Session X: Lecturer: Dr. Jose Olmo. Module: Economics of Financial Markets. MSc. Financial Economics. Department of Economics, City University, London

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1 Session X: Options: Hedging, Insurance and Trading Strategies Lecturer: Dr. Jose Olmo Module: Economics of Financial Markets MSc. Financial Economics Department of Economics, City University, London Option contracts can be used for many different purposes. There are three main objectives pursued by investors using these derivatives: hedging, insurance and speculation. Options are attractive to investors because permit them to get free of downside risk. Options used for hedging arguments allow investors to minimize the variation in portfolio s value. The use of options for insurance purposes is an attempt to obtain the impossible. To benefit from rises in stock markets but limiting losses derived from downfall markets. Finally the mechanics and design of these financial instruments permits to devise complicated instruments tailored for specific objectives that search to exploit investor s beliefs (speculation) about stock price fluctuations but without taking exaggerated amounts of risk. Hedging with options In order to diversify risk an investment portfolio usually consists on assets, bonds, and derivatives. Within the class of derivative instruments, options are widely employed due 1

2 2 to their capacity to provide coverage against negative fluctuations of the underlying asset. Alternatively stock options can also provide coverage against upwards movements of the underlying stock yielding adverse results for investor s portfolio. Consider a portfolio consisting on w 1 units of a stock and w 2 European call options on the same underlying stock. The value of the portfolio P at date t is expressed as P t = w 1 S t + w 2 c t, with c t and S t the prices of the option and the stock respectively. The variation in the value of the portfolio between two periods is P t+1 = w 1 S t+1 + w 2 c t+1, with representing the change in the variable s value. The sensitivity of the change in portfolio s value to changes in the asset is given by P t+1 S t+1 = w 1 + w 2 c t+1 S t+1. This result provides a way to choose w 1 and w 2 such that changes in the underlying stock are offset by changes in the option price. This is w 2 w 1 = S t+1 c t+1. If the price of the underlying stock is assumed to follow a geometric brownian motion the value of an option is given by the Black-Scholes formula: c t = Φ(x 1 )S t + Φ(x 2 )K exp R 0(T t), with x 1 and x 2 constants satisfying x 1 = log( K S t )+(R σ2 )(T t) σ T t, and x 2 = x 1 σ T t,

3 3 and Φ denoting a normal distribution function. be In order to have full coverage against fluctuations of the stock price the ratio w 1 w 2 should w 1 w 2 = 1 Φ(x 1 ). Suppose we add a put option to the portfolio to hedge against possible falls in the stock price. The variation in the value of the portfolio between two periods is P t+1 = w 1 S t+1 + w 2 c t+1 + w 3 p t+1, with p t the put option price. The sensitivity of the put price to changes in the stock price is derived from the Black- Scholes model and the put-call parity relationship. Then p t+1 S t+1 = Φ(x 1 ) 1. Hence the sensitivity of the portfolio to changes in the stock price is P t+1 S t+1 = w 1 + w 2 Φ(x 1 ) + w 3 (Φ(x 1 ) 1). This portfolio is fully hedged under variations of the price of the underlying asset if the weights corresponding to the different components satisfy the following expression. (w 1 w 3 ) + (w 2 + w 3 )Φ(x 1 ) = 0. If it is possible to construct a portfolio satisfying these restrictions and the stock options are well modelled by the Black-Scholes pricing formula the variations in stock price in both directions are offset by the option contracts.

4 4 Portfolio Insurance Portfolio insurance consists on combining the upside potential of increasing prices in stock markets with the limited downside potential provided by trading on options for declining markets. The goal is to construct portfolios that fully benefit from bull stock markets, in contrast to hedging strategies, but imposing a floor to the losses incurred by the portfolio in bear markets. Portfolio insurance is also distinguished from diversification in what the latter is an effort to minimize risk by finding portfolios consisting on independent assets. In this sense well diversified portfolios are free from idiosyncratic risk derived from negative asset-specific shocks but not from negative shocks affecting the market as a whole. These portfolios are affected by market risk. In fact the risk premium required by an investor is given by the relation between the portfolio and the market. Portfolio insurance pursues a more ambitious goal. To benefit from positive returns when asset prices increase without incurring large losses when they fall. There are different strategies for portfolio insurance involving options. Stop-loss selling and buying. The purchase of put options. Lending and the purchase of call options. The creation of synthetic put options. Stop-loss selling and buying In this strategy assets are sold when prices start falling and are bought when prices start increasing. This strategy is implemented by setting threshold levels such that in case these are exceeded the selling-buying mechanism is triggered. This strategy for portfolio

5 5 insurance works when the sequence of upturns or downturns in stock prices is continuous. Increases are followed by increases, or alternatively, decreases are followed by decreases. Portfolio insurance with put options An investor holding an stock can obtain insurance against downfalls of the stock by buying put options on the underlying stock. This strategy allows one to fully benefit from stock value increases but at the same time setting a floor for possible declines in the asset. The put option price is the price of the insurance. The value of an insured portfolio consisting on two assets (S t, p t ) is S T + max(0, K S T ) = max(s T, K). The net payoffs of the portfolio after considering the price of the insurance are max(s T, K) p t. Portfolio insurance with call options This strategy is used by investors interested in having insurance against rises in stock prices. Instead of buying the stock at a low price and waiting until the price rises, if this eventually occurs, or buying the stock when it has reached a high price and it is uncertain whether prices are going to increase more, investors can borrow money or invest in a bond at the risk-free interest rate, and lock in a long position in a call option. If K is the strike price of a call option on the underlying stock S t, borrowing K exp R 0(T t) permits one to obtain the stock at the expiration date when it is likely to rise but paying the strike price for it. The insurance premium is the price of the call option. The value of the portfolio is K + max(0, S T K) = max(s T, K).

6 6 The net payoffs of the portfolio are max(s T, K) c t. These strategies are useful for portfolios involving one single stock. In general portfolios consist on many more assets and these insurance strategies can be of limited use due to transactions costs derived from buying large numbers of options. These portfolios also incur in transaction costs derived from continuous changes in portfolio composition in order to maintain the level of insurance. This is due to the differences between options expiration dates and investors horizons for holding the portfolio. Synthetic put options. The idea is to replicate the payoffs of a put option on the portfolio, and in turn benefit of the insurance this derivative provides. This is implemented when trading in put options is not available, or the expiration dates of the available options do not suit the time horizon of investors s portfolio. Combinations and spreads Along with hedging and insurance strategies options can be also devised with the intention of matching specific investment objectives. Versions of these derivatives tailored to achieve these objectives are designed by using combinations of different options, combinations of options with different maturities, or combinations of long and short positions in different call and put options on the same underlying stock. These strategies are usually implemented for speculation purposes. These are further classified as combinations and spreads.

7 7 Combinations These portfolios combine bundles, either all of long positions or all of short positions in call and put options on the same asset. This class of portfolios can be further classified as Straddles. A long straddle (bottom straddle) consists of a long position in a call and a put option with the same strike price and expiration date. The reverse position is denoted top straddle. Strips. A long strip consists of buying one call and two puts with the same strike price and expiration date. Investors obtain profits from large variations in the stock price paying attention to negative departures. Straps. A long strap consists of buying two calls and one put with the same strike price and expiration date. Investors obtain profits from large variations of the stock price but considering more likely positive departures. Strangles (bottom vertical combinations). A long strangle consists on a long position in a call and a put option with the same expiration date but different strike prices. The strike price of the put option is lower than the strike corresponding to the call option. This trading strategy is used for large expected movements in the stock price. The coverage against slight variations of the stock price is higher than with a straddle position though depends on the spread between the strike prices. This also has an effect on the profit of the strangle strategy. Spreads This strategy involves trading on long and short positions in two or more options of the same type. That is, either a bundle of two or more call options based on the same underlying stock, or a bundle of two or more put options in the same stock. These trading strategies can be divided in two classes.

8 8 Horizontal (or calendar) spreads. A bundle of options in the same underlying stock with the same strike price but different expiration dates. An example of this spread is a short position on a call option with maturity t 1 and a long position on a call option with the same strike price and later expiration date t 2. The option with later expiration is sold when the first option matures. Vertical (or cylinder) spreads. These options involve trading on options with the same expiration date and different strike prices. Within this class we can distinguish the following. Bull spreads. It involves buying a call option with strike K 1 and selling a call with the same expiration date but higher strike price, K 1 < K 2. It is immediate to see the call option with lower strike price is always the most expensive. Bull spreads in put options are created by using the same strategy; buying a put with a low strike price and selling a put with a high strike price. We distinguish three types of spreads. Both calls are initially out of the money. One call is initially in the money and the other is initially out of the money. Both calls are initially in the money. These strategies are devised for scenarios where the stock is expected to rise. Alternatively, if the investor expects a decline in the stock price an appropriate strategy is bear spreads. Bear spreads. It involves buying a call with strike price K 1 and selling a call with the same expiration date but lower strike price, K 2 < K 1. Bear spreads constructed on put options consist on selling the put with maturity K 2 and buying the expensive put option. Box spreads. This strategy is a combination of a bull call spread with strike prices K 1 and K 2 and a bear put spread with the same strike prices.

9 REFERENCES 9 Butterfly spreads. It involves taking positions in options with three different strike prices. The strategy consists on a long position in a call option with low strike price K 1, and a call option with high strike price K 3, and short positions in two call options with strike price K 2 halfway between the other two strike prices. This strategy provides limited losses for departures of S T from the strike k 2 and profit when the stock price is close to the strike. Diagonal spreads. The options involved in this strategy may have different strike prices and expiration dates. This alternative provides higher freedom to benefit from the properties of trading in stock options. References [1] Bailey, R.E., (2005). The Economics of Financial Markets. Ed. Cambridge University Press, New York (Chapter 20). [2] Hull, J.C., (2006). Options, Futures and Other Derivatives. Ed. Prentice Hall (6th ed.), New Jersey (Chapter 10).