article
Deconstructing the Normal Swaption Model
October 26, 2014

## Overview

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:

Here,

P(Tα, Ti) is the price of a zero coupon bond at time Tα that terminates at Ti ;

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:

1. The distribution of the forward par swap rate is lognormal.
2. 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:

1. The standard deviation of basis point changes in forward swap rates is a constant normalized volatility.
2. 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.
3. 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.
4. 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.

## References:

1. Rebonato, Riccardo (2004). Volatility and Correlation: The Perfect Hedger and the Fox, John Wiley & Sons, England.
2. Brigo, D and Mercurio, F (2006). Interest Rate Models: Theory and Practise, Springer Finance, New York.