READING 37: RISK MANAGEMENT APPLICATIONS OF OPTIONS STRATEGIES A- Notation Review Time 0 is the time at which the strategy is initiated and time T is the time the option expires, stated as a fraction of a year. Accordingly, the amount of time until expiration is simply π β 0 = π, which is (π·ππ¦π π‘π ππ₯πππππ‘πππ)/365. The other symbols are π0 , ππ = price of the call option at time 0 and time π π0 , ππ = price of the put option at time 0 and time π 1 π = exercise price π0 , ππ = price of the underlying at time 0 and time π π0 , ππ = value of the position at time 0 and time π π± = profit from the transaction: π0 β ππ π = risk-free rate Quick reminder of basic option strategies
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B- Risk Management Strategies with Options and the Underlying 1- Covered Calls A covered call is a position in which you own the underlying and sell a call. This strategy is used when investors believe the price of the underlying they own will not move much over the period. They expect to pocket the premiums on the calls they sell. Selling a covered call as a strategy reduces not only the risk but also the expected return compared with simply holding the underlying. One should not expect to make a lot of money writing calls on the underlying. It should be apparent that in fact the covered call writer could miss out on significant gains in a strong bull market. The compensation for this willingness to give up potential upside gains, however, is that in a bear market the losses on the underlying will be cushioned by the option premium.
Payoff chart Value at expiration: Profit: Maximum profit: Maximum loss: Breakeven:
ππ = ππ β max(0, ππ β π) π± = ππ β π0 + π0 π β π0 + π0 π0 β π0 ππ β = π0 β π0
2- Protective Puts Holding an asset and a put on the asset is a strategy known as a protective put. The objective of the protective put is to protect the portfolio from a drop in prices while maintaining the potential to profit from a rise in prices. Buying a put to add to a long stock position is much better than selling a call. It provides downside protection while retaining the upside potential, but it does so at the expense of requiring the payment
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of cash up front. In contrast, a covered call generates cash up front but removes some of the upside potential. A protective put strategy seems appealing but it is typically expensive as the stock price movement that would recover the put price is quite substantial and therefore, the buyer would be giving up considerable upside potential.
Payoff chart Value at expiration: Profit: Maximum profit: Maximum loss: Breakeven:
ππ = ππ + max(0, πβππ ) π± = ππ β π0 β π0 β π0 + π0 β π ππ β = π0 + π0
C- Money Spreads A spread is a strategy in which you buy one option and sell another option that is identical to the first in all respects except either exercise price or time to expiration. ο·
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If the options differ by time to expiration, the spread is called a time spread. Time spreads are strategies designed to exploit differences in perceptions of volatility of the underlying. They are among the more specialized strategies, and we do not cover them here. Money spreads, are spreads in which the two options differ only by exercise price. The investor buys an option with a given expiration and exercise price and sells an option with the same expiration but a different exercise price.
1- Bull Spreads A bull spread is designed to make money when the market goes up. In this strategy we combine a long position in a call with one exercise price (π1 ) and a short position in a call with a higher exercise price (π2 ). The investor believes that the market is going up but not as much for the sold option to be 3
exercised. The point here is to offset the purchase price of one option with the sale of another which we don not expect to be exercised and to hold on to the asset to take advantage of any potential upside. The spread caps the profit.
Payoff chart Value at expiration: Profit: Maximum profit: Maximum loss: Breakeven:
ππ = max(0, ππ β π1 ) β max(0, ππ β π2 ) π± = ππ β π1 + π2 π2 β π1 β π1 + π2 π1 β π2 ππ β = π1 + π1 β π2 (there is an errata in the reading corrected here)
2- Bear Spreads If one uses the opposite strategy, selling a call with the lower exercise price and buying a call with the higher exercise price, the opposite results occur. The graph is completely reversed: The gain is on the downside and the loss is on the upside. This strategy is called a bear spread. The investor in this strategy expect prices to fall and benefits in this case. The more intuitive way of executing a bear spread, however, is to use puts. Specifically, we would buy the put with the higher exercise price and sell the put with the lower exercise price.
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Payoff chart Value at expiration: Profit: Maximum profit: Maximum loss: Breakeven:
ππ = max(0, π2 β ππ ) β max(0, π1 β ππ ) π± = ππ β π2 + π1 π2 β π1 β π2 + π1 π2 β π1 ππ β = π2 β π2 + π1
3- Butterfly Spreads A butterfly spread combines a bull and a bear spread of calls or puts. For a butterfly spread with calls: ο· Buy 2 calls/puts one at a low strike and the other at a high strike ο· Sell 2 calls/puts with an in the middle strike ο A main assumption here is that strike prices are equally spaced so that 2π2 β π1 β π3 = 0 ο Also, the premium net is always negative because the spread we buy is always more expensive than the spread we sell ο In this strategy the investor expects low volatility and that the underlying will trade near the middle strike. The maximum profit occurs if the underlying ends up precisely at the middle exercise price. ο If the investor believes the market to be more volatile than generally anticipated a butterfly spread should be sold with the reverse positions.
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Payoff chart Value at expiration: Profit: Maximum profit: Maximum loss: Breakeven:
ππ = max(0, ππ β π1 ) β 2 max(0, ππ β π2 ) +max(0, ππ β π3 ) π± = ππ β π1 + 2π2 β π3 π2 β π1 β π1 + 2π2 β π3 π1 β 2π2 + π3 ππ β = π1 + π1 β 2π2 + π3 and ππ β = 2π2 β π1 β π1 + 2π2 β π3
D- Combinations of Calls and Puts 1- Collars A collar is the combination of a covered call (short call/long underlying) and a protective put (long put/long underlying). A typical collar has the call and the put premiums offset. When this offsetting occurs no premium is paid up front. With a collar strategy, the holder of the asset gains protection below a certain level, the exercise price of the put, and pays for it by giving up gains above a certain level, the exercise price of the call. When premiums offset it is called a zero-cost collar even though the cost here is foregoing the upside potential. Collars are also known as range forwards and risk reversals. Asset managers often use them to guard against losses without having to pay cash up front for the protection. Clearly, however, they are virtually the same as bull spreads. The latter has a cap on the gain and a floor on the loss but does not involve actually holding the underlying.
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Payoff chart Value at expiration: Profit: Maximum profit: Maximum loss: Breakeven:
ππ = ππ + max(0, π1 β ππ ) β max(0, ππ β π2 ) π± = ππ β π0 π2 β π0 π0 β π1 ππ β = π0
2- Straddle What should an investor do if he believes the market will be volatile but does not feel particularly strongly about the direction? We discussed earlier that a short butterfly spread is one strategy. It benefits from extreme movements, but its gains are limited. There are other, more complex strategies, such as time spreads, that can benefit from high volatility; however, one simple strategy, the straddle, also benefits from high volatility. A straddle consists of a call and a put with the same exercise price on the same underlying with the same expiration. This strategy enables the investor to profit from upside or downside moves. Maximum loss occurs when the price of the underlying remains unchanged. As we have noted, a straddle would tend to be used by an investor who is expecting the market to be volatile but does not have strong feelings one way or the other on the direction. An investor who leans one way or the other might consider adding a call or a put to the straddle. Adding a call to a straddle is a strategy called a strap, and adding a put to a straddle is called a strip. It is even more difficult to make a gain from these strategies than it is for a straddle, but if the hoped-for move does occur, the gains are leveraged. Another variation of the straddle is a strangle, in which the put and call have different exercise prices.
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Payoff chart Value at expiration: Profit: Maximum profit: Maximum loss: Breakeven:
ππ = max(0, ππ β π) + max(0, π β ππ ) π± = ππ β (π0 + π0 ) β π0 + π0 π Β± (π0 + π0 )
3- Box Spreads A box spread is a combination of a bull spread and a bear spread. The objective of a box spread is to exploit arbitrage opportunities in the case of option mispricing. It is only possible to profit from this strategy if the present value of the payoff is different from the initial outlay: (πΏπ β πΏπ ) = ππ β ππ + ππ β ππ (π + π)π
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Payoff chart Value at expiration: Profit: Maximum profit: Maximum loss: Breakeven:
ππ = π2 β π1 π± = π2 β π1 β (π1 β π2 + π2 β π1 ) π πππ ππ ππππππ‘ ππ πππ π ππ πππ π ππππ πππ£ππ ππππ πππ‘πππ ππππππ ππ πππππππ£ππ, πππππ π
πΉ ππ πππ‘ππππ ππππππ¦ ππππππ
E- Interest Rate Option Strategies Interest rate call and put options are usually purchased to protect against changes in interest rates. For dollar-based interest rate derivatives, the underlying is usually LIBOR but is always a specific rate, such as the rate on a 90- or 180-day underlying instrument. An interest rate option is based on a specific notional principal, which determines the payoff when the option is exercised. Traditionally, the payoff does not occur immediately upon exercise but is delayed by a period corresponding to the life of the underlying instrument from which the interest rate is taken, an issue we review below. π«πππ ππ πππ
πππππππ ππππ (π΅πππππππ π·ππππππππ) π¦ππ±(π, πππ
πππππππ ππππ ππ ππππππππππ β π¬πππππππ ππππ) ( ) πππ
where βdays in underlyingβ refers to the maturity of the instrument from which the underlying rate is taken. The most important point, however, is that the rate is determined on one day, the option expiration, and payment is made m days later. This practice is standard in floating-rate loans and thus is used with interest rate options, which are designed to manage the risk of floating-rate loans. Likewise, the payoff of an interest rate put is π«πππ ππ πππ
πππππππ ππππ (π΅πππππππ π·ππππππππ) π¦ππ±(π, π¬πππππππ ππππ β πππ
πππππππ ππππ ππ ππππππππππ) ( ) πππ
1- Using Interest Rate Calls and Puts with Borrowing and Lending ο· ο·
An interest rate call, establishes a maximum interest rate for a loan to be taken out in the future by the borrower. A lender can buy a put that pays off if the interest rate falls below a chosen level. The put payoff then compensates the bank for the lower interest rate on the loan.
ο Donβt forget to capitalize the option premium and include it in the loan to calculate exact interest. If borrower, reduce amount by premium, id lender add premium to amount.
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2- Using an Interest Rate Cap with a Floating-Rate Loan An interest rate cap is typically used by corporation to protect a floating rate borrower from an increase in interest rates. The rate for each period is set the prior period as per prevailing rates and the caplet payoffs offset any deviation from the purchased cap strike. 3- Using an Interest Rate Floor with a Floating-Rate Loan An interest rate floor is typically used by floating rate lenders to protect them from a drop in interest rates. The rate for each period is set the prior period as per prevailing rates and the floorlet payoffs offset any deviation from the purchased cap strike. 4- Using an Interest Rate Collar with a Floating-Rate Loan An interest rate collar is the combination of a floor and a cap. ο· The borrower will buy a cap to protect it from an increase in rates and offset the purchase price by selling a floor thus giving up any benefits from a drop in rates. ο· Or a bank could buy a floor to protect it from a decrease in rates and sell a cap to offset the premium but will also be giving up the benefit of a rise in rates. F- Option Portfolio Risk Management Strategies 1- Delta Hedging an Option over Time Dealers delta hedge their positions with long position of a certain number of units of the underlying. The size of that long position will be related to the optionβs delta. π·πππ‘π =
πΆβππππ ππ πππ‘πππ πππππ πΆβππππ ππ π’ππππππ¦πππ πππππ
So if the price of the underlying changes by xtimes the price of the option will change by about Ξxtimesβ keep in mind that delta is just an approximationβ and so being short one and long the other will result in a neutral strategy. ο·
The delta will change, moving toward 1.0 for in-the-money calls (β1.0 for puts) and 0.0 for outof-the-money options as expiration approaches. Any change in the underlying price will also change the delta. These changes in the delta necessitate buying and selling options or the underlying to maintain the delta-hedged position. Any additional funds required to buy the underlying or other options are obtained by issuing risk-free bonds. Any additional funds released from selling the underlying or other options are invested in risk-free bonds.
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The delta of an option changes as the underlying changes and as time elapses. The delta will change more rapidly with large movements in the underlying and when the option is approximately at-the-money and near expiration. These large changes in the delta will prevent a delta-hedged position from being truly risk free. Dealers usually monitor their gammas 10
(
πΆβππππ ππ ππππ‘π ) πβππππ ππ π‘βπ π’ππππππ¦πππ
and in some cases hedge their gammas by adding other options to
their positions such that the gammas offset.
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The sensitivity of an option to volatility is called the vega (
πΆβππππ ππ πππ‘πππ πππππ ). πΆβππππ ππ π£ππππ‘ππππ‘π¦
An optionβs
volatility can change, resulting in a potentially large change in the value of the option. Dealers monitor and sometimes hedge their vegas so that this risk does not impact a delta-hedged portfolio.
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