What is option based portfolio insurance

Table of contents

I List of Figures

II List of Tables

III List of Abbreviations

1 Introduction

2 Overview of Portfolio Insurance
2.1 Origins
2.2 Using Portfolio Insurance to Manage Exposure to Market Risk

3 3 Theoretical Framework for Protective and Synthetic Put
3.1 Option Pricing
3.2 Protective Put
3.3 Synthetic Put

4 Evaluation of Option-Based Portfolio Insurance
4.1 Refinements and Considerations
4.2 Advantages and Disadvantages of Protective Put and Synthetic Put

5 Conclusion

IV Bibliography

I. List of Figures

Figure 1: Payoff Structure of Portfolio Insurance Strategies

Figure 2: Return distribution of Portfolio Insurance Strategies vs. Normal Distribution

Figure 3: Composition and Payoff Structure of a Protective Put Portfolio

Figure 4: Portfolio Composition for the Synthetic Put Strategy

Figure 5: Exposure of synthetic put portfolio over the investment period

II. List of Tables

Table 1: Statistical Comparison of an Insured Portfolio vs. Uninsured Portfolio (based on a Monte Carlo Simulation with 10 000 paths)

III. List of Abbreviations

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1 Introduction

Risk aversion is a common trait among investors. While it is possible to reduce risk attributed to specific industries and regions by diversifying among different securities, market risk affects all securities on the market. Even a perfectly diversified portfolio is subject to systematic or market risk. It can be managed through diversification across asset classes, for example by shifting some of the funds invested into risk-free assets. For some investors, this yields unsatisfactory results as the expected return directly decreases linearly with an increase in the position in the risk-free asset. Portfolio insurance (PI) describes an alternative set of strategies that allows investors to reduce their exposure to market risk by guaranteeing the value of the portfolio to be above a certain value at the end of the investment period while allowing for participation in rising stock markets.

Option-based portfolio insurance (OBPI) refers to a set of strategies in which either a conventional put option (protective put) or a replicated put option (synthetic put) is used to insure a portfolio against adverse price movements. In theory and assuming perfect market conditions, protective put (PP) and synthetic put (SP) yield identical payoffs and have the same cost. In practice, there are several important differences between the two strategies. On the one hand, PP seems to be an easy and uncomplicated strategy to implement, but the unavailability of listed options with desired maturities and strike prices are major issues. SP strategies, on the other hand, can suffer from obstacles like high transaction costs and jumps in stock prices.

In this thesis, the efficiency and performance of OBPI is analyzed. On the theoretical basis of the Black-Scholes-Merton Model, important practical refinements and considerations for OBPI in general are pointed out and examined in detail. Furthermore, the practical differences between conventional and synthetic puts used for the respective PP and SP strategies are explained. This helps to evaluate the optimal application of the strategies.

Section 2 presents the origins and main idea behind PI. Section 3 introduces the option pricing theory relevant to OBPI. Furthermore, a detailed theoretical analysis of the PP and SP highlighting the respective characteristics and complications. In Section 4 important considerations when implementing an OBPI strategy are discussed. Moreover, advantages and disadvantages of the PP and SP are presented. A Monte Carlo simulation supports these. Section 5 concludes.

2 Overview of Portfolio Insurance

2.1 Origins

The idea for PI was first conceptualized by Leland in the mid-1970s. He realized that an equity portfolio can be insured by purchasing a put option on it.[1] This strategy, a protective put (PP), prevents the value of the portfolio to end below the strike price of the option at the maturity date. If the portfolio is out of the money at the end of the investment period the option is exercised, thereby guaranteeing the floor value. Given that suitable exchange-traded put options did not exist at the time, Leland used the same arbitrage argument underlying the Black-Scholes option pricing formula to replicate options. Entering a long position in the equity portfolio to be insured and a long position in a risk-free asset can replicate the payoff of put option. The composition of the replication portfolio or synthetic put (SP) depends on the option delta or hedge ratio and must be adjusted continuously.

The theoretical foundation for PI lies in option pricing theory. Black and Scholes were the first to propose a model for option pricing that did not depend on one or more arbitrary parameters.[2] Instead, all necessary inputs for the model are either observable or exogenously given. Subsequently, extensions and further applications to the model were developed, for instance by Merton.[3] This created possibilities for new and innovative investment strategies, including OBPI. With the introduction of personal computers to provide the computing power necessary for option pricing and dynamic replication, first steps towards offering PI were undertaken by Leland and Rubinstein, starting 1976. Together with O’Brien they founded LOR Associates in 1981 to offer a portfolio insurance product based on the theoretical framework they had established so far.[4] The product was accepted among large investors, such as pension funds, and the volume of insured portfolios increased to around $ 100 billion in 1987.[5]

2.2 Using Portfolio Insurance to Manage Exposure to Market Risk

In its effect, PI is comparable to conventional insurance. Investors can eliminate their exposure to downside risk and must pay a certain price or insurance premium in return. However, the means of attaining PI differs from conventional insurance contracts. This is because an insurance company cannot pool risk from different contracts together as there is no stochastic independence between them.[6] The insurer would be exposed to substantial risk as risk pooling or diversification cannot eliminate market risk. This risk can be eliminated by fully investing in risk-free assets or through traditional hedging strategies with futures. The latter essentially locks the portfolio at its current terminal value through a long position in a futures contract on the underlying. The return on these strategies is fixed and equates to the risk-free rate, meaning that there is no participation in rising stock markets.

PI uses hedging procedures or dynamic rebalancing strategies to remove the exposure to market risk. The main goal behind these strategies is to provide downside protection against market risk for an investment while retaining some upside potential. The performance of an uninsured portfolio is proportional to the performance of the market. PI causes the payoff structure for a portfolio to become convex.[7]

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Figure 1: Payoff Structure of Portfolio Insurance Strategies[8]

No matter how low the price for the underlying portfolio falls, the payoff is at least equal to the floor return. If the price of the underlying increases over the investment period, the payoff increases as well. The difference to the payoff from an uninsured portfolio in the case of favorable price movements can be interpreted as the cost of insurance or insurance premium. The level of the floor is similar to the deductible of conventional insurance. Choosing a lower floor decreases the cost of insurance but also decreases the guaranteed amount for the investor at the end of the investment period.

Another way to illustrate the effect of PI strategies is by looking at the return distribution for insured and uninsured portfolios.

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Figure 2: Return distribution of Portfolio Insurance Strategies vs. Normal Distribution

The dashed line represents the distribution of an insured portfolio. Since the left-tail is eliminated, it becomes skewed to the right. Compared to the distribution of an unaltered equity portfolio, depicted as the solid line, investors incur a cost reflected in the lower mean of return.

The cost of portfolio insurance can be divided into two components. Implicit costs arise from the loss of upside capture or decreased participation in rising stock prices. Explicit costs arise from the increased transaction costs of constructing and maintaining an insured portfolio.[9]

The term PI is used for many investment strategies. Some may not yield the exact payoff profile and return distribution shown in figures 1 and 2. This broad definition only requires the strategy to guarantee a minimum value of the portfolio at the end of the investment period. There are static and dynamic approaches. Static models include buy and hold (allocating funds between risky and risk-free assets), stop-loss (selling the entire risky position as soon as the floor is breached) and PP. Dynamic models are, besides SP, modified stop-loss and constant proportion portfolio insurance (CPPI). This thesis focuses on OBPI, therefore the other will not be analyzed further.

Since PI aims at reducing exposure market risk, it is most relevant for insuring equity portfolios. Other applications in areas like credit and exchange rate risk are also conceivable and especially appealing to financial institutions.[10] Primary users of PI strategies are pension funds who have fixed obligations to fulfil and institutions offering absolute return products to individual investors.

3 Theory Protective Put and Synthetic Put

3.1 Option Pricing

European put options are a fundamental component to OBPI. They give the holder the right (but not obligation) to sell a stock at the predetermined strike price at the date of maturity. American options differ from their European counterparts in that they can be exercised prior to maturity. This additional versatility increases the option price which is unnecessary for the purposes of PI. For that reason, European options should be used when insuring a portfolio and are most relevant for the evaluation of PI strategies.[11]

This chapter briefly describes the approach to option pricing conceived by Black and Scholes, including the enhancements added by Merton. The so-called Black-Scholes-Merton model will be denoted as B-S-M. Additionally, the assumptions for the PP and SP strategies are introduced.

The valuation of European put options can be done using the B-S-M pricing formulas. The following assumptions are necessary:

1) Stock prices are lognormally distributed; this more closely resembles movements of stock prices than normally distributed returns (as shown by empirical studies); volatility and expected return of these stocks are known and constant.
2) Short-selling of stocks is not restricted.
3) Perfect markets conditions - meaning that there are no transaction costs, taxes and opportunities for arbitrage; financial instruments are perfectly divisible.
4) Continuous trading is possible.
5) The risk-free rate is known and constant for all maturities.

These assumptions apply to both the PP and SP strategies.

The first step to derive the formulas is setting up the B-S-M differential equation using an arbitrage argument. A hypothetical portfolio, consisting of positions in the derivative and underlying stock, can be set up as such that it yields the risk-free return. It requires the proportion of the portfolio positions to satisfy the hedge ratio or option delta. This is the case when any gain or loss in the stock position will be offset by a gain or loss in the derivative position. Such a portfolio must earn the risk-free rate, otherwise opportunities for arbitrage exist.[12] After a short period of time, movements in the stock price cause the hedge ratio of the portfolio to change. That means that the risk-free property of the portfolio is only provided for a short time period. In order to remain risk-free, the hypothetical portfolio needs to be rebalanced continuously. Using Itô’s Lemma, the composition of the hypothetical portfolio and thus the differential equation can be derived.

The second step is to solve the equation with respect to the boundary conditions of a particular derivative. For European put options this is with denoting the put price, denoting the strike price and denoting the initial stock price. A put option is never worth less than zero or more than the strike price minus stock price.

The resulting pricing formula for put options is usually written as follows: with denoting the present value factor for the strike price and denoting the cumulative value of the standard normal distribution.

Certain problems arise when the B-S-M Model is used in practice. Many of the assumptions are quite restrictive and violated in reality. For instance, interest rate and volatility of the stock price are assumed to be constant over the investment period. This is rarely the case. Still, the model has proven to deliver accurate results for option prices.[14]

3.2 Protective Put

This approach to PI utilizes the properties of option contracts to create an asymmetric payoff structure of the portfolio. The strategy is implemented by holding a long position in a stock portfolio and a long position in a put option contract on the same stock portfolio. Figure 3 illustrates the separate and combined payoff structures of these positions.

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Figure 3: Composition and Payoff Structure of a Protective Put Portfolio[15]

S0 denotes the initial investment. The payoff from the stock position moves linearly with changes in stock price S. The value of the put option with strike price K is subject to the boundary conditions max(K - S, 0). If the stock price at maturity lies above the strike, the option expires worthless. If the stock trades below the strike, the value of the position is (K - S). The option value minus the option premium p gives the payoff from the put. Combining both positions leads to the PP strategy. Should the stock position expire below the floor, the value of the option position compensates for the loss. This is downside protection. If the stock position ends up above the floor, the option value is zero and upside participation is granted. Although upside participation is unlimited it is always lower than an uninsured stock investment by the option premium p.

Characteristics

The payoff from a PP depends on the value of the stock position at maturity. This property is called path independence since the path taken by the stock price during the investment period is irrelevant.[16] Because the PP strategy is static, only requiring the initial investment in the respective stock and option positions, transaction costs are low.

There are three important specifications in the option contract that need to be considered when setting up the PP strategy. First, the maturity date T of the option determining the investment period or expiration of the PP. This date is the only date for which the portfolio is defined by the payoff structure shown above. Second, the strike price K representing the floor value, the lowest value the portfolio can be worth at expiration. In the case shown above, where S0 = K, the entire initial value of the portfolio minus the option premium is guaranteed at expiration.[17] This constellation is referred to as a 100% floor. K < S0 implies the guaranteed amount from the put is lower but upside participation in rising stock prices is larger. K > S0 offers a greater guaranteed amount but lower upside participation. Third, the option price or premium p can be interpreted as the cost of insurance. It can be derived using the B-S-M pricing formula and depends on several factors, some of which can be controlled by the investor. Selecting a lower strike and shorter maturity decreases the price of a put option. This is reasonable since there is a corresponding decrease of the value of insurance. An uncontrollable component in the option price is the volatility in the underlying. Put options can be interpreted as discounting the effect of future volatility in the underlying, thus become more expensive the higher the expected volatility of the market is. The interest rate has a negative effect on the option price because future payoffs from the option contract become less valuable the higher the interest rate is and vice versa.

Obstacles

The theoretical description of the PP suggests that it is a straightforward investment strategy which can transform the portfolio payoff from a linear into a convex function with respect to the stock price. In reality, implementing a PP is more challenging.

The main issue lies in the fact that the array of put options on offer is limited. To achieve complete PI, an investor requires a put option on his entire portfolio. Stock exchanges do not offer options on specific portfolios an investor might hold. As an alternative, put options on each individual stock in the portfolio could be used. However, assuming that all required options are available, this approach yields an inferior outcome to a theoretical PP. Due to diversification, the volatility of an entire portfolio is generally lower compared to that of individual stocks. Therefore, options on an entire portfolio are less expensive than individual stocks in a portfolio.[18] Fortunately for investors, options on market indices such as the DAX or the S&P 500 are available. While well-diversified portfolios tend to mirror these indices and can thus be insured using the respective index put options, any deviation of the insured portfolio from the index underlying the option generates a tracking error.[19] This inflicts costs on a PP due to uncertainty and unreliability. Consequently, investors can only use the PP to insure portfolios closely resembling an index for which options are available.

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[1] cf. Leland, Rubinstein, 1988, p.4.

[2] cf. Black, Scholes, 1973, p.639.

[3] cf. Merton, 1973, p.142.

[4] cf. Leland, Rubinstein, 1988, p.4-8.

[5] cf. Bouye, 2009, p.3.

[6] cf. Brennan, Solanki, 1981, p.279.

[7] cf. Kluß, Bayer, Cremers, 2005, p.7.

[8] cf. Kluß, Bayer, Cremers, 2005, p.10.

[9] cf. Zhu, Kavee, 1988, p.52.

[10] cf. Cluß, Mayer, Bremers, 2005, p.6.

[11] Assuming the value of the portfolio only needs to be at least at floor level at the end of the investment period; cf. Rubinstein, 1985, p.20f.

[12] cf. Black, Scholes, 1973, p.643.

[13] with and where denotes the stock price volatility, denotes the time to maturity and denotes the continuously compounded risk-free rate

[14] cf. Abken, 1987, p.6.

[15] cf. Cluß, Meyer, Bremers, 2005, p.13.

[16] cf. Rubinstein, 1985, p.13.

[17] The appropriate strike price K to achieve a required rate of return , considering the option premium, can be calculated as , not considering dividend payments; cf. Clarke, Arnott, 1987, p. 36+47.

[18] cf. Cluß, Mayer, Bremers, 2005, p.15.

[19] cf. Cluß, Mayer, Bremers, 2005, p.14.

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