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## Chapter 15

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**Chapter 15**Interest Rate Options: Hedging Applications and Pricing**Topics**• Hedging fixed income and debt positions with interest rate options • Hedging a series of cash flows with OTC caps, floors, and other interest-rate derivatives • Valuation of interest-rate options with the binomial interest rate tree • Pricing interest rate options with the Black Futures Model**Hedging**• By hedging with either exchange-traded futures call options on Treasury or Eurodollar contracts or with an OTC spot call option on a debt security or an interest rate put (floorlet), a fixed-income manager planning to invest a future inflow of cash, can obtain protection against adverse price increases while still realizing lower costs if security prices decrease.**Hedging**• For cases in which bond or money market managers are planning to sell some of their securities in the future, a manager, for the costs of buying an interest rate option, can obtain downside protection if bond prices decrease while earning greater revenues if security prices increase. • Similar downside protection from hedging positions using interest rate put options can be obtained by bond issuers, borrowers, and underwriters.**Hedging Risk**• Like futures-hedged position, option-hedged positions are subject to quality, quantity, and timing risk. • For OTC options, some of the hedging risk can be minimized, if not eliminated, by customizing the contract. • For exchange-traded option-hedged positions the objective is to minimize hedging risk by determining the appropriate number of option contracts.**Hedging Risk**• For direct hedging cases (cases in which the future value of the asset or liability to be hedged, VT, is the same as the one underlying the option contract) the number of options can be determined by using the naïve hedge where: n= VT/X. • For cross hedging cases (cases in which the asset or liability to be hedge is not the same as the one underlying the option contract), the number of options can be determined by using the price-sensitivity model defined in Chapter 13.**Future CD Purchase**• Consider the case of a money market manager who is expecting a cash inflow of approximately $985,000 in September that he plans to invest in a 90-day jumbo CD with a face value of $1M and yield tied to the LIBOR. • Assume: • CDs are currently trading at a spot index of 94.5 (S0 = $98.625 per $100 face value), for YTM = ($100/$98.625)(365/90) ‑ 1 = .057757).**Future CD Purchase**• Suppose the manager would like to earn a minimum rate on his September investment that is near the current rate, with the possibility of a higher yield if short‑term rates increase. • To achieve these objectives, the manager could take a long position in a September Eurodollar futures call. • Suppose the manager buys one September 94 Eurodollar futures call option priced at 4: • RD = 6% • X = $985,000 • C0 = 4($250) =$1,000 • nc = VT/X = $985,000/$985,000 = 1 call contract**Future CD Purchase**• Assume the manager will close the futures call at expiration at its intrinsic value (fT – X) if the option is in the money and then buy the $1M, 90-day CD at the spot price. • The exhibit shows the effective investment expenditures at expiration (costs of the CD minus the profit from the Eurodollar futures call) and the hedged YTM earned from the hedged investment: YTM = [$1,000,000/Effective Investment](365/90) –1)**Future CD Purchase**Profit = [Max(fT- $985,000), 0] - $1,000 YTM = [$1M]/Column 4]365/90 - 1**Future CD Purchase**• If spot rates (LIBOR) are lower than RD = 6% at expiration, the Eurodollar futures call will be in the money and the manager will be able to profit from the call position, offsetting the higher costs of buying the 90-day, $1M CD. • As a result, the manager will be able to lock in a maximum purchase price of $986,000 and a minimum yield on his 90-day CD investment of 5.885% when RD is 6% or less.**Future CD Purchase**• If spot discount rates (LIBOR) are 6% or higher, the call will be worthless. In these cases, though, the manager’s option losses are limited to just the $1,000 premium he paid, while the prices of buying CDs decrease, the higher the rates. • As a result, for rates higher than 6%, the manager is able to obtain lower CD prices and therefore higher yields on his CD investment as rates increase.**Future CD Purchase**• Thus, by hedging with the Eurodollar futures call, the manager is able to obtain at least a 5.885% YTM, with the potential to earn higher returns if rates on CDs increase.**Future 91-Day T-Bill Investment**• In Chapter 13, we examined the case of a corporate treasurer who was expecting a $5 million cash inflow in June, which she was planning to invest in T‑bills for 91 days. • In that example, the treasurer locked in the yield on the T‑bill investment by going long in June T‑bill futures contracts. • Suppose the treasurer expected higher short-term rates in June but was still concerned about the possibility of lower rates. • To be able to gain from the higher rates and yet still hedge against lower rates, the treasurer could alternatively buy June call options on T-bill futures.**Future 91-Day T-Bill Investment**• Suppose there is a June T-bill futures call with • Exercise price = $987,500 (index = 95, RD = 5) • Price = $1,000 (quote = 4; C = (4)($250) = $1,000) • June expiration (on both the underlying futures and futures option) occurring at the same time the $5M cash inflow is to be received.**Future 91-Day T-Bill Investment**• To hedge the 91-day investment with this call, the treasurer would need to buy 5 calls at a cost of $5,000:**Future 91-Day T-Bill Investment**• If T-bill rates were lower at the June expiration, then the treasurer would profit from the calls and could use the profit to defray part of the cost of the higher priced T-bills. • As shown in the exhibit, if the spot discount rate on T-bills is 5% or less, the treasurer would be able to buy 5.058 spot T-bills (assume perfect divisibility) with the $5M cash inflow and profit from the futures calls, locking in a YTM for the next 91 days of approximately 4.75% on the $5M investment.**Future 91-Day T-Bill Investment**Profit = 5[Max(fT- $987,500), 0] - $5,000 YTM = [(Number of Bill)($1M)]/$5M]365/91 - 1**Future 91-Day T-Bill Investment**• If T-bill rates are higher, then the treasurer would benefit from lower spot prices while her losses on the call would be limited to just the $5,000 costs of the calls. • For spot discount rates above 5%, the treasurer would be able to buy more T-bills, the higher the rates, resulting in higher yields as rates increase.**Future 91-Day T-Bill Investment**• Thus, for the cost of the call options, the treasurer is able to establish a floor by locking in a minimum YTM on the $5M June investment of approximately 4.75%, with the chance to earn a higher rate if short-term rates increase.**Hedging a CD Rate with an OTC Interest Rate Put**• Suppose the ABC manufacturing company was expecting a net cash inflow of $10M in 60 days from its operations and was planning to invest the excess funds in a 90-day CD from Sun Bank paying the LIBOR. • To hedge against any interest rate decreases occurring 60 days from the now, suppose the company purchases an interest rate put (corresponding to the bank's CD it plans to buy) from Sun Bank for $10,000.**Hedging a CD Rate with an OTC Interest Rate Put**• Suppose the put has the following terms: • Exercise rate = 7% • Reference rate = LIBOR • Time period applied to the payoff = 90/360 • Day Count Convention = 30/360 • Notional principal = $10M • Payoff made at the maturity date on the CD (90 days from the option’s expiration) • Interest rate put’s expiration = T = 60 days (time of CD purchase) • Interest rate put premium of $10,000 to be paid at the option’s expiration with a 7% interest: Cost = $10,000(1+(.07)(60/360)) = $10,117**Hedging a CD Rate with an OTC Interest Rate Put**• As shown in the exhibit, the purchase of the interest rate put makes it possible for the ABC company to earn higher rates if the LIBOR is greater than 7% and to lock in a minimum rate of 6.993% if the LIBOR is 7% or less.**Hedging a CD Rate with an OTC Interest Rate Put**• For example, if 60 days later the LIBOR is at 6.5%, then the company would receive a payoff (90 day later at the maturity of its CD) on the interest rate put of $12,500: • The $12,500 payoff would offset the lower (than 7%) interest paid on the company’s CD of $162,500: • At the maturity of the CD, the company would therefore receive CD interest and an interest rate put payoff equal to $175,000: $12,500 = ($10M)[.07-.065](90/360) $162,500 = ($10M)(.065)(90/360) $175,000 = $162,500 + $12,500**Hedging a CD Rate with an OTC Interest Rate Put**• With the interest-rate put’s payoffs increasing the lower the LIBOR, the company would be able to hedge any lower CD interest and lock in a hedged dollar return of $175,000. • Based on an investment of $10M plus the $10,117 costs of the put, the hedged return equates to an effective annualized yield of 6.993%: • On the other hand, if the LIBOR exceeds 7%, the company benefits from the higher CD rates, while its losses are limited to the $10,117 costs of the puts. 6.993% = [(4)($175,000)]/[$10M + $10,117]**Hedging Future T-Bond Sale with an OTC T-Bond Put**• Consider the case of a trust-fund manager who plans to sell ten $100,000 face value T‑bonds from her fixed income portfolio in September to meet an anticipated liquidity need. • The T‑bonds the manager plans to sell pay a 6% interest and are currently priced at 94 (per $100 face value), and at their anticipated selling date in September, they will have exactly 15 years to maturity and no accrued interest. • Suppose the manager expects long-term rates in September to be lower and therefore expects to benefit from higher T-bond prices when she sells her bonds, but she is also concerned that rates could increase and does not want to risk selling the bonds at prices lower than 94.**Hedging Future T-Bond Sale with an OTC T-Bond Put**• As a strategy to lock in minimum revenue from the September bond sale if rates increase, while obtaining higher revenues if rates decrease, suppose the manager decides to buy spot T-bond puts from an OTC Treasury security dealer who is making a market in spot T-bond options. • Suppose the manger pays the dealer $10,000 for a put option on ten 15-year, 6% T-bonds with an exercise price of 94 per $100 face value and expiration coinciding with the manager’s September sales date. • The exhibit shows the manager's revenue from either selling the T‑bonds on the put if T-bond prices are less than 94 or on the spot market if prices are equal to or greater than 94.**Hedging Future T-Bond Sale with an OTC T-Bond Put**• If the price on a 15‑year T‑bond is less than 94 at expiration (or rates are approximately 6.60% or more) the manager would be able to realize a minimum net revenue of $930,000 by selling her T-bonds to the dealer on the put contract at X = $940,000 and paying the $10,000 cost for the put; • If T-bond prices are greater than 94 (below approximately 6.60%), her put option would be worthless, but her revenue from selling the T-bond would be greater, the higher T-bond prices, while the loss on her put position would be limited to the $10,000 cost of the option.**Hedging Future T-Bond Sale with an OTC T-Bond Put**• Thus, by buying the put option, the trust-fund manager has attained insurance against decreases in bond prices. Such a strategy represents a bond insurance strategy.**Hedging Future T-Bond Sale with an OTC T-Bond Put**• Note: If the portfolio manager were planning to buy long‑term bonds in the future and was worried about higher bond prices (lower rates), she could hedge the future investment by buying T‑bond spot or futures calls.**Managing the Maturity Gap with a Eurodollar Futures Put**• In Chapter 13, we examined the case of a small bank with a maturity gap problem resulting from making $1M loans in June with maturities of 180 days, financed by selling $1M worth of 90-day CDs at the current LIBOR of 5% and then 90 days later selling new 90-day CDs to finance its June CD debt of $1,012,103: • To minimize its exposure to market risk, the bank hedged its $1,012,103 CD sale in September by going short in 1.02491 ($1,012,103/$987,500) September Eurodollar futures contract trading at quoted index of 95 ($987,500). $1,012,103 = $1M(1.05)90/365**Managing the Maturity Gap with a Eurodollar Futures Put**• Instead of hedging its future CD sale with Eurodollar futures, the bank could alternatively buy put options on Eurodollar futures. • By hedging with puts, the bank would be able to lock in or cap the maximum rate it pays on it September CD.**Managing the Maturity Gap with a Eurodollar Futures Put**• For example, suppose the bank decides to hedge its September CD sale by buying a September Eurodollar futures put with • Expiration coinciding with the maturity of its September CD • Exercise price of 95 (X = $987,500) • Quoted premium of 2 (P = $500)**Managing the Maturity Gap with a Eurodollar Futures Put**• With the September debt from the June CD of $1,012,103, the bank would need to buy 1.02491 September Eurodollar futures puts (assume perfect divisibility) at a total cost of $512.46 to cap the rate it pays on its September CD:**Managing the Maturity Gap with a Eurodollar Futures Put**• If the LIBOR at the September expiration is greater than 5%, the bank will have to pay a higher rate on its September CD, but it will profit from its Eurodollar futures put position, with the put profits being greater, the higher the rate. • The put profit would serve to reduce part of the $1,012,103 funds the bank would need to pay on the maturing June CD, in turn, offsetting the higher rate it would have to pay on its September CD.**Managing the Maturity Gap with a Eurodollar Futures Put**• As shown in the exhibit, if the LIBOR is at discount yield of 5% or higher, then the bank would be able to lock in a debt obligation 90 days later of $1,025,435 (allow for slight rounding differences), for an effective 180-day rate of 5.225%. • If the rate is less than or equal to 5%, then the bank would be able to finance its $1,012,615.46 debt (June CD of $1,012,103 and put cost of $512.46) at lower rates, while its losses on its futures puts would be limited to the premium of $512.46. • As a result, for lower rates the bank would realize a lower debt obligation 90 days later and therefore a lower rate paid over the 180-day period.**Managing the Maturity Gap with a Eurodollar Futures Put**• Thus, for the cost of the puts, hedging the maturity gap with puts allows the bank to lock in a maximum rate on its debt obligation, with the possibility of paying lower rates if interest rates decrease.**Hedging a Future Loan Rate with an OTC Interest Rate Call**• Suppose a construction company plans to finance one of its project with a $10M, 90-day loan from Sun Bank, with the loan rate to be set equal to the LIBOR + 100 BP when the project commences 60 day from now. • Furthermore, suppose that the company expects rates to decrease in the future, but is concerned that they could increase.**Hedging a Future Loan Rate with an OTC Interest Rate Call**• To obtain protection against higher rates, suppose the company buys an interest rate call option from Sun Bank for $20,000 with the following terms: • Exercise rate = 7% • Reference rate = LIBOR • Time period applied to the payoff = 90/360 • Notional principal = $10M • Payoff made at the maturity date on the loan (90 days after the option’s expiration) • Interest rate call’s expiration = T = 60 days (time of the loan) • Interest rate call premium of $20,000 to be paid at the option’s expiration with a 7% interest: Cost = $20,000(1+(.07)(60/360)) = $20,233**Hedging a Future Loan Rate with an OTC Interest Rate Call**• The exhibit shows the company's cash flows from the call, interest paid on the loan, and effective interest costs that would result given different LIBORs at the starting date on the loan and the expiration date on the option. • As shown in Column 6 of the table, the company is able to lock in a maximum interest cost of 8.016% if the LIBOR is 7% or greater at expiration, while still benefiting with lower rates if the LIBOR is less than 7%.**Hedging a Bond Portfolio with T-Bond Puts – Cross Hedge**• In Chapter 13, we defined the price sensitivity model for hedging debt positions in which the underlying futures contract was not the same as the debt position to be hedge. • The model determines the number of futures contracts that will make the value of a portfolio consisting of a fixed‑income security and an interest rate futures contract invariant to small changes in interest rates.**Hedging a Bond Portfolio with T-Bond Puts – Cross Hedge**• The model also can be extended to hedging with put or call options. The number of options (calls for hedging long positions and puts for short positions) using the price-sensitivity model is: where: Durs = duration of the bond being hedged. Duroption = duration of the bond underlying the option contract. V0 = current value of bond to be hedged. YTMs = yield to maturity on the bond being hedged. YTMoption = yield to maturity on the option’s underlying bond.**Hedging a Bond Portfolio with T-Bond Puts – Cross Hedge**• Suppose a bond portfolio manager is planning to liquidate part of his portfolio in September. • The portfolio he plans to sell consist of a mix of A to AAA quality bonds with a weighted average maturity of 15.25 years, face value of $10M, weighted average yield of 8%, portfolio duration of 10, and current value of $10M. • Suppose the manager would like to benefit from lower long-term rates that he expects to occur in the future but would also like to protect the portfolio sale against the possibility of a rate increase.**Hedging a Bond Portfolio with T-Bond Puts – Cross Hedge**• To achieve this dual objective, the manager could buy a spot or futures put on a T-bond. • Suppose there is a September 95 (X = $95,000) T-bond futures put option trading at $1,156 with the cheapest-to-deliver T-bond on the put’s underlying futures being a bond with a current maturity of 15.25 years, duration of 9.818, and currently priced to yield 6.0%.