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# swaptionbylg2f

Price European swaption using Linear Gaussian two-factor model

## Syntax

• Price = swaptionbylg2f(ZeroCurve,a,b,sigma,eta,rho,
Strike,ExerciseDate,Maturity)
example
• Price = swaptionbylg2f(___, Name,Value) example

## Description

example

Price = swaptionbylg2f(ZeroCurve,a,b,sigma,eta,rho,
Strike,ExerciseDate,Maturity)
returns the European swaption price for a two-factor additive Gaussian interest-rate model.

example

Price = swaptionbylg2f(___, Name,Value) returns the Eurpoean swaption price for two-factor additive Gaussian interest-rate model using optional name-value pairs.

## Examples

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### Price a European Swaption Using a Linear Gaussian Two-Factor Model

Define the ZeroCurve, a, b, sigma, eta, and rho parameters to compute the price of the swaption.

```Settle = datenum('15-Dec-2007');

ZeroTimes = [3/12 6/12 1 5 7 10 20 30]';
ZeroRates = [0.033 0.034 0.035 0.040 0.042 0.044 0.048 0.0475]';

irdc = IRDataCurve('Zero',Settle,CurveDates,ZeroRates);

a = .07;
b = .5;
sigma = .01;
eta = .006;
rho = -.7;

Reset = 1;
Strike = .05;

Price = swaptionbylg2f(irdc,a,b,sigma,eta,rho,Strike,ExerciseDate,Maturity,'Reset',Reset)
```
```Price =

1.1870
1.5633

```

## Input Arguments

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### ZeroCurve — Zero-curve for Linear Gaussian two-factor modelstructure

Zero-curve for the Linear Gaussian two-factor model, specified using IRDataCurve or RateSpec.

Data Types: struct

### a — Mean reversion for first factor for Linear Gaussian two-factor modelscalar

Mean reversion for first factor for the Linear Gaussian two-factor model, specified as a scalar.

Data Types: single | double

### b — Mean reversion for second factor for Linear Gaussian two-factor modelscalar

Mean reversion for second factor for the Linear Gaussian two-factor model, specified as a scalar.

Data Types: single | double

### sigma — Volatility for first factor for Linear Gaussian two-factor modelscalar

Volatility for first factor for the Linear Gaussian two-factor model, specified as a scalar.

Data Types: single | double

### eta — Volatility for second factor for Linear Gaussian two-factor modelscalar

Volatility for second factor for the Linear Gaussian two-factor model, specified as a scalar.

Data Types: single | double

### rho — Scalar correlation of the factorsscalar

Scalar correlation of the factors, specified as a scalar.

Data Types: single | double

### Strike — Swaption strike pricenonnegative integer | vector of nonnegative integers

Swaption strike price, specified as a nonnegative integer using a NumSwaptions-by-1 vector.

Data Types: single | double

### ExerciseDate — Swaption exercise datesnonnegative integer | vector of nonnegative integers

Swaption exercise dates, specified as a NumSwaptions-by-1 vector of serial date numbers or date strings.

Data Types: single | double

### Maturity — Underlying swap maturity datenonnegative integer | vector of nonnegative integers | string of dates

Underlying swap maturity date, specified using a NumSwaptions-by-1 vector of serial date numbers or date strings.

Data Types: single | double | cell

### Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside single quotes (' '). You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: Price = swaptionbylg2f(irdc,a,b,sigma,eta,rho,Strike,ExerciseDate,Maturity,'Reset',1,'Notional',100,‘OptSpec','call')

### 'Reset' — Frequency of swaption payments per year2 (default) | positive integer from the set[1,2,3,4,6,12] | vector of positive integers from the set [1,2,3,4,6,12]

Frequency of swaption payments per year, specified as positive integers for the values 1,2,4,6,12] in a NumSwaptions-by-1 vector.

Data Types: single | double

### 'Notional' — Notional value of swaption100 (default) | nonnegative integer | vector of nonnegative integers

Notional value of swaption, specified as a nonnegative integer using a NumSwaptions-by-1 vector of notional amounts.

Data Types: single | double

### 'OptSpec' — Option specification for the swaption'call' (default) | string with value of 'call' or 'put' | cell array of string with values of 'call' or 'put'

Option specification for the swaption, specified as a string or a NumSwaptions-by-1 cell array of strings with a value of 'call' or 'put'. A 'call' swaption entitles the buyer to pay the fixed rate. A 'put' swaption entitles the buyer to receive the fixed rate.

Data Types: char | cell

## Output Arguments

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### Price — Swaption pricescalar | vector

Swaption price, returned as a scalar or an NumSwaptions-by-1 vector.

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### Algorithms

The following defines the swaption price for a two-factor additive Gaussian interest-rate model, given the ZeroCurve, a, b, sigma, eta, and rho parameters:

where is a two-dimensional Brownian motion with correlation ρ and ϕ is a function chosen to match the initial zero curve.

## References

[1] Brigo, D. and F. Mercurio, Interest Rate Models - Theory and Practice, Springer Finance, 2006.