capbylg2f

Price cap using Linear Gaussian two-factor model

Syntax

CapPrice = capbylg2f(ZeroCurve,a,b,sigma,eta,rho,Strike,Maturity)

CapPrice = capbylg2f(___,Name,Value)

Description

CapPrice = capbylg2f(ZeroCurve,a,b,sigma,eta,rho,Strike,Maturity) returns cap price for a two-factor additive Gaussian interest-rate model.

CapPrice = capbylg2f(___,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 cap.

Examples

Price a Cap Using a Linear Gaussian Two-Factor Model

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

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);
  
irdc = IRDataCurve('Zero',Settle,CurveDates,ZeroRates);
  
a = .07;
b = .5;
sigma = .01;
eta = .006;
rho = -.7;
  
CapMaturity = 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 = capbylg2f(irdc,a,b,sigma,eta,rho,Strike,CapMaturity)

Price = 11×1

    0.0316
    0.3225
    0.7761
    1.3240
    1.9394
    3.1247
    4.8451
    7.3752
    9.8582
   11.4673
      ⋮

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

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

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 caps
CapMaturity = 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]';
Notional = {{'15-Dec-2010' 100;'15-Dec-2014' 70;'15-Dec-2022' 40;'15-Dec-2037' 10}};

% Price the amortizing caps
Price = capbylg2f(irdc,a,b,sigma,eta,rho,Strike,CapMaturity, 'Notional', Notional)

Price = 11×1

    0.0316
    0.3225
    0.7761
    1.1313
    1.5362
    2.3213
    2.8297
    3.6878
    3.7297
    3.8906
      ⋮

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 numeric

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

Data Types: single | double

b — Mean reversion for second factor for Linear Gaussian two-factor model

scalar numeric

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

Data Types: single | double

sigma — Volatility for first factor for Linear Gaussian two-factor model

scalar numeric

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

Data Types: single | double

eta — Volatility for second factor for Linear Gaussian two-factor model

scalar numeric

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

Data Types: single | double

rho — Scalar correlation of the factors

scalar numeric

Scalar correlation of the factors, specified as a scalar numeric.

Data Types: single | double

Strike — Cap strike price

nonnegative integer | vector of nonnegative integers

Cap strike price, specified as a nonnegative integer using a NumCaps-by-1 vector.

Data Types: single | double

Maturity — Cap maturity date

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

Cap maturity date, specified using a NumCaps-by-1 vector of serial date numbers or date character vectors.

Data Types: single | double | char | 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 quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: Price = capbylg2f(irdc,a,b,sigma,eta,rho,Strike,CapMaturity,'Reset',1,'Notional',100)

'Reset' — Frequency of cap 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 cap payments per year, specified as the comma-separated pair consisting of 'Reset' and a positive integers for the values [1,2,4,6,12] in a NumCaps-by-1 vector.

Data Types: single | double

'Notional' — Notional value of cap

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

Notional value of cap, specified as the comma-separated pair consisting of 'Notional' and a 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

CapPrice — Cap price

scalar | vector

Expected prices of cap, returned as a scalar or an NumCaps-by-1 vector.

More About

Cap

A cap is a contract that includes a guarantee that sets the maximum interest rate to be paid by the holder, based on an otherwise floating interest rate.

The payoff for a cap is:

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

For more information, see Cap.

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.