# capbylg2f

Price cap using Linear Gaussian two-factor model

## Syntax

• `CapPrice = capbylg2f(ZeroCurve,a,b,sigma,eta,rho,Strike,Maturity)` example
• `CapPrice = capbylg2f(___, Name,Value)` example

## Description

example

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

example

````CapPrice = capbylg2f(___, Name,Value)` returns cap price for a two-factor additive Gaussian interest-rate model using optional name-value pairs. Note:   Use the optional name-value pair argument, `Notional`, to pass a schedule to compute the price for an amortizing cap.```

## Examples

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### 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 = 0.0316 0.3225 0.7761 1.3240 1.9394 3.1247 4.8451 7.3752 9.8582 11.4673 12.7850 ```

### 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 = 0.0316 0.3225 0.7761 1.1313 1.5362 2.3213 2.8297 3.6878 3.7297 3.8906 4.0223 ```

## 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` — Cap strike pricenonnegative 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 datenonnegative integer | vector of nonnegative integers | string of dates

Cap maturity date, specified using a `NumCaps`-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 = 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 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

`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 pricescalar | vector

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

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### Algorithms

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

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

$dx\left(t\right)=-a\left(x\right)\left(t\right)dt+\sigma \left(d{W}_{1}\left(t\right),x\left(0\right)=0$

$dy\left(t\right)=-b\left(y\right)\left(t\right)dt+\eta \left(d{W}_{2}\left(t\right),y\left(0\right)=0$

where $d{W}_{1}\left(t\right)d{W}_{2}\left(t\right)=\rho dt$ 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.