A payer (receiver) swaption is an option to enter into an interest rate swap wherein a fixed coupon rate is paid (received) upon exercising the option. In case of a European payer swaption, the expiry of swaption coincides with the first rate fixing date of the underlying swap of length ( Tβ - Tα ) where Tα is the swap's first fixing date and Tβ is the swap's termination date. The payer-swaption's discounted payoff to time, t , is given by:
P(Tα, Ti) is the price of a zero coupon bond at time Tα that terminates at Ti ;
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The probability distribution of the forward par swap rate is characterized by the mean and the volatility of the underlying forward rates. A causal chain of reasoning is as follows:
Volatility → Probability distribution of forward par swap rates → Payoff → Swaption price
Different specifications of the forward par swap rate's probability distribution enable one to extract differing swaption prices.
Lognormal Forward Swaption Model:
When the underlying forward par swap rate, Sαβ, is assumed to have a lognormal distribution, the relevant stochastic differential equation is:
It is useful to quickly summarize some key issues arising in the lognormal forward model:
- The distribution of the forward par swap rate is lognormal.
- The standard deviation of the percentage changes in the forward par swap rate is the constant Black-Scholes volatility.
The Normal Forward Swaption Model:
Normalized volatility is the market convention - primarily because normalized volatility deals with basis point changes in rates rather than, as in lognormal volatility, with percentage changes in rates.
The underlying par-swap rate is given by:
Here, the basis point changes in the forward par swap rates are independent of the level of forward par swap rates involved. The features of the Normal Model are:
- The standard deviation of basis point changes in forward swap rates is a constant normalized volatility.
- There are some realizations of the par-swap rate that can be negative in the normal model; as opposed to the lognormal model where par swap rates are non-negative.
- Since the volatility input for the normal model refers to the volatility for the actual basis point changes - it tends to be smaller than volatility for the lognormal case.
- A useful translation between the lognormal volatility and normal volatility is the following thumbrule:
FINCAD Analytics Suite's implementation of Normal Swaption Model:
In FINCAD Analytics Suite, a function named aaNormal_Swaption_fs has been introduced which enables a user to price swaptions where the underlying forward rate is normally distributed. The resultant screen, from a workbook created, is shown here:
The user is expected to input, over and above the standard variables, the rate payment table applicable for the fixed leg and the floating.
- Rebonato, Riccardo (2004). Volatility and Correlation: The Perfect Hedger and the Fox, John Wiley & Sons, England.
- Brigo, D and Mercurio, F (2006). Interest Rate Models: Theory and Practise, Springer Finance, New York.
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