Swaptions
The Valuation of Swaption using the Black (1976) model
An interest rate swaption is an option that provides the holder with the right but not the obligation to enter into an interest rate swap on an agreed date(s) in the future on terms protected by the swaption. The nominal amount is also established in advance. The buyer of the option pays a premium. A swaption is an option on a swap which affords the buyer the right to either pay or receive a fixed rate. The buyer who earns the right to pay fixed and receive floating is said to have acquired a "payer" swaption. Conversely, the swaption that confers the right to pay floating (i.e. receive fixed) is termed a "receiver" swaption. The buyer of a payer (interest rate) swaption has an option to pay a fixed rate (the strike) and receive a floating rate typically/historically LIBOR . With a receiver swaption, the buyer has the option to receive in the future the previously set fixed interest rate in exchange for the payment of 3- or 6-month Libor. With a payer swaption, the buyer has the option to pay in the future the previously set fixed interest rate in exchange for the receipt of 3- or 6-month Libor. The buyer has the right to enter into the interest-rate swap, but importantly has no obligation.
The underlying swap can be modeled as two streams of cash flows. The right to receive fixed is logically equivalent to the right to pay floating. The option terminology of calls and puts is generally not applied to swaptions but the right to pay fixed strongly resembles the payoff a call or cap. Likewise, the right to received fixed conversely resembles a put or floor for interest rates payments. Swaptions are typically quoted as N x M, where N indicates the option expiry in years and M refers to the underlying swap tenor in years. Hence a 1 x 5 Swaption would refer to 1 year option to enter a 5 year swap1.
Swaptions can be cash or physically settled meaning that on option expiry if exercised we can specify to enter into the underlying swap or receive the cash equivalent on expiry. Swaptions can distinguished by option style with the main flavours being European, American and Bermudan, which refer to the option exercise date(s). Conferring on the the holder the right to exercise at option expiry only would be European. Exercising at any date up to expiry is American in style. A Bermudan swaption can be exercised on a defined number of pre-specified dates, making it more flexible than the European equivalent contract.
Swaptions can be employed varyingly as hedging vehicles for fixed debt, floating debt , swaps, and bonds etc. Some motivation to enter into a swaption position include:
to hedge call or put positions in bond issues
to change the tenor of an underlying swap
to assist in the engineering of structured notes
to change the payoff profile of the firm
to protect a borrower from future costs of borrowing without being locked into a fixed rate.
The valuation of swaptions has to be different from to standard interest-rate swaps. Swaption embed options. and are non-linear in relation to underlying interest rates. With an interest-rate swap, there is a linear connection between the underlying value (the level of the interest rates) and the value of the interest-rate swap. There are several factors that affect the valuation of a swaption. We can summarise these using the parameter inputs of the Black (1976) model. The value of premium for European swaptions depends on the contract interest rate, the market (forward) interest rate, interest-rate volatility, the tenor of the option, the exercise and risk free rate of interest.
Please see video below where the Black (1976) model is applied to value a European style swaption. The example is taken from John C Hull "Option , Futures and Other Derivatives"