OPTIONS ON BONDS ∗

8.6 OPTIONS ON BONDS ∗

Options on bonds are compound options, traded popularly in financial markets. The valuation of these options requires both an interest-rate model and the valua- tion of term structure bond prices (which depend on the interest rates for various maturities of the bond). For instance, say that there is a T bond call option, which confers the right to exercise it at time S < T . The procedure we adopt in valuing

a call option on a bond consists then in two steps. First we evaluate the term structure for a T and an S bond. Then we can proceed to value the call on the T bond with exercise at time S (used to replace the spot price at time S in the plain vanilla option model of Black–Scholes). The procedure is explicitly given by the following. First we construct a hedging portfolio consisting of the two bonds ma- turities S and T (S < T ). Such a portfolio can generate a synthetic rate, equated to the spot interest rate so that no arbitrage is possible. In this manner, we value the option on the bond uniquely. An extended development is considered in the Math- ematical Appendix while here we summarize essential results. Let, for example, the interest process:

dr = µ(r, t) dt + σ (r, t) dw

A portfolio (n S , n T ) of these two bonds has a value and a rate of return given by: dV dB(t, S)

dB(t, T )

B (t, T ) The rates of return on T and S bonds are assumed given as in the previous section.

Each bond with maturity T and S has at its exercise time a $1 denomination, thus the value of each of these (S and T ) bonds is given by B T (t, r ) and B S (t, r ). Given these two bonds, we define the option value of a call on a T bond with S < T and strike price K , to be:

X = Max [B(S, T ) − K , 0]

with B(S, T ) the price of the T bond at time S. The bond value B(S, T ) is of course found by solving for the term structure equation and equating B(r, S, T ) =

B (S, T ). To simplify matters, say that the solution (value at time t) for the T bond is given by F(t, r, T ), then at time S, this value is: F(S, r, T ) to which we equate

B (S, T ). In other words,

X = Max [F(S, r, T ) − K , 0]

Now, if the option price is B(.), then, as we have seen in the plain vanilla model in Chapter 6, the value of the bond is found by solving for P(.) in the following partial differential equation:

0= ∂ t + µ(r, t) ∂ r +σ

− r B, B(S, r) = Max [F(S, r, T ) − K , 0]

OPTIONS ON BONDS

A special case of interest consists again in using an affine term structure (ATS) model as shown above in which case:

F A (t, r, T ) = e (t,T )−r D(t,T )

where A(.) and D(.) are calculated by the term structure model. The price of an option of the bond is thus given by the solution of the bond partial differential equation, for which a number of special cases have been solved analytically. When this is not the case, we must turn to numerical or simulation techniques.

8.6.1 Convertible bonds

Convertible bonds confer the right to the bond issuer to convert the bond into stock or into certain amounts of money that include the conversion cost. For example, if the bond can be converted against m shares of stock, whose price dynamics is:

dS = µS dt + σ S dw

Then, the bond price is necessarily a function of the stock price and given by

V (S, t). To value such a bond we proceed ‘as usual’ by constructing an equivalent risk-free and replicating portfolio. Let this risk-free portfolio be:

π = V + αS and therefore dπ = dV + α dS

For this portfolio to be risk-free, we equate it to a portfolio whose rate of return is the risk-free rate R f . Thus, dπ = dV + α dS = R f π dt = R f (V + αS) dt and dV = R f (V + αS) dt − α dS. Using Ito’s Lemma, we calculate dV leading to:

which can be rearranged to:

∂ t + µS V−αR f S ∂ dt S + + αµS − R f

+σS ∂ S +α dw = 0

A risk-free portfolio has no volatility and therefore we require: ∂ V

dw = 0 or α = − ∂ S Inserting α = − ∂V /∂ S yields the following partial differential equation:

where at redemption the bond equals $1. If the conversion cost is C(S, t) = mS, the least cost is Min {V (S, t), C(S, t)}. Therefore, in the continuation region (i.e. as long as we do not convert the bonds into stocks), we have: V (S, t) ≥ C(S, t) = mS while in the stopping region (i.e. at conversion) we have: V (S, t) ≤ mS. In

FIXED INCOME , BONDS AND INTEREST RATES

other words, the convertible option has the value of an American option which we solve as indicated in Chapter 6. It can also be formulated as a stopping time problem, but this is left as an exercise for the motivated reader.

8.6.2 Caps, floors, collars and range notes

A cap is a contract guaranteeing that a floating interest rate is capped. For example, let r ℓ

be a floating rate and let r c be an interest rate cap. If we assume that the floating rate equals approximately the spot rate, r ≈ r ℓ then a simple caplet is

priced by:

− r V, V (r, T ) = Max [r − r with the cap being a series of caplets. By the same token, a floor ensures that the

∂ t + [µ(r, t) − λσ ] ∂ r +σ

c , 0]

interest rate is bounded below by the rate floor: r f . Thus, the rate at which a cash flow is valued is: Max (r f −r ℓ , 0), r ℓ ≥r c . Again, if we assume that the floating rate equals the short rate, we have a floorlet price given by:

− r V, V (r, T ) = Max f − r, 0 ∂ t

+ [µ(r, t) − λσ ]

while the floor is a series of floorlets. A collar, places both an upper and a lower bound on interest payments, however. A collar can thus be viewed as a long

position on a cap, with a given strike r c and a short position on a floor with a lower strike r f . If the interest rate falls below r f , the holder is forced into paying the higher rate of r f . The strike price of the call is often set up so that the cost of the cap is exactly subsidized by the revenue from the sale of the floor. When the interest rate on a notional principal is bounded above and below, then we have a range note. In this case, the value of the range note can be solved by using the differential equations framework as follows:

2 1 ∂ 2 V ∂ t + [µ(r, t) − λσ ] ∂ r +σ 2 ∂ r 2

#r if r < r < ¯r

0 otherwise

This is only an approximation since, in practice, the relevant interest rate will have a finite maturity (Wilmott, 2000).

8.6.3 Swaps

An interest rate swap is a private agreement between two parties to exchange one stream of cash flow for another on a specific amount of principal for a specific period of time. Investors use swaps to exchange fixed-rate liabilities/assets into floating-rate liabilities/assets and vice versa.

Interest rate swaps are most important in practice. They emerged in the 1980s and their growth has been spectacular ever since. They are essentially customized

263 commodity exchange agreements between two parties to make periodic payments

OPTIONS ON BONDS

to each other according to well-defined rules. In the simplest of interest rate swaps, one part periodically pays a cash flow determined by a fixed interest rate and receives a cash flow determined by a floating interest rate (Ritchken, lecture notes, 2002).

For example, consider Company A with $50 000 000 of floating-rate debt out- standing on which it is paying LIBOR plus 150 bps (basis points), i.e. if LIBOR is 4 %, the interest rate would be 5.5 %. The company thinks that interest rates will rise, i.e. company’s interest expense will rise, and the company decides to convert its debt from floating-rate into fixed-rate debt. Now consider Company

B which has $50 000 000 of fixed-rate 6 % debt. The company thinks that inter- est rates will fall, which would benefit the company if it has floating-rate debt instead of fixed-rate debt, since its interest expense will be reduced. By entering into an interest rate swap with Company A, both parties can effectively convert their existing liabilities into the ones they truly want. In this swap, Company A might agree to pay Company B fixed-rate interest payments of 5 % and Com- pany B might agree to pay Company A floating-rate interest payments of LIBOR. Therefore Company A will pay LIBOR + 150 to its original lender and 5 % in the swap, giving a total of LIBOR + 6.5 %; it receives LIBOR in the swap. This leaves an all-in cost of funds of 6.5 %, a fixed rate. In the case of Company B, it pays 6 % to its original lender and LIBOR in the swap, giving a total of LIBOR + 6 %. In return it receives 5 % in the swap, leaving an all-in cost of LIBOR +

1 %, a floating rate (see Figure 8.7). There are four major components to a swap: the notional principal amount,

the interest rate for each party, the frequency of cash exchange and the duration of the swap. A typical swap in swap jargon might be $20m, two-year, pay fixed, receive variable, semi. Translated, this swap would be for $20m notional principal, where one party would pay a fixed interest-rate payment for every 6 months based on the $20m and the counterparty would pay a variable rate payment every

6 months based on the $20m. The variable-rate payment is usually based on a specific short-term interest rate index such as the 6-months LIBOR. The time period specified by the variable rate index usually coincides with the frequency of swap payments. For example, a swap that is fixed versus 6 months LIBOR would have semiannual payments. Of course there can be exceptions to this rule.

Figure 8.7

A swap contract.

FIXED INCOME , BONDS AND INTEREST RATES

For example, the variable-rate payment could be linked to the average of all T-bill auction rates during the time period between settlements.

Most interest rate swaps have payment date arrears. That is, the net cash flow between parties is established at the beginning of the period, but is actually paid out at the end of the period. The fixed rate for a generic swap is usually quoted as some spread over benchmark US treasuries. For example a quote of ‘20 over’ for a 5-year swap implies that the fixed rate on a 5-year swap will be set at the 5-year Treasury yield that exists at the time of pricing plus 20 basis point. Usually, swap spreads are quoted against the two, three, five, seven and ten benchmark maturities. The yield used for other swaps (such as a 4-year swap) is then obtained by averaging the surrounding yields.

Finally, a ‘swaption’ is an option to swap. It confers the right to enter into a swap contract at a predetermined future date at a fixed-rate and be a payer at the fixed rate. ‘Captions’ and ‘floortions’ are similarly options on caps and floors respectively.

A swap price can be defined by a cap–floor parity, where

Cap = Floor + Swap

In other words, the price of a swap equals the price difference between the cap and the floor. A caplet can be shown to be equivalent to the price of a put option expiring

at a time t i −1 prior to a bond’s maturity at time t i . Set the payoff Max(r ℓ −r c , 0)

a period hence, which is discounted to t i −1 and yielding:

However (1 + r c )/(1 + r ℓ ) is the price of paying 1 + r c a period hence. Thus a caplet is equivalent to a put option expiring at t i −1 on a bond with maturity t i .