European Swaptions

 Consider first the value of a payer swaption. Clearly the option will have value at expiration if the swap rate prevailing in the market at that time is greater than the fixed rate prescribed in the swaption contract. If the uncertain future swap rate at maturity is denoted as and the contract swap rate is , then the cashflows to the party that pays fixed consist of a series of cashflows equal to: This is simply the payoff to a standard call option where the underlying 'asset' is the equilibrium swap rate. The most common way to determine the current value of this payoff is to assume that the expected value of is equal to the forward swap rate ( ), and then to apply the Black model. Thus, the current value of the swaption (v0) relating to the i'th swap payment can be expressed as: where, FV = notional swap principal ti = tenor of the i'th swap payment DFi = discount factor with the same maturity as swap payment i   volatility of the forward swap rate T = maturity (in years) of the swaption N(.) = cumulative normal distribution function Because the payment will accrue to the swaption holder at each of the N swap payment dates, the total value of the payer swaption is just the aggregate value of all the expected payments. That is: This expression can be simplified by noting that the term is constant for all N swap payments. If we define , then the pay-swaption can be equivalently expressed as: All that remains in order to operationalise the above equation is to derive an expression for the equilibrium forward swap rate, Fs. This rate is defined as the fixed swap rate that will equate the present value of the fixed and floating legs of the underlying swap, and can be thought of as today's forecast of the swap rate that will prevail at the maturity date of the swaption. Using the definitions given in the background to interest rate swaps, this means we wish to solve for the fixed rate ( ) such that: By rearranging the above equation, an expression for the forward swap rate can be derived as: Now, using similar logic as that used to derive an expression for the payer swaption, it is straight forward to show that the receiver swaption can be thought of as a standard put option written on the forward swap rate, and can be valued as:  