an uncomplete picture of its performance. We recall from the risk-adjusted performance analysis that the collar 95/105 strategy’s Sharpe ratio in the 12m portfolio was 0.10. Figure 6.8 in Appendix, indeed, shows that its drawdowns are signicantly smaller than the other protection strategies, but so is its total return over the whole investment horizon.

e two expensive strategies, namely the protective put and the put spread 100/90, performed signicantly worse versus divesting in the 1m portfolio and had more or less the worst peak- to-trough drawdowns across all windows. For example, Table 6.5 reports that 99th percentile peak-to-trough drawdown, in the 1m portfolio, for the protective put over the 20-day window was -15.4% versus -6.4% achieved by divesting. is result is in line with the conclusions drawn by Israelov (2017), but dierent from our Monte Carlo results. In the idealized environment, the protective put was, on average, beer than the divested equity strategy at the 99th percentile peak-to-trough drawdowns over the 20-day and 63-day windows. Also, the fact that the put spread 100/90 fared the worst against divesting in Figure 4.10 contradicts the Monte Carlo results. e backtesting was performed on a period in which the long-term return was positive, while the Monte Carlo simulation was overweight on periods with negative long-term return. is is an important dierence that has a material impact on the performance of both the protective put

77 and the put spread 100/90.

However, we note, in Figure 4.12, that the size of the 99th percentile drawdowns for the put spread 100/90 in the 1m portfolio was not the worst. It outperformed the protective put over all windows and was the second best performer over the shorter windows (5, 10 and 20 days). is contradicts the Monte Carlo results, where all put spreads consistently underperformed at the mean 99th percentile. is is evidently not the case when the implied volatility is non-constant and the long-term return is positive.

For the longer maturities, the performance of both the protective put and the put spread 100/90 versus divesting improved signicantly. e protective put, in particular, outperforms the divested equity strategy over all windows in the 12m portfolio and is fairly competitive in the 6m portfolio. For example, the 99th percentile peak-to-trough drawdown over the 20-Day window decreased by 42% (or 6.5 percentage points) from -15.4% to -8.9% by switching from 1m to 3m puts. Conversely, in absolute terms, the peak-to-trough drawdowns for the put spread 100/90 did not improve much and remained one of the most unaractive alternatives. is is in agreement with our earlier conclusion that the put spreads are not ideal for tail protection when longer-dated options are used.

e fractional protective put fared marginally beer against divesting compared to the regular protective put. Figures 6.5- 6.8 shows that the return on the S&P 500 in the rst two years is positive. Since the fractional protective put is partially hedged in the rst months, this would have a positive eect on the performance of the strategy. However, Figure 4.10 shows that the dierence between the fractional protective put and the regular protective put in absolute terms is small and is only noticeable in the 12m portfolio, where the former is beer over the longer windows.

e other put spreads consistently underperformed against the divested equity strategy across all windows and maturities. ey did, however, outperform the protective put in absolute terms over the longer windows in the 1m portfolio. Since protective put is a relatively more expensive strategy for short-dated options, this demonstrates the detrimental eect of higher cost not just on risk-adjusted performance, but also on the tail protection. As seen in Figure 4.12, the improvement of the peak-to-trough drawdowns of the put spreads with increasing maturity in

78 absolute terms is modest. For example, between the 1m and 12m portfolios the put spread 95/85 saw an improvement of 10.2 percentage points over the 250-day window in the 99th percentile peak-to-trough drawdown, while the protective put improved as much as 33.5 percentage points.

e fractional put spread 95/85 only showed a marginal improvement in the peak-to-trough drawdowns over the regular put spread 95/85 for the longer-dated options. Like the put spreads, its tail protection is uninspiring.

Based on these results, having full downside protection when a tail risk event occurs is generally beer than being partially protected. Shorter-dated options, in particular 1m options, should be avoided.

Modest market declines

Figure 4.11 displays that no option-based strategy, besides from the collar 95/105, outperforms the divested equity strategy at 50th percentile drawdowns. e collar 95/105 is a slightly beer alternative over the 5, 10, 20 and 63 days in the 3m and 6m portfolios. Overall, the investor should divest its equity exposure instead of employing an option-based strategy for protection against modest market declines.

In Figure 4.13, the absolute 50th percentile peak-to-trough drawdowns are universally best in the collar 95/105 strategy. e fractional protective put and the regular protective put are the worst, in agreement with our ndings in the Monte Carlo simulation. e high cost of full downside protection makes itself visible in the peak-to-trough drawdowns during modest declines. e put spreads do generally a beer job in protecting against modest market declines than tail risk events. ey benet from their reduced cost prole, due to the partial protection. We see, in particular, that, among the put spread strategies, the put spread 100/90 is the best option for the longer maturities. Evidently, when the cost burden is not too high, it is beer to protect against modest declines from the ATM level. We recall that the total cost of the put spread 100/90 decreases signicantly with increasing maturity and becomes comparable to the other strategies (see Figure 6.14).

79 5.2.3 Financial crisis

Figures 6.9- 6.12 in Appendix shows that, over the examined period of the nancial crisis, the protection strategies outperformed the underlying S&P 500 index, but their risk-adjusted performance versus the divested equity strategy, as documented in Table 4.5, shows mixed results. e protective put, the fractional protective put and the collar 95/105 improved on a risk-adjusted basis with increasing maturity, where the laer overall exhibits the most aractive Sharpe ratios. is is expected in a prolonged bear market, because the longer maturities puts the investor outside the falling market for a longer period of time. e Sharpe ratios of the put spreads, conversely, worsen with increasing maturity. e market has more time to fall beyond the extent of their spreads, leading to worse returns and increased volatility.

e risk-adjusted performance of the protective put was superior to the divested equity strategy across all four portfolios. e fractional protective put was, likewise, superior in the 1m and 3m portfolios, but comparable to divesting in the 6m and 12m portfolios. e collar 95/105 was only superior to divesting in the 1m portfolio, but on par with divesting for the longer-dated options. e put spreads were only competitive in the 1m portfolio. We see that, in general, option-based strategies that buy 1m options made a beer alternative to divesting during the nancial crisis. Furthermore, the put spread 100/90, despite being inferior to divesting, yielded a more aractive risk-adjusted return in the 1m portfolio than the protective put and the fractional protective put. In fact, it would be the best choice for 1m options. We will discuss why.

It should be noted that all but the put spread 100/90 among the 1m portfolios underperformed against the S&P 500 until October 2008 (see Figure 6.9 in Appendix). Evidently, all protection strategies paid o when the bigger crash occurred. Before the big crash, protecting from the 5% OTM level had signicantly greater drag on returns than the rolling cost of ATM puts. e outperformance of the put spread 100/90 is further validated in Table 4.6 by its annualized alpha of 4.9%, which is the only positive alpha among all 1m portfolios. e collar 95/105 did not benet from its comparably lower equity exposure - compare its beta value of 0.26 versus 0.53 for the put spread 100/90. e collar 95/105 realized an annualized alpha of -19.4%. Likewise, the alphas for the protective put and the fractional protective put amounted to -21.5% and -21.7%. Although, they might have been a beer choice than divesting, their weak performance in absolute terms

80 in the 1m portfolio is striking.

Table 4.2 reports the maximum drawdowns for the dierent strategies. Since the divested equity strategy was sized to achieve the same geometric return as the other protection strategies, the dierence between their maximum drawdowns is near zero. is happens because the largest drawdown aligns with the whole examined period. e maximum drawdown, however, sheds light on the relative performance of the dierent option-based protection strategies. In agreement with the previous paragraph, we see that the put spread 100/90 yielded the lowest maximum drawdown among the 1m portfolios. e dierence is striking, compare its maximum drawdown of -32.5% to -39.9% for the collar 95/105 and -55.6% for the protective put. e other put spreads showed maximum drawdowns between -53.0 % and -60.0%. As documented in Table 4.5, the median 20-day peak-to-trough drawdowns for the put spread 100/90 were also beer than those for the other strategies. ese results underscores that, if the investor considers 1m options for protection, what is most important during a prolonged bear market is not a lower total net cost nor a full downside protection, but the chosen strike level. Indeed, the put spread 100/90 only protects up to a 10% dip, but prices seldom fall more than 10% over the short-term. A 10% protection is therefore sucient as long as a sudden crash does not occur, in which case a full protection is certainly the more important factor. We see in Figure 6.9 that the protective put, the fractional protective put and the collar 95/105 managed the crash in October 2008 with less volatility, as conrmed in Table 4.5.

It should be noted that almost all option-based protection strategies in the 1m portfolio, except for the collar 95/105, exhibit worse median 20-day peak-to-trough drawdown compared to divesting. Figure 4.14 show that, in the 1m portfolio, divesting is more likely to yield a beer drawdown over the shorter windows. However, over longer windows, all strategies but the put spread 100/90 have a signicantly higher probability to provide beer protection. In particular, over the 250-day window, the protective put, the fractional protective put and the other put spreads are expected to yield a beer protection 100% of the time.

With increasing maturity, the protective put, the fractional protective put and the collar 95/105 are by far the best performers, see Figures 6.10- 6.12. eir equity exposure decreased signicantly as the maturity was increased and led to improved maximum drawdowns. e fractional protective put, however, saw its beta value increase between the 6m and 12m portfolios and pro-

81 vided comparably less eective protection in terms of its drawdown characteristics, due to its partial protection in the rst months. It should be noted that it slightly outperformed the regular protective put in the 1m portfolio in terms of both risk-adjusted return and peak-to-trough drawdowns. Hence, to fully benet from pursuing the fractional protective put instead of the regular protective put, the portfolio must gain full protection before the bear market commences.

Conversely, the maximum and median 20-day peak-to-trough drawdowns for the put spreads worsened with increasing maturity. As it is more likely that prices will fall more than 10-20% in a prolonged bear market, as evidenced by the nancial crisis, the protective armor of the put spreads was more easily penetrated. e shortcomings of the put spreads and the fractional protective put are combined in the fractional put spread 95/85. e partial protection, and thus more exposure to the market, makes this the worst performing strategy during the nancial crisis.

Figure 4.14, interestingly, shows that all option-based protection strategies exhibit best drawdown characteristics in the 12m portfolio. ey are collectively - with no exceptions - more likely to provide beer protection than divesting over the longer drawdown windows. is implies nothing about the tail protection and should not be confused with the results discussed in the preceding paragraphs.

Overall, for the longer-dated options, a full downside protection is more important than the strike level. Indeed, the collar 95/105 is perfectly suitable for a prolonged bear market, which, in addition to its full downside protection, allows it to collect the premiums from the sold calls that expire worthless.

5.3 Conclusions

e use of derivatives for portfolio insurance is just one approach of a myriad of protection strategies. Likewise, option-based protection strategies can be designed in many dierent ways to provide the desired volatility exposure. However, almost all of them require some degree of sacrice in terms of returns, either by reduction in expected upside capture or an explicit cost for protection. What makes a suitable protection strategy, whether it is to statically reduce

82 equity exposure or to use an overlay of options, was shown to depend on the degree of realized volatility in the underlying asset, the desired time frame for provision of downside protection and ultimately the type of drawdown that the investor is more concerned with. For example, the eectiveness of the protection strategies were shown to be dierent between a sudden crash and modest drawdowns. Moreover, the choice of maturity was shown to have a material impact on both the risk-adjusted return and the drawdown characteristics.

e choice of maturity for a rolling protection depends on the time horizon. As the time horizon increases, the aggregate cost of protection increases and at the same time the need for a protection decreases, since equity returns are expected to be positive in the long-term. If protection is only needed for the next month to hedge against an expected drawdown, then buying a 1m put option would do the job. However, if the investor seeks protection against volatility over the next few years, then perhaps longer maturities should be preferred. We report that rolling protection strategies become cheaper for longer maturities. e investor should, in general, refrain from buying 1m options across all strategies.

Based on our discussion around the results, an uninformed investor would generally be beer o divesting its equity position than pursuing an option-based protection strategy. However, if the uninformed investor seeks the comfort in options’ convex payo prole, which automatically reduces the equity exposure as markets decline, she should employ a collar 95/105 strategy for options with maturities 3m or 6m. e longer the maturity, the higher is the potential for better risk-adjusted return and drawdown characteristics. A problem with the collar 95/105 arises when the expected long-term return on the equity market is positive. In such case, for too long maturities, e.g. 12m, the sold call options are more likely to expire ITM and thus reduce returns. It should therefore be preferred when a rally in the equity market is unlikely. e low cost and low beta of the collar 95/105 makes it a suitable protection strategy for investors who are more concerned with the downside risk than the upside potential. In contrast to divesting and the other protection strategies, it displayed, on average, superior protection against both tail risk events and modest drawdowns.

e put spread strategies are generally not ideal for portfolio insurance. eir strongest performance was documented when the long-term equity return was positive, both in the backtesting and the Monte Carlo simulation, in which case the need for protection decreases. It is therefore

83 harder to justify the application of a put spread as a protection strategy. e strategies were shown to provide beer protection against modest drawdowns than the protective put, but ex- hibited very poor tail protection and were oen a worse alternative than divesting across all maturities. e only exception is a put spread 100/90 strategy that buys and sells 1m options, which showed an eective protection against protracted drawdowns and a promising risk-adjusted return during the nancial crisis. We nd, in particular, that during protracted drawdowns, e.g. a crisis, the importance of higher strike level outweighs the cost of the strategy. Buying 1m ATM puts is, conversely, far from promising when returns are positive due to their high cost, which makes the put spread 100/90 not feasible as a long-term rolling protection strategy. We conclude that there is very lile to gain from the put spread strategies as far as tail protection is concerned.

Usually, an investor who seeks uncapped upside potential and full downside protection - the ideal portfolio insurance structure - would refer to the popular protective put strategy. Our ndings show that the risk-adjusted performance and drawdown characteristics of the protective put are only aractive during tail risk events. Compared to divesting and the other option-based protection strategies, the protective put is less aractive during modest drawdowns and never when excess return is positive, due to the high cost burden from expensive premiums. Its risk- adjusted performance and eectiveness against tail protection improves with longer maturities, as demonstrated in the Monte Carlo simulation. As such, a protective put that buys 12m options did fairly well during the nancial crisis and over the whole backtesting period.

e fractional strategies were introduced to mitigate path dependency. As they are only partially protected in the rst months, their performance was, in general, weaker for longer maturities during periods of negative excess return. is was demonstrated in both the Monte Carlo simulation, which was overweight on negative returns, and the backtesting on the nancial crisis. It takes more time for the strategies to become fully protected for longer-dated options and are therefore more exposed to market risk. However, the fractional strategies showed a slight improvement over their regular counterparts during periods of positive excess return and, in particular, when no signicant drawdowns had occurred before they were fully protected. Overall, the results were not too convincing.

Similar to the previous studies, we conclude that no option-based protection strategy can dominate the other in all market situations. It is, though, encouraging to report that the divested equity

84 strategy is not completely unbeatable. Our results show that, although marginally, both beer risk-adjusted return and drawdown characteristics can be achieved by employing an option- based strategy.

Lastly, as a proposal for future research, the fractional strategies introduced in this paper need further examination. Also, it would be interesting to contrast the protective put to protection strategies that employ linear derivatives, such as swaps, futures and forwards.

85 Appendix

6.1 Modelling examples

6.1.1 Protective Put

Below is a description of how the protective put portfolio is modelled based on the assumptions and delimitations in Section 3.2 with some additional simplications. e risk-free rate and the expected dividend yield are set to zero for simplicity.

Let Sindex(t) represent the price of the S&P 500 Index (including dividends) and p(t,Tk) the Black-

Scholes-Merton price of a put option at time t with strike price K and time of maturity Tk.

During the investment period, the portfolio will buy n put options at times t = τ0,τ1,...,τn 1 − with corresponding maturities T1,T2,...,Tn, where Tk τk 1 = T (for k = 1,...,n) is constant, i.e. − − the time to maturity, T , is the same for all puts. Since a new put option is purchased every time the current one expires, it follows that τ1 = T1,τ2 = T2,...,τn 1 = Tn 1. e rst put option with − − maturity T1 is bought at time t = τ0 = 0 and is rolled over into a new one with maturity T2, as it matures at time t = τ1 = T1. Hence, at each maturity Tk the portfolio rolls over into a new put option with maturity Tk+1, except at time t = Tn, which is the end of the investment horizon. e Pn Pn investment horizon for the portfolio is thus given by T ∗ = i=1(Tk τk 1) == i=1(Tk Tk 1) = − − − − n T . ·

Assume further that the total net asset value (NAV) of the portfolio is invested in the S&P 500 Index. For simplicity, the put options are nanced with a nancing rate of 0% and their cost is incurred at the time of maturity - not at the purchase date. us, for i = 2,...,n, the total cost of

86 the put options incurred by time t is

  0, 0 t < T1  ≤  C(t) = Pi 1  j−=1 p(Tj 1,Tj ),Ti 1 t < Ti  − − ≤  Pn  j=1 p(Tj 1,Tj ), t = Tn − and the accrued payos from the puts by time t is

  0, 0 t < T1  ≤  Pi 1 + Φ(t) =  − (K S (T )) ,T t < T  j=1 index j i 1 i  − − ≤ P  n (K S (T ))+, t = T j=1 − index j n

en, the value of a portfolio, Π(t), at time t, which implements a protective put strategy, follows the price process

  Sindex(t) + p(t,T1), 0 t < T1  ≤  Π(t) =  Pi 1 + Pi 1 Sindex(t) + p(t,Ti) + j−=1(K Sindex(Tj )) j−=1 p(Tj 1,Tj ),Ti 1 t < Ti  − − − − ≤   Pn + Pn Sindex(Tn) + j=1(K Sindex(Tj )) j=1 p(Tj 1,Tj ), t = Tn − − − for i = 2,...,n.

If, for example, the investment horizon is 5 years, T ∗ = 5, and the time to maturity Ti Ti 1 is 1 − − month, then the investor will buy in total 12 5 = 60 put options during the entire period. ×

e modelling of the other option strategies in Table 3.1 is similar conceptually, but some of them also include a short position in an option, which would result in negative accrued payos and positive costs (income).

6.1.2 Divested Equity

For the modelling of the divested equity portfolio, let wr and wc represent the proportion of the

NAV invested in the risky asset and cash, respectively. Moreover, it holds that wr + wc = 1 and

87 thus 0 w ,w 1. e concept of the divested equity portfolio is based on keeping a certain ≤ r c ≤ percentage of the NAV invested in the risky asset, e.g. 75% in the risky asset (wr = 0.75) and the remaining 25% in cash (w = 1 0.75 = 0.25). is eectively requires frequent rebalancing c − of the portfolio at predetermined intervals to recapture the original exposure to the risky asset.

For simplicity, let the NAV be equally sized to one unit of the risky asset at time t = 0, i.e. S0.

If Vt,r and Vt,c represent the value of the position in the risky asset, St, and the cash position at time t, respectively, then the initial value of the portfolio is:

V0,r = wr S0 V = S w S = (1 w )S = w S 0,c 0 − r 0 − r 0 c 0

Π0 = V0,r + V0,c where Π is the portfolio value at time t = 0. Between the rebalancing dates T = 0,T ,...,T , 0 { 0 1 n} the dynamics of this portfolio follows the process dΠt = dVt,r = wr dSt and dVt,c = 0, where t (Ti 1,Ti) for i = 1,...,n. At every rebalancing date, t = Ti > 0, the portfolio is rebalanced to ∈ − recapture its original exposure to the risky asset:

Π VTi ,r = wr Ti V = Π V Ti ,c Ti − Ti ,r for i = 1,...,n.

6.2 Tables

6.2.1 Monte Carlo Simulation

88 Table 6.1: 1m portfolios: Mean value of the 99th, 95th and 50th percentiles of the peak-to-trough drawdowns for the simulated protection strategies, excluding cases where the geometric return on the divested equity portfolio could not be matched.

5-Day 10-Day 20-Day 63-Day 125-Day 250-Day 99th 95th 50th 99th 95th 50th 99th 95th 50th 99th 95th 50th 99th 95th 50th 99th 95th 50th Simulated stocks -7.4% -4.8% -1.2% -12.7% -7.0% -2.0% -23.8% -10.2% -3.0% -30.5% -25.1% -5.9% -34.7% -30.7% -8.8% -39.3% -35.9% -13.8% Protective Put -5.4% -3.9% -1.0% -7.6% -5.6% -1.7% -10.4% -7.9% -2.8% -17.3% -14.3% -5.6% -22.8% -19.7% -8.2% -28.3% -25.6% -12.0% Divested Equity -3.8% -2.5% -0.6% -6.8% -3.6% -1.0% -15.4% -5.4% -1.6% -18.8% -15.1% -3.1% -21.0% -18.4% -4.7% -23.8% -21.5% -7.5% Di -1.7% -1.4% -0.4% -0.8% -1.9% -0.7% +5.0% -2.5% -1.3% +1.5% +0.7% -2.5% -1.8% -1.3% -3.5% -4.6% -4.1% -4.5% Put Spread 95/80 -6.0% -3.9% -1.0% -10.4% -5.6% -1.8% -18.5% -8.1% -2.9% -24.5% -20.0% -5.6% -29.0% -25.5% -8.2% -33.9% -30.8% -12.8% Divested Equity -5.2% -3.2% -0.8% -10.1% -4.6% -1.3% -18.7% -6.9% -2.0% -23.1% -19.0% -3.9% -26.0% -23.0% -5.9% -29.5% -26.9% -9.6% Di -0.8% -0.8% -0.3% -0.3% -1.0% -0.5% +0.2% -1.2% -0.9% -1.4% -1.0% -1.7% -3.0% -2.5% -2.3% -4.4% -3.9% -3.1% Put Spread 95/85 -6.1% -4.0% -1.0% -10.7% -5.6% -1.8% -19.0% -8.1% -2.9% -24.9% -20.7% -5.6% -29.4% -25.9% -8.2% -34.2% -31.1% -12.7% 89 Divested Equity -5.3% -3.2% -0.8% -9.9% -4.6% -1.3% -17.4% -6.9% -1.9% -21.8% -18.0% -3.9% -24.8% -21.8% -5.9% -28.3% -25.7% -9.6% Di -0.9% -0.8% -0.3% -0.8% -1.0% -0.5% -1.6% -1.2% -0.9% -3.1% -2.7% -1.8% -4.6% -4.1% -2.3% -6.0% -5.4% -3.0% Put Spread 100/90 -4.8% -2.8% -0.7% -8.3% -4.0% -1.3% -16.2% -5.8% -2.2% -20.3% -17.2% -4.5% -23.8% -21.3% -6.4% -28.0% -25.7% -9.9% Divested Equity -4.5% -2.8% -0.7% -8.2% -4.0% -1.1% -14.7% -6.1% -1.8% -18.8% -15.5% -3.6% -21.7% -19.1% -5.5% -25.3% -22.9% -9.0% Di -0.3% -0.1% 0.0% -0.1% 0.0% -0.2% -1.5% +0.3% -0.5% -1.5% -1.7% -0.9% -2.1% -2.2% -0.9% -2.8% -2.8% -0.9% Collar 95/105 -4.4% -3.1% -0.6% -6.0% -4.4% -1.0% -8.1% -6.3% -1.7% -13.9% -11.2% -3.7% -18.0% -15.3% -5.5% -21.9% -19.6% -8.3% Divested Equity -3.2% -2.3% -0.6% -5.1% -3.3% -0.9% -10.1% -4.8% -1.4% -13.8% -10.9% -2.8% -16.2% -13.9% -4.2% -18.9% -16.9% -6.4% Di -1.1% -0.8% 0.0% -0.9% -1.1% -0.1% +2.0% -1.5% -0.3% -0.1% -0.3% -0.9% -1.8% -1.4% -1.4% -2.9% -2.6% -1.9% Fractional Protective Put -5.3% -3.9% -1.0% -7.3% -5.5% -1.8% -9.8% -7.8% -2.9% -16.8% -14.0% -5.7% -22.5% -19.4% -8.3% -28.1% -25.4% -11.9% Divested Equity -3.7% -2.5% -0.6% -6.7% -3.6% -1.0% -15.2% -5.3% -1.5% -18.7% -14.9% -3.1% -20.9% -18.3% -4.7% -23.6% -21.3% -7.5% Di -1.6% -1.4% -0.4% -0.7% -1.9% -0.8% +5.4% -2.5% -1.3% +1.8% +0.9% -2.6% -1.6% -1.1% -3.6% -4.5% -4.1% -4.5% Fractional Put Spread 95/85 -0.06 -3.9% -1.1% -10.7% -5.6% -1.8% -19.1% -8.0% -3.0% -25.1% -20.9% -5.8% -29.7% -26.2% -8.4% -34.7% -31.6% -12.9% Divested Equity -5.5% -3.3% -0.8% -10.5% -4.9% -1.3% -18.5% -7.3% -2.1% -23.2% -19.0% -4.1% -26.3% -23.1% -6.3% -29.9% -27.3% -10.3% Di -0.5% -0.6% -0.3% -0.2% -0.7% -0.5% -0.7% -0.7% -0.9% -2.0% -1.8% -1.7% -3.4% -3.1% -2.1% -4.8% -4.4% -2.6% Table 6.2: 3m portfolios: Mean value of the 99th, 95th and 50th percentiles of the peak-to-trough drawdowns for the simulated protection strategies, excluding cases where the geometric return on the divested equity portfolio could not be matched.

5-Day 10-Day 20-Day 63-Day 125-Day 250-Day 99th 95th 50th 99th 95th 50th 99th 95th 50th 99th 95th 50th 99th 95th 50th 99th 95th 50th Simulated stocks -7.4% -4.8% -1.2% -12.7% -7.0% -2.0% -23.8% -10.2% -3.0% -30.5% -25.1% -5.9% -34.7% -30.7% -8.8% -39.3% -35.9% -13.8% Protective Put -5.1% -3.5% -0.8% -7.4% -5.0% -1.4% -10.6% -7.2% -2.3% -15.5% -12.7% -5.1% -19.9% -17.5% -7.7% -25.5% -23.3% -11.3% Divested Equity -3.7% -2.4% -0.6% -6.7% -3.5% -1.0% -14.3% -5.2% -1.5% -17.8% -14.4% -3.0% -20.2% -17.7% -4.6% -23.0% -20.9% -7.4% Di -1.5% -1.1% -0.2% -0.8% -1.5% -0.4% +3.7% -2.0% -0.8% +2.3% +1.7% -2.1% +0.3% +0.2% -3.1% -2.5% -2.4% -3.9% Put Spread 95/80 -5.8% -3.6% -0.8% -10.0% -5.2% -1.4% -18.0% -7.6% -2.4% -22.9% -18.7% -5.2% -26.7% -23.7% -7.8% -31.4% -28.8% -11.8% Divested Equity -5.0% -3.1% -0.8% -9.4% -4.5% -1.2% -17.4% -6.7% -1.9% -21.9% -17.9% -3.8% -25.0% -22.0% -5.9% -28.7% -26.1% -9.6% Di -0.7% -0.5% -0.1% -0.6% -0.7% -0.2% -0.6% -0.9% -0.5% -1.0% -0.7% -1.4% -1.6% -1.6% -1.9% -2.7% -2.7% -2.2% Put Spread 95/85 -6.0% -3.7% -0.9% -10.3% -5.4% -1.5% -18.9% -7.9% -2.5% -24.1% -19.8% -5.3% -27.9% -24.7% -7.8% -32.5% -29.8% -11.9% 90 Divested Equity -5.3% -3.2% -0.8% -9.5% -4.7% -1.3% -17.3% -7.0% -2.0% -21.9% -18.0% -0.04 -25.2% -22.1% -6.1% -28.9% -26.2% -10% Di -0.7% -0.5% -0.1% -0.8% -0.7% -0.2% -1.7% -0.9% -0.5% -2.1% -1.8% -1.3% -2.7% -2.6% -1.7% -3.7% -3.6% -2.0% Put Spread 100/90 -5.3% -3.3% -0.7% -9.3% -4.7% -1.3% -17.6% -6.9% -2.1% -22.6% -18.4% -4.4% -25.8% -22.9% -6.7% -29.6% -27.1% -10.3% Divested Equity -4.9% -3.0% -0.7% -8.7% -4.4% -1.2% -15.6% -6.5% -1.9% -20.1% -16.4% -3.8% -23.1% -20.3% -5.9% -26.8% -24.3% -9.5% Di -0.5% -0.3% 0.0% -0.6% -0.3% 0.0% -2.0% -0.4% -0.2% -2.5% -2.0% -0.6% -2.7% -2.6% -0.8% -2.8% -2.8% -0.8% Collar 95/105 -3.2% -2.0% -0.3% -4.7% -3.0% -0.5% -6.4% -4.4% -0.9% -9.4% -7.6% -2.2% -12.1% -10.5% -3.5% -15.2% -13.7% -5.5% Divested Equity -2.5% -1.7% -0.4% -4.0% -2.5% -0.7% -7.7% -3.6% -1.1% -10.5% -8.3% -2.1% -12.3% -10.5% -3.2% -14.4% -12.8% -4.8% Di -0.7% -0.3% +0.1% -0.7% -0.5% +0.2% +1.2% -0.7% +0.2% +1.1% +0.6% 0.0% +0.1% +0.1% -0.3% -0.8% -0.9% -0.6% Fractional Protective Put -5.2% -3.6% -0.8% -7.4% -5.1% -1.4% -10.8% -7.4% -2.4% -15.2% -12.7% -5.4% -19.7% -17.2% -8.1% -25.5% -23.2% -11.7% Divested Equity -3.9% -2.5% -0.6% -6.7% -3.7% -0.01 -14.4% -5.5% -1.6% -18.2% -14.5% -3.2% -20.7% -18% -4.8% -23.8% -21.3% -7.7% Di -1.3% -1.0% -0.2% -0.7% -1.4% -0.4% +3.7% -1.9% -0.8% +3.0% +1.8% -2.2% +1.0% +0.8% -3.3% -1.8% -1.8% -4.0% Fractional Put Spread 95/85 -5.9% -3.8% -0.9% -10.4% -5.4% -1.6% -19.4% -7.9% -2.6% -24.4% -20.1% -5.5% -28.2% -25.0% -8.2% -32.9% -30.1% -12.3% Divested Equity -5.4% -3.4% -0.8% -10.0% -4.9% -1.4% -18.0% -7.3% -2.1% -23.0% -18.9% -4.2% -26.3% -23.2% -6.4% -30.2% -27.5% -10.4% Di -0.5% -0.4% -0.1% -0.5% -0.5% -0.2% -1.3% -0.6% -0.4% -1.5% -1.3% -1.3% -1.9% -1.8% -1.8% -2.7% -2.7% -1.9% Table 6.3: 6m portfolios: Mean value of the 99th, 95th and 50th percentiles of the peak-to-trough drawdowns for the simulated protection strategies, excluding cases where the geometric return on the divested equity portfolio could not be matched.

5-Day 10-Day 20-Day 63-Day 125-Day 250-Day 99th 95th 50th 99th 95th 50th 99th 95th 50th 99th 95th 50th 99th 95th 50th 99th 95th 50th Simulated stocks -7.4% -4.8% -1.2% -12.7% -7.0% -2.0% -23.8% -10.2% -3.0% -30.5% -25.1% -5.9% -34.7% -30.7% -8.8% -39.3% -35.9% -13.8% Protective Put -5.0% -3.3% -0.6% -7.2% -4.8% -1.2% -11.0% -6.9% -1.9% -15.6% -12.7% -4.4% -19.0% -16.8% -7.1% -23.9% -22.0% -10.8% Divested Equity -3.9% -2.5% -0.6% -7.0% -3.6% -1.0% -14.9% -5.4% -1.6% -18.6% -14.9% -3.1% -21.1% -18.4% -4.8% -24.2% -21.9% -7.7% Di -1.1% -0.8% 0.0% -0.2% -1.1% -0.1% +3.9% -1.5% -0.4% +3.0% +2.2% -1.3% +2.1% +1.6% -2.3% +0.3% -0.1% -3.1% Put Spread 95/80 -5.8% -3.6% -0.8% -10.0% -5.2% -1.4% -18.1% -7.7% -2.2% -23.1% -18.7% -4.7% -26.6% -23.5% -7.5% -30.9% -28.4% -11.7% Divested Equity -5.2% -3.2% -0.8% -9.5% -4.6% -1.3% -17.7% -6.9% -2.0% -22.3% -18.2% -4.0% -25.3% -22.4% -6.1% -29.1% -26.6% -9.9% Di -0.6% -0.4% 0.0% -0.4% -0.6% -0.1% -0.4% -0.8% -0.2% -0.8% -0.5% -0.8% -1.3% -1.2% -1.5% -1.8% -1.9% -1.8% Put Spread 95/85 -6.1% -3.8% -0.9% -10.4% -5.5% -1.5% -19.0% -8.1% -2.4% -24.3% -19.9% -5.0% -28.1% -24.8% -7.7% -32.4% -29.7% -12.0% 91 Divested Equity -5.5% -3.4% -0.8% -9.8% -4.9% -1.4% -17.7% -7.2% -2.1% -22.6% -18.5% -4.2% -25.9% -22.8% -6.4% -29.8% -27.2% -10.4% Di -0.6% -0.5% -0.1% -0.7% -0.6% -0.1% -1.2% -0.8% -0.3% -1.8% -1.4% -0.8% -2.2% -2.0% -1.3% -2.6% -2.5% -1.6% Put Spread 100/90 -5.8% -3.6% -0.8% -10.0% -5.2% -1.4% -18.4% -7.6% -2.2% -23.6% -19.2% -4.5% -27.2% -23.8% -6.9% -31.2% -28.3% -10.9% Divested Equity -5.4% -3.3% -0.8% -9.5% -4.8% -1.3% -17.1% -7.1% -2.1% -21.8% -17.8% -4.1% -24.9% -21.9% -6.3% -28.7% -26.0% -10.2% Di -0.5% -0.3% 0.0% -0.5% -0.4% -0.1% -1.3% -0.5% -0.1% -1.8% -1.4% -0.4% -2.3% -1.9% -0.6% -2.5% -2.3% -0.7% Collar 95/105 -2.4% -1.4% -0.2% -3.6% -2.1% -0.3% -5.5% -3.2% -0.5% -7.5% -6.1% -1.3% -9.0% -7.9% -2.3% -11.2% -10.1% -3.8% Divested Equity -2.0% -1.4% -0.4% -3.2% -2.1% -0.6% -6.2% -3.0% -0.9% -8.5% -6.7% -1.8% -10.0% -8.5% -2.6% -11.7% -10.4% -3.9% Di -0.4% 0.0% +0.2% -0.4% 0.0% +0.3% +0.8% -0.2% +0.4% +1.0% +0.6% +0.5% +1.0% +0.7% +0.3% +0.5% +0.2% 0.1% Fractional Protective Put -5.1% -3.5% -0.7% -7.5% -5.0% -1.3% -11.7% -7.3% -2.1% -16.3% -13.3% -4.7% -19.4% -17.1% -7.6% -24.4% -22.2% -11.6% Divested Equity -3.9% -2.5% -0.6% -6.8% -3.7% -1.0% -14.1% -5.4% -1.6% -17.8% -14.3% -3.1% -20.4% -17.7% -4.8% -23.5% -21.2% -7.6% Di -1.2% -0.9% -0.1% -0.7% -1.4% -0.3% +2.4% -1.9% -0.5% +1.6% +1.0% -1.6% +1.0% +0.6% -2.8% -0.9% -1.1% -4.0% Fractional Put Spread 95/85 -6.1% -3.9% -0.9% -10.8% -5.6% -1.6% -19.9% -8.3% -2.5% -25.3% -20.7% -5.1% -28.9% -25.5% -8.0% -33.3% -30.4% -12.4% Divested Equity -5.6% -3.5% -0.9% -10.1% -5.1% -1.4% -18.4% -7.5% -2.2% -23.4% -19.2% -4.3% -26.8% -23.6% -6.6% -30.9% -28.1% -10.7% Di -0.5% -0.4% -0.1% -0.7% -0.6% -0.2% -1.5% -0.7% -0.3% -1.9% -1.5% -0.8% -2.1% -1.9% -1.4% -2.4% -2.3% -1.7% Table 6.4: 12m portfolios: Mean value of the 99th, 95th and 50th percentiles of the peak-to-trough drawdowns for the simulated protection strategies, excluding cases where the geometric return on the divested equity portfolio could not be matched.

5-Day 10-Day 20-Day 63-Day 125-Day 250-Day 99th 95th 50th 99th 95th 50th 99th 95th 50th 99th 95th 50th 99th 95th 50th 99th 95th 50th Simulated stocks -7.4% -4.8% -1.2% -12.7% -7.0% -2.0% -23.8% -10.2% -3.0% -30.5% -25.1% -5.9% -34.7% -30.7% -8.8% -39.3% -35.9% -13.8% Protective Put -4.9% -3.1% -0.5% -7.2% -4.6% -0.9% -11.2% -6.6% -1.6% -15.8% -12.7% -3.7% -18.7% -16.6% -6.0% -22.8% -21.0% -10.0% Divested Equity -3.9% -2.5% -0.6% -7.3% -3.6% -1.0% -15.1% -5.4% -1.6% -18.7% -15.0% -3.1% -21.2% -18.5% -4.8% -24.3% -22% -7.8% Di -0.9% -0.6% +0.1% +0.1% -0.9% +0.1% +3.9% -1.2% 0.0% +2.9% +2.3% -0.6% +2.5% +1.9% -1.3% +1.6% +1.0% -2.2% Put Spread 95/80 -6.0% -3.7% -0.8% -10.4% -5.3% -1.4% -18.5% -7.9% -2.2% -23.7% -19.1% -4.6% -27.2% -23.8% -7.1% -31.3% -28.6% -11.5% Divested Equity -5.4% -3.3% -0.8% -10.0% -4.8% -1.3% -18.1% -7.1% -2.1% -22.8% -18.6% -4.1% -25.9% -22.8% -6.3% -29.7% -0.27 -10.3% Di -0.6% -0.4% 0.0% -0.4% -0.6% -0.1% -0.4% -0.8% -0.1% -0.9% -0.5% -0.5% -1.2% -1.0% -0.8% -1.6% -1.6% -1.2% Put Spread 95/85 -6.3% -3.9% -0.9% -11.0% -5.7% -1.5% -19.8% -8.4% -2.4% -25.4% -20.7% -4.9% -29.1% -25.6% -7.5% -33.3% -30.3% -12.0% 92 Divested Equity -5.7% -3.5% -0.9% -10.4% -5.1% -1.4% -19.0% -7.6% -2.2% -24.0% -19.6% -4.4% -27.2% -24.0% -6.6% -31.0% -28.2% -10.8% Di -0.6% -0.4% -0.1% -0.5% -0.6% -0.1% -0.8% -0.8% -0.2% -1.5% -1.1% -0.5% -1.9% -1.6% -0.8% -2.3% -2.1% -1.2% Put Spread 100/90 -6.2% -3.8% -0.9% -10.5% -5.6% -1.5% -19.0% -8.2% -2.3% -24.6% -20.0% -4.7% -28.2% -24.7% -7.2% -32.3% -29.3% -11.5% Divested Equity -5.6% -3.4% -0.8% -9.9% -5.0% -1.4% -17.9% -7.4% -2.2% -22.8% -18.7% -4.3% -26.0% -22.9% -6.6% -29.8% -27.1% -10.7% Di -0.6% -0.4% -0.1% -0.6% -0.5% -0.1% -1.1% -0.7% -0.2% -1.7% -1.3% -0.4% -2.2% -1.8% -0.6% -2.5% -2.2% -0.8% Collar 95/105 -1.7% -0.9% -0.1% -2.7% -1.4% -0.2% -4.2% -2.1% -0.3% -5.8% -4.5% -0.7% -6.7% -5.9% -1.2% -7.9% -7.2% -2.4% Divested Equity -1.7% -1.2% -0.3% -2.7% -1.7% -0.5% -5.3% -2.5% -0.8% -7.1% -5.7% -1.5% -8.4% -7.2% -2.1% -9.8% -8.7% -3.2% Di 0.0% +0.3% +0.2% 0.0% +0.4% +0.3% +1.0% +0.4% +0.5% +1.3% +1.1% +0.8% +1.7% +1.3% +1.0% +1.9% +1.5% +0.8% Fractional Protective Put -5.2% -3.4% -0.7% -7.8% -5.0% -1.2% -13.3% -7.3% -1.9% -18.2% -14.7% -4.2% -21.3% -18.9% -6.7% -24.9% -23.0% -10.9% Divested Equity -4.0% -2.5% -0.6% -7.0% -3.7% -1.0% -14.1% -5.5% -1.6% -17.9% -14.4% -3.2% -20.4% -17.8% -4.8% -23.6% -21.3% -7.6% Di -1.2% -0.9% -0.1% -0.8% -1.3% -0.1% +0.9% -1.9% -0.3% -0.3% -0.4% -1.0% -0.9% -1.1% -1.9% -1.4% -1.7% -3.3% Fractional Put Spread 95/85 -6.4% -4.1% -1.0% -11.2% -6.0% -1.7% -20.8% -8.8% -2.6% -26.6% -21.7% -5.2% -30.4% -26.8% -7.9% -34.7% -31.7% -12.6% Divested Equity -5.9% -3.7% -0.9% -10.5% -5.4% -1.5% -19.4% -8.0% -2.3% -24.7% -20.3% -4.6% -28.3% -24.9% -7.0% -32.4% -29.6% -11.2% Di -0.5% -0.4% -0.1% -0.7% -0.5% -0.1% -1.4% -0.7% -0.2% -1.8% -1.4% -0.5% -2.1% -1.8% -0.9% -2.3% -2.1% -1.3% 6.2.2 Backtesting

93 Table 6.5: Summary of peak-to-trough drawdown returns for the 1m portfolios.

5-Day 10-Day 20-Day 63-Day 125-Day 250-Day 99th 95th 50th 99th 95th 50th 99th 95th 50th 99th 95th 50th 99th 95th 50th 99th 95th 50th S&P 500 -7.8% -4.3% -0.4% -11.4% -5.7% -0.7% -16.8% -8.1% -1.0% -33.8% -13.9% -1.5% -48.6% -16.5% -1.9% -58.1% -39.6% -2.2% Protective Put -6.8% -4.5% -0.6% -10.6% -6.1% -1.0% -15.4% -9.1% -1.5% -32.7% -18.7% -2.7% -46.1% -23.6% -3.4% -57.4% -43.1% -4.4% Divested Equity -3.0% -1.7% -0.2% -4.4% -2.2% -0.3% -6.4% -3.1% -0.4% -11.9% -5.2% -0.6% -17% -6.0% -0.6% -20.2% -12.7% -0.8% Di -3.7% -2.9% -0.5% -6.1% -3.9% -0.8% -9.0% -6.0% -1.1% -20.7% -13.5% -2.1% -29.1% -17.6% -2.7% -37.2% -30.5% -3.6% Put Spread 95/80 -6.7% -4.3% -0.6% -10.5% -5.8% -1.0% -14.9% -8.5% -1.4% -28.2% -17.6% -2.5% -42.9% -22.6% -3.1% -54.7% -39.5% -3.9% Divested Equity -3.8% -2.1% -0.2% -5.5% -2.7% -0.3% -7.9% -3.9% -0.5% -15.0% -6.5% -0.7% -21.5% -7.6% -0.8% -25.7% -16.4% -1.0% Di -2.9% -2.2% -0.4% -5.0% -3.1% -0.6% -7.0% -4.7% -0.9% -13.3% -11.1% -1.8% -21.5% -15.0% -2.3% -29.0% -23.1% -2.9% Put Spread 95/85 -7.1% -4.4% -0.6% -11.3% -5.8% -0.9% -15.6% -8.5% -1.3% -29.7% -17.6% -2.3% -47.6% -21.3% -2.9% -58.6% -42.3% -3.5%

94 Divested Equity -4.3% -2.3% -0.2% -6.2% -3.1% -0.4% -9.0% -4.4% -0.5% -17% -7.4% -0.8% -24.6% -8.6% -1.0% -29.5% -19.0% -1.1% Di -2.8% -2.0% -0.3% -5.1% -2.7% -0.5% -6.7% -4.1% -0.8% -12.7% -10.2% -1.5% -23.0% -12.7% -2.0% -29.1% -23.4% -2.4% Put Spread 100/90 -6.7% -3.5% -0.5% -10.0% -4.8% -0.8% -13.2% -6.8% -1.4% -31.3% -11.5% -2.4% -44.4% -13.9% -3.1% -52.6% -34.6% -3.5% Divested Equity -1.8% -1.0% -0.1% -2.7% -1.3% -0.1% -3.8% -1.9% -0.2% -6.8% -3.0% -0.3% -9.8% -3.4% -0.4% -11.4% -7.0% -0.4% Di -4.9% -2.5% -0.4% -7.3% -3.5% -0.7% -9.4% -4.9% -1.2% -24.5% -8.5% -2.1% -34.6% -10.5% -2.7% -41.2% -27.7% -3.1% Collar 95/105 -4.8% -3.2% -0.4% -6.7% -4.4% -0.7% -9.2% -6.5% -1.1% -16.3% -13.0% -1.9% -23.1% -17.4% -2.5% -32.8% -24.9% -3.2% Divested Equity -4.6% -2.5% -0.2% -6.7% -3.3% -0.4% -9.6% -4.7% -0.6% -18.3% -7.9% -0.9% -26.5% -9.3% -1.0% -31.7% -20.5% -1.2% Di -0.2% -0.7% -0.2% -0.1% -1.1% -0.3% +0.4% -1.8% -0.5% +2.0% -5.1% -1.0% +3.4% -8.1% -1.5% -1.1% -4.4% -2.0% Fractional Protective Put -6.6% -4.4% -0.6% -10.4% -5.9% -1.0% -15.0% -8.8% -1.5% -31.9% -18.4% -2.7% -45.0% -23.2% -3.3% -56.1% -42.4% -4.3% Divested Equity -3.2% -1.8% -0.2% -4.7% -2.3% -0.3% -6.8% -3.3% -0.4% -12.8% -5.6% -0.6% -18.1% -6.5% -0.7% -21.7% -13.7% -0.8% Di -3.4% -2.7% -0.4% -5.7% -3.6% -0.7% -8.2% -5.5% -1.1% -19.1% -12.9% -2.1% -26.9% -16.7% -2.6% -34.4% -28.7% -3.5% Fractional Put Spread 95/85 -6.9% -4.3% -0.5% -11.0% -5.7% -0.9% -15.3% -8.3% -1.3% -29.0% -17.3% -2.3% -46.5% -21.0% -2.9% -57.4% -41.5% -3.5% Divested Equity -4.4% -2.4% -0.2% -6.5% -3.2% -0.4% -9.3% -4.6% -0.6% -17.8% -7.7% -0.8% -25.7% -9.0% -1.0% -30.9% -19.9% -1.2% Di -2.5% -1.8% -0.3% -4.6% -2.5% -0.5% -6.0% -3.7% -0.7% -11.2% -9.6% -1.4% -20.8% -12.0% -1.9% -26.5% -21.5% -2.3% Table 6.6: Summary of peak-to-trough drawdown returns for the 3m portfolios.

5-Day 10-Day 20-Day 63-Day 125-Day 250-Day 99th 95th 50th 99th 95th 50th 99th 95th 50th 99th 95th 50th 99th 95th 50th 99th 95th 50th S&P 500 -7.8% -4.3% -0.4% -11.4% -5.7% -0.7% -16.8% -8.1% -1.0% -33.8% -13.9% -1.5% -48.6% -16.5% -1.9% -58.1% -39.6% -2.2% Protective Put -4.8% -3.2% -0.5% -6.4% -4.5% -0.8% -8.9% -6.3% -1.3% -17.7% -10.7% -2.4% -24.0% -14.4% -3.1% -29.7% -21.2% -4.0% Divested Equity -3.4% -1.9% -0.2% -4.9% -2.4% -0.3% -7.0% -3.5% -0.4% -13.4% -5.8% -0.6% -19.0% -6.7% -0.7% -22.7% -14.3% -0.9% Di -1.5% -1.4% -0.3% -1.5% -2.1% -0.5% -1.9% -2.9% -0.9% -4.3% -4.9% -1.7% -5.0% -7.7% -2.4% -7.0% -6.9% -3.1% Put Spread 95/80 -5.7% -3.5% -0.5% -8.3% -4.6% -0.8% -12.3% -6.7% -1.2% -20.7% -11.4% -2.1% -32.7% -15% -2.8% -41.3% -30.3% -3.5% Divested Equity -4.2% -2.3% -0.2% -6.1% -3.0% -0.4% -8.8% -4.3% -0.5% -16.7% -7.3% -0.8% -24.1% -8.5% -0.9% -29.0% -18.6% -1.1% Di -1.5% -1.2% -0.3% -2.2% -1.6% -0.4% -3.5% -2.4% -0.7% -3.9% -4.1% -1.3% -8.6% -6.5% -1.9% -12.4% -11.7% -2.4% Put Spread 95/85 -6.2% -3.6% -0.5% -9.1% -4.8% -0.8% -14.0% -6.9% -1.1% -25.5% -11.6% -2.0% -37.5% -14.7% -2.5% -44.9% -33.2% -3.2%

95 Divested Equity -4.9% -2.7% -0.3% -7.2% -3.6% -0.4% -10.3% -5.1% -0.6% -20.0% -8.6% -0.9% -28.8% -10.0% -1.1% -34.5% -22.5% -1.3% Di -1.3% -0.9% -0.2% -1.9% -1.2% -0.3% -3.7% -1.8% -0.5% -5.5% -3.0% -1.0% -8.7% -4.7% -1.4% -10.4% -10.7% -1.8% Put Spread 100/90 -6.0% -3.3% -0.4% -8.9% -4.4% -0.6% -12.8% -6.2% -1.0% -27.2% -9.5% -1.9% -38.3% -12.0% -2.4% -41.5% -31.9% -3.0% Divested Equity -3.5% -2.0% -0.2% -5.2% -2.6% -0.3% -7.4% -3.7% -0.4% -14.1% -6.1% -0.7% -20.1% -7.1% -0.8% -24.1% -15.3% -0.9% Di -2.4% -1.3% -0.2% -3.7% -1.8% -0.3% -5.3% -2.5% -0.6% -13.1% -3.4% -1.2% -18.2% -4.8% -1.6% -17.5% -16.7% -2.1% Collar 95/105 -3.2% -2.0% -0.2% -4.3% -2.8% -0.4% -5.8% -3.9% -0.6% -8.6% -6.8% -1.0% -12.1% -9.1% -1.4% -15.3% -11.8% -2.0% Divested Equity -5.1% -2.8% -0.3% -7.4% -3.7% -0.4% -10.7% -5.2% -0.6% -20.7% -8.8% -1.0% -29.7% -10.3% -1.1% -35.7% -23.3% -1.4% Di +1.9% +0.8% 0.0% +3.1% +0.9% 0.0% +4.8% +1.3% 0.0% +12.1% +2.0% 0.0% +17.7% +1.2% -0.3% +20.4% +11.5% -0.6% Fractional Protective Put -4.8% -3.2% -0.5% -6.6% -4.4% -0.8% -8.9% -6.3% -1.3% -17.5% -10.6% -2.3% -23.7% -14.3% -3.0% -29.4% -21.0% -4.0% Divested Equity -3.6% -2.0% -0.2% -5.3% -2.6% -0.3% -7.6% -3.7% -0.4% -14.3% -6.2% -0.7% -20.4% -7.3% -0.8% -24.5% -15.6% -0.9% Di -1.2% -1.2% -0.3% -1.3% -1.8% -0.5% -1.4% -2.6% -0.8% -3.2% -4.4% -1.6% -3.2% -7.0% -2.2% -4.9% -5.4% -3.0% Fractional Put Spread 95/85 -6.3% -3.5% -0.4% -9.0% -4.8% -0.7% -14.0% -6.9% -1.1% -25.4% -11.5% -1.9% -37.3% -14.6% -2.5% -44.7% -33.0% -3.1% Divested Equity -5.1% -2.8% -0.3% -7.4% -3.7% -0.4% -10.7% -5.2% -0.6% -20.8% -8.8% -1.0% -29.8% -10.4% -1.2% -35.8% -23.4% -1.4% Di -1.2% -0.7% -0.2% -1.6% -1.1% -0.3% -3.3% -1.6% -0.5% -4.6% -2.7% -1.0% -7.4% -4.2% -1.4% -8.8% -9.6% -1.7% Table 6.7: Summary of peak-to-trough drawdown returns for the 6m portfolios.

5-Day 10-Day 20-Day 63-Day 125-Day 250-Day 99th 95th 50th 99th 95th 50th 99th 95th 50th 99th 95th 50th 99th 95th 50th 99th 95th 50th S&P 500 -7.8% -4.3% -0.4% -11.4% -5.7% -0.7% -16.8% -8.1% -1.0% -33.8% -13.9% -1.5% -48.6% -16.5% -1.9% -58.1% -39.6% -2.2% Protective Put -4.3% -2.9% -0.4% -5.6% -4.0% -0.7% -8.0% -5.7% -1.1% -13.1% -9.7% -1.9% -23.0% -12.6% -2.5% -30.4% -22.1% -3.0% Divested Equity -4.0% -2.2% -0.2% -5.9% -2.9% -0.3% -8.5% -4.2% -0.5% -16.1% -7.0% -0.8% -23.2% -8.2% -0.9% -27.8% -17.8% -1.1% Di -0.3% -0.7% -0.2% +0.3% -1.1% -0.3% +0.5% -1.5% -0.6% +3.0% -2.7% -1.2% +0.2% -4.4% -1.6% -2.6% -4.3% -1.9% Put Spread 95/80 -6.0% -3.4% -0.4% -8.7% -4.6% -0.7% -12.4% -6.5% -1.0% -23.8% -11.2% -1.7% -35.2% -13.5% -2.1% -43.5% -29.1% -2.7% Divested Equity -5.2% -2.8% -0.3% -7.6% -3.8% -0.4% -10.9% -5.4% -0.7% -21.3% -9.0% -1.0% -30.6% -10.6% -1.2% -36.7% -24.0% -1.4% Di -0.8% -0.5% -0.1% -1.1% -0.8% -0.2% -1.5% -1.1% -0.4% -2.6% -2.1% -0.7% -4.6% -2.9% -1.0% -6.8% -5.1% -1.2% Put Spread 95/85 -6.5% -3.5% -0.4% -9.6% -4.7% -0.7% -14.0% -6.6% -1.0% -26.6% -11.4% -1.7% -39.0% -13.8% -2.0% -46.9% -32.5% -2.5%

96 Divested Equity -5.9% -3.2% -0.3% -8.6% -4.3% -0.5% -12.4% -6.1% -0.7% -24.2% -10.3% -1.1% -35.1% -12.1% -1.3% -42.1% -27.8% -1.6% Di -0.7% -0.3% -0.1% -1.1% -0.5% -0.1% -1.6% -0.6% -0.3% -2.4% -1.1% -0.5% -4.0% -1.7% -0.7% -4.8% -4.7% -0.9% Put Spread 100/90 -6.6% -3.3% -0.4% -9.6% -4.4% -0.6% -14.2% -6.1% -0.9% -26.7% -10.7% -1.5% -36.5% -13.2% -2.0% -41.3% -29.2% -2.5% Divested Equity -5.1% -2.8% -0.3% -7.5% -3.7% -0.4% -10.8% -5.3% -0.6% -21.0% -8.9% -1.0% -30.2% -10.5% -1.2% -36.2% -23.6% -1.4% Di -1.4% -0.5% -0.1% -2.1% -0.7% -0.2% -3.4% -0.8% -0.3% -5.7% -1.8% -0.6% -6.4% -2.7% -0.8% -5.1% -5.5% -1.1% Collar 95/105 -2.7% -1.6% -0.1% -3.6% -2.3% -0.3% -5.4% -3.1% -0.4% -7.9% -6.0% -0.6% -12.8% -7.4% -0.8% -16.4% -12.0% -1.1% Divested Equity -3.5% -1.9% -0.2% -5.2% -2.6% -0.3% -7.4% -3.6% -0.4% -14.0% -6.1% -0.7% -20.0% -7.1% -0.8% -24.0% -15.2% -0.9% Di +0.8% +0.3% 0.0% +1.5% +0.3% +0.1% +2.1% +0.5% +0.1% +6.1% +0.1% +0.1% +7.2% -0.3% 0.0% +7.6% +3.2% -0.2% Fractional Protective Put -4.4% -2.8% -0.4% -5.7% -4.0% -0.7% -8.0% -5.7% -1.0% -12.8% -9.7% -1.8% -22.7% -12.5% -2.4% -30.2% -21.7% -3.0% Divested Equity -4.4% -2.4% -0.2% -6.5% -3.2% -0.4% -9.3% -4.6% -0.6% -17.8% -7.7% -0.8% -25.7% -9.0% -1.0% -30.9% -19.9% -1.2% Di +0.1% -0.4% -0.2% +0.8% -0.8% -0.3% +1.3% -1.1% -0.5% +5.1% -2.0% -1.0% +3.1% -3.5% -1.4% +0.6% -1.7% -1.8% Fractional Put Spread 95/85 -6.7% -3.5% -0.4% -9.7% -4.8% -0.7% -14.0% -6.8% -1.0% -26.9% -11.4% -1.6% -39.0% -14.0% -2.0% -47.2% -32.5% -2.5% Divested Equity -6.1% -3.3% -0.3% -8.8% -4.4% -0.5% -12.8% -6.3% -0.8% -25.1% -10.6% -1.2% -36.4% -12.5% -1.4% -43.6% -28.9% -1.7% Di -0.7% -0.2% -0.1% -0.9% -0.4% -0.1% -1.1% -0.5% -0.2% -1.8% -0.8% -0.5% -2.7% -1.4% -0.6% -3.6% -3.5% -0.8% Table 6.8: Summary of peak-to-trough drawdown returns for the 12m portfolios.

5-Day 10-Day 20-Day 63-Day 125-Day 250-Day 99th 95th 50th 99th 95th 50th 99th 95th 50th 99th 95th 50th 99th 95th 50th 99th 95th 50th S&P 500 -7.8% -4.3% -0.4% -11.4% -5.7% -0.7% -16.8% -8.1% -1.0% -33.8% -13.9% -1.5% -48.6% -16.5% -1.9% -58.1% -39.6% -2.2% Protective Put -4.1% -2.5% -0.4% -5.9% -3.6% -0.6% -8.1% -5.1% -0.9% -11.1% -8.3% -1.7% -15.4% -11.3% -2.1% -23.9% -17.3% -2.5% Divested Equity -4.6% -2.5% -0.2% -6.7% -3.3% -0.4% -9.7% -4.8% -0.6% -18.6% -8% -0.9% -26.8% -9.4% -1.0% -32.2% -20.8% -1.2% Di +0.5% 0.0% -0.1% +0.9% -0.3% -0.2% +1.6% -0.3% -0.3% +7.5% -0.2% -0.8% +11.4% -1.9% -1.1% +8.3% +3.5% -1.3% Put Spread 95/80 -6.0% -3.4% -0.4% -8.7% -4.6% -0.6% -12.6% -6.8% -0.9% -23.3% -11.2% -1.5% -34.5% -13.7% -1.9% -43.3% -28.9% -2.3% Divested Equity -5.9% -3.2% -0.3% -8.6% -4.3% -0.5% -12.5% -6.1% -0.7% -24.4% -10.4% -1.1% -35.3% -12.2% -1.4% -42.4% -28% -1.6% Di 0.0% -0.2% -0.1% -0.1% -0.3% -0.1% -0.1% -0.6% -0.2% +1.1% -0.9% -0.4% +0.9% -1.5% -0.6% -0.8% -0.9% -0.7% Put Spread 95/85 -6.7% -3.6% -0.4% -9.5% -4.9% -0.6% -14.3% -7.1% -0.9% -27.1% -12.0% -1.5% -39.9% -14.6% -1.8% -48.4% -33.1% -2.2%

97 Divested Equity -6.4% -3.5% -0.3% -9.4% -4.7% -0.6% -13.6% -6.7% -0.8% -26.9% -11.3% -1.2% -38.8% -13.3% -1.5% -46.5% -31.0% -1.8% Di -0.3% -0.1% 0.0% -0.2% -0.3% -0.1% -0.7% -0.4% -0.1% -0.2% -0.7% -0.3% -1.1% -1.3% -0.4% -1.9% -2.0% -0.5% Put Spread 100/90 -6.7% -3.4% -0.4% -9.5% -4.5% -0.6% -13.9% -6.4% -0.9% -27.5% -10.5% -1.4% -39.4% -12.4% -1.7% -46.3% -33.1% -2.0% Divested Equity -6.5% -3.5% -0.3% -9.5% -4.7% -0.6% -13.8% -6.7% -0.8% -27.2% -11.4% -1.3% -39.2% -13.4% -1.5% -46.9% -31.4% -1.8% Di -0.2% +0.1% 0.0% 0.0% +0.2% 0.0% -0.1% +0.3% -0.1% -0.3% +0.9% -0.1% -0.2% +1.0% -0.2% +0.6% -1.7% -0.2% Collar 95/105 -2.5% -1.3% -0.1% -3.8% -1.7% -0.2% -5.5% -2.6% -0.2% -10.0% -5.1% -0.4% -10.8% -7.5% -0.6% -13.3% -9.9% -0.9% Divested Equity -0.9% -0.5% 0.0% -1.3% -0.6% -0.1% -1.8% -0.9% -0.1% -3.2% -1.4% -0.1% -4.4% -1.6% -0.1% -4.9% -2.8% -0.2% Di -1.6% -0.8% -0.1% -2.5% -1.1% -0.1% -3.7% -1.7% -0.2% -6.8% -3.7% -0.3% -6.4% -5.9% -0.4% -8.4% -7.1% -0.7% Fractional Protective Put -4.0% -2.4% -0.3% -5.3% -3.4% -0.6% -7.1% -4.8% -0.8% -9.5% -7.7% -1.5% -13.4% -10.3% -2.0% -21.8% -15.4% -2.6% Divested Equity -4.8% -2.6% -0.3% -7.0% -3.5% -0.4% -10.1% -5.0% -0.6% -19.5% -8.4% -0.9% -28.1% -9.8% -1.1% -33.7% -21.9% -1.3% Di +0.8% +0.2% -0.1% +1.7% +0.1% -0.1% +3.0% +0.1% -0.2% +10.0% +0.7% -0.6% +14.6% -0.5% -0.9% +11.9% +6.5% -1.3% Fractional Put Spread 95/85 -6.8% -3.5% -0.4% -9.4% -4.8% -0.6% -14.1% -6.9% -0.9% -27.2% -11.6% -1.5% -38.5% -13.8% -1.9% -46.7% -30.4% -2.3% Divested Equity -6.6% -3.6% -0.4% -9.7% -4.8% -0.6% -14.1% -6.9% -0.8% -27.8% -11.7% -1.3% -40.2% -13.7% -1.5% -48.0% -32.2% -1.8% Di -0.2% +0.1% 0.0% +0.2% 0.0% -0.1% -0.1% 0.0% -0.1% +0.6% +0.1% -0.2% +1.7% -0.1% -0.3% +1.3% +1.8% -0.5% 6.3 Figures

6.3.1 Monte Carlo Simulation

5-Day 10-Day 60 45 Protective Put, (-5.4%, 0.8%) Protective Put, (-7.6%, 1.2%) Put Spread 95/80, (-6%, 2.4%) 40 Put Spread 95/80, (-10.4%, 6%) Put Spread 95/85, (-6.1%, 3.3%) Put Spread 95/85, (-10.7%, 7.4%) 50 Put Spread 100/90, (-4.8%, 3.2%) Put Spread 100/90, (-8.3%, 7%) Collar 95/105, (-4.4%, 0.7%) 35 Collar 95/105, (-6%, 1%) Frac. Protective Put, (-5.3%, 0.8%) Frac. Protective Put, (-7.3%, 1%) 40 Frac. Put Spread 95/85, (-6%, 3.4%) 30 Frac. Put Spread 95/85, (-10.7%, 7.8%)

25 30 20

20 15

Probability Density Function Probability Density Function 10 10 5

0 0 -35% -30% -25% -20% -15% -10% -5% 0% -45% -40% -35% -30% -25% -20% -15% -10% -5% 0% Peak-to-trough drawdown Peak-to-trough drawdown

20-Day 63-Day 35 18 Protective Put, (-10.4%, 1.8%) Protective Put, (-17.3%, 2.9%) Put Spread 95/80, (-18.5%, 9.3%) 16 Put Spread 95/80, (-24.5%, 8.8%) 30 Put Spread 95/85, (-19%, 11.1%) Put Spread 95/85, (-24.9%, 10.2%) Put Spread 100/90, (-16.2%, 12%) Put Spread 100/90, (-20.3%, 11.7%) Collar 95/105, (-8.1%, 1.4%) 14 Collar 95/105, (-13.9%, 2.6%) 25 Frac. Protective Put, (-9.8%, 1.3%) Frac. Protective Put, (-16.8%, 2.4%) Frac. Put Spread 95/85, (-19.1%, 11.6%) 12 Frac. Put Spread 95/85, (-25.1%, 10.6%)

20 10

15 8

6 10

Probability Density Function Probability Density Function 4 5 2

0 0 -60% -50% -40% -30% -20% -10% 0% -70% -60% -50% -40% -30% -20% -10% 0% Peak-to-trough drawdown Peak-to-trough drawdown

125-Day 250-Day 12 8 Protective Put, (-22.8%, 4.3%) Protective Put, (-28.3%, 6.2%) Put Spread 95/80, (-29%, 8.6%) Put Spread 95/80, (-33.9%, 9.9%) Put Spread 95/85, (-29.4%, 9.9%) 7 Put Spread 95/85, (-34.2%, 11.1%) 10 Put Spread 100/90, (-23.8%, 11.1%) Put Spread 100/90, (-28%, 11.6%) Collar 95/105, (-18%, 4%) 6 Collar 95/105, (-21.9%, 5.7%) Frac. Protective Put, (-22.5%, 3.9%) Frac. Protective Put, (-28.1%, 6%) 8 Frac. Put Spread 95/85, (-29.7%, 10.2%) Frac. Put Spread 95/85, (-34.7%, 11.3%) 5

6 4

3 4

2 Probability Density Function Probability Density Function 2 1

0 0 -80% -70% -60% -50% -40% -30% -20% -10% 0% -80% -70% -60% -50% -40% -30% -20% -10% 0% Peak-to-trough drawdown Peak-to-trough drawdown

Figure 6.1: 1m Portfolios: Probability density function of the 99th percentile peak-to-trough drawdowns for the dierent strategies.

98 5-Day 10-Day 60 40 Protective Put, (-5.1%, 0.9%) Protective Put, (-7.4%, 1.4%) Put Spread 95/80, (-5.8%, 2.5%) Put Spread 95/80, (-10%, 6%) Put Spread 95/85, (-6%, 3.4%) 35 Put Spread 95/85, (-10.3%, 7.1%) 50 Put Spread 100/90, (-5.3%, 3.2%) Put Spread 100/90, (-9.3%, 7.2%) Collar 95/105, (-3.2%, 0.7%) 30 Collar 95/105, (-4.7%, 1.1%) Frac. Protective Put, (-5.2%, 0.8%) Frac. Protective Put, (-7.4%, 1.2%) 40 Frac. Put Spread 95/85, (-5.9%, 3.3%) Frac. Put Spread 95/85, (-10.4%, 7.5%) 25

30 20

15 20

10 Probability Density Function Probability Density Function 10 5

0 0 -35% -30% -25% -20% -15% -10% -5% 0% -40% -35% -30% -25% -20% -15% -10% -5% 0% Peak-to-trough drawdown Peak-to-trough drawdown

20-Day 63-Day 30 20 Protective Put, (-10.6%, 2.7%) Protective Put, (-15.5%, 3.1%) Put Spread 95/80, (-18%, 9.4%) 18 Put Spread 95/80, (-22.9%, 9.5%) Put Spread 95/85, (-18.9%, 11.1%) Put Spread 95/85, (-24.1%, 10.9%) 25 Put Spread 100/90, (-17.6%, 11.5%) 16 Put Spread 100/90, (-22.6%, 11.5%) Collar 95/105, (-6.4%, 1.6%) Collar 95/105, (-9.4%, 2.1%) Frac. Protective Put, (-10.8%, 3.1%) Frac. Protective Put, (-15.2%, 2.6%) 14 20 Frac. Put Spread 95/85, (-19.4%, 11.6%) Frac. Put Spread 95/85, (-24.4%, 11.4%) 12

15 10

8 10 6 Probability Density Function Probability Density Function 4 5 2

0 0 -60% -50% -40% -30% -20% -10% 0% -70% -60% -50% -40% -30% -20% -10% 0% Peak-to-trough drawdown Peak-to-trough drawdown

125-Day 250-Day 16 12 Protective Put, (-19.9%, 3.5%) Protective Put, (-25.5%, 4.9%) Put Spread 95/80, (-26.7%, 9.4%) Put Spread 95/80, (-31.4%, 10.1%) 14 Put Spread 95/85, (-27.9%, 10.7%) Put Spread 95/85, (-32.5%, 11.4%) Put Spread 100/90, (-25.8%, 11.5%) 10 Put Spread 100/90, (-29.6%, 12.1%) 12 Collar 95/105, (-12.1%, 2.6%) Collar 95/105, (-15.2%, 3.9%) Frac. Protective Put, (-19.7%, 2.7%) Frac. Protective Put, (-25.5%, 4.3%) Frac. Put Spread 95/85, (-28.2%, 11%) 8 Frac. Put Spread 95/85, (-32.9%, 11.5%) 10

8 6

6 4

4 Probability Density Function Probability Density Function 2 2

0 0 -70% -60% -50% -40% -30% -20% -10% 0% -80% -70% -60% -50% -40% -30% -20% -10% 0% Peak-to-trough drawdown Peak-to-trough drawdown

Figure 6.2: 3m Portfolios: Probability density function of the 99th percentile peak-to-trough drawdowns for the dierent option-based protection strategies.

99 5-Day 10-Day 70 40 Protective Put, (-5%, 1%) Protective Put, (-7.2%, 1.5%) Put Spread 95/80, (-5.8%, 2.8%) Put Spread 95/80, (-10%, 6.2%) 60 Put Spread 95/85, (-6.1%, 3.6%) 35 Put Spread 95/85, (-10.4%, 7.2%) Put Spread 100/90, (-5.8%, 3.7%) Put Spread 100/90, (-10%, 7.4%) Collar 95/105, (-2.4%, 0.6%) 30 Collar 95/105, (-3.6%, 1%) 50 Frac. Protective Put, (-5.1%, 0.8%) Frac. Protective Put, (-7.5%, 1.5%) Frac. Put Spread 95/85, (-6.1%, 3.6%) Frac. Put Spread 95/85, (-10.8%, 7.7%) 25 40 20 30 15

20 10 Probability Density Function Probability Density Function

10 5

0 0 -40% -35% -30% -25% -20% -15% -10% -5% 0% -40% -35% -30% -25% -20% -15% -10% -5% 0% Peak-to-trough drawdown Peak-to-trough drawdown

20-Day 63-Day 25 20 Protective Put, (-11%, 3.5%) Protective Put, (-15.6%, 3.8%) Put Spread 95/80, (-18.1%, 9.6%) 18 Put Spread 95/80, (-23.1%, 9.6%) Put Spread 95/85, (-19%, 11.2%) Put Spread 95/85, (-24.3%, 10.9%) 20 Put Spread 100/90, (-18.4%, 11.5%) 16 Put Spread 100/90, (-23.6%, 11.3%) Collar 95/105, (-5.5%, 1.8%) Collar 95/105, (-7.5%, 2%) Frac. Protective Put, (-11.7%, 4.3%) Frac. Protective Put, (-16.3%, 3.8%) Frac. Put Spread 95/85, (-19.9%, 11.8%) 14 Frac. Put Spread 95/85, (-25.3%, 11.4%)

15 12

10 8

6 Probability Density Function Probability Density Function 5 4

0 0 -60% -50% -40% -30% -20% -10% 0% -70% -60% -50% -40% -30% -20% -10% 0% Peak-to-trough drawdown Peak-to-trough drawdown

125-Day 250-Day 18 15 Protective Put, (-19%, 4.1%) Protective Put, (-23.9%, 4.6%) 16 Put Spread 95/80, (-26.6%, 9.9%) Put Spread 95/80, (-30.9%, 10.7%) Put Spread 95/85, (-28.1%, 11%) Put Spread 95/85, (-32.4%, 12%) Put Spread 100/90, (-27.2%, 11.5%) Put Spread 100/90, (-31.2%, 12.5%) 14 Collar 95/105, (-9%, 2.3%) Collar 95/105, (-11.2%, 2.8%) Frac. Protective Put, (-19.4%, 3.5%) Frac. Protective Put, (-24.4%, 3.6%) 12 Frac. Put Spread 95/85, (-28.9%, 11.3%) 10 Frac. Put Spread 95/85, (-33.3%, 12%)

6 5

Probability Density Function 4 Probability Density Function

0 0 -80% -70% -60% -50% -40% -30% -20% -10% 0% -80% -70% -60% -50% -40% -30% -20% -10% 0% Peak-to-trough drawdown Peak-to-trough drawdown

Figure 6.3: 6m Portfolios: Probability density function of the 99th percentile peak-to-trough drawdowns for the dierent option-based protection strategies.

100 5-Day 10-Day 70 40 Protective Put, (-4.9%, 1%) Protective Put, (-7.2%, 1.8%) Put Spread 95/80, (-6%, 3.1%) Put Spread 95/80, (-10.4%, 6.6%) 60 Put Spread 95/85, (-6.3%, 3.7%) 35 Put Spread 95/85, (-11%, 7.7%) Put Spread 100/90, (-6.2%, 3.8%) Put Spread 100/90, (-10.5%, 7.4%) Collar 95/105, (-1.7%, 0.6%) 30 Collar 95/105, (-2.7%, 1.1%) 50 Frac. Protective Put, (-5.2%, 0.9%) Frac. Protective Put, (-7.8%, 2.2%) Frac. Put Spread 95/85, (-6.4%, 3.8%) Frac. Put Spread 95/85, (-11.2%, 7.9%) 25 40 20 30 15

20 10 Probability Density Function Probability Density Function

10 5

0 0 -35% -30% -25% -20% -15% -10% -5% 0% -40% -35% -30% -25% -20% -15% -10% -5% 0% Peak-to-trough drawdown Peak-to-trough drawdown

20-Day 63-Day 25 20 Protective Put, (-11.2%, 4.4%) Protective Put, (-15.8%, 4.7%) Put Spread 95/80, (-18.5%, 10.1%) 18 Put Spread 95/80, (-23.7%, 9.9%) Put Spread 95/85, (-19.8%, 11.5%) Put Spread 95/85, (-25.4%, 11.1%) 20 Put Spread 100/90, (-19%, 11.3%) 16 Put Spread 100/90, (-24.6%, 11%) Collar 95/105, (-4.2%, 1.9%) Collar 95/105, (-5.8%, 2%) Frac. Protective Put, (-13.3%, 6.3%) Frac. Protective Put, (-18.2%, 5.7%) Frac. Put Spread 95/85, (-20.8%, 12.2%) 14 Frac. Put Spread 95/85, (-26.6%, 11.6%)

15 12

10 8

6 Probability Density Function Probability Density Function 5 4

0 0 -60% -50% -40% -30% -20% -10% 0% -70% -60% -50% -40% -30% -20% -10% 0% Peak-to-trough drawdown Peak-to-trough drawdown

125-Day 250-Day 20 18 Protective Put, (-18.7%, 5%) Protective Put, (-22.8%, 5.3%) 18 Put Spread 95/80, (-27.2%, 10.3%) 16 Put Spread 95/80, (-31.3%, 11.5%) Put Spread 95/85, (-29.1%, 11.3%) Put Spread 95/85, (-33.3%, 12.5%) 16 Put Spread 100/90, (-28.2%, 11.3%) Put Spread 100/90, (-32.3%, 12.6%) Collar 95/105, (-6.7%, 2.1%) 14 Collar 95/105, (-7.9%, 2.5%) Frac. Protective Put, (-21.3%, 5.4%) Frac. Protective Put, (-24.9%, 4.9%) 14 Frac. Put Spread 95/85, (-30.4%, 11.5%) 12 Frac. Put Spread 95/85, (-34.7%, 12.5%) 12 10 10 8 8 6 6

Probability Density Function Probability Density Function 4 4

2 2

0 0 -80% -70% -60% -50% -40% -30% -20% -10% 0% -80% -70% -60% -50% -40% -30% -20% -10% 0% Peak-to-trough drawdown Peak-to-trough drawdown

Figure 6.4: 12m Portfolios: Probability density function of the 99th percentile peak-to-trough drawdowns for the dierent option-based protection strategies.

6.3.2 Backtesting

101 350 350 S&P 500 S&P 500 Protective Put Protective Put Put Spread 95/80 Put Spread 95/80 Put Spread 95/85 Put Spread 95/85 300 Put Spread 100/90 300 Put Spread 100/90 Collar 95/105 Collar 95/105 Fractional Protective Put 1/4 1m Fractional Protective Put 1/3 3m Fractional Put Spread 95/85 1/4 1m Fractional Put Spread 95/85 1/3 3m 250 250

200 200 Portfolio value Portfolio value

150 150

100 100

50 50 2002 2005 2007 2010 2012 2015 2017 2020 2002 2005 2007 2010 2012 2015 2017 2020

Figure 6.5: Performance of the 1m portfolios. Figure 6.6: Performance of the 3m portfolios.

350 350 S&P 500 S&P 500 Protective Put Protective Put Put Spread 95/80 Put Spread 95/80 Put Spread 95/85 Put Spread 95/85 300 Put Spread 100/90 300 Put Spread 100/90 Collar 95/105 Collar 95/105 Fractional Protective Put 1/6 6m Fractional Protective Put 1/12 12m Fractional Put Spread 95/85 1/6 6m Fractional Put Spread 95/85 1/12 12m 250 250

200 200 Portfolio value Portfolio value

150 150

100 100

50 50 2002 2005 2007 2010 2012 2015 2017 2020 2002 2005 2007 2010 2012 2015 2017 2020

Figure 6.7: Performance of the 6m portfolios. Figure 6.8: Performance of the 12m portfolios.

102 100 110

100 90

90 80

70 Portfolio value Portfolio value

60 S&P 500 60 S&P 500 Protective Put Protective Put Put Spread 95/80 Put Spread 95/80 50 Put Spread 95/85 Put Spread 95/85 Put Spread 100/90 50 Put Spread 100/90 Collar 95/105 Collar 95/105 Fractional Protective Put 1/4 1m Fractional Protective Put 1/3 3m Fractional Put Spread 95/85 1/4 1m Fractional Put Spread 95/85 1/3 3m 40 40 10-2007 01-2008 04-2008 07-2008 10-2008 01-2009 04-2009 10-2007 01-2008 04-2008 07-2008 10-2008 01-2009 04-2009

Figure 6.9: Financial crisis: Performance of the Figure 6.10: Financial crisis: Performance of the 1m portfolios. 3m portfolios.

100 110

100 90

90 80

70 Portfolio value Portfolio value

60 S&P 500 60 S&P 500 Protective Put Protective Put Put Spread 95/80 Put Spread 95/80 50 Put Spread 95/85 Put Spread 95/85 Put Spread 100/90 50 Put Spread 100/90 Collar 95/105 Collar 95/105 Fractional Protective Put 1/6 6m Fractional Protective Put 1/12 12m Fractional Put Spread 95/85 1/6 6m Fractional Put Spread 95/85 1/12 12m 40 40 10-2007 01-2008 04-2008 07-2008 10-2008 01-2009 04-2009 10-2007 01-2008 04-2008 07-2008 10-2008 01-2009 04-2009

Figure 6.11: Financial crisis: Performance of the Figure 6.12: Financial crisis: Performance of the 6m portfolios. 12m portfolios.

103 6.3.3 Total cost of strategies

140% 450% Protective Put Protective Put Put Spread 95/80 400% Put Spread 95/80 120% Put Spread 95/85 Put Spread 95/85 Put Spread 100/90 Put Spread 100/90 350% 100% Collar 95/105 Collar 95/105 Fractional Protective Put Fractional Protective Put Fractional Put Spread 95/85 300% Fractional Put Spread 95/85 80% 250% 60% 200% 40% 150%

20% 100% Total cost as % of initial investment Total cost as % of initial investment 0% 50%

-20% 0% 0 2 4 6 8 10 12 0 2 4 6 8 10 12 Maturity (months) Maturity (months)

Figure 6.13: Monte Carlo: Total cost for the Figure 6.14: Backtesting, whole period: Total option-based protection strategies. cost for the option-based protection strategies.

30% Protective Put Put Spread 95/80 25% Put Spread 95/85 Put Spread 100/90 Collar 95/105 20% Fractional Protective Put Fractional Put Spread 95/85

15%

10%

Total cost as % of initial investment 0%

-5% 0 2 4 6 8 10 12 Maturity (months)

Figure 6.15: Backtesting, nancial crisis: Total cost for the option-based protection strategies.

104 Bibliography

[1] Annaert, J., Osselaer, S.V., Verstraete, B., 2009. Performance evaluation of portfolio insurance strategies using stochastic dominance criteria, Journal of Banking & Finance 33 (2), 272–280

[2] Bennet C., 2014, Trading Volatility, 1st ed, CreateSpace Independent Publishing Platform

[3] Bertrand, P. , Prigent, J.-L., 2005 Portfolio insurance strategies: OBPI versus CPPI, Finance 26 (1), 5–32

[4] Bertrand, P. , Prigent, J.-L., 2011, Omega performance measure and portfolio insurance, J. Bank Finance. 35 (7), 1811–1823

[5] Bjork¨ T., 2009, Arbitrage eory in Continuous Time, 3rd ed, Oxford University Press

[6] Black, F. & Jones, R., 1987, Simplifying portfolio insurance, e Journal of Portfolio Manage- ment, 48-51.

[7] Black, F. & Perold, A.R., 1992, eory of constant proportion portfolio insurance, e Journal of Economics, Dynamics and Control, 16, 403-426.

[8] Brooks, C., Davies, R., and Kim, S.S., 2006, Cross Hedging with Single Stock Futures

[9] Chaerjee R., 2014, Practical Methods of Financial Engineering and Risk Management, New York: Springer

[10] Chicago Board Options Exchange, 2012a Equity Options Strategies — Equity Collar, hps://www.cboe.com/strategies/equityoptions/equitycollars/partl.aspx. Who Should Use Equity Collars?, hps://www.cboe.com/strategies/pdf/equitycollarstrategy.pdf.

[11] Hull J., 2012, Options, futures, and other derivatives, 8th ed., England: Pearson

[12] El Karoui, N., Jeanblanc, M., Lacoste, V., 2005. Optimal portfolio management with American capital guarantee, Journal of Economic Dynamics & Control 25 (3), 449–468

[13] Figlewski S, Chidambaran N. K. and Kaplan S., 1993, Evaluating the Performance of the Pro- tective Put Strategy, Financial Analysts Journal, Vol. 49, No. 4 (Jul. - Aug., 1993), pp. 46-56+69

105 [14] Fodor, A., Doran, J. S., Carson, J. M. and Kirch, D. P., 2010, On the Demand for Portfolio Insurance,

[15] Israelsen, C, 2005 A renement to the Sharpe ratio and information ratio Journal of Asset Management, April 2005, Volume 5, Issue 6, pp 423–427

[16] Israelov R., 2017, Pathetic Protection: e Elusive Benets of Protective Puts

[17] Israelov R., Klein M Risk and Return of Equity Index Collar Strategies e Journal of Alter- native Investments

[18] Leland, H.E. , Rubinstein, M. , 1976, e evolution of portfolio insurance, In: Luskin, D.L. (Ed.), Portfolio Insurance: A Guide to Dynamic Hedging. Wiley, New York

[19]P ezier,´ J. , Scheller, J. , 2013, Best portfolio insurance for long-term investment strategies in realistic conditions Insurance 52 (2), 263–274

[20] Scholz, H., 2007, Renements to the Sharpe ratio: Comparing alternatives for bear markets Journal of Asset Management, January 2007, Volume 7, Issue 5, pp 347–357

[21] Zagst, R., Kraus, J., 2011, Stochastic dominance of portfolio insurance strategies, Annals of Operations Research 185 (1), 75–103.

[22] Zhu, Y., Kavee, R.C., 1988, Performance of portfolio insurance strategies, e Journal of Port- folio Management 14, 48–54

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