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

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

The Brownian Motion (BM) model (`bm`

) derives directly from the linear drift
(`sdeld`

) class:

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

Create a univariate Brownian motion (`bm`

)
object to represent the model using the `bm`

constructor:

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

`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`

class 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.

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

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

)
class:

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

The `cev`

class 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*).

Create a univariate `cev`

object to represent
the model using the `cev`

constructor:

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

`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`

.

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`

class 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}

Create a univariate `gbm`

object to represent
the model using the `gbm`

constructor:

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

`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

The `sdemrd`

class
derives directly from the `sdeddo`

class.
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:

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

Create an `sdemrd`

object using the `sdemrd`

constructor
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`

.

The Cox-Ingersoll-Ross (CIR) short rate class, `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.

Create a `cir`

object using the `cir`

constructor
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`

.

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

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

) class:

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

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

object using the `hwv`

constructor
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`

constructors 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`

class
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 `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})*,

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`

class encompasses
the models.

The Heston (`heston`

) class derives directly from
SDE from Drift and Diffusion (`sdeddo`

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

$$d{X}_{1t}=B(t){X}_{1t}dt+\sqrt{{X}_{2t}}{X}_{1t}d{W}_{1t}$$ | (17-5) |

$$d{X}_{2t}=S(t)[L(t)-{X}_{2t}]dt+V(t)\sqrt{{X}_{2t}}d{W}_{2t}$$ | (17-6) |

`heston`

are typically used to price
equity options.Create a `heston`

object using the `heston`

constructor
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

`bm`

| `cev`

| `cir`

| `diffusion`

| `drift`

| `gbm`

| `heston`

| `hwv`

| `interpolate`

| `sde`

| `sdeddo`

| `sdeld`

| `sdemrd`

| `simByEuler`

| `simBySolution`

| `simBySolution`

| `simulate`

| `ts2func`

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