Despite being philosophically similar, unlike futures contracts that only trade on the exchanges (e.g., CME), forward contracts trade on the OTC markets. Although some market participants transact in options on forward contracts in order to exercise into the underlying spot contract, more often than not this is done for mathematical convenience.^{[1]} Given this backdrop, one can use the same diffusion equation given by equation (3.9)

where

F is the forward rate.

σ is the annualized volatility of forward rate returns. dz is the random variable drawn from a standard normal probability density function.

dF is the small change in the forward rate over a small time interval dt.

As a consequence, the implementation laid out on Table 3.6 still holds, although adjustments need to be made to take into consideration settlement

TABLE 3.6 Valuing European Style Options on Futures Contracts

mechanisms associated with the underlyings of the options. To help shed more light on this comment, I present three such examples.

Interest Rate Swaptions An interest rate swaption (i.e., an option on an interest rate swap) is one example of commonly traded interest rate options in OTC capital markets around the world. Upon exercise of the swaption, the holder has the right to enter into an interest rate swap so as to either receive or pay a fixed rate^{[2]} on the swap. This is akin to the swaption holder taking physical delivery of the underlying swap upon exercise. Because these contracts trade in the OTC market, the holder has the flexibility of receiving the in-the-money payoff in cash instead of taking physical delivery of the underlying swap, provided this mode of settlement is prespecified prior to the maturity of the option. When trading swaptions, practitioners use the terms^{[3]} right-to-pay (RTP) option and right-to-receive (RTR) option. These refer to the right to pay a fixed rate on the swap underlying the option or the right to receive a fixed rate on the swap underlying the option, respectively.

The RTP-option payoff to the owner is slightly different from the payoff associated with the purchase of a futures call option contract. The reason for this stems from the fact that once the RTP option is in the money, the owner of the option exercises into a swap. As a consequence, on option maturity date, the owner of the RTP option on a notional principal of a dollar receives a payoff that is of the form

where

FT is the swap rate on option maturity.

K is the strike rate.

n is the number of cash-flow exchanges in the swap.

τi is the tenor (time between the setting of the i floating rate and settling of the cash flows arising from this setting).

is the discount factor that is used to discount the cash flow arising from the settlement of the i floating rate using the zero curve at time T.^{[4]}

The resulting RTP (call) and RTR (put) option-pricing formulae can be simplified, as in equations (3.12a) and (3.12b).

(3.12a)

(3.12b)

where

is the forward start swap rate^{[5]} computed using the zero-rate curve at time t (ZRC) for a swap whose first floating rate is set at time T and whose last floating rate is set at time

is the discount factor used to discount the cash flow arising from the setting of the /th floating rate using the zero-rate curve at time t (ZRC).

is assumed to be given.^{[6]}

Table 3.7 shows the implementation of equations (3.12a) and (3.12b) to price options on interest rate swaps.

Interest Rate Gaps/Floors In addition to swaptions, caps and floors are another class of interest rate options that get traded quite widely in the OTC markets. Since the terms cap and floor are used to refer to a collection of caplets and floorlets respectively, the price of a cap (floor) is simply the sum of the prices of all the underlying caplets (floorlets). Hence to price a cap or a floor, one needs to know how to price a caplet and a floorlet.

Practitioners use the term caplet (floorlet) to refer to a call (put) option on LIBOR. More precisely, on option maturity the value of the LIBOR (e.g.,

1-month LIBOR, 3-month LIBOR) is compared with a strike rate. As a consequence, the owner of each caplet (floorlet) is essentially purchasing an RTP (RTR) or a call (put) option on a single-period swap (also called a forward rate agreement).

Since one is dealing with single-period swaps, to value a caplet on a notional principal of a dollar, it is straightforward to write down the caplet owner's payoff on the option's maturity date as

is the value of the LIBOR rate that sets on f, (option maturity) and settles at time ().

K is the strike rate of caplet.

is the tenor of the ith floating rate.

is the discount factor that is used to discount the cashflow arising from the settling of the /th floating rate using the zero curve at time .

Taking the present value of the expectations allows one to arrive at equations (3.13a) and (3.13b).

is the forward rate applied from time () computed using the zero-rate curve at time t (ZRC).

is the discount factor that is used to discount the cash flow arising from the settling of /th floating rate using the zero rate curve at time t (ZRC) and assumed to be given for our purposes.

Table 3.8 shows the implementation of equations (3.12a) and (3.12b) to evaluate both a caplet and a floorlet.

where

(3.12a)

(3.12b)

where

TABLE 3.9 Valuation of Cap and Floor

Table 3.9 shows an example of how three caplets (floorlets) are put together to arrive at the value of a cap (floor).

Bond Options In addition to swaptions and caps/floors, bond options (or bondtions) represent the third class of interest-rate options that get traded quite widely in the OTC markets. Unlike the earlier examples, however, trading bond options can be trickier since bonds get quoted by both price and yield. As a consequence, it becomes imperative to ensure that when a call option on a bond is being transacted, the transaction also contains details whether the underlying is a bond price or a bond yield.^{[7]}

Since most bond options that trade in the OTC market tend to be European-style in nature, where the underlying is bond-yield based,^{[8]} one would need to know how to adapt the models of the earlier examples to price bond options. To do this, one has to tweak the payoffs – just like what was done for swaptions, caps, and floors.

Although there are a few ways of defining an option payoff associated with a bond yield, one common form of representation is that the owner of a call option gets $10,000 for every basis point the option finishes in-the- money – which can be alternatively rewritten as follows:^{[9]}^{[10]}

where

BYT is the value of the bond yield at T (option maturity date).

K is the strike rate of the option.

Taking the relevant expectations and present value yields the following result in equations (3.13a) and (3.13b)

(3.13a)

(3.13b)

where

FBYt is the forward bond yield^{[11]} of the bond applicable from time T to bond maturity that is computed at time t.

is the discount factor used to discount the cash flow arising at time T using the zero-rate curve at time t (ZRC).

Table 3.10 shows the implementation of equations (3.13a) and (3.13b).

TABLE 3.10 Valuation of Bond Options

As the reader will note, to convert the values in cells B9 and BIO to a dollar amount to be consistent with the payoff to the owner of the bond options, one still needs to multiply the values in cells B9 and BIO by 10,000 * 10,000.

[1] To model an interest rate or commodity-spot price at some time in the future, practitioners assume that the future interest rate (or spot price) movements are centered at the current forward rate (or price) that is implied by the market and follows the diffusion process used for modeling futures prices/rates. In addition, one also assumes that the option would mature at the same time as the forward. The purpose of this is to ensure that upon the exercise of the option, the spot-interest rates (or prices) at the time of option expiry are compared with the strike rate – as the forward rates (or prices) converge to the spot rates (or prices) on the day of the forward contract maturity. The reader is again referred to Hull (2012) for details.

[3] The term RTP (RTR) option can be thought of as a call (put) option on interest rate swap.

[4] The term is sometimes known as the swap delta.

[5] This can be in practice calculated using equation (2.6).

[6] This is in practice computed from the same zero rate curve that is used to calculate the forward start swap rate.

[7] Given the inverse relationship between a bond price and yield, a call option on a bond price is in fact a put option on the yield of the same bond (with some adjustments made to the notional size of the options).

[8] If the underlying is a bond price instead of a bond yield, there are certain limitations on how much stochasticity a bond price can exhibit. See Hull (2012).

[9] The first 10,000 refers to the conversion required to convert the decimal-based yield differential to one that is denominated in basis points. The second 10,000 refers to the dollar payoff associated with each basis point the option finishes in the money.

[10] Bond options are also traded on the CBOE under the trading symbols FVX, TNX, and TYX. Similar to the example, these options are also written on bond yields with each basis point in-the-money providing the option holder $100 instead of the $10,000 per the example.

[11] Since the calculation of the forward bond yield is a little complicated, I have included it as an Appendix to this chapter.

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