## Parametric Models

### Creating Brownian Motion (BM) Models

The Brownian Motion (BM) model (`bm`

) derives directly from the linear
drift (`sdeld`

) model:

$$d{X}_{t}=\mu (t)dt+V(t)d{W}_{t}$$

### Example: BM Models

Create a univariate Brownian motion (`bm`

) object to represent the model
using `bm`

:

$$d{X}_{t}=0.3d{W}_{t}.$$

`obj = bm(0, 0.3) % (A = Mu, Sigma)`

obj = Class BM: Brownian Motion ---------------------------------------- Dimensions: State = 1, Brownian = 1 ---------------------------------------- StartTime: 0 StartState: 0 Correlation: 1 Drift: drift rate function F(t,X(t)) Diffusion: diffusion rate function G(t,X(t)) Simulation: simulation method/function simByEuler Mu: 0 Sigma: 0.3

`bm`

objects display the parameter `A`

as
the more familiar `Mu`

.

The `bm`

object also provides an overloaded Euler simulation method that
improves run-time performance in certain common situations. This specialized method
is invoked automatically only if *all* the following conditions
are met:

The expected drift, or trend, rate

`Mu`

is a column vector.The volatility rate,

`Sigma`

, is a matrix.No end-of-period adjustments and/or processes are made.

If specified, the random noise process

`Z`

is a three-dimensional array.If

`Z`

is unspecified, the assumed Gaussian correlation structure is a double matrix.

### Creating Constant Elasticity of Variance (CEV) Models

The Constant Elasticity of Variance (CEV) model (`cev`

) also derives directly from the
linear drift (`sdeld`

) model:

$$d{X}_{t}=\mu (t){X}_{t}dt+D(t,{X}_{t}^{\alpha (t)})V(t)d{W}_{t}$$

The `cev`

object constrains
*A* to an `NVars`

-by-`1`

vector of zeros. *D* is a diagonal matrix whose elements are the
corresponding element of the state vector *X*, raised to an
exponent *α*(*t*).

#### Example: Univariate CEV Models

Create a univariate `cev`

object to represent the model using
`cev`

:

$$d{X}_{t}=0.25{X}_{t}+0.3{X}_{t}^{\frac{1}{2}}d{W}_{t}.$$

`obj = cev(0.25, 0.5, 0.3) % (B = Return, Alpha, Sigma)`

obj = Class CEV: Constant Elasticity of Variance ------------------------------------------ Dimensions: State = 1, Brownian = 1 ------------------------------------------ StartTime: 0 StartState: 1 Correlation: 1 Drift: drift rate function F(t,X(t)) Diffusion: diffusion rate function G(t,X(t)) Simulation: simulation method/function simByEuler Return: 0.25 Alpha: 0.5 Sigma: 0.3

`cev`

and `gbm`

objects display the parameter
`B`

as the more familiar `Return`

.

### Creating Geometric Brownian Motion (GBM) Models

The Geometric Brownian Motion (GBM) model (`gbm`

) derives directly from the CEV (`cev`

) model:

$$d{X}_{t}=\mu (t){X}_{t}dt+D(t,{X}_{t})V(t)d{W}_{t}$$

Compared to the `cev`

object, a `gbm`

object constrains all elements of the *alpha*
exponent vector to one such that *D* is now a diagonal matrix with
the state vector *X* along the main diagonal.

The `gbm`

object also provides two simulation
methods that can be used by separable models:

An overloaded Euler simulation method that improves run-time performance in certain common situations. This specialized method is invoked automatically only if

*all*the following conditions are true:The expected rate of return (

`Return`

) is a diagonal matrix.The volatility rate (

`Sigma`

) is a matrix.No end-of-period adjustments/processes are made.

If specified, the random noise process

`Z`

is a three-dimensional array.If

`Z`

is unspecified, the assumed Gaussian correlation structure is a double matrix.

An approximate analytic solution (

`simBySolution`

) obtained by applying a Euler approach to the transformed (using Ito's formula) logarithmic process. In general, this is*not*the exact solution to this GBM model, as the probability distributions of the simulated and true state vectors are identical*only*for piecewise constant parameters. If the model parameters are piecewise constant over each observation period, the state vector*X*is lognormally distributed and the simulated process is exact for the observation times at which_{t}*X*is sampled._{t}

#### Example: Univariate GBM Models

Create a univariate `gbm`

object to represent the model using
`gbm`

:

$$d{X}_{t}=0.25{X}_{t}dt+0.3{X}_{t}d{W}_{t}$$

`obj = gbm(0.25, 0.3) % (B = Return, Sigma)`

obj = Class GBM: Generalized Geometric Brownian Motion ------------------------------------------------ Dimensions: State = 1, Brownian = 1 ------------------------------------------------ StartTime: 0 StartState: 1 Correlation: 1 Drift: drift rate function F(t,X(t)) Diffusion: diffusion rate function G(t,X(t)) Simulation: simulation method/function simByEuler Return: 0.25 Sigma: 0.3

### Creating Stochastic Differential Equations from Mean-Reverting Drift (SDEMRD) Models

The `sdemrd`

object derives directly from
the `sdeddo`

object. It provides an
interface in which the drift-rate function is expressed in mean-reverting drift
form:

$$d{X}_{t}=S(t)[L(t)-{X}_{t}]dt+D(t,{X}_{t}^{\alpha (t)})V(t)d{W}_{t}$$

`sdemrd`

objects provide a parametric
alternative to the linear drift form by reparameterizing the general linear drift
such that:

$$A(t)=S(t)L(t),B(t)=-S(t)$$

#### Example: SDEMRD Models

Create an `sdemrd`

object using `sdemrd`

with a square root
exponent to represent the model:

$$d{X}_{t}=0.2(0.1-{X}_{t})dt+0.05{X}_{t}^{\frac{1}{2}}d{W}_{t}.$$

obj = sdemrd(0.2, 0.1, 0.5, 0.05)

obj = Class SDEMRD: SDE with Mean-Reverting Drift ------------------------------------------- Dimensions: State = 1, Brownian = 1 ------------------------------------------- StartTime: 0 StartState: 1 Correlation: 1 Drift: drift rate function F(t,X(t)) Diffusion: diffusion rate function G(t,X(t)) Simulation: simulation method/function simByEuler Alpha: 0.5 Sigma: 0.05 Level: 0.1 Speed: 0.2

` % (Speed, Level, Alpha, Sigma)`

`sdemrd`

objects display the familiar `Speed`

and `Level`

parameters instead of `A`

and
`B`

.

### Creating Cox-Ingersoll-Ross (CIR) Square Root Diffusion Models

The Cox-Ingersoll-Ross (CIR) short-rate object, `cir`

, derives directly from the SDE
with mean-reverting drift (`sdemrd`

) class:

$$d{X}_{t}=S(t)[L(t)-{X}_{t}]dt+D(t,{X}_{t}^{\frac{1}{2}})V(t)d{W}_{t}$$

where *D* is a diagonal matrix whose elements are the square root
of the corresponding element of the state vector.

#### Example: CIR Models

Create a `cir`

object using `cir`

to represent the same model
as in Example: SDEMRD Models:

`obj = cir(0.2, 0.1, 0.05) % (Speed, Level, Sigma)`

obj = Class CIR: Cox-Ingersoll-Ross ---------------------------------------- Dimensions: State = 1, Brownian = 1 ---------------------------------------- StartTime: 0 StartState: 1 Correlation: 1 Drift: drift rate function F(t,X(t)) Diffusion: diffusion rate function G(t,X(t)) Simulation: simulation method/function simByEuler Sigma: 0.05 Level: 0.1 Speed: 0.2

Although the last two objects are of different classes, they represent the
same mathematical model. They differ in that you create the `cir`

object by specifying only
three input arguments. This distinction is reinforced by the fact that the
`Alpha`

parameter does not display – it is defined to be
`1/2`

.

### Creating Hull-White/Vasicek (HWV) Gaussian Diffusion Models

The Hull-White/Vasicek (HWV) short-rate object, `hwv`

, derives directly from SDE with
mean-reverting drift (`sdemrd`

) class:

$$d{X}_{t}=S(t)[L(t)-{X}_{t}]dt+V(t)d{W}_{t}$$

#### Example: HWV Models

Using the same parameters as in the previous example, create an
`hwv`

object using `hwv`

to represent the model:

$$d{X}_{t}=0.2(0.1-{X}_{t})dt+0.05d{W}_{t}.$$

`obj = hwv(0.2, 0.1, 0.05) % (Speed, Level, Sigma)`

obj = Class HWV: Hull-White/Vasicek ---------------------------------------- Dimensions: State = 1, Brownian = 1 ---------------------------------------- StartTime: 0 StartState: 1 Correlation: 1 Drift: drift rate function F(t,X(t)) Diffusion: diffusion rate function G(t,X(t)) Simulation: simulation method/function simByEuler Sigma: 0.05 Level: 0.1 Speed: 0.2

`cir`

and `hwv`

share the same interface and
display methods. The only distinction is that `cir`

and `hwv`

model objects constrain
`Alpha`

exponents to `1/2`

and
`0`

, respectively. Furthermore, the `hwv`

object also provides an
additional method that simulates approximate analytic solutions (`simBySolution`

) of separable
models. This method simulates the state vector
*X _{t}* using an approximation of
the closed-form solution of diagonal drift

`HWV`

models. Each
element of the state vector *X*is expressed as the sum of

_{t}`NBrowns`

correlated Gaussian random
draws added to a deterministic time-variable drift. When evaluating expressions, all model parameters are assumed piecewise
constant over each simulation period. In general, this is
*not* the exact solution to this `hwv`

model, because the probability distributions of the simulated and true state
vectors are identical *only* for piecewise constant
parameters. If *S(t,X _{t})*,

*L(t,X*, and

_{t})*V(t,X*are piecewise constant over each observation period, the state vector

_{t})*X*is normally distributed, and the simulated process is exact for the observation times at which

_{t}*X*is sampled.

_{t}#### Hull-White vs. Vasicek Models

Many references differentiate between Vasicek models and Hull-White models.
Where such distinctions are made, Vasicek parameters are constrained to be
constants, while Hull-White parameters vary deterministically with time. Think
of Vasicek models in this context as constant-coefficient Hull-White models and
equivalently, Hull-White models as time-varying Vasicek models. However, from an
architectural perspective, the distinction between static and dynamic parameters
is trivial. Since both models share the same general parametric specification as
previously described, a single `hwv`

object encompasses the
models.

### Creating Heston Stochastic Volatility Models

The Heston (`heston`

) object derives directly from
SDE from the Drift and Diffusion (`sdeddo`

) class. Each Heston model is a
bivariate composite model, consisting of two coupled univariate models:

$$d{X}_{1t}=B(t){X}_{1t}dt+\sqrt{{X}_{2t}}{X}_{1t}d{W}_{1t}$$ | (1) |

$$d{X}_{2t}=S(t)[L(t)-{X}_{2t}]dt+V(t)\sqrt{{X}_{2t}}d{W}_{2t}$$ | (2) |

`heston`

are typically used to price
equity options.#### Example: Heston Models

Create a `heston`

object using `heston`

to represent the model:

$$\begin{array}{l}d{X}_{1t}=0.1{X}_{1t}dt+\sqrt{{X}_{2t}}{X}_{1t}d{W}_{1t}\\ d{X}_{2t}=0.2[0.1-{X}_{2t}]dt+0.05\sqrt{{X}_{2t}}d{W}_{2t}\end{array}$$

obj = heston (0.1, 0.2, 0.1, 0.05)

obj = Class HESTON: Heston Bivariate Stochastic Volatility ---------------------------------------------------- Dimensions: State = 2, Brownian = 2 ---------------------------------------------------- StartTime: 0 StartState: 1 (2x1 double array) Correlation: 2x2 diagonal double array Drift: drift rate function F(t,X(t)) Diffusion: diffusion rate function G(t,X(t)) Simulation: simulation method/function simByEuler Return: 0.1 Speed: 0.2 Level: 0.1 Volatility: 0.05

## See Also

`sde`

| `bm`

| `gbm`

| `merton`

| `bates`

| `drift`

| `diffusion`

| `sdeddo`

| `sdeld`

| `cev`

| `cir`

| `heston`

| `hwv`

| `sdemrd`

| `ts2func`

| `simulate`

| `simByEuler`

| `simByQuadExp`

| `simBySolution`

| `simBySolution`

| `interpolate`