Swaptions

Below are links to the following topics:

• Credit Swaps
• Swaptions II
• Extendable and Cancelable Swaps
• Swaption Pricing and Applications
• Swaption Valuation at Expiration
• The Value of a Receiver Swaption at Expiration
• Swaption Valuation before Expiration
• A Swaption Pricing Example
• A Swaption Application

Credit Swaps

A credit swap is a privately negotiated, over-the-counter derivative, designed to transfer credit risk from one counterparty to another. The payoff of a credit swap is linked to the credit characteristics of an underlying reference asset, also called a reference credit. Credits swaps enable financial institutions and corporations to manage credit risks. The market for credit swaps has been growing rapidly in recent years. The ISDA survey of the market shows that the size of the market, which was less than $1 trillion in notional amounts as of December 2001, grew to exceed $17 trillion by the end of 2005.

Credit swaps take many forms. In a credit default swap, two parties enter into a contract whereby Company A pays Company B a fixed periodic payment for the life of the agreement. Company B makes no payments unless a specified credit event occurs. Credit events are typically defined to include a failure to make payments when due, bankruptcy, debt restructuring, a change in external credit rating, or a rescheduling of payments for a specified reference asset. If such a credit event occurs, the party makes a payment to the first party, and the swap then terminates. The size of the payment is usually linked to the decline in the reference asset's market value following the credit event.

In a total return swap, two companies enter an agreement whereby they swap periodic payment over the life of the agreement. Company C (called the protection buyer) makes payments based upon the total return - coupons plus capital gains or losses - of a specified reference asset or group of assets. Company C (the protection seller) makes fixed or floating payments as with a plain vanilla interest rate swap. Both companies' payments are based upon the same notional amount. The reference asset can be almost any asset, index or group of assets. Among the underlying assets of a total return swap are loans and bonds.

Total return swaps have numerous application. For example, total return swaps enable banks to manage the credit exposure resulting from their lending activities. Consider a Milwaukee bank that lends $10 million to a local brewing company at a fixed interest rate of 7 percent. This interest rate charged by the bank includes a built-in risk premium to account for expected increase in the brewing company's credit risk over the life of the loan. If credit risk unexpectedly increases, the market value of the loan (an asset to the bank) will fall. To hedge this credit risk, the bank can enter into a total return swap. Assume that the life of the swap is one year, with a single exchange of cash flows at maturity and a notional principal of $10 million. Assume also that the swap is structured so that the bank pays the swap dealer a fixed rate of 9 percent plus the change in the loan's market value. In return, the bank receives one-year US dollar LIBOR. Assume that over the course of the following year an increase in credit risk causes the market value of the loan to fall so that on the swap's maturity date the loan is worth only 95 percent of its initial value. Under the terms of the swap, the bank owes the swap dealer the fixed rate of 9 percent minus the 5 percent capital loss on the market value of the loan, for a net total of 4 percent. In return, the bank receives a floating payment of one-year LIBOR, (8 percent minus 4 percent) multiplied by the swap's notional principal. This gain can be used to offset the loss of market value on the loan over the period.

Total return swaps provide protection against loss in value of the underlying asset irrespective of cause. For example, if the interest rate changes, then the net cash flows of the total return swap will also change even though the credit risk of the underlying loans has not necessarily changed. In other words, the swap's cash flows are influenced by market risk as well as credit risk. The credit default swap enables the bank to avoid the interest-sensitive element of total return swap provides protection against loss due to market risk and credit risk. Finally, either credit default swaps or total return swaps entail two sources of credit exposure: one from the underlying reference asset and another from possible default by the counterparty to the transaction.

Swaptions

A swaption is an option on a swap that can be either American or European in form. A receiver swaption gives the holder the right to enter a particular swap agreement as the fixed-rate receiver. A payer swaption gives the holder the right to enter a particular swap agreement as the fixed-rate payer. As with all options, ownership gives a right but not an obligation. The swap agreement underlying the swaption is defined with respect to all of its terms, such as its tenor, notional principal, and fixed-rate and floating-rate index. For the swaption, there is a stated expiration date.

receiver swaption = fixed-rate receiver = call option on a bond = higher bond prices, lower interest rates , you would exercise swaption; lower bond prices, higher interest rates, let swaption expire worthless = floating rate payer

payer swaption = fixed-rate payer = put option on a bond = (lower bond prices, higher interest rates) = floating rate receiver lower bond prices, higher interest rates, exercise swaption; higher prices, lower interest rates, you would let swaption expire worthless; = floating rate receiver

We may think of a receiver swaption as being a call option on a bond that pays a fixed rate of interest. If the owner of a receiver swaption exercises, she will be in a position of the owner of a bond who receives fixed interest payments. The floating payments that she makes play the role of the exercise price (betting interest rates will move lower).

Similarly, a payer swaption is analogous to a put option on a bond. When the owner of a payer swaption exercises, he will be in the position of an issuer of a fixed rate bond, because the exercise of the payer swaption requires a sequence of fixed-rate interest payments in exchange for inflows at a floating rate.

To understand these features more closely, consider a payer swaption. The purchase of the payer swaption pays an agreed premium to the seller at the inception of the transaction. Usually the premium is stated as some number of basis points on the notional principal of the swap underlying the option. It is typical for this premium to lie in the range of 20 - 40 basis points, but the premium depends upon the exercise price, the time to expiration, and the volatility of the underlying rates. The exercise price of the payer swaption is a fixed rate specified in the swaption agreement. Assuming that the swaption is European, the owner will exercise the payer swaption if the contractual fixed rate (the exercise price) is lower than the fixed rate prevailing in the open market for swaps with the same tenor as that underlying the payer swaption. Upon exercise by the owner of the payer swaption, the seller of the payer swaption is obligated to make the series of floating-rate payments specified in the swap agreement.

The receiver and payer swaptions come into-the-money in different circumstances. A receiver swaption pays off when interest rates fall, and a payer swaption pays off when interest rates rise.

The owner of a receiver swaption will want to exercise when interest rates fall, because the owner of the receiver swaption will then receive a sequence of fixed interest payments stipulated in the swaption contract in exchange for making a sequence of floating-rate payments at the new lower rate of interest. By contrast, the owner of a payer swaption will exercise when interest rates rise. Upon the exercise of a payer swaption, the owner of the payer swaption will be obligated to make a sequence of fixed interest payments at the old lower rate stipulated in the swaption contract and will receive a sequence of floating-rate payments at the now higher rate.

As an example consider a European payer swaption on a five-year swap with annual payments and a notional principal of $10 million. Let us assume that the fixed rate specified in the swap agreement is 8 percent and floating rate is LIBOR. For this payer swaption, the premium might be 30 basis points. With a principal of $10 million, the premium would then be $30,000. Six months after contracting, at the expiration date of the option, the owner of the payer swaption can either exercise or let the option expire worthless. If the owner exercises he will pay a fixed rate of 8 percent and receive a floating rate of LIBOR for the five years of the swap. The owner of the payer swaption will exercise if the fixed rate on swaps of the type underlying the option is greater than 8 percent. For example, at expiration assume that the market for this type of swap calls for a fixed-rate payment of 8.5 percent in exchange for LIBOR for a five-year tenor. The holder of the payer swaption will exercise because he can enter the swap agreement at terms more favorable than those prevailing in the market. Conversely, if the market fixed rate for such a swap is less than 8 percent, say 7.75 percent, the owner of the payer swaption will allow the swaption to expire because it is worthless. The swaption is worthless because it is at expiration and gives the owner the right to enter a swap to pay 8 percent and receive LIBOR. However, the prevailing market rate allows anyone to enter a swap to pay 7.75 percent and receive LIBOR.

Consider now the choices facing the holder of a European receiver swaption. To acquire this swaption the owner pays a premium to purchase the swaption. At the expiration date, the owner may exercise. If she exercises, she will enter a swap to receive-fixed and pay-floating. The owner should exercise if the fixed rate on the swap underlying the swaption exceeds the market fixed rate on swaps of the same type as the swap underlying the swaption. For example, assume that the swap underlying this receiver swaption has a seven-year tenor with semiannual payments, a notional principal of $50 million, and a fixed rate of 6.5 percent in exchange for LIBOR. At expiration, assume that the market fixed rate for this type of swap is 7 percent. In this circumstance the receiver swaption is worthless and its holder will allow it to expire. The swaption is worthless because it allows the holder to enter a swap to receive 6.5 percent, but the market rate for this type of swap is to receive a fixed rate of 7 percent. By contrast, if the prevailing fixed rate for this type of swap is 6 percent at the expiration of the swaption, the holder of the receiver swaption should exercise, because the swaption entitles her to enter a swap to receive a 6.5 percent fixed rate when the prevailing market offers only a 6 percent fixed rate on this type of swap.

A Corporate Application of a Swaption

Swaptions offer the same kinds of speculative and hedging opportunities as all other options. As an example, consider a firm that has issued a callable bond. From the perspective of the issuer, we may analyze a bond with a call provision as consisting of a noncallable bond plus the purchase of a call option on the bond. From the issuing firm's point of view, the firm has paid for the option by promising a higher coupon rate for the callable bond than the rate necessary for a noncallable bond. We assume that the first call date for this bond lies in the future and the firm has decided that it will not want to call the bond, because the firm anticipates a continuing need for funds and does not particularly wish to incur the transaction cost of call the bond and going through the registration procedures to issue another bond to meet its financing requirements Having made this determination, the call feature of the bond represents an unwanted option in the firm's portfolio that will be losing value through time decay as expiration approaches. Yet the call provision has a real value. The firm's problem is to find a way to capture the value of the call provision without actually calling the bond. To capture the value of the call provision, the firm could sell a receiver swaption with terms that match the call feature of the bond. In effect, this transaction would unwind the call feature inherent in the original bond, because the firm receives a swaption premium that equals the value of the call provision on the firm's bond.

For the firm in this situation, consider the following data. The call date is one year away, the semi-annual coupon rate on the bond is 9%. The principal amount of the bond issue is $150 million, and the present maturity of the bond is seven years. To implement its strategy, the firm sells a European receiver swaption with an expiration of one year. The swap that underlies the swaption has a tenor of six years, a notional principal of $150 million, calls for semiannual payments, a fixed rate of 9 percent, and a floating rate equal to LIBOR. The price of this swaption depends on the current fixed swap rates relative to the terms specified for the swap underlying the swaption. The lower rates are today, when the firm sells the swaption, the greater is the value of the call provision on the bond and the greater is the value of the swaption that the firm sells. After selling the swaption, the swaption premium belongs to the firm, but the firm has a potential obligation in one year.

At expiration, the holder of this receiver option will make its exercise decision depending on previous market rates for this kind of swap. At expiration, if the fixed rate for this kind of swap is above the fixed rate specified in the swap agreement, say 11 percent, the owner of the swaption will not exercise, because exercising and the swaption will expire worthless The swaption holder will not exercise, because exercising entails entering a swap to receive a fixed rate of 9 percent and pay LIBOR, when it could simply go to the market and enter an analogous swap to receive a fixed rate of 11 percent and pay LIBOR. In this case, the obligation of the firm is extinguished without performance, and the firm has simply gained the swaption premium. Note also, that with current fixed interest rates of 11 percent, the firm has no incentive to call its bond.

By contrast, if market interest rates at expiration are lower than the fixed rate specified for the swap underlying the swaption, say 7 percent, the owner of the swaption will exercise. By exercising, the owner of the swaption can enter a swap to receive a fixed rate of 9 percent and pay LIBOR at a time when the market only offers a fixed rate of 7 percent in exchange for LIBOR. If the swaption is exercised against the firm, it will be obligated to make a sequence of interest payments at 9 percent and to receive LIBOR. In this situation, the firm will also call its bond, because market rates are at 7 percent and the coupon rate on its bond is 9 percent. The firm then might issue a floating-rate note for its financing need. Assuming that it issues a floating-rate note at LIBOR for a term of six years to match the tenor of the swap, the firm's cash flows for the life of the swap are as follows:

Swap

Pay 9 percent fixed rate on $150 million notional principal; receive LIBOR on $150 million notional principal

Debt Financing

Issue a $150 million floating-rate note with a maturity of six years at LIBOR; use the proceeds of the $150 million floating-rate note to call the existing bond and repay the $150 million principal.

Netting out the receipt and payment of LIBOR and the call and refinancing of the original bond, the firm's net position is to continue paying the fixed rate of 9 percent, equal to the coupon rate on its original bond. However, the firm has captured the value of its call provision by the receipt of the swaption premium.

Extendable and Cancelable Swaps

In an extendible pay-fixed swap, the pay-fixed counterparty has the option to extend the tenor of the swap. If she exercises that option, the existing swap agreement remains in force for the additional time specified in the agreement. The swap underlying the payer swaption, a new swap is created on which payments would begin at the end of the existing swap. We may analyze this extendible pay-fixed swap as consisting of two distinct elements - a plain vanilla pay-fixed swap plus a payer swaption. The plain vanilla pay-fixed swap portion of the extendible swap has a tenor that extends to the expiration date of the embedded payer swaption.

The swaption gives the holder the right to extend the swap agreement for an additional period.

A cancelable swap gives a party the right, but not the obligation, to cancel an existing plain vanilla swap at a specific time during the originally contemplated tenor of the swap. The embedded swaption has an exercise date falling some time during the tenor of the underlying plain vanilla swap, and the swap underlying the swaption would have a tenor that covers the period from the exercise date through the tenor of the original plain vanilla swap. For example, a cancelable receive-fixed swap agreement consists of a plain vanilla receive-fixed swap plus a pay-fixed swaption. The plain vanilla swap has a tenor that extends over a given period. The swaption has an expiration date during the tenor of the plain vanilla swap, and the swap underlying the swaption extends from the expiration date of the swaption to the end of the tenor of the plain vanilla swap. Exercise of the swaption creates a new swap that exactly offsets the remaining payments on the plain vanilla swap.

To make this more concrete, consider the following example of a cancelable receive-fixed swap. A firm enters a cancelable receive-fixed swap with a notional amount of 20 million, a tenor of seven years, receipt of annual interest payments of 6.60 percent, and payments of LIBOR. The agreement also includes a swaption, the right to cancel the swap immediately after the fourth payment. This cancelable receive-fixed swap may be analyzed as consisting of two elements: a plain vanilla receive-fixed swap with a tenor of seven years and a European payer swaption that expires in four years and has a tenor of three years. If the owner of the cancelable swap cancels the swap, this is equivalent to exercising the payer swaption, because the swap underlying the payer swaption would have exactly the cash flows that offset the remaining tenor of the original plain vanilla swap.

Analyzing the extendable and cancelable swaps as plain vanilla swaps plus swaptions makes pricing these complex swaps simpler, because it is relatively easy to price plain vanilla swaps and swaption. Thus, there are four possible extendible and cancelable swap forms that may be analyzed as follows:

• extendible pay-fixed swap = plain vanilla pay-fixed swap plus a payer swaption
• extendible receive-fixed swap = plain vanilla receive-fixed swap plus a receiver swaption
• cancelable pay-fixed swap = plain vanilla pay-fixed swap plus a receiver swaption
• cancelable receive-fixed swap = plain vanilla receive-fixed swap plus a payer swaption

Thus the extendible swaps have an embedded swaption of the same type as the swap itself. An extendible pay-fixed swap has an embedded payer swaption, and an extendible receive-fixed swap has an embedded receiver swaption. Cancelable swaps have embedded swaptions of the opposite type that offset the terms of the swap itself. A cancelable pay-fixed swap has an embedded receiver swaption, and a cancelable receive-fixed swap has an embedded payer swaption.

Swaption Pricing and Applications [viewable here in Excel]

A swaption is an option on a swap. A receiver swaption gives the owner the right to enter a swap at a specified future date as the received-fixed party. A payer swaption gives the owner the right to enter a swap at a specified future date as the pay-fixed party. We continue to focus on European-style swaptions. Second, because a swaption is an option on a swap to begin at a future date, the swap underlying the swaption is a forward swap. We have explored the pricing of forward swaps and saw on that pricing a forward swap is similar to pricing a plain vanilla swap. Finally, the Black options model can be used to price these kinds of interest rate options. We will show how to find the price of a swaption and explore the ways in which swaptions and swaps can be used together for financial risk management.

The Exercise Decision

When a swaption expires, the owner can exercise and enter a swap. If the owner of a payer swaption exercises, she will make fixed payments at the strike price on the swaption, SR, and receive floating payments on the swap. If the owner of a receiver swaption exercises, she will receive fixed payments at the strike rate on the swaption, SR, and make floating payment. In both cases, the exercise decision depends on the value of the payments to be made at the strike rate compared to the anticipated value of the floating-rate payments.

The most basic principle of swap pricing is that the present value of the fixed and floating payments must be equal, when the fixed payments are discounted at the SFR and the floating payments are discounted at the forward rates implied by the yield curve. Since these two sides are equal, the SFR summaries the present value of the floating-side payments in a single number. Therefore, the exercise decision on the swap can be made by comparing the strike rate on the swaption with the fixed rate on the underlying swap. (In other words, the SFR serves as a proxy for the value of the floating-rate payments on the swap.)

Upon exercise of a payer swaption, the swaption owner will enter the swap to pay the strike rate on the swap, SR, and receive the floating-rate payments on the swap, whose value is reflected by the SFR on the swap. Similarly, upon exercise of a receiver swaption, the swaption owner will enter the swap to receive the strike rate on the swap and make the floating payments, with a value reflected by the SFR on the underlying swap. Summarizing, we can give the appropriate exercise rules as follows:

Table 22.13 Term structure data for swaption valuation ad examples

Semi-annual period	Par Yield(annual terms)	Par Yield (semi-annual terms)	Zero-coupon Factor	Forward rate factor (FCF)	1/Z0,t	FRAt-1,t/Z0,t
1	0.0609	0.03045	1.030450	1.030450	0.9704498	0.02955
2	0.0633	0.03165	1.064341	1.032889	0.93954855	0.03090
3	0.0657	0.03285	1.101993	1.035376	0.90744692	0.03210
4	0.0671	0.03355	1.141445	1.035801	0.87608215	0.03136
5	0.0691	0.03455	1.185882	1.038930	0.84325419	0.03283
6	0.0701	0.03505	1.230758	1.037842	0.81250742	0.03075
7	0.0703	0.03515	1.274858	1.035832	0.7844008	0.02811
8	0.0717	0.03585	1.327830	1.041551	0.75310829	0.03129
9	0.0719	0.03595	1.376825	1.036898	0.72630881	0.02680
10	0.0724	0.0362	1.430414	1.038922	0.69909817	0.02721
11	0.0734	0.0367	1.491779	1.042900	0.67034057	0.02876
12	0.0741	0.03705	1.554339	1.041937	0.64336019	0.02698
13	0.0745	0.03725	1.617077	1.040363	0.61839962	0.02496
14	0.0758	0.0379	1.696634	1.049198	0.58940247	0.02900
15	0.0762	0.0381	1.767774	1.041930	0.5656832	0.02372
16	0.0776	0.0388	1.862638	1.053663	0.53687295	0.02881
17	0.0776	0.0388	1.934908	1.038800	0.51682032	0.02005
18	0.0788	0.0394	2.040646	1.054647	0.49004086	0.02678
19	0.0795	0.03975	2.141559	1.049451	0.46694953	0.02309
20	0.0803	0.04015	2.253429	1.052237	0.4437682	0.02318
21	0.0805	0.04025	2.351470	1.043508	0.42526587	0.01850
22	0.0813	0.04065	2.480371	1.054817	0.40316554	0.02210
23	0.0827	0.04135	2.650501	1.068591	0.37728713	0.02588
24	0.0828	0.0414	2.765752	1.043483	0.36156535	0.01572
25	0.0837	0.04185	2.937885	1.062237	0.3403809	0.02118
26	0.0850	0.0425	3.157790	1.074851	0.31667714	0.02370
27	0.0850	0.0425	3.291996	1.042500	0.30376704	0.01291
28	0.0859	0.04295	3.518782	1.068890	0.28418923	0.01958
29	0.0872	0.0436	3.817729	1.084957	0.26193584	0.02225
30	0.0875	0.04375	4.023761	1.053967	0.24852372	0.01341

Payer swaption: exercise if SFR > SR
Receiver swaption: exercise if SR > SFR

Swaption Valuation at Expiration

In this section, we briefly explore the value of a payer and receive swaption at expiration. As we will see, if a swaption is in-the-money, the benefit is essentially an interest savings on each payment that is a function of the difference between the SFR on the swap and the SR on the swaption. We also use this discussion to lay the foundation for the pricing of swaptions prior to expiration.

The Value of a Payer Swaption at Expiration

Upon exercising a European swaption at its expiration, the underlying swap will commence. We know from our study of swap pricing that, at inception, the floating-rate payments and the fixed-rate payments must have the same present value. By exercising a payer swaption, the holder of the swaption enters a swap and pays the strike rate on the swaption, SR, instead of the currently prevailing swap fixed rate, SFR. On each payment in the swap initiated by exercising the swaption, the cash outflow is reduced by the difference between the SFR and the SR:

Payoff on each swap payment = (SFR - SR ) x NP x FRAC

where FRAC is the fraction of the year covered by the maturity of the underlying LIBOR instrument and NP is the notional principal. Because the payer swaption will be exercised only when SFR > SR, it is also the case that the payoff from the option in any payment in the swap equals the following:

Equation (22.25) Payoff on each swap payment = MAX{0, (SFR - SR ) x NP x FRAC}

We will use the fact expressed by Equation 22.25 when we turn to pricing the option before expiration.

Viewed from the expiration of the swaption and the initiation of the swap, the present value of all the payoffs from the T payments in the swap is as follows:

Equation (22.26) Value of payoff swaption at expiration = Σ^T_t=1 x (SFR - SR ) x NP x FRAC

Assuming no default on the swap this is a risk-free value. As we see by comparing Equations 22.26 and 22.25, the value of the swaption at expiration is just the value of the portfolio of payoffs on all of the individual payments.

The Value of a Receiver Swaption at Expiration

The holder of a receiver swaption will exercise at expiration if the strike price on the swaption, SR, exceeds the swap fixed rate, SFR. Exercising allows the holder to receive a cash flow each period that is larger than it otherwise would be. The additional periodic cash flow is a function of the difference between SR and SFR, and is as follows:

Payoff on each swap payment = (SR - SFR ) x NP x FRAC

Because the holder of the swaption will exercise only if the strike rate on the option exceeds the swap fixed rate, it must also be the case that:

Equation (22.28) Payoff on each swap payment = MAX{0, (SR - SFR ) x NP x FRAC}

At the expiration of the swaption and the initiation of the swap, the present value of all of the payoffs from the T payments in the receiver swap is as follows:

Value of receiver swaption at expiration = Σ^T_t=1 x (SR - SFR ) x NP x FRAC

Swaption Valuation Before Expiration

We now show that a swaption can be understood as a portfolio of calls or puts on LIBOR.

We have seen that the payoff at expiration for a call and a put on LIBOR is given by Equations 19.10 and 19.11:

Call = payer swaption

Put = receiver swaption

Equation (19.10) = MAX{0, (Observed LIBOR - SR ) x NP x FRAC}

Equation (19.11) = MAX{0, (SR - Observed LIBOR) x NP x FRAC}

Equation 19.10 gives the payoff for a call on LIBOR, while Equation 19.11 gives the payoff for a put on LIBOR. The payoff for a payer swaption (Equation 22.25) has the same form as the payoff for a call on LIBOR (Equation 19.10). Similarly, the payoff for a receiver swaption (Equation 22.28) has the same form as the payoff of the form of Equation 22.25 (or 19.10) for each swap payment date; similarly, a receiver swaption consists of one payoff of the form of Equation 22.28 (or 19.11) for each swap payment date. Therefore, a payer swaption is a portfolio of calls on LIBOR while a receiver swaption is a portfolio of puts on LIBOR.

We have seen that the Black model could be used to value options on LIBOR, assuming the LIBOR rate at the expiration if the option is log-normally distributed. In pricing a swaption, we apply the identical technique to value each call or put that pertains to a single payment on the swaption and that together constitute the value of a payer or receiver option.

We first focus on a single payment on a swap that underlies a swaption. Assume that the swaption is being valued at time t and that the option expires at time T. We consider the nth payment on the swap, which occurs at time T_n, which occurs some time after time T. Thus, at the time the swaption is being valued, the swaption has time T - t remaining until expiration and time T_n - t remaining until the nth payment on the swap. This difference between the time to expiration and the time until the payment is received is important for pricing. For example, in pricing calls and puts on LIBOR, we took account of the fact that payments are determined in advance and paid in arrears. This required taking into account the full amount of time from the present valuation date until the time the payment is actually received. Keeping in mind the distinction between times T - t and T_n - t, the values of the call and put pertaining to a single payment on the swap are as follows:

Equation (22.30) C_t = NP x FRAC x exp^-r(Tn-t)[SFR x N(d₁) - SR N(d₂)]

Pt = NP x FRAC x exp^-r(Tn-t)[SR x N(-d₂) - SR N(-d₁)]

where

d₁ = ln (SFR_t/SR) + 0.5σ²(T-t) /σ√T-t

d₂ = d₁ - σ√T-t

As equation 22.30 shows, the d₁ and d₂ terms use the time period T - t. This time period pertains to the expiration on the swaption, which determines the probability that the swaption will finish in the money. By contrast, the discounting runs over the period T_n - t, because this is the time from the valuation date to the payment on the underlying swap.

Because a swaption is simply a portfolio of call or puts, the value of a payer swaption is simply a portfolio of calls of the form of Equation 22.30, and a receiver swaption is a similar portfolio.

Equation (22.30) Payer swaption_t = Σ^N_n=1NP x FRAC x exp^-r(T_n-t)[SFR x N(d₁) - SR N(d₂)]

Equation (22.31) Receiver swaption_t = Σ^N_n=1NP x FRAC x exp^-r(Tn-t)[SR x N(-d₁) - SR N(-d₁)]

where d₁ and d₂ are as described in Equation 22.30, t is the valuation date, and T_n is the time at which the nth payment will be received. In addition to using the continuous discounting at a constant rate of Equation 22.31, we can also value the option using the zero-coupon factors that reflect the shape of the term structure:

Payer swaption_t = NP x FRAC x [SFR x N(d₁) - SR x N(d₁)] x Σ^N_n=1 1/Z_t,Tn

Receiver swaption_t = NP x FRAC x [SR x N(-d₂) - SFR x N(-d₁)] x Σ^N_n=1 1/Z_t,Tn

A Swaption Pricing Example [viewable here in Excel]

Table 22.13 presents term structure information that we will use for several swaption examples. Consider the yield curve environment for the point of view of time zero. A forward swap begins in four years and has semiannual payments, a tenor of three years, and a notional principal of $75 million. The SFR for this forward swap is as follows:

SFR = (¹⁴Σ_t=9 FRA_(t-1),t/Z_0,t) / (¹⁴Σ_t=9 1/Z_0,t)

Treating the numerator and denominator separately for convenience, we have the following:

numerator = 0.0369898/1.376828 +0.038922/1.430414 + 0.042900/1.491779 + 0.041937/1.554339 + 0.040363/1.617077 + 0.049108/1.696634 = 0.163706

denominator = 1/1.376828 + 1/1.430414 + 1/1.491779 + 1/1.554339 + 1/1.617077 + 1/1.696634 = 3.946910

Based on these calculations: SFR = 0.163706/3.946910 = 0.041477 on a semiannual basis, or 4.148% x 2 = 0.082954 in annual terms.

Consider now a payer and receiver swaption on this forward swap from the point of view of time zero. The payer swaption has a strike rate of 8 percent in annual terms, and the receiver swaption has a strike rate of 9 percent. The standard deviation of the forward swap rate is 0.11. First, for the payer swaption, the d₁ and d₂ terms are as follows:

d₁ = ln (SFR_t/SR) + 0.5Σ²(T-t) / σ√T-t = ln(0.082954/.08) + 0.5 x 0.112 x 4 / 0.11 x √4 = 0.274816

d₂ = d₁ - σ √T-t = 0.274816 - 0.11 x 2 = 0.054816

The cumulative normal values are N(d₁) = 0.608272 and N(0.274816) and N(d₂) = N(0.054816) = 0.521858. In finding the SFR for the forward swap, we have already determined that:

N (d₁) = NORMSDIST(0.0) = NORMSDIST (0.274816) = .608271197

N (d₁) = is the delta ratio. The call option will change price about .6082 as fast as the stock for a small price change by the stock.

N (d₂) = NORMSDIST(0.0) = NORMSDIST (0.054816) = .521857473

D₂ is the probability that the option will expire in the money i.e. spot above strike for a call.

N(D₂) gives the expected value (i.e. probability adjusted value) of having to pay out the strike price for a call.

ΣN_n=1 1/Z_t,Tn = 1/1.37685+1/1.430414+1/1.491779+1/1.554339+1/1.617077+1/1.696634 = 3.946910

Applying Equation 22.32, the value of the payer swaption is as follows:

Payer swaption_t = NP x FRAC x [SFR x N(d₂) - SR x N(d₂)] x ΣN_n=1 1/Z_t,Tn

= $75,000,000 x 0.5 x [.0082954 x 0.608271 - 0.08 x 0.521858] x 3.946910 = $1,289,142

The cost of the payer swaption is $1,289,142. The cost of swaptions is often expressed in basis points of the notional principal. This swaption costs 171.89 basis points = $1,289,142/$75,000,000 x 1,000 = 171.89 basis points.

We turn now to the receiver swaption, which has its own d₁ and d₂ terms. For the receiver swaption:

d₁ = ln (SFR_t/SR) + 0.5Σ²_(T-t) / Σ√T-t = ln(0.082954/.09) + 0.5 x 0.112 x 4 / 0.11x √4 = -0.266044

d₂ = d₁ - σ√T-t = -0.2660561059 - 0.11 x √4 = -0.480561059

The cumulative normal values are N(-d₁) = N(0.266044) = 0.604897 and N(-d₁) = N(0.486044) = 0.686532. Applying Equation 22.32 for a receiver swaption we have the following:

N (d₁) = NORMSDIST(0.0) = NORMSDIST (0.266044) = .604897339

N (d₁) = is the delta ratio. The call option will change price about .697 as fast as the stock for a small price change by the stock.

N (d₂) = NORMSDIST(0.0) = NORMSDIST (0.4860561059) = .686536307

D₂ is the probability that the option will expire in the money i.e. spot above strike for a call.

N(D₂) gives the expected value (i.e. probability adjusted value) of having to pay out the strike price for a call.

Payer swaption_t = NP x FRAC x [SR x N(-d₂) - SFR x N(-d₁)] x Σ^N_n=1 1/Z_t,Tn

= $75,000,000 x 0.5 x [.09 x 0.6865321 - 0.082954 x 0.604897] x 3.946910 = $1,718,329

The cost of the receiver swaption is $1,718,329. The cost of swaptions is often expressed in basis points of the notional principal. This swaption costs 229.11 basis points = $1,718,329/$75,000,000 x 1,000 = 229.11 basis points.

For this forward swap, the option to receive 9 percent instead of the market swap rate of 8.2954 percent costs $1,718,276 or 229.11 basis points.

A Swaption Application [viewable here in Excel]

Continuing with the term structure environment of table 22.13, we now consider a swaption application. Viewing the term structure data of Table 22.13 from time zero, a bond portfolio manager is considering entering a pay-fixed plain vanilla interest rate swap with a tenor of five years, a notional principal of $50 million, and semiannual payments. The manager is somewhat uneasy about the tenor of the swap and would like to know how much it would cost to have the option to cancel the swap after three years or extend it for another two years at the same rate as the plain vanilla swap.

Extendable and cancelable swaps can be created by combining a plain vanilla swap and the appropriate swaption. Our manager will enter a pay-fixed swap. The relevant rules for this situation are as follows:

Extendable pay-fixed swap = plain vanilla pay-fixed swap + a payer swaption

Cancelable pay-fixed swap = plain vanilla pay-fixed swap + a receiver swaption

Specifically, to extend the swap for two additional years the manager needs a payer swaption with a maturity of five years, an underlying swap of two years, and a strike rate equal to the rate on the initial five-year plain vanilla swap that she is contemplating. To cancel the swap after three years, the manager will need a receiver swaption with a maturity of three years, an underlying swap with a tenor of two years, and a strike rate that equals the fixed rate on the five-year plain vanilla swap.

To evaluate the issue fully, the manager needs to know the swap rate for three swaps: the five-year plain vanilla swap, the two-year swap to begin in three years that underlies the swaption for the cancelable swap (ranging from year 3 to year 5), and the two-year swap to begin in five years that underlies the swaption for the extendable swap (ranging from year 5 to year 7). Based on the data in Table 22.13, where the subscripts range over the semiannual periods, we have the following:

¹⁰Σ_t=1 1/Z_0,t = 8.312206, ¹⁰Σ_t=1 1/Z_0,t = 2.962917, ¹⁴Σ_t=11 1/Z_0,t = 2.521503

1/Z_0,t
Z11= 1.04289998657955
Z12 = 1.04193667320907
Z13 = 1.04036316681335
Z14 = 1.04919754103925
Sum = 2.521502849

¹⁰Σ_t=1 FRA_t-1,t/Z_0,t = 0.300901, ¹⁰Σ_t=7 FRA_t-1,t/Z_0,t = 0.113409, ¹⁴Σ_t=11 FRA_t-1,t/Z_0,t = .109696

FRA_t-1,t/Z_0,t
FRA11 = 0.0287576015950996
FRA12 = 0.0269803859303206
FRA13 = 0.0249605670300663
FRA14 = 0.0289971521154259
Sum = 0.109695707

Based on these values, the fixed rates of the various swaps in the analysis are as follows:	Semiannual	Annual
five-year plain vanilla SFR = (10Σt=1 FRAt-1,t/Z0,t) / (10Σ t=1 1/Z0,t) = 0.300901 / 8.312206	3.6200%	7.2400%
forward swap (years 3 - 5) SFR = (10Σt=7 FRAt-1,t/Z0,t) / (10Σt=7 1/Z0,t) = 0.113409 / 2.962917	3.8276%	7.6552%
forward swap (years 5 - 7) SFR = (14Σt=11 7 FRAt-1,t/Z0,t) / (14Σt=11 1/Z0,t) = .109696 / 2.521503	4.3504%	8.7008%

Based on these values, the fixed rates of the various swaps in the analysis are as follows: five-year plain vanilla SFR = (¹⁰Σ_t=1 FRA_t-1,t/Z_0,t) / (¹⁰Σ_t=1 1/Z_0,t) = 0.300901 / 8.312206 forward swap (years 3 - 5) SFR = (¹⁰Σt=7 FRA_t-1,t/Z_0,t) / (¹⁰Σ_t=7 1/Z_0,t) = 0.113409 / 2.962917 forward swap (years 5 - 7) SFR = (¹⁴Σ_t=11 ⁷FRA_t-1,t/Z_0,t) / (¹⁰Σ_t=11 1/Z_0,t) = .109696 / 2.521503.

These are semiannual rates and they correspond to annual rates of 0.072400 for the five year plain vanilla swap, .076522 for the forward swap years 3-5, and 0.087008 for the forward swap covering years 5-7. The key rate is the SFR = 0.072400 for the five-year plain vanilla swap, because this is the rate on the swap that the manager would like to be able to either cancel at year 3 or extend from year 5 to 7.

To make the plain vanilla swap extendable, we have seen that the manager will require a payer swaption. Given her situation, the necessary payer swaption will have a maturity of five years, a strike rate equal to the SFR on the plain vanilla swap of 0.072400, an underlying swap with a tenor of two years, semiannual payments, and a notional principal of $50 million. We have seen that for this forward swap, the swap rate is 0.087008, and we assume that the volatility of this forward rate is 0.15. This gives the information necessary to value the payer swaption.

The d₁ and d₂ terms are as follows:

d₁ = ln (SFR_t/SR) + 0.5Σ²(T-t) / σ√T-t = ln(0.087008/0.072400) + 0.5 x 0.152 x 5 / 0.15 x √5 = 0.715672

d₂ = d₁ - Σ√T-t = 0.715672247 - 0.15 x √5 = 0.380262

The cumulative normal values are N(d₁) = 0.608272 and N(0.274816) and N(d₂) = N(0.054816) = 0.521858. In finding the SFR for the forward swap, we have already determined that:

N (d₁) = NORMSDIST(.715672247) = 0.715672247

N (d₂) = NORMSDIST(0.38026205) = .648124549

payer swaptiont = NP x FRAC x [SFR x N(d₁) - SR x N(d₂)] x 14σ_n=11 1/Z_t,Tn

= $50,000,000 x 0.5 x [.087008 x 0.762903124 - 0.0724008 x 0.64812549] x 2.521503 = $1,226,325

The cost of the payer swaption is $1,226,325. The cost of swaptions is often expressed in basis points of the notional principal. This swaption costs 245.27 basis points = $1,226,325/$50,000,000 x 1,000 = 245.27.

To make the swap cancelable after three years, the manager will require a receiver swaption with a maturity of three years and a strike rate equal to the rate of 0.072400 on the plain vanilla swap. The underlying swap will have a notional principal of $50 million, semiannual payments, and a tenor of two years. We assume that the standard deviation of this forward swap rate is 0.13. For the receiver swaption, the d₁ and d₂ terms are as follows:

d₁ = ln (SFR_t/SR) + 0.5Σ²(T-t) / σ√T-t = ln(0.076552/0.072400) + 0.5 x 0.132 x 5 / 0.13 x √3 = 0.360240

d₂ = d₁ - σ√T-t = 0.360239695 - 0.13 x √3 = 0.135073

The cumulative normal values are N(-d₁) = N(-0.360240) = 0.359333946 and N(-d₂) = N(-0.13507309) = 0.446277043. In finding the SFR for the forward swap, we have already determined that:

N (d₁) = NORMSDIST(0.360239695) = .359333946

N (d₂) = NORMSDIST(0.13507309) = 0.446277043

payer swaption_t = NP x FRAC x [SFR x N(d₁) - SR x N(d₂)] x ¹⁰Σ_n=7 1/Z_t,Tn = 2.62916069

1/Z_0,t
Z7 = 0.784400801832365
Z8 = 0.753108286854812
Z9 = 0.726308805484142
Z=10 = 0.699098174820481
Sum = 2.962916069

= $50,000,000 x 0.5 x [.072400 x 0.446277 - 0.076552 x 0.359334] x 2.962917 = $355,751

The cost of the payer swaption is $355,751. The cost of swaptions is often expressed in basis points of the notional principal. This swaption costs 71.15 basis points $355,751/$50,000,000 x 1,000 = 71.15.

In sum, our manager plans to enter a five-year plain vanilla swap at a swap rate of 7.24 percent with a notional principal of $50,000,000. She would like to be able to cancel the swap after three years, which will cost $355,751, or 71.15 basis points. She would also like to be able to extend the swap for two additional years at the end of the five-year tenor of the plain vanilla swap. The payer swaption necessary to meet this desire will cost $1,226,359 or 245.27 basis points. Together, these two swaptions will cost $1,582,110 or 31.42 basis points.

Paying over 3 percent of the notional principal of a swap is expensive insurance for not being able to make up your mind about the necessary tenor of the swap. Both options cannot be used, as they are mutually exclusive. The manager might be able to salvage some value from the (at least) one swaption that will not be exercised, but it might be better for her to rethink the strategy of the swap and the flexibility that is really necessary in canceling or extending the swap.

Sources: Damodaran, Hull, Basu, Kolb, Brueggeman, Fisher, Bodie, Merton, White, Case, Pratt, Agee.