Product nature Swaptions The buyer of a swaption has the right to enter into an interest rate swap by some specified date. The swaption also specifies the maturity date of the swap. The buyer can be the fixed-rate receiver (put swaption) or the fixed-rate payer (call swaption). The writer becomes the counterparty to the swap if the buyer exercises. The strike rate indicates the fixed rate that will be swapped versus the floating rate. The buyer of the swaption either pays the premium upfront or can be structured into the swap rate. 1
Uses of swaptions Used to hedge a portfolio strategy that uses an interest rate swap but where the cash flow of the underlying asset or liability is uncertain. Uncertainties come from (i) callability, eg, a callable bond or mortgage loan, (ii) exposure to default risk. Example Consider a S & L Association entering into a 4-year swap in which it agrees to pay 9% fixed and receive LIBOR. The fixed rate payments come from a portfolio of mortgage pass-through securities with a coupon rate of 9%. One year later, mortgage rates decline, resulting in large prepayments. The purchase of a put swaption with a strike rate of 9% would be useful to offset the original swap position. 2
Management of callable debt Three years ago, XYZ issued 15-year fixed rate callable debt with a coupon rate of 12%. 3 0 2 12 today bond call date original bond issue bond maturity Strategy The issuer sells a two-year receiver option on a 10-year swap, that gives the holder the right, but not the obligation, to receive the fixed rate of 12%. 3
Call monetization By selling the swaption today, the company has committed itself to paying a 12% coupon for the remaining life of the original bond. The swaption was sold in exchange for an upfront swaption premium received at date 0. Company XYZ Swap Counterparty Pay 12% Coupon Swaption Premium Bondholders 4
Call-Monetization Cash Flow: Swaption Expiration Date Interest Rates 12% Company XYZ Swap Counterparty Bondholders Pay 12% Coupon Interest Rates < 12% Company XYZ Pay FRN Coupon at LIBOR New Bondholders LIBOR 12% Swap Counterparty 5
Disasters for the issuer The fixed rate on a 10-year swap was below 12% in two years but its debt refunding rate in the capital market was above 12% (due to credit deterioration) The company would be forced either to enter into a swap that it does not want or liquidate the position at a disadvantage and not be able to refinance its borrowing profitably. 6
Example on the use of swaption In August 1992 (two years ago), a corporation issued 7-year bonds with a fixed coupon rate of 10% payable semiannually on Feb 15 and Aug 15 of each year. The debt was structured to be callable (at par) offer a 4-year deferment period and was issued at par value of $100 million. In August 1994, the bonds are trading in the market at a price of 106, reflecting the general decline in market interest rates and the corporation s recent upgrade in its credit quality. 7
Question The corporate treasurer believes that the current interest rate cycle has bottomed. If the bonds were callable today, the firm would realize a considerable savings in annual interest expense. Considerations The bonds are still in their call protection period. The treasurer s fear is that the market rate might rise considerably prior to the call date in August 1996. Notation T = 3-year Treasury yield that prevails in August, 1996 T + BS = refunding rate of corporation, where BS is the company specific bond credit spread T + SS = prevailing 3-year swap fixed rate, where SS stands for the swap spread 8
Strategy I. Enter an off-market forward swap as the fixed rate payer Agreeing to pay 9.5% (rather than the at-market rate of 8.55%) for a three-year swap, two years forward. Initial cash flow: Receive $2.25 million since the the fixed rate is above the at-market rate. Assume that the corporation s refunding spread remains at its current 100 bps level and the 3-year swap spread over Treasuries remains at 50 bps, then the annual reduction in interest rate expense after refunding 10% - (T + 1.0) if the firm is able to refund 0 if it is not. The gain (or loss) on unwinding the swap is the fixed rate at that time = [T + 0.5% - 9.5%]. The two effects offset each other, given the assumed spreads. The corporation has effectively sold the embedded call option for $2.25m. 9
Gains and losses August 1996 decisions: Gain on refunding (per settlement period): [10 percent (T + BS)] if T + BS < 10 percent, 0 if T + BS 10 percent. Gain (or loss) on unwinding the swap (per settlement period): [9.50 percent (T + SS)] if T + SS < 9.50 percent, [(T + SS) 9.50 percent] if T + SS 9.50 percent. Assuming that BS = 1.00 percent, SS = 0.50 percent, these gains and losses in 1996 are: 10
Callable Debt Management with a Forward Swap Gains Gain on Refunding Gain on Unwinding Swap If BS goes up 9% T If SS goes down Losses 11
Net Gain Gains 9% T If SS goes down or BS goes up Losses 12
Comment on the strategy Since the company stands to gain in August 1996 if rates rise, it has not fully monetized the embedded call options. This is because a symmetric payoff instrument (a forward swap) is used to hedge an asymmetric payoff (option) problem. 13
Strategy II. Buy payer swaption expiring in two years with a strike rate of 9.5%. Initial cash flow: Pay $1.10 million as the cost of the swaption (the swaption is out-of-the-money) August 1996 decisions: Gain on refunding (per settlement period): [10 percent (T + BS)] if T + BS < 10 percent, 0 if T + BS 10 percent. Gain (or loss) on unwinding the swap (per settlement period): [(T + SS) 9.50 percent] if T + SS > 9.50 percent, 0 if T + SS 9.50 percent. With BS = 1.00 percent and SS = 0.50 percent, these gains and losses in 1996 are: 14
Gain on Refunding Gain on Exercising Payer Swaption Gains If BS goes up If SS goes down 9% T Losses 15
Net Gain Gains If SS goes down or BS goes up 9% T Losses 16
Comment on the strategy The company will benefit from Treasury rates being either higher or lower than 9% in August 1996. However, the treasurer had to spend $1.1 million to lock in this straddle. 17
Strategy III. Sell a receiver swaption at a strike rate of 9.5% expiring in two years. Initial cash flow: Receive $2.50 million (in-the-money swaption) August 1996 decisions: Gain on refunding (per settlement period): [10 percent (T + BS)] if T + BS < 10 percent, 0 if T + BS 10 percent. Loss on unwinding the swap (per settlement period): [9.50 percent (T + SS)] if T + SS < 9.50 percent, 0 if T + SS 9.50 percent. With BS = 1.00 percent SS = 0.50 percent, these gains and losses in 1996 are: 18
Gain on Refunding Gains If BS goes up 9% T Losses Loss on selling receiver swaption If SS goes down 19
Gains Net Gain 9% T If SS goes down or BS goes up Losses 20
Comment on the strategy By selling the receiver swaption, the company has been able to simulate the sale of the embedded call feature of the bond, thus fully monetizing that option. The only remaining uncertainty is the basis risk associated with unanticipated changes in swap and bond spreads. 21
Black s model to price European options Consider a European call option on a variable whose value is V. Define T: Time to maturity of the option F: Forward price of V for a contract with maturity T F 0 Value of F at time zero K: Strike price of the option P(t, T): Price at time t of a zero-coupon bond paying $1 at time T V T : Value of V at time T σ: Volatility of F Call payoff = max(v T K, 0) at time T 22
The expected payoff from the option is calculated by assuming 1. V T has a lognormal distribution with the standard deviation of ln V T equal to σ T. 2. The expected value of V T is F 0. The expected payoff is E[V T ] N(d 1 ) KN(d 2 ), where 2 ln( E( VT / K) + σ T / 2 d1 =, d2 = d1 σ T. σ T Taking E(V T ) = F 0, then call value = P(0, T)[F 0 N(d 1 ) - KN(d 2 )], where 2 ln( F0 / K) + σ T / 2 d1 =, d2 = d1 σ T. σ T Recall that N(d 2 ) = probability that V T is greater than K (under the risk neutral pricing probability measure). 23
Numerical example 10-month European call option on a 9.75-year bond with a face value of $1,000. current cash bond price = $960 strike price = $1,000 10-month riskfree interest rate = 10% per annum Volatility of the forward bond price in 10 months = 9% per annum Coupon payments of $50 are expected in 3 months and 9 months. This would mean that the accrued interest is $25 and the quoted bond price is $935. 3-month and 9-month riskfree interest rates are 9% and 9.5%, respectively. 24
Present value of the coupon payment is 50e 0.25 0.09 + 50e 0.75 0.095 = 95.45. Bond forward price is F 0 = (960 95.45)e 0.1 0.8333 = 939.68. (a) If the strike price is the cash price that would be paid for bond on exercise, then F 0 = 939.68, K = 1,000, P(0, T) = e 0.1 10/12 = 0.9200, σ = 0.09, T = 10/12, then price of call option = $9.49. (b) If the strike price is the quoted price that would be paid for the bond on exercise, one month s accrued interest must be added to K since the maturity of the option is one month after a coupon date. Then K = 1,000 + 50 0.16667 = 1,008.33 The price of the option becomes $7.97. 25
Caplet A caplet is a contract which pays at time t i+1 the difference between the τ i -period rate resetting at time t i, (call it R i ) and a strike price K multiplied by the year fraction τ i, if this difference is positive, and zero otherwise. Caplet ( t 1) = max( R i+ i K,0) τ i 26
Valuation of a caplet For valuation purpose, the unknown future value of the rate R i must be set equal to today s implied forward rate F i. Accepting the lognormal assumption for F i, we have Caplet value at t0 = [ F( t0, ti, ti+ 1) N( h1 ) KN( h2 )] P( t0, ti+ 1) where h 1,2 = ln F K σ ± 2 σ 2 t i ( t i t t 0 0 ). 27
Interest rate caps Used to provide insurance against the floating rate going above a certain level cap rate. Example The rate on a loan, where the principal is $10 million, is reset every 3 months equal to 3-month LIBOR. The writer provides an interest rate cap of 10% per annum. To fulfill the obligations, the writer must pay at the end of each quarter (in million of dollars) 0.25 x 10 x max(r 0.1,0) where R is the 3-month LIBOR rate at the beginning of the quarter. A cap is a portfolio of caplets. 28
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30 Cap as a portfolio of bond options Consider a cap with a total life of T, a principal of L and a cap rate of K. Let t 1, t 2,, t n be the reset dates and t n+1 = T. Define R k be the interest rate for the period between t k and t k+1 as observed at time t k, k = 1, 2,, n. The cap s payoff at time t k+1 is,0). ) max( ( 1 K R t t L k k k + The above payoff is equivalent to + + = + + + + + 0, ) ( 1 )] ( [1 max,0) max( ) ( 1 ) ( 1 1 1 1 k k k k k k k k k k k t t R t t K L L K R t t R t t L at time t k.
L[1 + K( tk + 1 tk )] The term is the value at time t k of a zero-coupon 1+ Rk ( tk + 1 tk ) bond that pays off L[1 + R k (t k+1 t k )] at time t k+1. This is a put option with maturity t k on a zero-coupon bond with maturity t k+1 and face value is L[1 + K(t k+1 t k )]. 31