The relative efficiency of option hedging strategies using the third-order stochastic dominance

Abstract

The selection of an appropriate portfolio hedging strategy is a concern for both investment theory and practice. Options are believed to be flexible and useful hedging instruments, but they are also complicated to manage and costly to implement in a strategy, intensifying the financial leverage, and thus increasing the riskiness of a portfolio. This paper investigates the hedging strategies with options as efficient hedging tools and the possibility to identify an efficient hedging strategy with options that will not result in smaller utility for investors, compared to other strategies under defined criteria. The empirical analysis is based on simulated returns of the following trading strategies: a covered call, protective put, collar, ratio covered call strategy and a buy-and-hold unhedged stock. The return distributions of these strategies are compared using the stochastic dominance criteria up to the third degree, which is an appropriate approach for investors who prefer the greater return, are risk-averse in terms of downside volatility and in terms of a loss, and who prefer greater positive skewness. The results obtained from the simulated returns indicate that portfolio hedging strategies with options are never dominated by an unhedged portfolio. This finding confirms that hedging strategies with options are useful tools for risk hedging. The methodology used in the paper also presents the general framework that can be used in investment decision-making in the presence of uncertainty.

1 Introduction

Risk management of a financial portfolio is a necessary and challenging task in turbulent financial markets for investors at all scales. Therefore, numerous researches investigate potential approaches and strategies for efficient decision-making and portfolio risk management. Generally, research in risk assessment and risk management is getting more and more sophisticated, and new methods and techniques are being developed for various fields and sectors (see Aven 2016). In a multitude of approaches, one possible approach of portfolio risk management relies heavily on different financial instruments, which are used, for example, to transfer risk from one period to another or provide a distribution of risks across a number of (almost) independent risks (Tapiero 2004). Financial derivatives, such as options, are one of these instruments.

This paper specifically focuses on investigating the effectiveness of risk hedging strategies using options. Usually, the payoffs of portfolios with options are modelled in a simplified manner as functions of the price of an underlying asset and illustrated with payoff diagrams and profit functions graphs. In this simplified approach, it is easy to construct specific strategies, evaluate their possible outcomes regarding the price change of an underlying asset, and finally choose a strategy with the desired effect on a portfolio payoff. Following such findings, one could conclude that using options for portfolio hedging is straightforward and that strategies with options are generally useful tools for hedging risk. However, these findings should be reconsidered for several reasons.

The main reason is that the idealistic scenarios of trading options, as assumed in simplified models, are not usually achievable in real financial markets. First of all, market conditions change all the time, and portfolio strategies should be frequently reassessed and rebalanced, which is time-consuming and potentially unprofitable due to transaction costs. This aspect of trading options is completely ignored in simplified static analyses. Secondly, there would not always be an option on the market with ideal characteristics for constructing the desired hedging strategy, meaning that setting up a theoretical hedging strategy using options may not be possible. Thirdly, options have a leverage effect. It is well known that options in a portfolio tend to intensify and multiply the portfolio’s payoffs caused by the price changes of an underlying asset. In the case of favourable market movements, this is a positive feature, but in the case of unexpected and unfavourable market events, the (inappropriate) use of options will multiply the loss. These are only some of the existing problems connected with trading options and applying trading strategies with options. Thus, this paper questions if strategies with options can be justifiably characterised as efficient and useful risk hedging instruments.

Although this problem was considered in some previous researches, this paper further develops the research methodology and tools for investigating this problem, thus providing a new research framework that delivers new findings. Furthermore, the evaluation of the strategies with options as risk hedging tools is based on the concept of relative efficiency. Certain strategies can be attributed as (relatively) efficient, given a set of criteria, if and only if there is no other strategy that is better according to at least one criterion and is no worse in all other criteria. This is consistent with the definition of a Pareto-Koopmans efficiency (Koopmans 1951). The efficiency criteria should be meaningful and arise from the preferences of the decision-maker. Hence, to determine whether some risk hedging strategies should be preferred by some investors over other portfolio strategies, the attitude of the investors should be known. However, specifying these attitudes is, in most cases, impossible due to the limited knowledge of the researchers. Also, such an approach limits the findings of the analysis to a small group of investors with the same attitudes. In cases where there exists only some general knowledge about the preferences of the decision-makers and the empirical distribution of the return of financial portfolios, finding efficient portfolios for a group of investors with the same set of preferences can be carried out using the stochastic dominance (SD) criteria. The SD methodology has also been recommended in cases where the return distributions are not normal, as in the case of options portfolio (Faias and Santa-Clara 2017) and where there is limited knowledge about the utility functions of the decision-makers (Levy 2016). If we assume that our target investors prefer greater return and are risk-averse, we can use the second-order stochastic dominance (SSD) criteria in order to find efficient portfolios for these investors. Following the findings of Scott and Horvath (1980), who concluded that investors should prefer greater values of odd moments of distribution and smaller absolute values of even moments, we assume that our target investors also prefer greater positive values of the third moment, i.e. skewness. Positive preference for the greater third moment is confirmed empirically by Friedman and Savage (1948) and Levy (2006). The addition of the assumption about the investors’ preferences towards the greater positive skewness requires the use of the third-order stochastic dominance (TSD) criteria. Therefore, the analysis investigates the TSD between five different portfolio strategies: passive buy-and-hold long stock portfolio, covered call, protective put, covered ratio call and collar. These strategies are simulated using the empirical data from Optionmetrics database in the period of 1996–2014, which is described in Sect. 4.

The paper is organised as follows. The second section reviews the literature of the previous research regarding option-based hedging strategies, and the third section provides the methodological background of the SD criteria. The dataset and the results of the empirical analysis are described in the fourth section. The conclusions are given in the final, fifth section.

2 Literature review

The majority of the previous research investigating portfolio hedging strategies with options concentrated on two basic strategies: covered call and protective put. A covered call (CC) includes buying an underlying asset (mostly stocks or indexes) and selling a (slightly) out-of-the-money (OTM) call option with one month to maturity. Therefore, the covered call portfolio has a long asset and a short OTM call position. Protective put strategy includes buying an asset and selling an underlying (slightly) in-the-money (ITM) put option.

A study of the performance of portfolios applying a covered call strategy can be found in Bookbinder (1976), Kassouf (1977), Pounds (1978), Grube and Panton (1978), or Yates and Kopprasch (1980) [originally in Lhabitant (2000)]. They all concluded that covered call achieved greater return and had lower risk in terms of variance (Lhabitant 2000), compared to a traditional buy-and-hold strategy. Merton, Scholes and Gladstein (1978, 1982) reached different conclusions (Lhabitant 2000). Boostaber and Clarke (1984) found that a covered call is dominant over a protective put strategy.

Noting that variance is not an adequate risk measure for portfolios with options, Booth et al. (1985), Clarke (1987), and Brooks et al. (1987) applied more appropriate tools in their analysis, using the stochastic dominance criteria. Booth et al. (1985) concluded in favour of the dominance of covered call strategies. However, when they conducted their analysis, the time series of empirical data was limited, so their research was based on simulated data for portfolios with 10% OTM options with six months to expiration; and where options were priced with Black and Scholes (BS) model (Black and Scholes 1973). However, it is known that BS formula proved invalid for option valuation. Their approach included holding all positions until expiration over a period of 1963–1978. Brooks et al. (1987) did not find any evidence of stochastic dominance up to the third degree when comparing buy-and-hold and covered call strategies. The main pitfall of the analysis was testing stochastic dominance based on the assumption of population data, without the use of valid statistical tests for stochastic dominance.

Lhabitant (2000) compared the Sharpe ratios of the covered call, protective put, and buy-and-hold strategies. The data included stock prices and simulated BS option prices. The conclusion of this research was that protective put strategy is always dominated by the buy-and-hold strategy, whereas a covered call, using a slightly OTM call option, dominated over a buy-and-hold portfolio. Finally, Lhabitant (2000) concluded that, in terms of mean and variance, the dominant strategy was a covered call with slightly OTM call options and with the longest time to maturity possible. This conclusion is somewhat predictable given that options with longer time to maturity have greater prices compared to those with a shorter time to maturity, ceteris paribus.

Dash and Goel (2014) analysed ITM and OTM covered call and protective put strategies based on their returns. Their research concluded that using OTM options is a better solution in covered calls and protective puts for protecting stocks with relatively greater returns and smaller standard deviations of return, while ITM options are better in the opposite cases. Besides, they found that a protective put is a relatively better strategy for protecting stocks with greater return and smaller standard deviation, and covered calls are better for stocks with relatively smaller return and greater standard deviation. Finally, Dash and Goel (2014) concluded that the efficiency of strategies depends on the distribution of portfolio returns.

Castellano and Giacometti (2001) compared covered call and protective put strategies using the GARCH methodology. They analysed the multi-currency portfolio and concluded that protective put strategies outperform covered call strategies in terms of a greater utility, using the Value at Risk as a measure of risk. Bollen (1999) suggests using the Monte Carlo method for simulating strategies with options and obtaining vivid illustrations of the impact of options trading on portfolio returns in many different scenarios. Trennepohl and Dukes (1979) investigated the long call strategy, the long stock, (buy-and-hold) and the covered call using weekly prices in the period 1973–1976. They simulated seven portfolios of OTM call options with different maturity times. Using the average return, variance and skewness as comparison criteria, they concluded that portfolios with options achieve smaller standard deviation and a greater return. Their results also show the dominance of the covered call strategy initially formed with a short 9-months call. Moreover, investors who are confident of their predictive abilities should use call options with nine months to expiration when the market is stable, and options with shorter maturity on a growing market. In case of a declining market, they suggest that investors should not invest in stocks. Isakov and Morard (2001) analysed a portfolio insurance strategy that combines diversification and hedging with OTM call option. They concluded that this strategy was dominant compared to a traditional buy-and-hold strategy using stochastic dominance criteria and the Treynor index with a modified beta coefficient.

Hemler and Miller (2015) simulated a long stock, covered call, protective put, collar (long stock, short call and a long put) and a covered combination (long stock and the simultaneous sale of a covered OTM call and an OTM cash-secured put) portfolios using a sample of ten stocks from the pension plan 401(k). They concluded that portfolios with options achieve better results in terms of the Sharpe, Jensen, Treynor and Sortino ratios when compared to a long stock portfolio. This was especially evident in the cases of covered combination and covered call. However, Hemler and Miller (2015) evaluated the performances of these portfolios using ratios which are based on Capital Asset Pricing Model (CAPM). In addition, Hemler and Miller (2015) did not consider the possibility of an early exercise nor the existence of transaction costs.

Szado and Schneeweis (2012) investigated different versions of the collar strategy using different underlying assets, starting from 2007 (2008) up to 2011. Their collar strategy assumed that a short call with one month to expiration and a long put with six months to expiration are rolled each month on the maturity date of a short call. Szado and Schneeweis (2012) also included transaction costs in terms of a bid-ask spread, and they calculated the returns using a daily average of bid-ask prices. The conclusion of their research is that the collar strategy reduces the volatility and shortfalls of a portfolio, and that in some cases, it even increases the return for most assets. A recent accomplishment in computationally efficient options’ risk measurement, using the idea of a replicating portfolio and coherent risk measurement is presented in Mitra (2017).

3 Research methodology based on stochastic dominance criteria

Finding an efficient alternative for the decision-makers in cases where their utility functions are known can be done by modelling the problem as a utility maximisation problem given a set of constraints. In cases where the utility functions of decision-makers are not explicitly known, finding alternatives which are efficient (in terms of their expected utility) for all the decision-makers who have the same preferences can be done using stochastic dominance (SD) criteria. SD methodology assumes that any investment alternative can be represented by its distribution function and that some general preferences of the decision-makers are known. The SD criteria are useful and commonly applied for finding efficient alternatives in investment decision-making where some preferences of investors can be logically assumed, but their utility functions, as functions of the expected return or the final wealth, are not explicitly known and are hard to estimate. Moreover, even if there is some knowledge about the individual utility functions of a group of investors, in most cases their utility functions differ, and a utility maximisation models will not be able to provide a general conclusion. An additional problem that needs to be addressed in modelling problems concerning investment decision-making is the non-normal return distribution of financial series. In such cases, the selection of appropriate risk measures is decisive and the inclusion of the higher moments of the return distributions in models can significantly complicate the problem-solving. However, all these issues can be overcome using the stochastic dominance criteria.

Generally, the nth degree SD criteria can identify the efficient alternatives (investments) for all decision-makers (investors) whose utility functions U are such that $(-1)^{k} {U^{(k)}} \le 0,k \le n,$ and for distribution functions F(x) that have the nth integral $F_n(x)$ defined (if the first n moments of the distribution of a variable x exist). In this context, the SD - efficiency of nth degree is an attribute of an alternative if and only if the alternative is not dominated by any other possible alternative due to the stochastic dominance criteria of nth degree. Let us have alternatives F and G. It is said that distribution F stochastically weakly dominates over distribution G in a closed interval $\left[ a,b\right]$ in nth degree of SD if and only if the following $(n-1)$ conditions are met (Ingersoll 1987):

$$\begin{aligned} {G_{n - 1}}(x) - {F_{n - 1}}(x) \ge 0,\quad \forall x\; \in \left[ {a,b} \right] \end{aligned}$$

(1)

and

$$\begin{aligned} {G_k}(b) - {F_k}(b) \ge 0,\quad k \in \left\{ {1,...,n - 2} \right\} . \end{aligned}$$

(2)

The dominance is strict if $\exists x_0 \in [a,b]$ such that any of the inequalities (1) and (2) is strict. Figure 1 presents a graphical exposition of SD relations.

Fig. 1

A graphical exposition of the stochastic dominance relations

If a certain set of criteria is met and distribution F stochastically dominates distribution G in nth degree, we denote $F\,{D_n}\,G$. Otherwise we denote , but this does not imply that G dominates F. The dominance of G over F can be concluded only by checking the inversed inequalities (1) and (2). All distributions which are not stochastically dominated by any other distribution are efficient according to the nth degree SD.

The nth degree SD criteria are relevant for all investors who prefer greater values of all odd moments and smaller absolute values of even moments of the distribution, up to the nth moment. In investment theory, usually only the first three degrees of stochastic dominance are used. The investment theory assumes that an average investor prefers greater return. Since the expected return is calculated as the arithmetic mean, i.e. the first moment of the return distribution, then investors have a positive preference towards the first moment. Also, the assumption is that investors are risk-averse. Assuming that risk-aversion implies aversion towards greater volatility and towards greater loss, investors should dislike greater variance and they should prefer greater positive skewness. Reminding that variance is the second central moment and skewness is the third central moment of the distribution (see definitions of distribution moments in, for example, Miller 2012), it follows that investors have a negative preference for the second moment and positive preference for the third moment. Therefore, the first three degrees of SD are in accordance with the investment theory. The utility functions U of such investors are members of a set ${\mathcal{U}_n} = \left\{ {U:{{\left( { - \mathrm{{1}}} \right) }^n}{U^{(n)}} \le 0} \right\}$, $n \in \left\{ {1,2,3} \right\}$.

Following the Eqs. (1) and (2), we can state the criteria for the first three degrees of stochastic dominance (Levy 2016), observing the interval [a, b] and $x \in [a,b]$, $x_0 \in [a,b]$.

$FD_1G$ for all $U \in {\mathcal{U}_1}$ if and only if:

$$\begin{aligned} F(x) \le G(x),\forall x\; \wedge \;\exists {x_0}\ {\mathrm{such ~~that}}\ F\left( {x_0}\right) < G\left( {x_0}\right) . \end{aligned}$$

(3)

$FD_2G$ for all $U \in {\mathcal{U}_2}$ if and only if:

$$\begin{aligned} {{F_1}(x) \le {G_1}(x),\forall x,\; \wedge \;\exists {x_0}\ {\mathrm{such ~~that }\ }{F_1}\left( {x_0}\right) < {G_1}\left( {x0}\right) }. \end{aligned}$$

(4)

$FD_3G$ for all $U \in {\mathcal{U}_3}$ if and only if:

$$\begin{aligned}&{F_2}(x) \le {G_2}(x),\forall x,\; \wedge \ {F_1}(b) \le {G_1}(b),\\&\mathrm{and} \nonumber \\&\exists {x}_0 \,\,such\,\, that\,\, F_2(x_0) < G_2(x_0)\nonumber \end{aligned}$$

(5)

Using higher degrees of SD $(k\ge 4)$, such as the fourth, fifth and so on, is not justified in investment theory so far. However, the criteria for the first three degrees of SD might be unable to differentiate efficient from inefficient alternative(s) in a set of possible alternatives. In such cases, higher degrees of SD can be used, but it is up to the researcher to decide whether this is justified given the nature of the research.

3.1 Stochastic dominance analysis assuming population data

When the distribution functions of populations are known, the SD criteria can be examined over pairs of distributions. The following SD model and the SD algorithm are hereafter referred to as a model and an algorithm SD1 and have been formed based on Levy (2016).

(SD1) Let ${K_G}$ be the empirical distribution of returns ${R^K}=\left\{ r_t^K\right\} ,\; t \in \left\{ 1,\dots ,{n^K} \right\}$, where t represents a time index, of a strategy K in year $G \in \mathbb {N}$. Let ${S_G}$ be the empirical distribution of returns ${R^S} = \left\{ r_t^S \right\} ,\; t \in \left\{ 1,\dots ,n^S \right\}$ of a strategy S. Also let $Z_G^0 = \left\{ z_k,k \in \left\{ 1,\dots ,l \right\} \right\}$ be the union of all unique observations from $\{r_t^K\}$ and $\{r_t^S\}$, $l\le n^S+n^K$. We denote a set of nth degree stochastically non-dominated (efficient) strategies in a year G as $ES_G^n$. For a given $n \in \left\{ {1,2,3} \right\}$, $ES_G^n$ is a set of strategies which are efficient for all investors whose utility functions U are members of a set ${\mathcal{U}_n} = \left\{ {U:{{\left( { - \mathrm{{1}}} \right) }^n}{U^{(n)}} \le 0} \right\}$, $n \in \left\{ {1,2,3} \right\}$. The SD of the nth degree is examined by checking necessary conditions (Levy 2016), until there is only one alternative left in the set $ES_G^n$ ($card\left( {ES_G^n} \right) = 1$), if this is possible for some $n\le 3$.

For each year in a sample, the returns are equiprobable and sorted in increasing order, such that $r_1^K \le r_2^K \le \dots \le r_{{n^K}}^K$, $r_1^S \le r_2^S \le \dots \le r_{{n^S}}^S$. Let $\overline{r^K}$ and $\overline{r^S}$ be the average returns of K and S, respectively. The value of the empirical cumulative distribution function of $K_G$ for $r_t^K$, $\mu ^{K_G}(r_t^K)$, is equal to the ratio of number of sorted elements smaller than $r_t^K$ to the number of observations ($n^K$). Further, let us define:

$$\begin{aligned} \mu _2^{K_G}\left( r^K\right) =\int _{-\infty }^{r^K}\mu ^{K_G}(u)du \end{aligned}$$

(6)

and

$$\begin{aligned} \mu _3^{K_G}\left( r^K\right) =\int _{-\infty }^{r^K}\mu _2^{K_G}(u)du. \end{aligned}$$

(7)

Expression in (6) is the integral of the cumulative distribution function. It is an accumulated cumulative distribution, which can be called a ’super-cumulative’ distribution function. The value of (6) at any point $r^K$ is the area under the $\mu ^{K_G}(u)$ for $u \in \langle - \infty , r^K ]$. We interpret (7) in a similar way. Furthermore, we define and interpret $\mu ^{S_G}(r^S)$, $\mu _2^{S_G}(r^S)$ and $\mu _3^{S_G}(r^K)$ in a similar way.

To determine if ${K_G}$ dominates ${S_G}$ in FSD (${K_G}{D_1}{S_G}$), SSD (${K_G}{D_2}{S_G}$) or TSD (${K_G}{D_3}{S_G}$), we need to check conditions for each degree. Let us assume here that $n^K=n^S$, so that the probability of achieving a return smaller than $r_t^K$ and $r_t^S$ is the same for a fixed $t\in \{ 1,\dots ,{n^K} \}$. These conditions are represented in a form of an algorithm (SD1), where point 1 is a necessary condition for ${K_G}{D_n}{S_G}$ (Levy 2016) and points 1.1, 1.2 and 1.3 represent conditions that need to be proved for ${K_G}{D_1}{S_G}$, ${K_G}{D_2}{S_G}$ and ${K_G}{D_3}{S_G}$, respectively:

• 1. If $\overline{r^K} \ge \overline{r^S}$, then

  • 1.1 If $r_t^K \ge r_t^S,\forall t \in \{ 1,\dots ,{n^K} \} \wedge \exists t \in \left\{ 1,\dots ,{n^K} \right\}$ such that $r_t^K > r_t^S$ then ${K_G}{D_1}{S_G}$ and $ES_G^1 = \{ K\}.$ Stop the algorithm.

    • 1.1.1. Else and $ES_G^1 = \{ K,S\}.$ (Go to next step).

  • 1.2. If $ES_G^1 = \{ K,S\}$ and $\overline{r^K} \ge \overline{r^S}$, check if

    • 1.2.1. $\sum _{t = 1}^{m} {r_t^K} \ge \sum _{t = 1}^{m} {r_t^S} ,\forall m \in \{ {1,\dots ,n^K} \}, \wedge \exists m \in \{ {1,\dots ,n^K} \}$ such that $\sum _{t = 1}^{m} {r_t^K} > \sum _{t = 1}^{m} {r_t^S}$, then ${K_G}{D_2}{S_G}$ and $ES_G^2 = \{ K\}.$ Stop the algorithm.

    • 1.2.2. Else and $ES_G^2=\{ K,S\}.$ (Go to next step).

  • 1.3. If $ES_G^2 = \{ K,S\}$ and $\overline{r^K} \ge \overline{r^S}$, check if $\forall z\in [z_k,z_{k+1}] \text { where } [z_k,z_{k+1}] \subseteq [r_i^K,r_{i+1}^K] \text { and } [z_k,z_{k+1}] \subseteq [r_j^S,r_{j+1}^S] \text { for some } i \text { and } j,$

    • 1.3.1. $\mu _3^{{K_G}}(z_k) \le \mu _3^{{S_G}}(z_k)$ and $\mu _3^{{K_G}}(z_{k+1}) \le \mu _3^{{S_G}}(z_{k+1})$, and

      • 1.3.1.1. $\mu _2^{K_G}\left( z_k\right) \le \mu _2^{S_G}\left( z_k\right)$ or

        $\mu _2^{K_G}\left( z_k\right) \ge \mu _2^{S_G}\left( z_k \right) , \mu _2^{K_G}\left( z_{k+1}\right) \ge \mu _2^{S_G}\left( z_{k + 1}\right)$ or

        $\mu _2^{K_G}\left( z_k\right) > \mu _2^{{S_G}}\left( z_k\right) , \mu _2^{K_G}\left( z_{k + 1}\right) \le \mu _2^{S_G}(z_{k + 1}),$

        $\mu _3^{K_G} \left( \frac{-b}{2a} \right) \le \mu _3^{S_G} \left( \frac{-b}{2a} \right) , \text {where}$

        $b=\frac{1}{n^K} \sum _{t = 1}^i {r_t^K - } \frac{1}{n^S} \sum _{t = 1}^j {r_t^S},$

        $a = \frac{j = i}{2l}, \left( i,j:r_i^K, r_j^S \in \left[ z_k,z_{k + 1} \right] \right)$,

      then ${K_G}{D_3}{S_G}$ and $ES_G^3 = \{ K\}.$ Stop the algorithm.

    • 1.3.2. Else . Stop the algorithm.

• 2. Elseif $\overline{r^K} \le \overline{r^S}$, change the inequalities in step 1 and run the modified algorithm 1.1–1.3.

If $ES_G^3 = \{ K,S\}$, $G \in \mathbb {N}$, then no SD up to the third degree can be concluded between strategies K and S in a year G. In that case, both strategy K and strategy S are efficient portfolio strategies for all investors whose utility functions are in $\mathcal{U}_3$, i.e., those who prefer greater return, are risk averse and prefer greater positive skewness. If $ES_G^n = \{ K\}$, $n \in \{ 1,2,3\}$, then a strategy K stochastically dominates over strategy S in a year G for all investors whose preferences are presented with utility functions in $\mathcal{U}_n$. Note that if $n^K \ne n^S$, then inequalites in 1.1 and 1.2 should be checked up to any given level of probability. These probability levels are usually defined as unique values from a union of $t/n^K, t\in \{1,\dots , n^K\}$, and $t/n^S, t\in \{1,\dots , n^S\}$. Using the presented algorithm, we compare empirical return distributions of hedged portfolios to the unhedged (buy and hold) portfolio on a per year basis (model (SD1)), on an industry level (model (SD2)) and for the whole period in all industries (model (SD3)). We also make pairwise comparisons only between hedged portfolios on a per year basis, on an industry level and for the whole period in all industries (models (SD-K1), (SD-K2) and (SD-K3), respectively).

3.2 Stochastic dominance model assuming sample data

Despite the fact that this research included a large dataset of options’ and stocks’ prices, the data used is a sample of a concerned population. Thus, there is a need to statistically test the SD between the two return distributions. In this research, we used the Davidson–Duclos (DD) (non)-dominance test using a minimal value of t-statistics (Davidson and Duclos 2006, 2013).¹ The hypotheses of the DD test are formed according to Wong et al. (2008), originally found in Bishop et al. (1992). Stochastic dominance of two trading strategies is based on the dominance functions from the portfolio return distribution. Let $D_{K,G}^n$ and $D_{S,G}^n$ be the dominance functions of nth degree, $n \in \{ 1,2,3\}$, for a return distribution of a strategy K and a return distribution of a strategy S in a year $G \in \{ 1,2,\dots , T\}$. In empirical research, $D_{K,G}^n$ for some chosen value of $x \in \mathbb {R}$ is estimated as

$$\begin{aligned} \widehat{D}_{K,G}^n(x) = \frac{1}{{{n^K}\left( {n - 1} \right) !}}\sum \limits _{t = 1}^{n^K} {\left( {x - r_t^K} \right) _ + ^{n - 1}} ,\quad G \in \mathbb {N}. \end{aligned}$$

(8)

The estimate of $D_{S,G}^n$ for some chosen value of $x\in \mathbb {R}$, denoted as $\widehat{D}_{S,G}^n(x)$ is estimated in a similar way.

For the purpose of testing the stochastic non-dominance between portfolio strategies S and K in a year $G \in \{ 1,2,\dots , T\}$, the following hypotheses are formed:

$$\begin{aligned} \begin{aligned} {H_0}\dots D_{K,G}^n - D_{S,G}^n&= 0, \\ {H_1}\dots D_{K,G}^n - D_{S,G}^n&\ne 0, \\ {H_{1A}}\dots D_{K,G}^n - D_{S,G}^n&\le 0 \wedge \exists \left( {D_{K,G}^n - D_{S,G}^n < 0} \right) ,\\ {H_{1B}}\dots D_{K,G}^n - D_{S,G}^n&\ge 0 \wedge \exists \left( {D_{K,G}^n - D_{S,G}^n > 0} \right) . \end{aligned} \end{aligned}$$

(DD1)

The null hypothesis ($H_{0}$) assumes that there is no difference between the dominance functions of two distributions. $H_{0}$ can be rejected if there is dominance of portfolio K over portfolio S ($H_{1A}$), or if there is dominance of a portfolio S over portfolio K ($H_{1B}$), or if the two distributions are not the same, but no dominance can be concluded according to the criterion used ($H_{1}$). If we reject the null in favour of $H_{1A}$ or $H_{1B}$ hypothesis, it is possible to conclude stochastic dominance. We used a similar set of hypotheses as (DD1) for testing the nth degree dominance functions of the returns of the strategies K and S in an industry $j \in \{ 1,2,\dots , J\}$ denoted as $D^n_{K,j}$ and $D^n_{S,j}$, and $D^n_{K}$ and $D^n_{S}$ for the dominance functions of the nth degree for the return distributions of strategies K or S for the whole period of analysis for all industries. We named them (DD2) and (DD3), respectively.

Furthermore, we question whether there is a hedging strategy with options which is not dominated by the others in any scenario. For that purpose, DD test is used to investigate the non-dominance between all pairs of return distributions (i.e. their dominance functions) of hedged portfolios on a per year level, on an industry level and for the whole period of analysis in all industries. Let K(i) and K(l) be return distributions of hedging strategies i and l, $i,l \in \left\{ 1,\dots ,4 \right\} ,l \ne i$. The dominance function of nth degree of distribution K(i) in year G is $D^n_{K(i),G}$, and $D^n_{K(l),G}$ represents the same for distribution K(l). When we observe dominance on an industry level for the whole period of analysis, then we introduce $D^n_{K(i),j}$ and $D^n_{K(l),j}$ as the dominance functions of distributions K(i) and K(i) in industry $j, j \in \left\{ 1,\dots ,5 \right\}$. Therefore, we adjust the (DD1) such that the analysed dominance functions are $D^n_{K(i),G}$ and $D^n_{K(l),G}$ (model DD-K1), $D^n_{K(i),j}$ and $D^n_{K(l),j}$ (model DD-K2) and $D^n_{K(i)}$ and $D^n_{K(l)}$ (model DD-K3), $i,l \in \left\{ 1,\dots ,4 \right\} ,l \ne i, j \in \left\{ 1,\dots ,5 \right\}$. Rejecting the null in favour of $H_{1}$, $H_{1A}$, or $H_{1B}$ is done according to value of t-test t(z) calculated for the estimator $\hat{D}_K^n\left( z \right) - \hat{D}_S^n\left( z \right)$ given the values of z. $\hat{D}_K^n$ and $\hat{D}_S^n$ are estimators for nth degree dominance functions of distributions K and S. We define $\hat{D}_K^n,n\in \{1,2,3\}$ (and $\hat{D}_S^n$ in a similar way):

$$\begin{aligned} \hat{D}_K^n\left( z \right) = \frac{1}{{{n^K}\left( {n - 1} \right) !}}\sum _{t = 1}^{{n^K}} {\left( {z - r_t^K} \right) _ + ^{n - 1}} \end{aligned}$$

(9)

for some $z \in {Z^R}$, where ${Z^R}$ is a set of values (thresholds) from the joint support of distribution functions K and S. For the purpose of the test strength, the set of thresholds should be trimmed (Hodder et al. 2013), and in this study the upper and the bottom 2% of values were ignored. For practical purposes, empirical distribution functions (3) are not estimated for all values of $z \in {Z^R}$, they are estimated just for a chosen number of values which are equally distributed between the greatest and the smallest value (Abid et al. 2014). The critical value should be used from the studentized maximum modulus (SMM) distribution (Ury et al. 1979, 1980), which depends on the arbitrarily chosen number of control values. If, as in our case $k\left( {{Z^R}} \right) = 100$, $p=5\%$ and degrees of freedom are $\nu =\infty$, the corresponding critical value is $M_{0.05,\infty }^{100} = \mathrm{{4.409}}$. The conclusions are made following the rules:

$$\begin{aligned} \begin{aligned} \left( 1\right) \;&\left| {t\left( z \right) } \right|< M_{0.05,\infty }^{100},\forall z \in {Z^R} \Rightarrow {H_0}, \\ \left( 2\right) \;&\exists \left( {t\left( z \right)> M_{0.05,\infty }^{100}} \right) \; \wedge \;\exists \left( {t\left( z \right)< - M_{0.05,\infty }^{100}} \right) ,z \in {Z^R} \Rightarrow {H_1},\\ \left( 3\right) \;&t\left( z \right)< M_{0.05,\infty }^{100},\forall z \in {Z^R}\; \wedge \exists \;\left( {t\left( z \right) < - M_{0.05,\infty }^{100}} \right) \Rightarrow {H_{1A}},\\ \left( 4\right) \;&t\left( z \right)> - M_{0.05,\infty }^{100},\forall z \in {Z^R} \wedge \exists \;\left( {t\left( z \right) > M_{0.05,\infty }^{100}} \right) \Rightarrow {H_{1B}}. \end{aligned} \end{aligned}$$

(10)

For any set of hypotheses ((DD1), (DD2) and (DD3), (DD-K1), (DD-K2) and (DD-K3)), a hedging strategy is found inefficient if and only if for any $n\le 3$ the t-statistics implies $H_{1B}$.

4 Data and results

The data is obtained by simulating hypothetical portfolios with stocks and options using their historical prices from the OptionMetrics database. Each portfolio in the sample contains one stock from the S&P100, and four of them contain a certain combination of options, forming five different strategies. One strategy is a buy-and-hold long stock, and the other four strategies combine a long stock with a certain combination of options:

• covered call (CC) including a long stock and a short 5% OTM call,

• protective put (PP) including a long stock and a long 5% OTM put,

• collar (CP) including a long stock, short 5% OTM call and 5% ITM long put, and

• ratio covered call (RC) including a long stock and two ITM calls.

The portfolios are held during one year within the period of 1996–2014 (19 years). The trading algorithm assumes long term investors that invest in value. Certain rules were applied for portfolio construction:

• 1. The portfolio is formed on the first possible date, starting on January 1st, when a strategy with options could be formed, and is held for the next 12 months (or until the data for the stock is available).

• 2. Portfolios are estimated and rebalanced, if necessary, every 28 days such that options always create protection for a stock in accordance with the strategy and the 5% moneyness. If options held in a portfolio are no longer in accordance with the strategy, their positions are closed and new ones are opened. The long stock position is being held constant.

• 3. If there were no options with the required characteristics available on the market at the time of portfolio rebalancing, then a stock portfolio is held unprotected for the next month (having in mind that forcing a poor implementation of an option’s strategy could increase the risk of a portfolio).

• 4. At the end of the year, returns from all strategies are evaluated.

• 5. The transaction costs are integrated in the analysis in terms of a bid-ask spread.

• 6. A long position is opened assuming ask prices, and short positions assuming bid prices (the difference is the transaction cost).

The stocks are also divided into five industry groups using the Fama-French 5-industry classification based on their SIC codes. The number of stocks per industry per year is given in Table 1.

Table 1 The number of stocks and options in the sample, per year and per industry

The descriptive statistics of these simulated portfolios in each year, in a certain industry and for the whole period are given in Tables 15, 16, 17 and 18 in Appendix. The negative returns are the consequence of including the bid-ask spread which is shown to have a significant negative impact on the final result of trading, thus implying that options market is still more attractive for “big players”. Also, Tables 15, 16, 17 and 18 in the Appendix show the values of risk measures (variance, Value at Risk, Expected shortfall (tail loss)), which indicate that hedging strategies with options realised a smaller variability of return and a smaller extreme loss, and in some cases even greater returns than the unhedged portfolio. However, in general, the smaller risk comes with the price of a smaller return and greater (positive) skewness of the return distribution. These findings indicate that it does make sense to further investigate strategies with options as efficient hedging strategies.

4.1 Results from testing SD assuming population data

First, we present the results of the models that assume population data. Table 2 shows the results of the model (SD1), which compared the returns of a hedged strategy K={CC, PP, CP, RC} to those of the unhedged portfolio S by nth degree SD, $n\in \{1,2,3\}$, in each year in the period 1996–2014, and vice versa. The upper left part of Table 2 shows in how many years (in $\%$) a hedged portfolio K dominated an unhedged portfolio S up to TSD. The column 6 ($\textstyle \sum$) in Table 2 shows in how many years (in %) a hedged portfolio K dominated over S. We observe that PP dominates S in most cases–57.89%. Column 7 shows in how many cases a hedged portfolio K is TSD–efficient. We found that PP and CP strategies are efficient in 100% of cases, CC is an efficient strategy in 89.47% cases, and RC in 73.68% cases. The lower part of Table 2 shows the results of testing the dominance of an unhedged portfolio S over a particular hedged portfolio K. We observe that S dominates RC strategy by TSD in 26.32% years, and the % of the dominance over other hedged strategies is negligible.

Table 2 Summary of the per year dominance, model (SD1)

Table 3 shows the degree of SD which could be concluded by comparing the returns from the strategy $K=\{\text {CC, PP, CP, RC}\}$ to strategy S (buy-and-hold unhedged stock portfolio) within a certain industry following the adjusted algorithm (SD2). It is found that the unhedged portfolio does not dominate any hedged portfolio in any industry according to TSD criteria. Moreover, PP is an efficient strategy in all industries, implying that the domination of PP is realised regardless of the portfolio’s exposure to a certain industry. Although the per year analysis (Table 2) suggests that CP could be dominant over the unhedged portfolio up to the TSD, these results do not support it, meaning that the returns from CP strategy were significantly fluctuating during the years in the analysed period.

Table 3 The degree of SD ($n\in \{1,2,3\}$) of K over S, and of S over K, in an industry Ind j, $j=1,\dots ,5$, model (SD2)

Table 4 shows the results from testing the dominance of a hedged portfolio $K=\{\text {CC, PP, CP, RC}\}$ over an unhedged stock portfolio S (and vice versa), for the whole period of analysis in all industries (model (SD3)). We observe that in this setting only PP stochastically dominates over S by TSD. In all other pairwise comparisons, both S and $K \in \{\text {CC,CP,RC} \}$ are efficient, implying that $ES^n = \{ K,S\} ,\;\;\forall n \in \left\{ {1,2,3} \right\}$.

Table 4 Results from testing SD over the whole period in all industries, model (SD3)

Table 5 The % of dominance of $K_1$ over $K_2$ per year in the period 1996–2014, model (SD-K1)

Secondly, we present the results from the pairwise comparisons of hedged strategies CC, PP, RC and CP. Table 5 shows the summary results of checking the FSD, SSD and TSD of a hedged strategy $K_1$ over $K_2$ in each year from 1996 to 2014. The column 6 ($\textstyle \sum$) shows in how many years (in $\%$) TSD can be concluded. Apparently, CC dominates RC in 42.11% years of the analysis, while RC dominates CC in approximately 26% years. However, looking year-by-year, both CC and RC are dominated by PP and CP more often than the other way around. The relation between PP and CP is not that clear even though PP dominates CP in a greater number of years (31.58–10.53%). According to these results, it can be concluded that PP is TSD-efficient in 89.47% of cases, CP in 68.42%, and CC and RC in 15.79% of cases. Thus, the efficiency ranking of hedging strategies in a per year based analysis (model SD-K1) is: (1) PP, (2) CP, (3) CC and RC.

The results of the model (SD-K2), which compares the return distributions between hedging strategies $K_1$ and $K_2$ in each of the five industries (Ind 1–Ind 5), are presented in Table 6. We observe that only TSD can be found. Precisely speaking, PP dominates by TSD over RC in all industries, over CC in Industries 1, 2, 3 and 4 and over CP in Industries 1 and 3. Also, RC dominates CC in Industry 1, and CP dominates both CC and RC in Industry 3. These results indicate that all investors with preferences that match TSD criteria have greater utility from PP than from all other strategies regardless of the industrial structure of the portfolio.

Table 6 The degree (n) of the determined SD from testing the hypotheses $K_1 D_n K_2$, $n\in \{1,2,3\}$, per industry Ind j, $j=1,..,5$; model (SD-K2)

Finally, when the return distributions of hedging strategies from the whole period in all industries were compared (model (SD-K3)), it was found that PP dominates CC and RC by TSD. This is evident from the results presented in Table 7. Other dominances up to TSD could not be concluded. If the investors’ preferences allow, higher degrees of SD could be applied to differentiate these strategies more strictly. However, we decided to use statistical tests for SD to analyse these strategies and obtain more precise results, which are presented in the following subsection.

Table 7 The degree (n) of the determined SD from testing hypothesis $K_1 D_n K_2$, $n\in \{1,2,3\}$, in the period of 1996-2014 in all industries, model (SD-K3)

4.2 Results from SD tests assuming sample data

Hereafter we present the results of the stochastic non-dominance DD tests, with hypotheses elaborated in the Sect. 3.2. Table 8 shows the final results of statistically testing the (non-)dominance between a hedged portfolio $K=${CC, PP, RC, CP} and an unhedged portfolio S, at $\alpha =5\%$. Explicitly, the null (the non-dominance by TSD) of a hedged strategy K over S can be rejected at 5% in favour of CC and CP strategies in 73.68% (14/19), for RC in 52.63% (10/19) and for PP strategy in 94.74% (18/19) years. The dominance of an unhedged portfolio S can be determined only over CC in 10.53% (2/19) and over RC in 31% (6/19) of cases. Given such results, it can be concluded that all investors who prefer greater return, are risk-averse, and prefer greater values of positive skewness, have greater utility from portfolio hedging strategies with options than from the buy-and-hold unprotected long stock portfolio. Moreover, the efficiency can be determined more frequently for strategies PP and CP.

Table 8 Frequency of rejection of the null in testing hypotheses (DD1) at $\alpha =5\%$

Table 9 presents the results of testing hypotheses of the (non-)dominance of K over S (DD2) within each of the five industries. Here, $H_1$ indicates that the null (non-dominance) can be rejected, but there is not enough evidence to conclude the dominance either way. $H_{1A}$ confirms that the null can be rejected in favour of the dominance of a strategy $K=${CC, PP, CP, RC} over S with $\alpha =5\%$, while $H_{1B}$ indicates the dominance of S over K. However, $H_{1B}$ does not hold in any industry up to TSD. Therefore, it can be concluded that the unhedged portfolio does not dominate any of the hedged portfolios with options in any industry in our sample. These hedging strategies with options dominate the buy-and-hold in all industries by TSD, except for the CC in Industry 1, where no dominance can be concluded either way. Interestingly, according to these results, the stochastic non-dominance at TSD of a hedged strategy over S can be rejected at $\alpha =5\%$ even for the CP strategy in all industries, which was not the case when algorithms for SD assuming population data were used (see Table 3).

Table 9 Results from testing the null within industries at $\alpha =5\%$, model (DD2)

Following the model (DD3), the comparisons of the dominance function estimators of hedging strategies and an unhedged strategy for the period of 1996–2014 in all industries were obtained. The results from these comparisons show that for $n=3$ (TSD), the null of non-dominance can be rejected at $\alpha =5\%$ in favour of dominance for all hedging strategies over an unhedged portfolio (Table 10).

Table 10 Results from testing the null , in the period of 1996–2014 in all industries at $\alpha =5\%$, model (DD3)

Table 11 Frequency of rejection of the null , in favour of $H_1, H_{1A}$ or $H_{1B}$ in a per year analysis at $\alpha =5\%$, model (DD-K1)

Table 12 Results from testing hypotheses , $\ n \in \{1,2,3\}$, in industry $j \in \{1,\dots ,5\}$ at $\alpha =5\%$, model (DD-K2)

Table 13 Efficiency ranking of hedging strategies within industries at $\alpha =5\%$, model (DD-K2)

Table 14 Results from testing hypotheses , $n \in \{1,2,3\}$, in all industries in the period of 1996-2014 at $\alpha =5\%$, model (DD-K3)

The final part of this analysis is focused on finding a hedging strategy which can be found efficient compared to the others by DD tests up to the TSD, and is based on hypotheses (DD-K1), (DD-K2) and (DD-K3). By examining hypotheses (DD-K1), pairwise comparisons of the dominance function estimators of each return distribution in each year were made, including only those of the hedged strategies. These results are summarised in Table 11. It is found that at $\alpha =5\%$, it can be concluded: (1) PP and CP dominate CC by TSD in 94,74% and 89,47% cases, respectively, (2) CC, PP and CP dominate RC by TSD in $>50\%$ cases, (3) CP is non-dominant over PP in more cases than PP is non-dominant over CP. Therefore, it can be found that the efficiency ranking of the hedging strategies using options at $\alpha =5\%$ is as follows: 1. PP, 2. CP, 3. CC , 4. RC.

The comparison of the returns of hedged portfolios within each sector using SD criteria up to the 3^rd degree, and following the (DD-K2), revealed the results presented in Table 12. As shown, strategies PP and CP dominate over CC and RC in all industries. Also, PP dominates CP in Industries 1 and 3, and the opposite is concluded for Industry 4. From these findings, we provide the ranking of the hedging strategies within sectors, shown in Table 13. These results show that strategies which are efficient according to TSD in per year analysis for all industries are also efficient within industries. This again confirms that the efficiency of a certain strategy does not depend on the structure of a portfolio, that is, to its exposure to a certain industry.

Finally, DD test and the set of rules (10) were also used to investigate the possibility to reject the null of non-dominance of one hedging strategy over the other in the whole period of the analysis in all industries.² In Table 14, $H_{1A}$ indicates that strategy $K_1$ dominates over strategy $K_2$ by FSD ($n=1$), SSD ($n=2$), or TSD ($n=3$) at $\alpha =5\%$, and $H_{1B}$ indicates the opposite. From these results, it is obvious that PP and CP are TSD-efficient strategies and are dominant over CC and RC. Between CC and RC we cannot conclude dominance up to TSD.

5 Conclusions

Finding an efficient option-based hedging strategy can be a demanding task given the non-normal return distributions of portfolios with options, the diversity of investors’ attitudes and the unknown characteristics of investors’ utility functions. For this reason, this research uses stochastic dominance criteria which are appropriate for non-normal distributions, and are aimed at finding efficient strategies (in terms of their expected utility) for all investors who prefer a greater return, are risk-averse in terms of downside volatility and in terms of loss, and who prefer greater positive skewness. The paper examines two main scenarios: the efficiency of hedging strategies with options compared to an unhedged portfolio and the efficiency of one hedging strategy relative to the other. Both scenarios are examined on a per year basis, within different industries and in general. The analysis is based on the return distributions of the hedging strategies: covered call, protective put, collar and ratio covered call, and a buy-and-hold unhedged stock portfolio, which are obtained from simulated portfolios using historical data. The results obtained with both standard stochastic dominance criteria and statistical tests indicate that portfolio hedging strategies with options are never dominated by a buy-and-hold portfolio.

It was also found that hedging strategies which had smaller returns, smaller risk and greater positive skewness were more frequently found to be efficient than the buy-and-hold portfolio, indicating that the loss of investors’ utility due to the smaller return could be compensated by greater skewness and smaller risk. This led to the conclusion that hedging strategies with options are indeed useful instruments for hedging risk. Moreover, given the mentioned criteria, strategies protective put and collar proved more efficient compared to the others. A protective put was found to be efficient regardless of the characteristics of return distributions within certain industries. Assuming that the large sample during a long period of analysis provides a relevant base for making general conclusions, we conclude that collar strategy created by buying a slightly (5%) out-of-the-money put option and selling (if the market conditions allow) a short out-of-the-money call, is a hedging strategy that will not be less efficient than the unhedged buy-and-hold portfolio, covered call, or ratio covered call. The limitations of the research stem from the simplification of the real market situation assumed in the model. In this sense, the portfolios are simulated following a proposed algorithm which follows certain rules which can deviate from a real-world situation and which does not include certain market data, such as dividends (due to lack of availability). Conclusions based on examining a long time series of a large number of historical data using appropriate analytic tools can give valuable guidelines for future decision-making, but this does not guarantee its realisation in the future. The recognition of the above-mentioned limitations opens a space for future research. The future research would consider more flexible algorithms for simulating portfolios at different levels of options’ moneyness, maturity and rebalancing. Besides, extending the time-series and investigating the strategies in different market regimes could also bring some interesting findings. The new research would also take into account issues of financing, and further development of the methodology of the research. In addition, a further application of the SD models would be investigated in similar problems concerning investment decision-making.

Appendix

See Tables 15, 16, 17 and 18.

Table 15 Descriptive statistics of simulated portfolios per year in the period of 1996-2014 for strategies covered call (CC) and protective put (PP)

Table 16 Descriptive statistics of simulated portfolios per year in the period of 1996–2014 for strategies covered call (CC) and protective put (PP)Table caption

Table 17 Descriptive statistics of simulated portfolios per industry in the period of 1996–2014

Table 18 Descriptive statistics of simulated portfolios for the whole period of 1996–2014