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.

Note

Alternatively, you can use the Cap object to price cap instruments. For more information, see Get Started with Workflows Using Object-Based Framework for Pricing Financial Instruments.

example

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.

example

Examples

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Price a Cap Using a Linear Gaussian Two-Factor Model

Open Live Script

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

Settle = datetime(2007,12,15);
  
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

    0.0218
    0.3167
    0.7640
    1.3055
    1.9152
    3.0909
    4.7998
    7.3122
    9.7917
   11.4568
      ⋮

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

Open Live Script

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

Settle = datetime(2007,12,15);
% 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 = {{datetime(2010,12,15) 100;datetime(2014,12,15) 70;datetime(2022,12,15) 40;datetime(2037,12,15) 10}};

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

Price = 11×1

    0.0218
    0.3167
    0.7640
    1.1150
    1.5162
    2.2952
    2.8006
    3.6532
    3.6963
    3.8628
      ⋮

Input Arguments

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`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: 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: 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: 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: double

`rho` — Scalar correlation of the factors
scalar numeric

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

Data Types: 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: double

`Maturity` — Cap maturity date
datetime array | string array | date character vector

Cap maturity date, specified using a NumCaps-by-1 vector using a datetime array, string array, or date character vectors.

To support existing code, capbylg2f also accepts serial date numbers as inputs, but they are not recommended.

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

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

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`CapPrice` — Cap price
scalar | vector

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

More About

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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:

$\max (C u r r e n t R a t e - C a p R a t e, 0)$

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:

$r (t) = x (t) + y (t) + ϕ (t)$

$d x (t) = - a (x) (t) d t + σ (d W_{1} (t), x (0) = 0$

$d y (t) = - b (y) (t) d t + η (d W_{2} (t), y (0) = 0$

where $d W_{1} (t) d W_{2} (t) = ρ d t$ 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.

Version History

Introduced in R2013a

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R2022b: Serial date numbers not recommended

Although capbylg2f supports serial date numbers, datetime values are recommended instead. The datetime data type provides flexible date and time formats, storage out to nanosecond precision, and properties to account for time zones and daylight saving time.

To convert serial date numbers or text to datetime values, use the datetime function. For example:

t = datetime(738427.656845093,"ConvertFrom","datenum");
y = year(t)

y =

        2021

There are no plans to remove support for serial date number inputs.

capbylg2f

Syntax

Description

Examples

Price a Cap Using a Linear Gaussian Two-Factor Model

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

Input Arguments

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

`a` — Mean reversion for first factor for Linear Gaussian two-factor model
scalar numeric

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

`sigma` — Volatility for first factor for Linear Gaussian two-factor model
scalar numeric

`eta` — Volatility for second factor for Linear Gaussian two-factor model
scalar numeric

`rho` — Scalar correlation of the factors
scalar numeric

`Strike` — Cap strike price
nonnegative integer | vector of nonnegative integers

`Maturity` — Cap maturity date
datetime array | string array | date character vector

Name-Value Arguments

`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]`

`Notional` — Notional value of cap
`100` (default) | nonnegative integer | vector of nonnegative integers

Output Arguments

`CapPrice` — Cap price
scalar | vector

More About

Cap

Algorithms

References

Version History

R2022b: Serial date numbers not recommended

See Also

Topics

capbylg2f

Syntax

Description

Examples

Price a Cap Using a Linear Gaussian Two-Factor Model

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

Input Arguments

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

a — Mean reversion for first factor for Linear Gaussian two-factor model scalar numeric

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

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

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

rho — Scalar correlation of the factors scalar numeric

Strike — Cap strike price nonnegative integer | vector of nonnegative integers

Maturity — Cap maturity date datetime array | string array | date character vector

Name-Value Arguments

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]

Notional — Notional value of cap 100 (default) | nonnegative integer | vector of nonnegative integers

Output Arguments

CapPrice — Cap price scalar | vector

More About

Cap

Algorithms

References

Version History

R2022b: Serial date numbers not recommended

See Also

Topics

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

`a` — Mean reversion for first factor for Linear Gaussian two-factor model
scalar numeric

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

`sigma` — Volatility for first factor for Linear Gaussian two-factor model
scalar numeric

`eta` — Volatility for second factor for Linear Gaussian two-factor model
scalar numeric

`rho` — Scalar correlation of the factors
scalar numeric

`Strike` — Cap strike price
nonnegative integer | vector of nonnegative integers

`Maturity` — Cap maturity date
datetime array | string array | date character vector

`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]`

`Notional` — Notional value of cap
`100` (default) | nonnegative integer | vector of nonnegative integers

`CapPrice` — Cap price
scalar | vector