CFA level 3 CFA level 3 CFA level 3 CFA level 3 CFA level 3 finquiz curriculum note, study session 15, reading 29

Risk Management Applications of Option Strategies It is important to note that options not always automatically increase risk 2.2 Risk Management Strategies with Options and the Underlying An investor can reduce exposure without selling the underlying by: 1) Selling a call on the underlying i.e covered call 2) Buying a put i.e protective put 2.2.1) Covered Calls Covered Call = Long stock position + Short call position Covered Call is appropriate to use when an investor: • Owns the stock and • Expects that stock price will neither increase nor decrease in near future Characteristics: • It is an imperfect form of portfolio protection It provides only limited downside protection • It exchanges “upside” potential for current income in the form of option premium • It generates cash up front in the form of option premium but removes some of the upside potential • It reduces both the overall risk and the expected return compared with simply holding the underlying This loss in potential upside gains is compensated by option premium received by selling a call • Writers of covered call options make small amounts of money, but make it often; because expected profits come from rare but large payoffs • Selling a call option on a stock already owned by an investor reduces the overall risk • Selling a call without owning the stock exposes the investor to unlimited loss potential Thus, covered call should not be viewed as a conservative strategy Relationship between exercise price and potential upside gains for the short call: • The higher the exercise price of the call option, the lower the price of an option and thus short call receives lower premium However, in this case, short call has a greater opportunity to gain from the upside NOTE: Current value of the asset should be viewed as an opportunity cost of an investor Practice: Example 3, Volume 5, Reading 29 2.2.2) Protective Puts Protective Put = Long stock position + Long Put position This provides protection against a decline in value It is similar to “insurance" i.e buying insurance in the form of the put, paying a premium to the seller of the insurance, the put writer Characteristics: To summarize: a) Value at expiration = Value of the underlying + Value of the short call = VT = ST – max (0, ST – X) b) Profit = Profit from buying the underlying + Profit from selling the call = VT – S0 + c0 c) Maximum Profit = X – S0 + c0 d) Max loss would occur when ST = Thus, Maximum Loss = S0 – c0 e) Breakeven =ST* = S0 – c0 • It provides downside protection while retaining the upside potential • It requires the payment of cash up front in the form of option premium • The higher the exercise price of a put option, the more expensive the put will be and consequently the more expensive will be the downside protection Protective put is appropriate to use when: • An investor owns a stock and does not want to sell it • An investor expects a decline in the value of the stock in the near future but wants to preserve upside potential To summarize: a) Value at expiration: VT = ST + max (0, X - ST) –––––––––––––––––––––––––––––––––––––– Copyright © FinQuiz.com All rights reserved –––––––––––––––––––––––––––––––––––––– FinQuiz Notes Reading 29 Reading 29 Risk Management Applications of Option Strategies b) Profit = VT – S0 - p0 c) Maximum Profit = ∞ d) The maximum loss would occur when underlying asset is sold at exercise price Thus, Maximum Loss = S0 + p0 – X e) In order to breakeven, the underlying must be at least as high as the amount paid up front to establish the position Thus, Breakeven =ST* = S0 + p0 FinQuiz.com but a different time to expiration Time spread can benefit from high volatility Money spreads: When the options have different exercise price, the spread is called a money spread e.g an investor buys an option with a given expiration and exercise price and sells an option with the same expiration but a different exercise price • Note that the options are on the same underlying asset Example: 2.3.1) Bull Spreads Strike price = X = $45 Option cost = p0 = $6 • The maximum possible loss is $6 • The potential gain is unlimited A Bull Call Spread: This strategy involves a combination of a long position in a call with a lower exercise price and a short position in a call with a higher exercise price i.e • Buy a call (X1) with option cost c1 and sell a call (X2) with option cost c2, where X1< X2 and c1 > c2 Note that the lower the exercise price of a call option, the more expensive it is Rationale to use Bull Call Spread: Bull call spread is used when investor expects that the stock price or underlying asset price will increase in the near future Characteristics: Put v/s Insurance: The exercise price of the put is like the insurance deductible because the magnitude of the exercise price reflects the risk assumed by the party who owns the underlying A higher exercise price of the put option is equivalent to a lower insurance deductible • The higher the exercise price, the higher the option premium and the less risk assumed by the holder of the underlying and the more risk assumed by the put seller • In insurance, the higher the deductible, the more risk assumed by the insured party and the less risk assumed by the insurer Practice: Example 4, Volume 5, Reading 29 2.3 Money Spreads A spread is a strategy that involves buying one option and selling another identical option but either with different exercise price or different time to expiration Time Spread: When the options have different time to expiration, the spread is called a time spread Time spread strategies are used to exploit differences in perceptions of volatility of the underlying e.g an investor buys an option with a given expiration and exercise price and sells an option with the same exercise price • This strategy gains when stock price rises/ market goes up • Like covered call, it provides protection against downside risk but provides limited gain i.e upside potential • It is similar to Covered call strategy i.e o In covered call, short position in call is covered by long position in underlying o In bull call spread, the short position in the call with a higher exercise price is covered by long position in the call with a lower exercise price To summarize: a) The initial value of the Bull call spread = V0 = c1 – c2 b) Value at expiration: VT = value of long call – Value of short call = max (0, ST – X1) - max (0, ST – X2) c) Profit = Profit from long call + profit from short call Thus, Profit = VT – c1 + c2 d) Maximum Profit = X2 – X1 – c1 + c2 e) Maximum Loss = c1 – c2 f) Breakeven =ST* = X1 + c1 – c2 Reading 29 Risk Management Applications of Option Strategies FinQuiz.com Maximum Profit = X2 – X1 – p2 + p1 e) Maximum Loss occurs when both puts expire outof-the-money and investor loses net premium i.e when ST> X2 Thus, Maximum Loss = p2 – p1 f) Breakeven =ST* = X2 – p2 + p1 B Bull Put spread: In bull put spread, investor buys a put with a lower exercise price and sells an otherwise identical put with a higher strike price • Buy a put (X1) and sell a put (X2), with X1< X2 • Since put with a higher exercise price (X2) is expensive than a put with a lower exercise price (X1), bull put spread generates cash inflow at initiation of the position • Profit occurs when both put options expire out-ofthe-money i.e investor will earn net premium • It is identical to the sale of Bear put spread • Bull put spread pay-off diagram is the mirror-image of the pay-off diagram of bear put spread Practice: Example 5, Volume 5, Reading 29 2.3.2) Bear Spreads A Bear Put Spread: This strategy involves a combination of a long position in a put with a higher exercise price and a short position in a put with a lower exercise price i.e • Buy a put (X2) with option cost p2 and sell a put (X1) with option cost p1, where X1< X2 and p1 < p2 Note that the higher the exercise price of a put option, the more expensive it is Rationale to use Bear Put Spread: Bear Put spread is used when investor expects that the stock price or underlying asset price will decrease in the future To summarize: a) The initial value of the bear put spread = V0 = p2 – p1 b) Value at expiration: VT = value of long put – Value of short put = max (0, X2 - ST) - max (0, X1 - ST) c) Profit = Profit from long put + profit from short put Thus, Profit = VT – p2 + p1 d) Maximum Profit occurs when both puts expire inthe-money i.e when underlying price ≤ short put exercise price (ST ≤ X1), • Short put is exercised and investor will buy an asset at X1 and • This asset is sold at X2 when long put is exercised Thus, B Bear Call Spread: In bear call spread, investor sells a call with a lower exercise price and buys an otherwise identical call with a higher strike price • Sell a call (X1) and buy a call (X2), with X1< X2 • Since call with a lower exercise price (X1) is expensive than a call with a higher exercise price (X2), bear call spread generates cash inflow at initiation of the position • Profit occurs when both call options expire out-ofthe-money i.e investor will earn net premium • It is identical to the sale of a bull call spread i.e it is used when investor expects a decline in stock price • Bear call spread pay-off diagram is the mirrorimage of the pay-off diagram of bull call spread Practice: Example 6, Volume 5, Reading 29 2.3.3) Butterfly Spreads Butterfly spread strategy is a combination of a bull and bear spread Butterfly spreads perform based on the volatility of the underlying A Long Butterfly Spread (Using Call): Long Butterfly Spread = Long Bull call spread + Short Bull call spread (or Long Bear call spread) Long Butterfly Spread = (Buy the call with exercise price of X1 and sell the call with exercise price of X2) + (Buy the call with exercise price of X3 and sell the call with exercise price of X2) where, X1< X2 < X3 Cost of X1 (c1) > Cost of X2 (c2) > Cost of X3 (c3) NOTE: Long Butterfly spread requires cash outlay at initiation because bull spread purchased by an investor is expensive than a bull spread that is sold Reading 29 Risk Management Applications of Option Strategies FinQuiz.com Rationale to use Long Butterfly Spread: It is used when investor expects that the volatility of the underlying will be relatively low compared to what market expects i.e the underlying asset will trade near the middle exercise price • When market is highly volatile, butterfly spread strategy is not profitable and generates losses To summarize: a) Value at expiration: VT = max (0, ST – X1) – max (0, ST – X2) + max (0, ST – X3) b) Profit = VT – c1 + 2c2 - c3 c) Maximum Profit occurs when price of underlying is close to the middle exercise price i.e when ST = X2 Thus, Maximum Profit = X2 – X1 – c1 + 2c2 – c3 d) Maximum Loss occurs when price of underlying < lower strike price or > upper strike price and investor loses net premium Thus, Maximum Loss = c1 – 2c2 + c3 e) There are two breakeven points i.e i Breakeven =ST* = X1 + net premium = X1 + c1 – 2c2 + c3 ii Breakeven = ST* = 2X2 – X1 – Net premium = 2X2 – X1 – (c1 – 2c2 + c3 ) = 2X2 – X1 – c1 + 2c2 - c3 C Long Butterfly Spread (Using Puts): Butterfly Spread = Long Bear put spread + Short bear put spread (or Long Bull put spread) Long Butterfly Spread = (Buy the put with exercise price of X3 and sell the put with exercise price of X2) + (Buy the put with exercise price of X1 and sell the put with exercise price of X2) where, X1< X2 < X3 Cost of X1 (p1) < Cost of X2 (p2) current value of the underlying • When price < X1, put provides protection against loss • When price > X2, short call reduces gains • When price lies between X1 and X2, both put and call are out-of-the-money • Like forwards, collar requires no initial outlay except the underlying price • Unlike forwards, collar payoff represents a range as it is shown in the figure above that it breaks at the two exercise prices • Collars represent directional strategies i.e their performance is based on the direction of the movement in the underlying Practice: Example 8, Volume 5, Reading 29 *NOTE: Typically, in a collar, the call and put premiums offset each other However, it is not necessarily always the case i.e call premium can be > put premium Important to note: • Put premium decreases when put exercise price is lowered • To offset this lower put premium, investor can sell call option with a higher exercise price • Decreasing put exercise price and increasing call exercise price results in increase in both the upside potential and downside risk Collar v/s Bull Spread: The collar is quite similar to a bull spread i.e both have a cap on the gain and a floor on the loss However, bull spread does not involve actually holding the underlying FinQuiz.com 2.4.2) Straddle A Long straddle: It involves buying a put and a call with same strike price on the same underlying with the same expiration; both options are at-the-money • In this strategy, an investor can make profit from upside or downside movement of the underlying price • Due to call option, the gain on upside is unlimited and due to put option, downside gain is quite large but limited • Straddle is a strategy that is based on the volatility of the underlying It benefits from high volatility • Straddle is a costly strategy Rationale to use Straddle: Straddle is to be used only when the investor expects that volatility of the underlying will be relatively higher than what market expects but is not certain regarding the direction of the movement of the underlying price To summarize: To summarize: (For zero-cost collar) a) Initial value of the position = value of the underlying asset = V0 = S0 b) Value at expiration: VT = Value of underlying ST + Value of the put option + Value of the short call option = ST + max (0, X1 - ST) – max (0, ST – X2) c) Profit = VT – V0 = VT –S0 d) Maximum Profit = X2 – S0 e) Maximum Loss = S0 – X1 f) Breakeven =ST* = S0 Range forwards and risk reversals: Collars are also known as range forwards and risk reversals a) Value at expiration: VT = max (0, ST -X) + max (0, X– ST) b) Profit = VT –p0 - c0 c) Maximum Profit = ∞ d) Maximum Loss occurs when both call and put options expire at-the money and investor loses premiums on both options i.e Maximum Loss = p0 + c0 e) Breakeven = ST* = X ± (p0 + c0) B Short Straddle: It involves selling a put and a call with same strike price on the same underlying with the same expiration; both options are at-the-money Reading 29 Risk Management Applications of Option Strategies FinQuiz.com • This strategy is preferably used when investor has neutral view of the volatility or when investor expects a decrease in volatility • This strategy gains when both the options expire at-the money i.e investor earns call and put premium • This strategy has unlimited loss potential Practice: Example 9, Volume 5, Reading 29 2.4.3) Box Spreads A box spread is a combination of a bull spread and a bear spread i.e Box-spread = Bull spread + Bear spread Variations of Straddle: When investor has any specific outlook regarding direction of price movement, then either a call or a put can be added to the straddle • Adding call option to a straddle is known as “Strap” • Adding put option to a straddle is known as “Strip” • These strategies generate greater gains when price movement occurs in the expected direction; however, these strategies are more complex than a straddle a) Long Strangle: It is a variation of the straddle This strategy involves buying the put and call on the same underlying with the same expiration but with different exercise prices This strategy is used if investor view is that volatility will increase A Long Box-spread= (buy the call with exercise price X1 and sell the call with exercise price X2) + (buy the put with exercise X2 and sell the put with exercise X1) • Box spread pay-off (i.e profit) is always the same i.e it is must be risk-free when the options are priced correctly * In simple words, box-spread always results in buying the underlying at X1 and selling it at X2 Since this outcome is known to an investor at the start, a box-spread can be viewed as a riskless strategy • Since transaction is risk free, the PV of the pay-off, discounted at risk-free rate should be equal to the initial outlay (net premium) i.e we should have: (X2 – X1) / (1 + r) r = c1 – c2 + p2 – p1 o When PV of the pay-off > net premium, the box spread is underpriced and it should be purchased Buying a box-spread is referred to as long box-spread o When PV of the pay-off < net premium, the box spread is overpriced and it should be sold It is referred to as short box-spread *Arbitrage opportunity is available when options are not priced correctly b) Short Strangle: This strategy involves selling the put and call on the same underlying with the same expiration but with different exercise prices This strategy is used if investor has a neutral view about volatility or he/she expects that volatility will decrease B Short Box-spread = (Sell the call with exercise price X1 and buy the call with exercise price X2) + (Sell the put with exercise X2 and buy the put with exercise X1) Advantages: • A box spread can be used to exploit an arbitrage opportunity • A box spread does not require the binomial or Black-Scholes-Merton model to hold • It does not require a volatility estimate and all the transactions associated with box-spread strategy can be executed within the options market • Box-spread is a simple strategy and has lower transaction costs Reading 29 Risk Management Applications of Option Strategies To summarize (for Long Box-spread): a) Initial value of the box spread = Net premium = c1 – c2 + p2 – p1 b) Value at expiration: VT = X2 –X1 c) Profit = X2 –X1 - (c1 – c2 + p2 – p1) d) Maximum Profit = same as profit e) Maximum Loss = no loss is possible given fair option prices f) Breakeven =ST* = no break-even; the transaction always earns the risk-free rate, given fair option prices Practice: Example 10, Volume 5, Reading 29 Volatility will increase Neutral view on volatility Volatility will decrease Price will decrease Buy Puts Sell Underlying Sell Calls Neutral view on price Buy Straddle Do nothing Sell Straddle Price will increase Buy Calls Buy Underlying Sell Puts INTEREST RATE OPTION STRATEGIES Interest rate call and put options are used to protect against changes in interest rates • For dollar based interest rate options, generally, the underlying rate is LIBOR • The underlying rate is always a specific rate i.e the rate on 90-day or 180-day underlying instrument • When the option is exercised, the pay-off is determined using a specific notional principal • Traditionally, the pay-off on interest rate option does not occur immediately upon exercise; rather, it is paid on the date when payment on the underlying instrument is due The pay-off of an interest rate Call Option= (Notional principal) × max (0, Underlying rate at expiration – ୈୟ୷ୱ ୧୬ ୳୬ୢୣ୰୪୷୧୬୥ ୰ୟ୲ୣ ቁ Exercise rate) × ቀ ଷ଺଴ • 180-day LIBOR can be used as the underlying rate and days in underlying could be 180 or perhaps 182, 183 etc • When an interest rate option is based on m-day LIBOR, it is important to note that the rate is determined on the day when the option expires and payment is made m days later The pay-off of an interest rate Put Option= (Notional principal) × max (0, Exercise rate - Underlying rate at ୈୟ୷ୱ ୧୬ ୳୬ୢୣ୰୪୷୧୬୥ ୰ୟ୲ୣ ቁ expiration) × ቀ ଷ଺଴ FinQuiz.com 3.1 Using Interest Rate Calls with Borrowing Interest rate call options are used by borrowers to manage interest rate risk on floating-rate loans In interest rate call options, the following factors must be considered 1) Option expiration date is the same as when loan starts 2) Option pay-offs occur at the time when borrower makes interest payments on loan 3) Option premium is paid by the borrower today (i.e at time t0) Example: A company plans to borrow $40 million in 128 days at 180-day LIBOR plus 200 basis points To manage the risk associated with higher interest rate on a loan, it buys a call option in which the underlying is the rate on 180-day LIBOR • • • • • The option expires in 128 days The exercise rate is 5% The notional principal is $40 million The company pays a premium of $100,000 Current LIBOR = 5.5% Reading 29 Risk Management Applications of Option Strategies Solution: loan, it buys a put option in which the underlying is the rate on 90-day LIBOR a) The company will pay $100,000 up front in the form of option premium • The rate the firm could earn if it invested the $100,000 would be 5.5% (i.e current LIBOR given) Thus, compounding premium at the original/current LIBOR of 5.5% + 200 bps for 128 days = $100,000[1 + (0.055+ 0.02) ì (128/360)] = $102,667 ã Thus, call premium of $100,000 is equivalent to $102,667 at the time the loan is taken out • This increases the cost of the loan because by paying this amount, the firm effectively receives = $40 million - $102,667 = $39,897,333 • Thus, effective loan proceeds = $39,897,333 b) The option expires on the date when the loan is taken out by the company and pays off = ଵ଼଴ ($40,000,000) × Max (0, LIBOR 5%) ì ã Note that whenever LIBOR is below 5%, the payoff is zero • Whenever LIBOR is > 5% e.g when LIBOR is 8%, the payoff is ଵ଼଴ ($40,000,000) × Max (0, 8% – 5%) × ቀ ቁ = ଷ଺଴ $600,000 c) Loan interest = ($40,000,000) × (LIBOR on the date ଵ଼଴ loan is taken out + 200 bps) × ቀ ቁ ଷ଺଴ For LIBOR = 8%, Loan interest = ($40,000,000) × (8% + 200 bps) ì = $2,000,000 ã ã ã • • • The option expires in 47days The exercise rate is 7% The notional principal is $50 million The company pays a premium of $62,500 Current LIBOR = 7.25% Assume LIBOR changes to 6% Solution: a) The bank will pay $62,500 up front in the form of option premium • The rate the bank could earn on if it invested the $62,500 would be 7.25% (i.e current LIBOR given) Thus, compounding premium at the original/current LIBOR of 7.25% + 250 bps for 47 days =$62,500 [1 + (0.0725+ 0.025) × (47 / 360)] = $63,296 • Thus, put premium of $62,500 is equivalent to $63,296 at the time the loan is made • Thus, by paying this amount, the bank loaned = $50 million + $63,296 = $50,063,296 • Thus, effective amount loaned = $50,063,296 b) The option expires on the date when the loan is made by the bank and pays off = ଽ଴ ($50,000,000) × Max (0, 0.07 LIBOR) ì ã Note that whenever LIBOR is above 7%, the payoff is zero • Whenever LIBOR is < 7% e.g when LIBOR is 6%, the payoff is ଽ଴ ($50,000,000) × Max (0, 0.07 – 0.06%) × ቀ ቁ = ଷ଺଴ d) Effective Interest paid = $2,000,000 - $600,000 = $1,400,000 e) Effective rate on the loan = {(NP + Effective interest) / effective loan proceeds} 365 / Days in underlying rate – = {($40m + $1,400,000) / $39,897,333}365 / 180 – = 0.0779 = 7.79% ଷ଺଴ $125,000 c) Loan interest = ($50,000,000) × (LIBOR on the date ଽ଴ loan is made + 250 bps) × ቀ ቁ ଷ଺଴ For LIBOR = 6%, Loan interest = ($50,000,000) × (6% + 250 bps) × ଽ଴ ቀ ቁ = $1,062,500 Source: Curriculum, Reading 29, Exhibit 13 Practice: Example 11, Volume 5, Reading 29 3.2 FinQuiz.com ଷ଺଴ d) Effective Interest received = $1,062,500 + $125,000 = $1,187,500 e) Effective rate on the loan = {(NP + Effective interest) / effective amount loan loaned} 365 / Days in underlying rate – = {($50m + $1,187,500) / $50,063,296}365 / 90 – = 0.0942 = 9.42% Using Interest Rate Puts with Lending Source: Reading 29, Exhibit 15 Interest rate put options can be used by lenders to manage interest rate risk on floating-rate loans i.e when interest rate falls below a specific level, interest rate put option generates a pay-off for the lender and thus compensates the lender (e.g bank) for the lower interest rate on the loan Example: A Bank plans to lend $50 million in 47 days at 90-day LIBOR plus 250 basis points To manage the risk associated with lower interest rate on a floating-rate Practice: Example 12, Volume 5, Reading 29 3.3 Using an Interest Rate Cap with a Floating-Rate Loan Interest rate cap is a combination of interest rate call options where each option’s pay-off occurs on the date Reading 29 Risk Management Applications of Option Strategies when the interest payments on a loan are due Each option in a cap is called a caplet • Each caplet has its own expiration date • Each caplet has the same Exercise rate • The cap seller makes payments to the borrower if interest rates > strike rate during the term of the cap • The pay-off of each caplet is determined on its expiration date, but the caplet pay-off (if any) is made on the next payment date i.e the date on which the loan interest is paid o This implies that if a loan has e.g interest payments, the cap will contain only five caplets because there will be only five risky payments as the first rate on the loan is already set o A cap will contain six caplets only when the borrower purchases the cap in advance of taking out the loan i.e the additional caplet can be used to protect the 1st rate setting on a loan Effect of Notional principal amount and exercise rate/strike rate on cost of cap: • The cap can be used to protect the entire loan amount or only a portion of the loan amount Reducing the dollar amount of the cap results in reduction of the cost of the cap • Reducing the strike rate of a cap results in increase in the cost of the cap 3.4 FinQuiz.com Using an Interest Rate Floor with a Floating-Rate Loan Interest rate floor is a combination of interest rate put options where each option’s pay-off occurs on the date when the interest payments on a loan are due to be received Each option in a floor is called a floorlet • Each floorlet has its own expiration date • Exercise rate on each floorlet is the same a) Loan Interest payment: It is computed as follows Loan interest = Notional Principal × (LIBOR on previous ୈୟ୷ୱ ୧୬ ୱୣ୲୲୪ୣ୫ୣ୬୲ ୮ୣ୰୧୭ୢ reset date + 100 bps) × ቀ ቁ ଷ଺଴ b) Floorlet Pay-off: It is computed as follows The floor pay-off = Notional Principal × (0, Exercise rate ୈୟ୷ୱ ୧୬ ୱୣ୲୲୪ୣ୫ୣ୬୲ ୮ୣ୰୧୭ୢ LIBOR on previous reset date) × ቀ ቁ ଷ଺଴ c) Effective Interest = Interest received on the loan + Floorlet pay-off For detail calculations, refer to Exhibit 18, Reading 29 Practice: Example 14, Volume 5, Reading 29 a) Loan Interest payment: It is computed as follows Loan interest = Notional Principal × (LIBOR on previous ୈୟ୷ୱ ୧୬ ୱୣ୲୲୪ୣ୫ୣ୬୲ ୮ୣ୰୧୭ୢ reset date + 100 bps) × ቀ ቁ ଷ଺଴ b) Cap Pay-off: It is computed as follows The cap pay-off = Notional Principal × (0, LIBOR on previous reset date – Exercise rate) × ୈୟ୷ୱ ୧୬ ୱୣ୲୲୪ୣ୫ୣ୬୲ ୮ୣ୰୧୭ୢ ቀ ቁ 3.5 A collar is a combination of a long (short) position in a cap and a short (long) position in a floor For borrower For lender The borrower can buy a cap to protect against rising interest rates and sell the floor to finance the premium paid to buy a cap The lender can buy a floor to protect against falling interest rates and sell the cap to finance the premium paid to buy a floor NOTE: NOTE: In this case, effective interest paid in each period will be: In this case, effective interest earned in each period will be: Effective interest paid = actual interest paid – cap pay-off + floor pay-off • The premium received by selling the floor can be used to offset the premium paid to buy a cap • Buying a cap provides Effective interest earned = actual interest earned + floor pay-off – cap pay-off • The premium received by selling the cap can be used to offset the premium paid to buy a floor • Buying a floor provides ଷ଺଴ c) Effective Interest = Interest due on the loan – Caplet pay-off NOTE: Since loan has multiple payments, the effective rate on a loan is similar to IRR on capital investment project or YTM on a bond For detail calculations, refer to Exhibit 17, Reading 29 Practice: Example 13, Volume 5, Reading 29 Using an Interest Rate Collar with a Floating-Rate Loan Reading 29 Risk Management Applications of Option Strategies For borrower For lender protection against rising interest rates but sale of the floor results in the borrower giving up any gains from interest rates falling below the exercise rate on the floor • Like Zero-cost collar in equity options, in Zerocost interest rate collar, first of all borrower selects exercise rate of the cap Then, the floor exercise rate is set such that the floor premium offsets the cap premium so that there is no initial outlay protection against falling interest rates but sale of the cap results in the lender giving up any gains from interest rates rising above the exercise rate on the cap • Like Zero-cost collar in equity options, in Zerocost interest rate collar, first of all lender selects exercise rate of the floor Then, the cap exercise rate is set such that the cap premium offsets the floor premium so that there is no initial outlay • Typically, in a collar, the call and put premiums offset each other However, it is not necessarily always the case • Zero-cost only means that there is no upfront cash outlay FinQuiz.com • Initial cost of the hedge can be reduced by increasing the cap exercise rate and decreasing the floor exercise rate; this will result in a decrease in cost of the cap and generate income from selling the floor However, it will expose the buyer of collar to more interest rate risk • Initial cost of the hedge can be reduced by having lower notional principal for the cap and higher notional principal for the floor A collar creates a band within which the buyer’s effective interest rate fluctuates i.e • The borrower will benefit when interest rate falls and will be hurt when interest rate increases within that range/band This implies that the borrower will face risk within that range • Change in interest rate will have no net effect when: a) Interest rate > cap exercise rate b) Interest rate < floor exercise rate Practice: Example 15, Volume 5, Reading 29 Effect of exercise rate and size of notional principal on cost of hedge: For example in case of long cap & short floor, OPTION PORTFOLIO RISK MANAGEMENT STRATEGIES By trading in options, dealers provide liquidity to the market and take risk To earn the bid-ask spread without taking risk, dealers can hedge their positions by using hedging strategies For example if a dealer has sold a call, he can hedge his/her risk either by: i Buying an identical call option or ii Buying a put with the same exercise price and expiration, buying the asset, and selling a bond or taking out a loan with face value equal to the exercise price and maturity equal to that of the option’s expiration (it refers to put-call parity) This hedge is static in nature i.e.no change in the position is required as time passes iii Using Delta Hedging: When necessary options are not available or are not favorably priced, then the dealer can hedge risk by taking a long position in a certain number of units of the underlying asset The size of that long position is determined using option’s delta i.e Delta = ۱‫ ܍܋ܑܚ۾ ܖܗܑܜܘ۽ ܖܑ ܍܏ܖ܉ܐ‬ ۱‫܍܋ܑܚ۾ ܏ܖܑܡܔܚ܍܌ܖ܃ ܖܑ ܍܏ܖ܉ܐ‬ = ∆۱ ∆‫܁‬ • Delta is used to measure the sensitivity of the price of an option to changes in the price of the underlying asset • The delta usually lies between and o Delta will be only at expiration and only if the option expires in-the-money o During the option’s life, if the option is in-themoney, delta will tend to be above 0.5 o As expiration approaches, the deltas of in-themoney options will move slowly towards 1.0 o Delta will be only at expiration and only if the option expires out-of-the money o During the option’s life, if the option is out-of-themoney, delta will tend to be below 0.5 o As expiration approaches, the deltas of out-ofthe-money options will move slowly toward o Delta moves quickly towards or when delta is at-the-money and/or near expiration o 0.5 is often viewed as an “average” delta o For calls: delta lies between and o For puts: delta lies between -1 and NOTE: The deltas of options that are very slightly in-the-money will temporarily move down as expiration approaches But eventually they will move up towards 1.0 How to determine size of the Long position: Delta can be used to determine how many units of the underlying are Reading 29 Risk Management Applications of Option Strategies needed to offset the risk associated with short position in option Nc / Ns = - / (∆C / ∆S) = -1 / Delta where, Nc = Number of call options Ns = Number of units of the underlying e.g stocks For example, if the dealer sells 100 calls, it will need to own number of shares = 100 ì (Delta) ã The loss on the underlying may be offset by the gain on the options Rule: Buy (sell) delta shares for each option short (long) Three complicating issues in delta hedging: 1) Delta represents only an approximate change in the call price for a (small) change in the underlying Delta is not a perfect hedge because (particularly for calls), the delta underestimates the effects of increases in the underlying and overestimates the effects of decreases in the underlying 2) Delta changes with the change in the price of the underlying and/or time The greater the change in the price of the underlying, the worse the deltabased approximation • Increase in the underlying price leads to an increase in delta • Delta decreases as time passes • Usually, effect of changes in underlying price dominates the effect of time 3) Rounding off the number of units of the underlying per option results in a small amount of imprecision in the balancing of the two offsetting positions • When we round up, we have more units of the underlying than needed It negatively affects the hedged position when the price of underlying decreases • When we round down, we have fewer units of the underlying than needed It negatively affects the hedged position when the price of underlying increases 4.1 Delta Hedging an Option over Time A delta-hedged position needs to be rebalanced whenever the underlying price changes and/or with the passage of time It is referred to as Dynamic hedging Guidelines for Perfectly hedged Short-call and long underlying Portfolio: • With the passage of time, both the delta & value of short-call position decrease In this situation (all else equal), the dealer should sell shares of stock and invest the proceeds at risk-free rate Thus, FinQuiz.com value of portfolio will grow at risk-free rate over time • A small decrease in the underlying price results in decrease in delta; however, there would be no change in the value of the portfolio In this case, dealer should sell some shares of stock and invest the proceeds at risk-free rate • A small increase in the underlying price results in increase in delta; however, there would be no change in the value of the portfolio In this case, dealer should buy some shares of stock by borrowing the required amount at risk-free rate • Delta of in-the-money call option will increase towards near expiration; whereas, delta of outof-the-money call option will decrease towards near expiration Example: Suppose a call option in which: • • • • • • • Underlying price = $1210 Exercise price = $1200 Continuously compounded risk-free rate = 2.75% Expiration time = 120 days Option price = $65.88 Delta = 0.5826 Number of call options sold = 1000 a) If underlying price changes to $1200, new option price is computed as follows: Option price + (∆S × delta) = 65.88 + (1200 – 1210) (0.5826) = 60.05 b) Number of shares of stock needed to delta hedge = delta × number of options sold = 0.5826 × 1000 = 583 c) Value of portfolio = 583 ($1210) – 1,000 ($65.88) = $639,550 • Thus, initially, we need to invest $639,550 to delta hedge d) One day later, this amount should grow at a riskfree rate i.e it should be = $639,550 exp (0.0275 / 365) = $639,598 Thus, our benchmark value = $639,598 Suppose after day, • • • • • • Underlying price = $1215 Exercise price = $1200 Continuously compounded risk-free rate = 2.75% Expiration time = 119 days New option price = $68.55 New Delta = 0.5966 a) Now value of portfolio = 583 ($1215) – 1,000 ($68.55) = $639,795 b) Compare this amount with the benchmark value: The amount we have in excess of benchmark value = $639,795 - $639,598 = $197 c) Number of shares of stock needed to delta hedge = delta × number of options sold = 0.5966 × 1000 = 597 • We have 583 shares, so now we need to buy 14 more shares (i.e 597 – 583) • To buy additional 14 shares, we need = 14 × Reading 29 Risk Management Applications of Option Strategies $1215 = $17,010 d) We will borrow $17,010 at risk-free rate e) Now value of portfolio is still = 597 ($1215) – 1,000 ($68.55) - $17,010 = $639,795 • This implies that, we cannot create or destroy any wealth by just rearranging the position e) One day later, this amount should grow at a riskfree rate i.e it should be = $639,795 exp (0.0275 / 365) = $639,843 Thus, our new benchmark value = $639,843 f) The loan will grow at a risk-free rate to = $17,010 exp (0.0275 / 365) = $17,011 Now suppose next day, • • • • • Underlying price = $1198 Exercise price = $1200 Continuously compounded risk-free rate = 2.75% New option price = $58.54 New Delta = 0.5479 g) New portfolio value = 597 ($1198) – 1,000 ($58.54) - $17,011* = $639,655 • *The loan will grow to $17,010 exp(0.0275/365) = $17,011 h) Compare this amount with the benchmark value i.e = $639,655 - $639,843 = -$188 f) Number of shares of stock needed to delta hedge = delta × number of options sold = 0.5479 ì 1000 = 548 ã We have 597 shares, so now we need to sell 49 shares (i.e 597 – 548) • By selling 49 shares, we will generate cash inflow = 49 ì $1198 = $58,702 ã Now we can pay back our debt i.e $58,702 $17,011 = $41,691 • This amount can be invested at risk-free rate i) New portfolio value is still = 548 ($1198) – 1,000 ($58.54) + $41,691 = $639,655 j) After one day, value of portfolio should grow at a risk-free rate to $639,655 exp (0.0275 / 365) = $693,703 Thus, our new benchmark value = $693,703 Source: Curriculum, Reading 29, Section 4.1 Example: Dealer has sold 500 call options on a stock currently priced at $125.75, 60 days until expiration, a price of $10.89 and a delta of 0.5649 Note that risk-free rate is 4% • Number of shares of stock needed to delta hedge 500 call options = 0.5649 × 500 = 282.45 Suppose after day, delta changes to 0.6564 • Number of shares of stock needed to delta hedge 500 call options = 0.6564 × 500 = 328.20 ≈ 328 FinQuiz.com Suppose after days, the following information applies: • • • • • • Stock price = $122.75 Option price = $9.09 Delta = 0.5176 Number of options sold = 500 Number of shares purchased = 328 Bond Balance = -$6,072 When stock price = $122.75, stocks worth = $122.75 × 328 = $40,262 When option price = $9.09, options worth = $9.09 × (-500) = -$4,545 Bond balance (given) = -$6,072 • Total market value of dealer’s total position = $40,262 + (-$4,545) + (-$6,072) = $29,645 • When delta = 0.5176, number of shares of stock needed to delta hedge 500 call options = 0.5176 × 500 = 258.80 ≈ 259 • Dealer now needs 259 shares instead of 328; thus, he must sell = 328 – 259 = 69 shares • Selling those 69 shares will generate = 69 ($122.75) = $8,470 • Dealer invests $8,470 in risk-free bonds • Since the bond balance was -$6,072, this amount can be used to pay-off this debt i.e $8,470 - $6,072 = $2,398 will be excess amount that can be invested in risk-free bonds Now the total market value of dealer’s total position is calculated as follows: • When stock price = $122.75, stocks worth = $122.75 ì 259 = $31,792 ã When option price = $9.09, options worth = $9.09 ì (-500) = -$4,545 ã Bond balance (given) = $2,398 • Total market value = $31,792 + $4,545 + $2,398 = $29,645 Benchmark value = $29,645 exp (0.04 / 365) = $29,648 Value of bond after day = $2,398 exp (0.04 / 365) = $2,398 Now suppose after days, the following information applies: Stock price = $120.50 Option price = $7.88 • When stock price = $120.50, stocks worth = $120.50 × 259 = $31,210 • When option price = $7.88, options worth = $7.88 ì (-500) = -$3,940 ã Bond balance (given) = $2,398 • Total market value = $31,210+ (-$3,940) + $2,398 = $29,668 • Market value ($29,668) > benchmark value Reading 29 Risk Management Applications of Option Strategies ($29,648) i.e market value is $20 more than the benchmark Source: Curriculum, Reading 29, example 16 NOTE: Delta hedges are most difficult to maintain for at-themoney options and/or near expiration Hedging using non-identical option: Suppose two options are available on the same underlying but are not identical i.e they differ by exercise price, expiration, or both • One option has a delta of ∆1 • Other option has a delta of ∆2 The value of the position is: V = N1 c1 + N2c2 FinQuiz.com • For a perfect delta hedge, the return to the delta hedge would be the risk-free rate and gamma = Note that gamma would be zero when delta would represent a perfectly linear slope i.e ∆c = ∆S • When gamma = 0, dealer does not need to adjust the delta hedge • When delta does not represent a perfectly linear slope, gamma ≠ • The larger the gamma, the more the deltahedged position deviates from being risk free and greater will be the risk In this case, dealer should gamma hedge his/her position • Gamma is largest for at-the-money options and/or near expiration Gamma hedge involves adding a position in another option so that both delta and gamma are zero i.e Gamma hedge = Position in underlying + Positions in two options where, N1 and N2 represent the quantity of each option in a portfolio that hedges the value of one of the options in a portfolio c1 = price of option c2 = price of option To delta hedge, we get: 4.3 Vega and Volatility Risk An option price is very sensitive to the changes in volatility of the underlying Vega is used to measure the sensitivity of the option price to the volatility Vega = େ୦ୟ୬୥ୣ ୧୬ ୓୮୲୧୭୬ ୮୰୧ୡୣ େ୦ୟ୬୥ୣ ୧୬ ୚୭୪ୟ୲୧୪୧୲୷ ୭୤ ୲୦ୣ ୳୬ୢୣ୰୪୷୧୬୥ Desired Quantity of option relative to option = ୈୣ୪୲ୟ ୭୤ ୭୮୲୧୭୬ ଶ ୈୣ୪୲ୟ ୭୤ ୭୮୲୧୭୬ ଵ N1 / N2 = - ∆c2 / ∆c1 • It is known as Ratio Spread • The negative sign indicates that a long position in one option will require a short position in the other • These deltas will change and will require monitoring and modification of the position over time 4.2 Gamma and the Risk of Delta Gamma measures the sensitivity of the delta to a change in the underlying In effect, it is the delta of the delta େ୦ୟ୬୥ୣ ୧୬ ୢୣ୪୲ୟ Gamma = େ୦ୟ୬୥ୣ ୧୬ ୳୬ୢୣ୰୪୷୧୬୥ ୮୰୧ୡୣ • Like delta, gamma represents only an approximate change in the delta for a (small) change in the underlying • Like delta and gamma, Vega represents only an approximate change in the option price for a (small) change in the volatility • Price of at-the-money option has greater sensitivity to changes in volatility • Volatility is the most critical variable because it is the only unobservable variable Thus, it is difficult to estimate Vega • Value of delta-hedged position with a zero or insignificant gamma can change by a large amount when the volatility changes e.g if dealer has sold options to delta hedge a long position in the underlying, then an increase in volatility leads to increase in value of options and results in large loss for the dealer How to manage Vega Risk: Vega risk cannot be managed independently The dealer is required to jointly monitor and manage the risk associated with the delta, gamma and Vega Reading 29 Risk Management Applications of Option Strategies FinQuiz.com FINAL COMMENTS Equity options v/s Bond options: Equity option strategies apply similarly to bond option strategies However, the major difference is that the bond options must expire before the bonds mature Interest rate options Equity or Bond Options 1) Bullish (bearish) investors buy puts (calls) on interest rates Because being bullish (bearish) on interest rates means that investor expects decrease (increase) in interest rates 2) Interest rate options pay-offs represent interest payments 3) Interest rate options can be used for hedging purposes e.g caps and floors can be used for hedging floating-rate loans 1) Bullish (bearish) equity or bond investors buy calls (puts) 2) Equity or bond options pay-offs represent sale or purchase of stocks or bonds by the option buyer 3) Standard option strategies i.e straddles and spreads can be used by investors for the purpose of hedging NOTE: Interest rate swaps represent the most widely used financial derivative by investors However, they are less widely used with currencies and equities when compared to forwards, futures and options Practice: End of Chapter Practice Problems for Reading 29 & FinQuiz Item-set ID# 9280  ... $ 639 ,655 j) After one day, value of portfolio should grow at a risk-free rate to $ 639 ,655 exp (0.0275 / 36 5) = $6 93, 7 03 Thus, our new benchmark value = $6 93, 7 03 Source: Curriculum, Reading 29, ... (47 / 36 0)] = $ 63, 296 • Thus, put premium of $62,500 is equivalent to $ 63, 296 at the time the loan is made • Thus, by paying this amount, the bank loaned = $50 million + $ 63, 296 = $50,0 63, 296 •... ($50,000,000) × (6% + 250 bps) × ଽ଴ ቀ ቁ = $1,062,500 Source: Curriculum, Reading 29, Exhibit 13 Practice: Example 11, Volume 5, Reading 29 3. 2 FinQuiz. com ଷ଺଴ d) Effective Interest received = $1,062,500