IN
DEGREE PROJECT MATHEMATICS,
SECOND CYCLE, 30 CREDITS ,
STOCKHOLM SWEDEN 2018
Portfolio Protection
Strategies
A study on the protective put and its extensions
GUSTAV ALPSTEN
Portfolio Protection Strategies
A study on the protective put and its extensions
GUSTAV ALPSTGEN
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
The 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. This has given rise to a liquid portfolio insurance mar-ket in the form of options, as well as attracted the attention of many researchers. Previous literature has, in particular, studied the effectiveness of the widely known protective put strategy, which serially buys a put option to protect a long position in the underlying asset. The results are often uninspiring, pointing towards few, if any, protective benefits with high option premiums as a main concern. This raises the question if there are ways to improve the protective put strategy or if there are any cost-efficient alternatives that provide a relatively better protection. This 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 modified protective put is introduced to mitigate the path dependency in a rolling protection strategy. The results show that no option-based protection strategy can dominate the other in all market situations. Although reducing the equity position is generally more effective than buying options, we report that a collar strategy that buys 5% OTM put options and sells 5% OTM call options has an attractive risk-reward profile and protection against drawdowns. We also show that the protective put becomes more effective, both in terms of risk-adjusted return and tail protection, for longer matu-rities.
Abstrakt
Hantering av volatilitet i finansiella marknader har under de senaste decennierna visat sig vara n¨odv¨andigt f¨or investerare, framf¨or allt i samband med krascher och l˚angdragna nedg˚angar i de globala aktiemarknaderna. Detta har gett upphov till en likvid derivatmarknad i form av optioner samt v¨ackt ett intresse f¨or forskning i omr˚adet. Tidigare studier har i synnerhet unders¨okt effektiviteten i den v¨alk¨anda protective put-strategin som kombinerar en l˚ang position i underliggande aktie med en put-option. Resultaten ¨ar ofta inte tilltalande och visar f˚a f¨ordelar med strategin, d¨ar dess h¨oga kostnader lyfts upp som ett stort problem. S˚aledes v¨acks fr˚agan om protective put-strategin kan f¨orb¨attras eller om det m¨ojligtvis finns n˚agra kostnad-seffektiva alternativ med relativt b¨attre s¨akerhet mot eventuella nedg˚angar i under-liggande. Denna studie utvidgar tidigare forskning i omr˚adet genom att unders¨oka f¨orb¨attringsm¨ojligheter f¨or och alternativ till protective put-strategin. S¨arskilt stud-eras tre olika put spread-strategier och en collar-strategi, samt en modifierad ver-sion av protective put som ¨amnar att minska p˚a v¨agberoendet i en l¨opande option-sstrategi.
Resultatet fr˚an denna studie pekar p˚a att ingen optionsbaserad strategi ¨ar universellt b¨ast. Generellt sett ger en avyttring av delar av aktieinnehavet ett mer effektivt skydd, men vi visar att det finns situationer d˚a en collar-strategi som k¨oper 5 % OTM put-optioner och s¨aljer 5% OTM call-optioner har en attraktiv risk-justerad profil och s¨akerhet mot nedg˚angar. Vi visar vidare att protective put-strategin blir mer effektiv, b˚ade i termer av en risk-justerad avkastning och som s¨akerhet mot svansrisker, f¨or l¨angre f¨orfallodatum p˚a optionerna.
Contents
Introduction 7
Theoretical Background 12
2.1 Option contracts . . . 12
2.1.1 Mathematical definition 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 Diffusion 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
3.3.3 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
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
Introduction
The watchword among traders of financial instruments has been, and is today, volatility. It is a measure that captures the degree of uncertainty in the price movements of financial assets. Following a sequence of unprecedented global economic events, such as the financial 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. The 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.
There 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. Derivaderiva-tives are contracts that can be used to reduce the portfolio’s equity exposure by providing downside protection during market drawdowns. The most common instruments used for hedging are futures and options. Both are traded on liq-uid and transparent markets. The distinctive difference between them, from a hedging point of view, resides in the investment flexibility and liability attributed to the contract parties. Sophisti-cated hedgers often seek flexibility to construct suitable pay-off profiles, which are best achieved through option contracts. Futures provide little flexibility in this matter and is generally inferior
The concept of hedging equity investments against market drawdowns, while allowing upside participation, is often 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 predefined floor, or a cushion, by dynamically rebalancing allocations between the risky and risk-less asset (Bertrand and Prigent, 2005).
The effectiveness of the OBPI and CPPI strategies have been thoroughly studied and compared in the literature. The 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. The 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 different types of strike methods: a fixed strike, a fixed percentage strike and a combination of both. Figlewski et al. (2013) argue that the fixed percentage strike method, which resets the strike price at a fixed 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 fixed percentage strike is much less protective than the fixed strike method in a prolonged bear market and provides very limited protection when out-of-the-money (OTM) puts are used. The fixed 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 drift away from the fixed strike price. The results from the study, in terms of combined limited downside risk and attractive mean return, favors a fixed percentage strategy using in-the-money (ITM) or
at-the-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 effectiveness of the protective put strategy by comparing its peak-to-trough drawdown characteristics to those of a static risk-reducing strategy. The 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) finds that unless the option purchases and their maturities are timed just right around market drawdowns, the protective put strategy may offer little downside protection. The 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. The combined effect 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 find out if there
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 offsets the negative alpha from the purchased put (Bennett, 2014). Israelov and Klein (2016) compares the protectiveness of the collar to the divested equity strategy. They find, in particular, that investors would be better off simply reducing their equity exposure rather than investing in a collar strategy. Their 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. These findings point to yet another inferior strategy compared to the divested equity strategy.
The 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 protecposi-tiveness. This raises the question if there are other portfolio insurance constructions that provide more attractive risk-adjusted return and tail protection relative to the previous findings. 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 often a more convenient construction for the investor who may not have the time to continuously reset the equity exposure. This creates incentives to search for protection strategies that are better alternatives than the divested equity strategy.
This study aims to draw upon the previous findings and add several more protection strategies to the analysis. The 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 financial crisis and other preceding comparable events. This study will also include more advanced options strategies, including a version of the Bear Put Spread and the Collar. In addition, we present a modified version of the protective put and the put spread strategies in an attempt to reduce the negative impact of path dependence, which was particularly emphasized as detrimental to returns by Israelov (2017). The strategies are benchmarked against each other and a divested equity strategy which does not include any use of options.
We formulate our research questions as follows:
• Is the protective put strategy really an effective 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-efficient?
– Reducing the path dependency of a rolling option strategy?
The 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). There 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 different strategies against periods of falling asset prices that persist over different time windows.
The study will be structured as follows: We begin by presenting, in Chapter 2, the preliminaries of option theory. This 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. The results of their performance are presented in tables and graphs in Chapter 4 and are discussed in detail in Chapter 5.
Theoretical Background
2.1
Option contracts
An option contract (also called an option) is a financial 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. There 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. The predetermined date and the predetermined price are commonly referred to as the maturity date and the strike price, respectively (Hull, 2012).
There are various styles of options, each with different terms specifying under what conditions they are allowed to be executed. These are broadly categorized into vanilla options and exotic options. Vanilla options are further classified 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).
The 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 hereafter implicitly refers to European stock options.
2.1.1 Mathematical definition of European options
The holder of a European option will claim the payoff Φ(ST)at maturity time T , where Φ is the
contract function and ST is the stock price at maturity.
The 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) = (S
T−K)+ (2.1.1)
where K is the strike price. The call option has a non-zero payoff at maturity when the stock price ST exceeds the strike price K, otherwise it expires worthless. Figure2.1shows the
payoff-diagram for a European call option (Bj¨ork, 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 price K must exceed the stock price ST at maturity to result in a non-zero payoff
(Bj¨ork, 2009). Figure2.2shows the payoff-diagram for a European put option.
The value calculated in equations2.1.1and2.1.2is 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) (Bennett, 2014).
0 10 20 30 40 50 60 70 80 90 100 Stock price, S T 0 5 10 15 20 25 30 35 40 45 50 Payoff
Figure 2.1: Payoff-diagram for a European call option with strike price K = 50.
0 10 20 30 40 50 60 70 80 90 100 Stock price, S T 0 5 10 15 20 25 30 35 40 45 50 Payoff
Figure 2.2: Payoff-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. The price is determined by the market, influenced by the participants’ attitude to risk and expectations about the future stock prices. These 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 (Bennett, 2014).
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. The 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), (Bennett, 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 (Bennett, 2014).
2.2
Black-Scholes-Merton Model
There 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 Scholes-Merton model. The Black-Scholes-Merton model is widely used by option market participants and is perhaps the world’s most well-known option pricing model. The model is both arbitrage free and complete, mak-ing the prices it produces unique. Furthermore, the model is developed based on the followmak-ing assumptions (Hull, 2012):
1. The stock price S(t) dynamics is described by the geometric Brownian motion:
where S0is the initial stock price.
2. Volatility is constant. 3. Short selling is permitted.
4. There are no transaction costs or taxes.
5. There are no dividends during the life of the option. 6. There are no arbitrage opportunities.
7. Trading is continuous.
8. The 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 differential equation problem:
∂F ∂t(t, St) + 1 2S 2 tσ2 ∂2F ∂St2(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 written to maturity, 0 ≤ t ≤ T . The boundary condition
F(T , ST) = Φ(ST)ensures that the price of the option at maturity is equal to the payoff of the
contract at maturity (Hull, 2012).
Solving equation 2.2.2for 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 h d1(t, St) i −e−r(T −t)KNhd2(t, St)i p(t, St) = Ke −r(T −t) Nh−d2(t, St)i−StNh−d1(t, St)i (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 S t K + r +1 2σ 2(T − t) √ (2.2.4)
(Hull, 2012).
The presence of an expected rate of return, µ, in equation2.2.1introduces a risk preference, which may vary among investors. The Black-Scholes-Merton differential equation2.2.2is 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 + σ Std ˜Wt (2.2.5)
where the expected rate of return is equal to the risk-free rate r and ˜Wtis a Wiener process under
the risk-neutral measure.
2.2.1 Black-Scholes-Merton model with dividends
The Black-Scholes-Merton formulas in2.2.3are derived based on non-dividend paying stocks. In reality, stocks often pay dividends and to take this into account the Black-Scholes-Merton formula must be modified. 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
Given that the Black-Scholes-Merton formula is derived based only on the risky component, the equations in2.2.3can 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:
St∗e−q(T −t)
where t is the current time and T is the time of maturity. The risk-neutral price process of a stock with dividend yield q can be written as
dSt= (r − q)Stdt + σ Std ˜Wt (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. The theory presented in the previous section2.2.1can 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∗
te
−q(T −t)and pays no dividends, the Black-Scholes-Merton
c(t, St∗e−q(T −t)) = St∗e−q(T −t)Nhd1(t, S ∗ te −q(T −t) )i −e−r(T −t)KNhd2(t, S∗ te −q(T −t) )i p(t, St∗e−q(T −t)) = Ke−r(T −t)Nh−d2(t, S∗ te −q(T −t) )i −S∗ te −q(T −t) Nh−d1(t, S∗ te −q(T −t) )i (2.2.7) d1(t, S ∗ te −q(T −t) ) = 1 σ √ T − t ln S∗ t K + r − q +1 2σ 2(T − t) d2(t, S ∗ te −q(T −t) ) = d1(t, S ∗ te −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 Diffusion Model
The Black-Scholes-Merton model assumes that stock returns have a lognormal distribution de-scribed by the geometric Brownian motion presented in section2.2. 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 Diffusion 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. The 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,
where d ˜Wtand dptare assumed to be independent.
The 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 modifications to the geometric Brownian motion and assumptions for the jumps, one can show that the price of the European option can be written as:
∞ X n=0 e−λ0T(λ0T )n n! fn (2.2.10) where λ0
= λ(1+k)and fnis the Black-Scholes-Merton option price obtained from equation2.2.7
for an underlying asset with dividend yield q, variance equal to:
σ2+nδ
2
T
and risk-free rate equal to:
r − λk +n ln(1 + k)
T
(Hull, 2012).
Merton’s Jump Diffusion 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. The price obtained in2.2.10is not unique, since there is no unique risk-neutral measure in an incomplete market. However, by a change of probability measure, and effectively by changing the drift according to equation2.2.9, Merton constructed an arbitrage-free model. Yet, due to the incompleteness, it is not possible to construct a replicating portfolio and no perfect hedge.
2.2.4 Convexity of options
The attractiveness of options for hedging revolves around their non-linear (”convex”) payoff structure. For example, put options provide downside protection while preserving upside po-tential. The value of put options rises as the price of the underlying stock falls. This is captured by the measure delta, ∆. It is given by the first derivative of the price equation for the put option
in2.2.3with 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 in2.2.3with respect to the underlying stock price - or the first derivative of ∆ with respect to the underlying stock price. The 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. This measure is known as the gamma, Γ , and quantifies the sensitivity of ∆ to changes in the stock price:
Γput=∂∆ ∂S = ∂2p ∂S2 = N0(d1) S0σ √ T (2.2.12)
which varies with respect to the underlying stock price, S, as a bell-shaped curve. The described relationship between gamma and the underlying stock price is typical for a long position in a put option. This 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).
2.3
Volatility
2.3.1 Realized versus implied volatility
The volatility, σ, measures the amount of variability in the stock returns. There are two volatility measures that are of interest to participants in the options market. The first one, realized volatility, also called historical volatility, is an ex-post estimate of stock return variation. It is defined as the annualized standard deviation of daily stock returns:
σrealized= v u t 252 N − 1 N X t=1 (rt−¯r)2 (2.3.1)
where N + 1 is the number of observations and
rt= ln St St−1 , ¯r = 1 N N X t=1 rt (2.3.2) (Hull, 2012).
The 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
St = µdt + σimplieddWt (2.3.3)
(Hull, 2012).
Understanding the difference between realized and implied volatility is fundamental to any in-vestor engaging in options trading. As can be seen in equations2.2.3and2.2.4, 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. The 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
implied volatility at which the option was bought (Bennett, 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 equation2.2.3to compensate for the risk of losses during periods when realized volatility suddenly increases, also called crash risk. Note in Figure2.3how realized volatility spiked higher than the implied volatility during the financial crisis 2007 - 2008. The implied volatility, however, tends to exceed the realized volatility on average. The discrepancy between implied volatility and the subsequent realized volatility is known as the volatility risk premium. The volatility risk premium indicates how expensive an option is and reduces realized returns for the buyer (Israelov, 2017).
The volatility risk premium over a certain period can, for example, be measured by the difference between the S&P 500 annualized 1-month ATM implied volatility and the S&P 500 annualized subsequent 1-month realized volatility. The volatility risk premium at time t is thus the difference 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 Figure2.3, the volatility risk premium is plotted over the period December 2002 - March 2018.
2002 2005 2007 2010 2012 2015 2017 2020 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Annualized volatility Implied Volatility Realized Volatility Volatility Risk Premium
Figure 2.3: The volatility risk premium measured as the difference between the S&P 500 annual-ized 1-month ATM implied volatility and the S&P 500 annualannual-ized subsequent 1-month realannual-ized 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 table2.1for more statistics.
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
The 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 often plotted as two separate two-dimensional graphs.
The 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 Figure2.4). Hence, the volatility used to price an option with low strike price (i.e. a deep OTM put or a deep ITM call) is significantly higher than that used to price options with higher strike price (i.e. a deep ITM put or a deep OTM call). By fixing the time to maturity, the volatility skew can be depicted in a two-dimensional graph (see Figure2.5), (Hull, 2012).
Plotting 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. The 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 Figure2.5). This reflects an expectation that the volatility will increase over time, leading to more richly priced long-dated options (Hull, 2012).
0 Q2-20 0.2 Q1-20 Q4-19 0.4 Q3-19Q2-19 2 Implied volatility 0.6 Q1-19 1.5 Q4-18 Time of maturity 0.8 Q3-18 Strike percentage Q2-18 1 1 Q1-18 Q4-17 0.5 Q3-17 Q2-17Q1-17 0
Figure 2.4: A volatility surface for options on the S&P 500 Index. The time of maturity spans from 2 April 2018 to 23 March 2021, where the latter is the longest dated option available in the dataset. The 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 Implied volatility 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 Implied volatility
Figure 2.5: Left: Volatility skew for a fixed time to maturity. Right: Term structure for a fixed strike price.
The 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. This information is useful for building suitable option strategies. For example, a flat (or flatter) 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 Figure2.5. Thus, the observed volatility skew in Figure2.5informs 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.
2.4
Protection strategies
A myriad of portfolio insurance strategies have been developed to manage volatility and tail risk exposure. This section presents some of the most widely employed option-based protection strategies.
2.4.1 Protective Put Strategy
The 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. The strategy protects the portfolio from losses below a certain strike level, while allowing unlimited profits as long as the underlying asset’s price rises. The 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. The 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. Figure2.6shows the payoff 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
• A fixed value, K = C, where C is a constant
• A fixed 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 latter is more common since its the cheapest alternative and is expected to reduce the portfolio return the least. The 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. The strategy takes advantage of the observed volatility skew in Figure2.5by selling a richly priced deeper OTM put and buying a relatively cheaper OTM put with a higher strike price, which effectively reduces the net premium paid. The left graph in Figure2.7illustrates a payoff diagram for a bear put spread. The right graph shows the payoff diagram when it is used as a hedge on a long position in an underlying asset, e.g. a stock (Bennett, 2014).
0 10 20 30 40 50 60 70 80 90 100 Stock price, S T -40 -30 -20 -10 0 10 20 30 40 50 Payoff 0 10 20 30 40 50 60 70 80 90 100 Stock price, S T -40 -30 -20 -10 0 10 20 30 40 50 Payoff
Figure 2.7: Left: A bear put spread consisting of a long position in a put option with strike price
K = 60and a short position in a put option with strike price K = 30. Right: A bear put spread
A bear put spread as a hedging strategy is more complex compared to its protective put counter-part. The maximum protection provided by the bear put spread is equal to the size of the spread less the net premium paid. The 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-off between keeping the net premium paid as low as possible while maximizing the coverage of potential losses. The right graph in Figure
2.7illustrates the extent of the hedge obtained by the bear put spread. The stock is hedged only down to ST = 30, below which any additional losses are not covered (Bennett, 2014).
2.4.3 Collar Strategy
The 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 offset) 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. The net cost of a collar is often less than the net cost of a put spread (see section2.4.2) and is therefore considered as a low-cost method for protection. However, to obtain this significant reduction in cost, the strategy gives up some upside. The 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 offsets the premium paid for the long put option, the strategy is called a zero cost collar (Bennett, 2014).
The left graph in Figure2.8illustrates a payoff diagram for the collar strategy. The right graph shows the payoff diagram when it is used as a hedge on a long position in an underlying asset, e.g. a stock (Bennett, 2014).
0 10 20 30 40 50 60 70 80 90 100 Stock price, ST -30 -20 -10 0 10 20 30 40 Payoff 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 2.8: Left: 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.
The collar is more exposed to skew compared to the bear put spread because of the parabola-like shape around the ATM level (see the left graph in Figure2.5).
Methodology
3.1
Overview
This study aims to evaluate the protective performance of different protection strategies involv-ing options. The 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. The study is split into two parts. First, a Monte Carlo simulation is performed to evaluate the different strategies under ideal conditions. The simulation aggregates the performance of the dif-ferent strategies over varying stock price developments, including rising markets, bear markets and extreme crashes. Section3.5covers this part in more detail.
The results from the Monte Carlo simulation cannot be generalized beyond the ideal conditions it was performed under. Therefore, 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 significant 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 specific market environment that prevailed during the backtesting.
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 different option strategies. In particular, the construction of the option strategies is based on the following:
1. The portfolio is fully invested in the underlying asset (S&P 500 index) 2. Rolling purchases and selling of options:
(a) The first option is financed 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) The following options are financed by borrowing at the risk-free interest rate if, at each rollover date, the accumulated payoffs from previous expired options are not sufficient to cover the cost of buying a new option
(c) Option payoffs are held as cash and grow with the risk-free interest rate between the rollover dates
(d) No reinvestments of option payoffs
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. The options are either funded by accumulated payoffs or by borrowing at the risk-free interest rate, where the latter would reduce returns with an extra amount equal to the interest expense.
3.3
Option-based protection strategies and portfolios
The 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. The 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 profits from the options. The 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 suffers 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.
Table3.1provides an overview of the selected option-based protection strategies. The 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 Section3.3.2.
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 call Full protection belowstrike, capped upside 5% + net premium Low cost Fractional Protective Put Long overlapping fof 5% OTM put Full protection aftera certain time 5% + (1 − f )·95% + f ·premium Reduced pathdependency Fractional Put Spread 95/85 Long/Short overlappingfof 5%/15% OTM put Full protection aftera certain time 5% + 85% + (1 − f )·10% +f ·net premium Low cost, Reducedpath dependency afrepresents 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 after 3 months.
bThe maximum loss of the Fractional Protective Put strategy occurs in the first 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
The option strategies in Table3.1will be implemented in four different portfolios. The portfolios are presented in Table3.2and are designed to show how the different option-based strategies perform across different maturities of the constituent options.
Table 3.2: Description of the four different 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.
aThe fractional strategies are exceptions. They 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
The 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 deficiencies. 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 Appendix6.1.1for a modelling example.
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. The 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 willunderly-ingness to assume risk against expected returns through the exposure to the risky asset. See Section6.1.2
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 Section3.4for the definition) as the option-based strategy it is compared to. This 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
The put spread 95/80, put spread 95/85, put spread 100/90 and collar 95/105 in Table3.1are all low-cost strategies compared to the protective put strategy. The put spreads are of special interest because they will shed light on the benefits, if there are any, of forgoing some protection for lower option premiums. The collar, on the other hand, caps the upside in exchange for lower premiums, which gives an equity exposure that is different from the protective put, but still interesting for comparative purposes. Ultimately, the comparison provides a perspective on what is worth sacrificing for improved risk-adjusted return and reduced tail risk.
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. The 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. The reason is to keep the costs low, while not giving up too much protection from the ATM level. In order to capture the effect 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 Section2.4.1, the protective put strategy is exposed to path dependency risk. When a put option with a fixed 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. Figure2.3shows how implied volatility spiked during the financial crisis, which at the time led to very high premiums for the put options.
As a measure to address the path dependency risk, a modified protective put strategy has been constructed. The 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. The 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.
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 twelfth month. The 1m portfolio is slightly different, but follows the same logic. It buys 1/4 of a 1m option every week and reaches full protection by the first month.
In the first months of the strategy, the portfolio is only partially hedged and therefore more exposed to market downturns. The 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
The fractional put spread 95/85 is designed to combine the benefits 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 ·h R1+ R2+ . . . RN −1 i · 1 N −1 Geometric return = h(1 + R1) · (1 + R2) · . . . (1 + RN −1)i 252 N −1− 1 (3.4.1)
Risk-adjusted performance
The risk-adjusted performance of a portfolio will be assessed on its excess return per unit of volatility. This measure is captured by the Sharpe ratio. It is defined as the excess return over the risk-free rate per unit of volatility:
Sp =
rp−rrf
σp (3.4.2)
where rpis the portfolio arithmetic return, rrf is the risk-free interest rate and σpis 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 often outperforms the portfolio returns. A negative excess return gives a negative Sharpe ratio, which makes the relative ranking between the different portfolios misleading. In order to address the ranking issue, Israelsen (2009) proposed an adjusted Sharpe ratio:
Sp,adj =
ep
σpep/abs(ep) (3.4.3)
where ep = rp−rrf is the excess return on the portfolio. This modification solves the ranking
issue. The less negative the adjusted Sharpe ratio is, the better is the risk-adjusted performance, whereas larger negative values indicate the opposite.
Peak-to-trough drawdowns
In order to measure the effectiveness 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. The 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 specific time window. Let VP(ti)and VL(ti)represent the value of the portfolio at its highest peak and lowest trough,
5, 10, etc.). Then 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.
The effectiveness of the protection is measured at the 99th, 95th and 50th (median) percentiles of the peak-to-trough drawdowns over the different windows. The 99th and 95th percentiles, in particular, quantify the effectiveness of the tail protection of the strategy.
Capital Asset Pricing Model: Beta and Alpha
The capital asset pricing model (CAPM) describes the relationship between systematic (undiver-sifiable) risk and expected return on an asset, e.g. a stock. The excess return on the asset over the risk-free rate is regressed against the excess return on the market over the risk-free rate. The 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 market RM−Rf is sometimes referred to as the equity risk premium. Often the linear relationship
in equation3.4.5has a non-zero intercept, α, which captures abnormal returns that cannot be explained by the correlation with the market returns (Hull, 2012).
3.5
Monte Carlo Simulation
A Monte Carlo simulation is performed to produce different sets of future stock prices following a distribution based on the Merton’s Jump Diffusion model presented in section2.2.3. The simu-lation enables an idealized environment and captures many different possible outcomes for the underlying stock. The different outcomes allow for more generality in the results, but their ap-plicability to the reality is restricted to the ideal conditions, which often are very different from the actual ones. It is, however, worthwhile to examine how the different protection strategies compare against each other and how they fare against the divested equity strategy in an ideal setting.
We simulate 1,000 stock paths with a 5 year investment horizon, i.e. T = 5 · 252 = 1,260 days. The stock prices are drawn from a lognormal distribution with the addition of a Poisson-driven jump process in accordance with the risk-neutral process2.2.9in the Merton’s Jump Diffusion model. The jump size is drawn from a lognormal distribution, N(−0.5,0.052), and with a jump
frequency of 0.1 per year. These 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. The jump component contributes with an amount of −λk = 0.0393 to the drift 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. The 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. The 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). Thus, 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 Diffusion model, equation2.2.10.
r = 0% (risk-free interest rate)
S0 = 100 (The price of the stock at the initial investment)
q = 0% (dividend yield)
σs = 20% (stock volatility)
σp = 22% (stock volatility + volatility risk premium) dt = 2521 (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)
The 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. This 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 different 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. Thus, as crashes occur in the simulated stock paths, the time value of the different protection strategies will not be affected. In reality, the cost of put options would rise significantly, making, for example, the protective put more expensive. This scenario is covered in the backtesting part of the study.
3.6
Backtesting
Their 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. Table3.3
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 different time to maturities
Dividend yield Expected S&P 500 dividend yield for the different time to maturities aThese are fixed values and do not vary by days.
Source: Bloomberg
3.6.2 Outline
The option-based protection strategies and portfolios presented in Table3.1and Table3.2 are modelled based on the assumptions presented in Section3.2and are performed on the data set in Table3.3. All four portfolios (1m, 3m, 6m and 12m) implement each of the different strategies and their performance is measured in terms of risk-adjusted return and peak-to-trough drawdowns. The backtesting consists of two parts:
The whole period: 20 December 2002 - 23 March 2018
As depicted in Figure 2.3, the volatility in the S&P 500 index varied significantly during the whole period. This part examines the performance of the protection strategies over a long-term investment horizon and how they fare during shifting market environments.
The financial crisis: 14 December 2007 - 5 March 2009
Backtesting the protection strategies on the data from the financial 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
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. Those cases (or stock paths) are omitted to enable a fair comparison between them.
In addition to the results presented in this section, Tables6.1-6.4in Appendix report the draw-down characteristics in more detail. Figure6.13in Appendix visualizes the change in the total cost as a percentage of the initial investment with respect to option maturity.
