swaptionbylg2f

Price European swaption using Linear Gaussian two-factor model

Syntax

Price = swaptionbylg2f(ZeroCurve,a,b,sigma,eta,rho,Strike,ExerciseDate,Maturity)

Price = swaptionbylg2f(___,Name,Value)

Description

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.

Price = swaptionbylg2f(___,Name,Value) adds optional name-value pair arguments.

Examples

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]';
CurveDates = daysadd(Settle,360*ZeroTimes,1);
 
irdc = IRDataCurve('Zero',Settle,CurveDates,ZeroRates);
 
a = .07;
b = .5;
sigma = .01;
eta = .006;
rho = -.7;
 
Reset = 1;
ExerciseDate = daysadd(Settle,360*5,1);
Maturity = daysadd(ExerciseDate,360*[3;4],1);
Strike = .05;
  
Price = swaptionbylg2f(irdc,a,b,sigma,eta,rho,Strike,ExerciseDate,Maturity,'Reset',Reset)

Price = 2×1

    1.1869
    1.5590

Input Arguments

ZeroCurve — Zero-curve for Linear Gaussian two-factor model

structure

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 model

scalar

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 model

scalar

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 model

scalar

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 model

scalar

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

Data Types: single | double

rho — Scalar correlation of the factors

scalar

Scalar correlation of the factors, specified as a scalar.

Data Types: single | double

Strike — Swaption strike price

nonnegative 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 dates

vector of serial date numbers | character vector of dates

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

Data Types: single | double | char | cell

Maturity — Underlying swap maturity date

vector of serial date numbers | character vector of dates

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

Data Types: single | double | char | cell

Name-Value 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 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 year

2 (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 the comma-separated pair consisting of 'Reset' and positive integers for the values 1,2,4,6,12 in a NumSwaptions-by-1 vector.

Data Types: single | double

Notional — Notional value of swaption

100 (default) | nonnegative integer | vector of nonnegative integers

Notional value of swaption, specified as the comma-separated pair consisting of 'Notional' and a nonnegative integer using a NumSwaptions-by-1 vector of notional amounts.

Data Types: single | double

OptSpec — Option specification for the swaption

'call' (default) | character vector with value of 'call' or 'put' | cell array of character vectors with values of 'call' or 'put'

Option specification for the swaption, specified as the comma-separated pair consisting of 'OptSpec' and a character vector or a NumSwaptions-by-1 cell array of character vectors with a value of 'call' or 'put'.

A 'call' swaption or Payer swaption allows the option buyer to enter into an interest-rate swap in which the buyer of the option pays the fixed rate and receives the floating rate.

A 'put' swaption or Receiver swaption allows the option buyer to enter into an interest-rate swap in which the buyer of the option receives the fixed rate and pays the floating rate.

Data Types: char | cell

Output Arguments

Price — Swaption price

scalar | vector

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

More About

Call Swaption

A call swaption or payer swaption allows the option buyer to enter into an interest-rate swap in which the buyer of the option pays the fixed rate and receives the floating rate.

Put Swaption

A put swaption or receiver swaption allows the option buyer to enter into an interest-rate swap in which the buyer of the option receives the fixed rate and pays the floating rate.

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:

$$r(t)=x(t)+y(t)+\varphi (t)$$

$$dx(t)=-ax(t)dt+\sigma d{W}_1(t),x(0)=0$$

$$dy(t)=-by(t)dt+\eta d{W}_2(t),y(0)=0$$

where $$d{W}_1(t)d{W}_2(t)=\rho dt$$ 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.