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:

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

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_t is lognormally distributed and the simulated process is exact for the observation times at which X_t is sampled.

Example: Univariate GBM Models

Create a univariate gbm object to represent the model using gbm:

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

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:

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

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_t is expressed as the sum of 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_t), and V(t,X_t) are piecewise constant over each observation period, the state vector X_t is normally distributed, and the simulated process is exact for the observation times at which X_t is sampled.

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)

Equation 1 is typically associated with a price process. Equation 2 represents the evolution of the price process' variance. Models of type heston are typically used to price equity options.

Example: Heston Models

Create a heston object using heston to represent the model:

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

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