2018 Study Session # 15, Reading # 29 “RISK MANAGEMENT APPLICATIONS OF OPTION STRATEGIES” IR = Interest Rate NP = Notional Principal 2.2 Risk Management Strategies with Options and the Underlying An investor can exposure without selling the underlying through: Covered call Protective put 2.2.1 Covered Calls Long stock + short call Appropriate when stock price neither nor in the near future Limited upside potential & downside protection Reduces both overall risk & the expected return ்ܸ = ்ܵ − ݔܽܯሺ0, ்ܵ − ܺሻ ܲ ்ܸ = ݐ݂݅ݎ− ܵ + ܥ ܵ = ݏݏ݈ ݔܽܯ − ܥ ܵ = ݊݁ݒ݁ ݇ܽ݁ݎܤ − ܥ the X, lower the option premium 2.2.2 Protective Puts Long stock + long put Provide protection against in value (similar to insurance) Requires upfront option cost Appropriate when an investor expects a in value of the stock in the near future ்ܸ = ்ܵ + ݔܽܯሺ0, ܺ − ்ܵ ሻ Profit =்ܸ − ܵ − ܲ& ∞ = ݐ݂݅ݎ ݔܽܯ Max loss =ܵ + ܲ − ܺ Break even = ܵ + ܲ 2.3 Money Spreads Spread ⇒ strategy that involves buying one option & selling another identical option but either with different X or different time to expiration Time spread ⇒ different time to expiration Money spread ⇒ different exercise price Copyright © FinQuiz.com All rights reserved 2018 Study Session # 15, Reading # 29 2.3.1 Bull Spreads Buying a call with a lower X & selling another with X Rationale ⇒when investor expects an in stock price in the near future Similar to covered call it provides protection against downside risk & limited upside potential ܸ = ܥଵ − ܥଶ Where ܥଵ &ܥଶ are option premiums for the lower X & higher X respectively ்ܸ =Value of long call-value of short call Profit =்ܸ = ܥଵ + ܥଶ Max profit = ܺଵ − ܺଶ − ܥଵ + ܥଶ Max loss = ܥଵ − ܥଶ Breakeven = ܺଵ + ܥଵ − ܥଶ Bull Put Spread Buys a put with a lower X & sells an identical put with a higher X Cash inflows at initiation of the position Identical to the sale of bear put spread Profit occurs when both put options expire out-of-the money 2.3.2 Bear Spreads Bear Put Spread Long position in a put with X & short position in a put with a X Rationale⇒ investor expects that stock price will in the future ܸ = ܲଶ − ܲଵ Where P2 = put premium on higher X ்ܸ = value of long put-value of short put Profit = ்ܸ − ܲଶ + ܲଵ Max profit = ܺଶ − ܺଵ − ܲଶ + ܲଵ Max loss = ܲଶ − ܲଵ Breakeven = ܺଶ − ܲଶ + ܲଵ Bear Call Spread Investor sells a call with a lower X & buys an otherwise identical call with a higher X Investor will earn net premium when both call options expire outof-the money Identical to the sale of a bull call spread Copyright © FinQuiz.com All rights reserved 2018 Study Session # 15, Reading # 29 2.3.3 Butterfly Spreads Long Butterfly Spread Long bull call spread + short bull call spread Require cash outlay at initiation because bull spread purchased by an investor is expensive than a bull spread sold Rationale ⇒ useful when investor expects that the volatility of underlying will be low relative to market expectations ்ܸ = ݔܽܯሺ0, ்ܵ − ܺଵ ሻ − 2ݔܽܯሺ0, ்ܵ − ܺଶ ሻ + ݔܽܯሺ0, ்ܵ − ܺଷ ሻ ܲ ்ܸ = ݐ݂݅ݎ− ܥଵ + 2ܥଶ − ܥଷ ܺ = ݐ݂݅ݎ ݔܽܯଶ − ܺଵ − ܥଵ + 2ܥଶ − ܥଷ ܥ = ݏݏ݈ ݔܽܯଵ − 2ܥଶ + ܥଷ Two breakeven points ܺଵ + ܥଵ − 2ܥଶ + ܥଷ 2ܺଶ + ܺଵ − ܥଵ + 2ܥଶ − ܥଷ Short Butterfly Spread Selling the calls with ܺଵ &ܺଷ& buying two calls withܺଶ Rationale ⇒ preferable when investor expects that the volatility of the underlying will be relatively high compared to market expectations Long Butterfly Spread (using puts) Long bear put spread + short bear put spread Cost of ܺଵ ሺܲଵሻ < ܿܺ ݂ ݐݏଶ ሺܲଶሻ < ܿܺ ݂ ݐݏଷ ሺܲଷሻ Long Butterfly Spread (using puts) Selling the puts with ܺଵ &ܺଷ& buying two puts with ܺଶ Max profit = ܲଷ + ܲଵ − 2ܲଶ If correctly priced, butterfly spread using calls will provide the same result as butterfly using puts 2.4 Combinations of calls and Puts 2.4.1 Collars Strategy in which cost of buying put option can be reduced by selling a call option Provide downside protection at the expense of giving up upside potential Zero cost collar ⇒ call option premium is equal to put option premium Put X & call X results is in both the upside & downside potential Quite similar to bull spread with respect to cap on gains & a floor on loss but no underlying holdings in bull spread ܸ = ܵ ்ܸ = ்ܵ + ݔܽܯሺ0, ܺଵ − ்ܵ ሻ − ݔܽܯሺ0, ்ܵ − ܺଶ ሻ ܲ ்ܸ = ݐ݂݅ݎ− ܸ ܺ = ݐ݂݅ݎ ݔܽܯଶ − ܵ ܵ = ݏݏ݈ ݔܽܯ − ܺଵ Breakeven = ܵ Collars are also known as range forwards & risk reversals Copyright © FinQuiz.com All rights reserved 2018 Study Session # 15, Reading # 29 2.4.2 Straddle Long Straddle Buying at-the-money put & a call with same X on same underlying & expiration Rationale ⇒ investor expects volatility than what market expects Costly strategy ்ܸ = ݔܽܯሺ0, ்ܵ − ܺሻ + ݔܽܯሺ0, ܺ − ்ܵ ሻ ܲ ்ܸ = ݐ݂݅ݎ− ܲ − ܥ & ∞ = ݐ݂݅ݎ ݔܽܯmax ݈ܲ = ݏݏ +ܥ Breakeven ܺ ± ሺܲ+ܥሻ Short Straddle Selling a put & a call with same X on the same underlying with the same expiration Preferable when neutral view about volatility Unlimited loss potential This strategy gains when both the options expire out-of-the money Variation of Straddle Adding call (put) to a straddle is known as strap (strip) Long strangle ⇒ variation of the straddle (buying put & calls with different (X) Short strangle ⇒ selling the put & call with different X 2.4.3 Box Spread Bull spread + bear spread Long Box Spread Long call with ܺଵ& short cal with ܺଶ+ long put with ܺଶ& short put withܺଵ If options are correctly priced, the box spread payoff is always RF (riskless strategy) PV of the payoff discounted at RF should be equal to initial outlay ܸ = ܥଵ − ܥଶ + ܲଶ − ܲଵ ்ܸ = ܺଶ − ܺଵ Profit & Max profit = ܺଶ − ܺଵ − ሺܥଵ − ܥଶ + ܲଶ − ܲଵሻ No breakeven & max loss Short box is also possible with opposite positions Benefits of box spread: To exploit an arbitrage opportunity Does not require a volatility estimate Hold lower transaction costs Copyright © FinQuiz.com All rights reserved 2018 Study Session # 15, Reading # 29 INTEREST RATE OPTION STRATEGIES IR call & put options are used to protect against IR IR call option pay-off = N.P × Max (0, underlying rate at expiration –exercise rate) × Days in underlying rate/360 180 day LIBOR can be used as the underlying rate & underlying days could be 180, 182 183 etc Rate is determined on the day when option expires & payment is made m days later IR put option pay-off =NP × Max (0, X –underlying rate at expiration) × days in underlying rate/360 3.1 Using Interest Rate Calls with Borrowing Used by borrowers to manage IR risk on floating rate loans Consider the following factors: Option expiration date is the same as when loan starts Option pay-offs must occur at the time when borrower makes IR payments on loan 3.2 Using Interest Rate Puts with Lending Used by lender to manage IR risk on floating rate loans 3.3 Using an Interest Rate Cap with a Floating-Rate Loan Interest rate cap ⇒combination of IR call options Each option in a cap is called a caplet Each caplet has same X but its own expiration date Cap seller makes payments if IR < strike rate Payoff is determined on its expiration date but made on the next payment date Cap pay-off = NP × (0, LIBOR on previous reset date – X) X days in settlement period / 360 Effective interest = interest due on the loan – caplet pay-off 3.4 Using an Interest Rate Floor with a Floating-Rate Loan Interest rate floor ⇒ combination of IR put options Floorlet pay-off = NP X (0,X –LIBOR on previous reset date) × days in settlement period/360 Effective interest = interest received on the loan + floorlet pay-off Copyright © FinQuiz.com All rights reserved 2018 Study Session # 15, Reading # 29 3.5 Using an Interest Rate Collar with a Floating-Rate Loan Combination of a long (short) position in a cap & a short (long) position in a floor The borrower (lender) can buy a cap (floor) to protect against rising (falling) IR & sell the floor (cap) to finance the premium paid to buy a cap Initial cost of the hedge can be by call exercise rate & floor exercise rate Cost can also be by having NP for the cap & NP for the floor Borrower will benefit when IR & will be hurt when IR within the collar OPTION PORTFOLIORISK MANAGEMENT STRATEGIES Dealers provide liquidity to the market & take risk by trading in options Dealers use different hedging strategies to avoid risk If a dealer has sold a call, he can hedge his/her risk by: Buying an identical call option Buying a put with same X & expiration, buying the asset & selling a bond (static hedge) Using delta hedging Size of the long position in underlying to offset the risk associated with short position in option = -1/ delta Three complicating issues in delta hedging: Delta is an approximate for small changes only Delta changes with the change in the price of the underlying & or time Small amount of imprecision due to rounding the no of units of underlying 4.1 Delta Hedging an Option over Time Dynamic hedging ⇒ delta-hedged position needs to be rebalanced as underlying price ∆ or with the passage of time Delta of in-the-money (out-of-the money) call option will ( ) towards (0) near expiration Delta hedges are most difficult to maintain for-at-the-money options & /or near expiration Hedging Using Non-Identical Option ܸ = ܰଵ ܥଵ + ܰଶ ܥଶ Where ܰଵ &ܰଶ = quantity of each option that hedges the value of one of the options in a portfolio ܥଵ &ܥଶ = Price of option 1&2 Desired quantity of option relative to option 2: ݊݅ݐ ݂ ܽݐ݈݁ܦ2 −∆ܥଶ ܰ ܴܽ = ݀ܽ݁ݎݏ ݅ݐ = ଵൗܰ = ଶ ݊݅ݐ ݂ ܽݐ݈݁ܦ1 ∆ܥଵ Copyright © FinQuiz.com All rights reserved 2018 Study Session # 15, Reading # 29 4.2 Gamma and the Risk of Delta = ܽ݉݉ܽܩ ∆ ௗ௧ ∆ ௨ௗ௬ Larger the gamma, greater will be the risk Gamma is largest for at-the-money options & /or near expiration Gamma hedge ⇒ position in underlying + position in two options 4.3 Vega and Volatility Risk ܸ݁݃ܽ = ∆ ௧ ∆ ௩௧௧௬ ௧ ௨ௗ௬ At-the-money option has greater sensitivity to ∆ in volatility Volatility is unobservable, so it is difficult to estimate Vega Delta is required to manage Vega risk jointly with delta & gamma FINAL COMMENTS Major difference b/w equity & bond option strategies are that bond options must expire before the bond matures Bullish (bearish) investor buys puts (calls) on IR Bullish (bearish) equity or bond investors buy calls (puts) Copyright © FinQuiz.com All rights reserved ... money Identical to the sale of a bull call spread Copyright © FinQuiz. com All rights reserved 2018 Study Session # 15, Reading # 29 2 .3. 3 Butterfly Spreads Long Butterfly Spread Long bull call spread... settlement period /36 0 Effective interest = interest received on the loan + floorlet pay-off Copyright © FinQuiz. com All rights reserved 2018 Study Session # 15, Reading # 29 3. 5 Using an Interest... are also known as range forwards & risk reversals Copyright © FinQuiz. com All rights reserved 2018 Study Session # 15, Reading # 29 2.4.2 Straddle Long Straddle Buying at-the-money put & a call