IN DEGREE PROJECT MATHEMATICS,SECOND CYCLE, 30 CREDITS

, STOCKHOLM SWEDEN 2018

Portfolio Protection StrategiesA study on the protective put and its extensions

GUSTAV ALPSTEN

SERCAN SAMANCI

KTH ROYAL INSTITUTE OF TECHNOLOGYSCHOOL 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.

1

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 mar-

ket 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 matu-

rities.

2

Abstrakt

Hantering av volatilitet i nansiella marknader har under de senaste decennierna

visat sig vara nödvändigt för investerare, framför allt i samband med krascher och

långdragna nedgångar i de globala aktiemarknaderna. Dea har ge upphov till

en likvid derivatmarknad i form av optioner samt väckt e intresse för forskning i

området. Tidigare studier har i synnerhet undersökt eektiviteten i den välkända

protective put-strategin som kombinerar en lång position i underliggande aktie med

en put-option. Resultaten är oa inte tilltalande och visar få fördelar med strategin,

där dess höga kostnader lys upp som e stort problem. Således väcks frågan om

protective put-strategin kan förbäras eller om det möjligtvis nns några kostnad-

seektiva alternativ med relativt bäre säkerhet mot eventuella nedgångar i under-

liggande. Denna studie utvidgar tidigare forskning i området genom a undersöka

förbäringsmöjligheter för och alternativ till protective put-strategin. Särskilt stud-

eras tre olika put spread-strategier och en collar-strategi, samt en modierad ver-

sion av protective put som ämnar a minska på vägberoendet i en löpande option-

sstrategi.

Resultatet från denna studie pekar på a ingen optionsbaserad strategi är universellt

bäst. Generellt se ger en avyring av delar av aktieinnehavet e mer eektivt

skydd, men vi visar a det nns situationer då en collar-strategi som köper 5 % OTM

put-optioner och säljer 5% OTM call-optioner har en araktiv risk-justerad prol och

säkerhet mot nedgångar. Vi visar vidare a protective put-strategin blir mer eektiv,

både i termer av en risk-justerad avkastning och som säkerhet mot svansrisker, för

längre förfallodatum på optionerna.

3

Contents

Introduction 7

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

6

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 meth-

ods 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 deriva-

tives 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 liq-

uid 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 posi-

tive 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 con-

vex 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 protec-

tion?

• 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 pro-

tection, 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 prede-

termined 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(ST ) = max(ST −K,0) = (ST −K)+ (2.1.1)

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 (Björk, 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 (ST ) = max(K − ST ,0) = (K − ST )+ (2.1.2)

where the strike priceK must exceed the stock price ST at maturity to result in a non-zero payo

(Björk, 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)

(K − St)+ (for a put option)(2.1.3)

(Benne, 2014).

13

0 10 20 30 40 50 60 70 80 90 100

Stock price, ST

0

5

10

15

20

25

30

35

40

45

50

Payoff

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

0 10 20 30 40 50 60 70 80 90 100

Stock price, ST

0

5

10

15

20

25

30

35

40

45

50

Payoff

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 com-

mon 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, mak-

ing 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:

St = S0e(µ− σ22 )t+σWt

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∂t

(t,St) +12S2t σ

2∂2F

∂S2t(t,St)− rF(t,St) = 0

F(T ,ST ) = Φ(ST )

(2.2.2)

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:

c(t,St) = StN[d1(t,St)

]− e−r(T−t)KN

[d2(t,St)

]p(t,St) = Ke

−r(T−t)N[− d2(t,St)

]− StN

[− d1(t,St)

] (2.2.3)where N is the cumulative distribution function for the N [0,1] distribution and

d1(t,St) =1

σ√T − t

ln(StK )+ (r + 12σ2)(T − t)

d2(t,St) = d1(t,St)− σ√T − t

(2.2.4)

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 + σStdW̃t (2.2.5)

where the expected rate of return is equal to the risk-free rate r and W̃t 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:

S∗t = 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:

St = S∗t −Dt

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:

S∗t e−q(T−t)

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

dSt = (r − q)Stdt + σStdW̃t (2.2.6)

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

stock index has the current price, S∗t e−q(T−t) and pays no dividends, the Black-Scholes-Merton

equations for call and put options on a stock index are:

18

c(t,S∗t e−q(T−t)) = S∗t e

−q(T−t)N[d1(t,S

∗t e−q(T−t))

]− e−r(T−t)KN

[d2(t,S

∗t e−q(T−t))

]p(t,S∗t e

−q(T−t)) = Ke−r(T−t)N[− d2(t,S∗t e−q(T−t))

]− S∗t e−q(T−t)N

[− d1(t,S∗t e−q(T−t))

](2.2.7)

d1(t,S∗t e−q(T−t)) =

1

σ√T − t

ln(S∗tK )+ (r − q+ 12σ2)(T − t)

d2(t,S∗t e−q(T−t)) = d1(t,S

∗t e−q(T−t))− σ

√T − t

(2.2.8)

where q is the anticipated annual dividend yield during the life of the stock index option, S∗t 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 de-

scribed 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 com-

pared to the lognormal distribution. Merton’s Jump Diusion model was introduced as an al-

ternative 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 num-

ber 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:

dSt = (r − q −λk)Stdt + σStdW̃t + dpt (2.2.9)

19

where dW̃t 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

size k = E(X) = eγ+δ2/2−1. 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:

∞∑n=0

e−λ′T (λ′T )n

n!fn (2.2.10)

where λ′ = 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:

σ2 +nδ2

T

and risk-free rate equal to:

r −λk + n ln(1 + k)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 po-

tential. 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:

∆put =∂p

∂S=N (d1)− 1 (2.2.11)

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:

Γput =∂∆∂S

=∂2p

∂S2=N ′(d1)

S0σ√T

(2.2.12)

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:

σrealized =

√√√252N − 1

N∑t=1

(rt − r̄)2 (2.3.1)

where N +1 is the number of observations and

rt = lnStSt−1

, r̄ =1N

N∑t=1

rt (2.3.2)

(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:

dStSt

= µdt + σimplieddWt (2.3.3)

(Hull, 2012).

Understanding the dierence between realized and implied volatility is fundamental to any in-

vestor 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, heav-

ily 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

2002 2005 2007 2010 2012 2015 2017 2020-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1A

nn

ua

lize

d v

ola

tilit

yImplied Volatility

Realized Volatility

Volatility Risk Premium

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

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 - March2018.

Percentiles99th 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 pro-

vides 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

0

Q2-20

0.2

Q1-20Q4-19

0.4

Q3-19 2Q2-19

Implie

d v

ola

tilit

y

0.6

Q1-191.5Q4-18

Time of maturity

0.8

Q3-18

Strike percentage

Q2-18 1

1

Q1-18Q4-17

0.5Q3-17Q2-17

Q1-17 0

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

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Strike percentage

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Imp

lied

vo

latilit

y

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

Time of maturity

0.14

0.145

0.15

0.155

0.16

0.165

0.17

0.175

0.18

0.185

0.19

Imp

lied

vo

latilit

y

Figure 2.5: Le: Volatility skew for a xed time to maturity. Right: Term structure for a xedstrike 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 10 20 30 40 50 60 70 80 90 100

Stock price, ST

-40

-30

-20

-10

0

10

20

30

40

50

Payoff

Figure 2.6: Protective put strategy: A long position in a put option with strike price K = 50 anda 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 · St , where p is given in

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).

0 10 20 30 40 50 60 70 80 90 100

Stock price, ST

-40

-30

-20

-10

0

10

20

30

40

50

Pa

yo

ff

0 10 20 30 40 50 60 70 80 90 100

Stock price, ST

-40

-30

-20

-10

0

10

20

30

40

50

Pa

yo

ff

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

0 10 20 30 40 50 60 70 80 90 100

Stock price, ST

-30

-20

-10

0

10

20

30

40

Pa

yo

ff

0 10 20 30 40 50 60 70 80 90 100

Stock price, ST

-40

-30

-20

-10

0

10

20

30

40

50

Pa

yo

ff

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

(c) Option payos are held as cash and grow with the risk-free interest rate between the

rollover dates

(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 pre-

miums 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

Protective Put Long 5% OTM put Full protection belowstrike 5% + premium Simple

Put Spread 95/80 Long 5% OTM put,short 20% OTM put Protection up to 20% dip 5% + 80% + net premium Low cost

Put Spread 95/85 Long 5% OTM put,short 15% OTM put Protection up to 15% dip 5% + 85% + net premium Low cost

Put Spread 100/90 Long ATM put,short 10% OTM put Protection up to 10% dip 90% + net premium Low cost

Collar 95/105 Long 5% OTM put,short 5% OTM callFull protection belowstrike, capped upside 5% + net premium Low cost

Fractional Protective Put Long overlapping fof 5% OTM putFull protection aera certain time 5% + (1− f )·95% + f ·premium

Reduced pathdependency

Fractional Put Spread 95/85 Long/Short overlappingf of 5%/15% OTM put

Full protection aera certain time

5% + 85% + (1− f )·10% +f ·net premium

Low cost, Reducedpath 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 inthe 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 perfor-

mance 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 down-

side risk). However, volatility can also be dampened via asset allocation. One such passive ap-

proach 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 underly-

ing 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 di-

vested 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 depen-

dency 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.

Arithmetic return = 252 ·[R1 +R2 + . . .RN−1

]· 1N−1

Geometric return =[(1 +R1) · (1 +R2) · . . . (1 +RN−1)

] 252N−1 − 1

(3.4.1)

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:

Sp =rp−rrfσp

(3.4.2)

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:

Sp,adj =ep

σep/abs(ep )p

(3.4.3)

where ep = rp − rrf is the excess return on the portfolio. is modication solves the ranking

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:

D(ti−k) =VP (ti)−VL(ti)

VP (ti)(3.4.4)

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:

Ra = Rf + β(RM −Rf ) (3.4.5)

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

marketRM−Rf is sometimes referred to as the equity risk premium. Oen the linear relationship

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 simu-

lation 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)

dt = 1252 (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)

(3.5.1)

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 period20 December 2002 - 23 March 2018.

Parameter Daily dataUnderlying asset S&P 500 indexStrike 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 maturityRisk-free interest rate Zero coupon rates for the dierent time to maturitiesDividend yield Expected S&P 500 dividend yield for the dierent time to maturitiesa 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 draw-

down 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

Tabl

e4.1:

Mea

nva

lueo

fann

ualiz

edre

turn

sand

perfo

rman

cem

easu

resf

orth

esim

ulat

edst

rate

gies

acro

ssth

efou

rpor

tfolio

s.

eadj

uste

dSh

arpe

ratio

isus

edon

stoc

kpa

thsw

ithne

gativ

ean

nual

ized

arith

met

icre

turn

.Eac

hst

rate

gyis

benc

hmar

ked

agai

nstt

hedi

vest

edeq

uity

.A

rithm

etic

retu

rnGe

omet

ricre

turn

Vola

tility

Shar

pera

tio1m

3m6m

12m

1m3m

6m12

m1m

3m6m

12m

1m3m

6m12

mPr

otec

tive

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

Div

este

dEq

uity

-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

PutS

prea

d95

/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

Div

este

dEq

uity

-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

PutS

prea

d95

/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

Div

este

dEq

uity

-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

PutS

prea

d10

0/90

-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

Div

este

dEq

uity

-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

Colla

r95/

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

Div

este

dEq

uity

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

Frac

tiona

lPro

tect

ive

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

Div

este

dEq

uity

-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

Frac

tiona

lPut

Spre

ad95

/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

%0.2

10.1

90.1

90.1

9D

ives

ted

Equi

ty-0

.9%-0

.9%-0

.8%-0

.7%-2

.2%-2

.2%-2

.2%-2

.3%16

.7%16

.7%17

.2%18

.3%0.2

70.2

40.2

30.2

1Di

+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

45

Tabl

e4.2:

Mea

nva

lueo

fthe

annu

aliz

edal

pha,

beta

and

tota

lcos

t(as

%of

initi

alin

vest

men

t)fo

rthe

simul

ated

stra

tegi

esac

ross

the

four

portf

olio

s.

eto

talc

osti

smea

sure

das

thea

ccum

ulat

edco

stov

erth

ewho

lein

vest

men

thor

izon

divi

ded

byth

eini

tiali

nves

tmen

t.

esiz

eoft

hein

itial

inve

stm

enti

seq

ualf

oral

lstra

tegi

es.

Ann

ualiz

edal

pha

Beta

Tota

lcos

tas%

ofin

itial

inve

stm

ent

1m3m

6m12

m1m

3m6m

12m

1m3m

6m12

mPr

otec

tive

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%

Div

este

dEq

uity

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%

PutS

prea

d95

/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%

Div

este

dEq

uity

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%

PutS

prea

d95

/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%

Div

este

dEq

uity

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%

PutS

prea

d10

0/90

-2.2%

-1.6%

-1.6%

-1.6%

0.60

0.67

0.73

0.79

139.8

%66

.6%39

.3%22

.8%D

ives

ted

Equi

ty0.0

%0.0

%0.0

%0.0

%0.5

60.6

20.6

80.7

10.0

%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%

Colla

r95/

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%

Div

este

dEq

uity

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%

Frac

tiona

lPro

tect

ive

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%

Div

este

dEq

uity

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%