floorbylg2f

Price floor using Linear Gaussian two-factor model

Syntax

FloorPrice = floorbylg2f(ZeroCurve,a,b,sigma,eta,rho,Strike,Maturity)

FloorPrice = floorbylg2f(___,Name,Value)

Description

FloorPrice = floorbylg2f(ZeroCurve,a,b,sigma,eta,rho,Strike,Maturity) returns the floor price for a two-factor additive Gaussian interest-rate model.

FloorPrice = floorbylg2f(___,Name,Value) adds optional name-value pair arguments.

Note

Use the optional name-value pair argument, Notional, to pass a schedule to compute the price for an amortizing floor.

Examples

Price a Floor Using a Linear Gaussian Two-Factor Model

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

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;
 
FloorMaturity = daysadd(Settle,360*[1:5 7 10 15 20 25 30],1);
 
Strike = [0.035 0.037 0.038 0.039 0.040 0.042 0.044 0.046 0.047 0.047 0.047]';
 
Price = floorbylg2f(irdc,a,b,sigma,eta,rho,Strike,FloorMaturity)

Price = 11×1

         0
    0.4190
    0.8485
    1.3365
    1.8671
    3.1091
    4.9807
    7.8518
    9.8297
   11.4578
      ⋮

Price an Amortizing Floor Using a Linear Gaussian Two-Factor Model

Define the ZeroCurve, a, b, sigma, eta, rho, and Notional parameters for the amortizing floor.

Settle = datenum('15-Dec-2007');
% Define ZeroCurve
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);

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

% Define a, b, sigma, eta, and rho
a = .07;
b = .5;
sigma = .01;
eta = .006;
rho = -.7;

% Define the amortizing floors
FloorMaturity = daysadd(Settle,360*[1:5 7 10 15 20 25 30],1);
Strike = [0.025 0.036 0.037 0.038 0.039 0.041 0.043 0.045 0.046 0.046 0.046]';
Notional = {{'15-Dec-2012' 100;'15-Dec-2017' 70;'15-Dec-2022' 40;'15-Dec-2037' 10}};

% Price the amortizing floors
Price = floorbylg2f(irdc,a,b,sigma,eta,rho,Strike,FloorMaturity,'Notional',Notional)

Price = 11×1

         0
    0.2776
    0.6630
    1.1062
    1.5938
    2.5589
    3.9582
    5.4985
    6.1113
    6.2670
      ⋮

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

Data Types: single | double

rho — Scalar correlation of factors

scalar

Scalar correlation of the factors, specified as a scalar.

Data Types: single | double

Strike — Floor strike price

nonnegative integer | vector of nonnegative integers

Floor strike price specified, as a nonnegative integer using a NumFloors-by-1 vector of floor strike prices.

Data Types: single | double

Maturity — Floor maturity date

serial date number | vector of serial date numbers | date character vector

Floor maturity date, specified using a NumFloors-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 = floorbylg2f(irdc,a,b,sigma,eta,rho,Strike,FloorMaturity,'Reset',1,'Notional',100)

Reset — Frequency of floor 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 floor 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 NumFloors-by-1 vector.

Data Types: single | double

Notional — Notional value of floor

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

NINST-by-1 of notional principal amounts or NINST-by-1 cell array where each element is a NumDates-by-2 cell array where the first column is dates and the second column is the associated principal amount. The date indicates the last day that the principal value is valid.

Data Types: single | double

Output Arguments

FloorPrice — Floor price

scalar | vector

Floor price, returned as a scalar or a NumFloors-by-1 vector.

More About

Floor

A floor is a contract that includes a guarantee setting the minimum interest rate to be received by the holder, based on an otherwise floating interest rate.

The payoff for a floor is:

$$\max (FloorRate-CurrentRate,0)$$

Algorithms

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

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

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

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

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.