# bates

Bates stochastic volatility model

## Description

The bates function creates a bates object, which represents a Bates model.

The Bates model is a bivariate composite model that derives from the heston object. The Bates model is composed of two coupled and dissimilar univariate models, each driven by a single Brownian motion source of risk and a single compound Poisson process representing the arrivals of important events over NPeriods consecutive observation periods. The Bates model approximates continuous-time Bates stochastic volatility processes.

The first univariate model is a GBM model with a stochastic volatility function and a stochastic jump process, and usually corresponds to a price process whose variance rate is governed by the second univariate model. The second model is a Cox-Ingersoll-Ross (CIR) square root diffusion model that describes the evolution of the variance rate of the coupled GBM price process.

Bates models are bivariate composite models. Each Bates model consists of two coupled univariate models:

• A geometric Brownian motion (gbm) model with a stochastic volatility function and jumps.

$d{X}_{1t}=B\left(t\right){X}_{1t}dt+\sqrt{{X}_{2t}}{X}_{1t}d{W}_{1t}+Y\left(t\right){X}_{1t}d{N}_{t}$

This model usually corresponds to a price process whose volatility (variance rate) is governed by the second univariate model.

• A Cox-Ingersoll-Ross (cir) square root diffusion model.

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

This model describes the evolution of the variance rate of the coupled Bates price process.

## Creation

### Description

example

Bates = bates(Return,Speed,Level,Volatility,JumpFreq,JumpMean,JumpVol) create a bates object with the default options.

Since Bates models are bivariate models composed of coupled univariate models, all required inputs correspond to scalar parameters. Specify required inputs as one of two types:

• MATLAB® array. Specify an array to indicate a static (non-time-varying) parametric specification. This array fully captures all implementation details, which are clearly associated with a parametric form.

• MATLAB function. Specify a function to provide indirect support for virtually any static, dynamic, linear, or nonlinear model. This parameter is supported by an interface because all implementation details are hidden and fully encapsulated by the function.

### Note

You can specify combinations of array and function input parameters as needed. Moreover, a parameter is identified as a deterministic function of time if the function accepts a scalar time t as its only input argument. Otherwise, a parameter is assumed to be a function of time t and state Xt and is invoked with both input arguments.

example

Bates = bates(___,Name,Value) sets Properties using name-value pair arguments in addition to the input arguments in the preceding syntax. Enclose each property name in quotes.

The bates object has the following Properties:

• StartTime — Initial observation time

• StartState — Initial state at time StartTime

• Correlation — Access function for the Correlation input argument

• Drift — Composite drift-rate function

• Diffusion — Composite diffusion-rate function

• Simulation — A simulation function or method

### Input Arguments

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Expected mean instantaneous rate of return of the GBM price process, specified as an array, or a deterministic function of time.

If you specify Return as an array, it must be scalar.

If you specify Return as a deterministic function of time, you call Return with a real-valued scalar time t as its only input, it must return a scalar.

If you specify Return as a deterministic function of time and state, it must return a scalar when you call it with two inputs:

• A real-valued scalar observation time t

• A 2-by-1 bivariate state vector Xt

Data Types: double | function_handle

Mean-reversion speed of the CIR stochastic variance process, specified as an array or deterministic function of time.

If you specify Speed as an array, it must be a scalar.

If you specify Speed as a deterministic function of time, you call Speed with a real-valued scalar time t as its only input, it must return a scalar.

If you specify Speed as a function of time and state, the function calculates the speed of mean reversion. This function must return a scalar of reversion rates when you call it with two inputs:

• A real-valued scalar observation time t

• A 2-by-1 bivariate state vector Xt

### Note

Although bates enforces no restrictions on Speed, the mean-reversion speed is nonnegative such that the underlying process reverts to some stable level.

Data Types: double | function_handle

Reversion level or long-run average of the CIR stochastic variance process, specified as an array, or deterministic function of time.

If you specify Level as an array, it must be a scalar.

If you specify Level as a deterministic function of time, you call Level with a real-valued scalar time t as its only input, it must return a scalar.

If you specify Level as a deterministic function of time and state, it must return a scalar of reversion levels when you call it with two inputs:

• A real-valued scalar observation time t

• A 2-by-1 bivariate state vector Xt

Data Types: double | function_handle

Instantaneous volatility of the CIR stochastic variance process (often called the volatility of volatility or volatility of variance), specified as a scalar, a deterministic function of time, or a deterministic function of time and state.

If you specify Volatility as a scalar, it represents the instantaneous volatility of the CIR stochastic variance model.

If you specify Volatility as a deterministic function of time, you call Volatility with a real-valued scalar time t as its only input, it must return a scalar.

If you specify Volatility as a deterministic function time and state, Volatility must return a scalar when you call it with two inputs:

• A real-valued scalar observation time t

• A 2-by-1 bivariate state vector Xt

### Note

Although bates enforces no restrictions on Volatility, the volatility is usually nonnegative.

Data Types: double | function_handle

Instantaneous jump frequencies representing the intensities (the mean number of jumps per unit time) of Poisson processes (Nt) that drive the jump simulation, specified as an array, a deterministic function of time, or a deterministic function of time and state.

If you specify JumpFreq as an array, it must be a scalar.

If you specify JumpFreq as a deterministic function of time, you call JumpFreq with a real-valued scalar time t as its only input, it must return a scalar.

If you specify JumpFreq as a function of time and state, JumpFreq must return a scalar when you call it with two inputs:

• A real-valued scalar observation time t

• A 2-by-1 bivariate state vector Xt

Data Types: double | function_handle

Instantaneous mean of random percentage jump sizes J, where log(1+J) is normally distributed with mean (log(1+JumpMean) - 0.5 × JumpVol2) and standard deviation JumpVol, specified as an array, a deterministic function of time, or a deterministic function of time and state.

If you specify JumpMean as an array, it must be a scalar.

If you specify JumpMean as a deterministic function of time, you call JumpMean with a real-valued scalar time t as its only input, it must return a scalar.

If you specify JumpMean as a function of time and state, JumpMean must return a scalar when you call it with two inputs:

• A real-valued scalar observation time t

• A 2-by-1 bivariate state vector Xt

Data Types: double | function_handle

Instantaneous standard deviation of log(1+J), specified as an array, a deterministic function of time, or a deterministic function of time and state.

If you specify JumpVol as an array, it must be a scalar.

If you specify JumpVol as a deterministic function of time, you call JumpVol with a real-valued scalar time t as its only input, it must return a scalar.

If you specify JumpVol as a function of time and state, JumpVol must return a scalar when you call it with two inputs:

• A real-valued scalar observation time t

• A 2-by-1 bivariate state vector Xt

Data Types: double | function_handle

## Properties

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Starting time of first observation, applied to all state variables, specified as a scalar.

Data Types: double

Initial values of state variables, specified as a scalar, column vector, or matrix.

If StartState is a scalar, bates applies the same initial value to all state variables on all trials.

If StartState is a bivariate column vector, bates applies a unique initial value to each state variable on all trials.

If StartState is a matrix, bates applies a unique initial value to each state variable on each trial.

Data Types: double

Correlation between Gaussian random variates drawn to generate the Brownian motion vector (Wiener processes), specified as a scalar, a 2-by-2 positive semidefinite matrix, or as a deterministic function Ct that accepts the current time t and returns an 2-by-2 positive semidefinite correlation matrix. If Correlation is not a symmetric positive semidefinite matrix, use nearcorr to create a positive semidefinite matrix for a correlation matrix.

A Correlation matrix represents a static condition.

If you specify Correlation as a deterministic function of time, Correlation allows you to specify a dynamic correlation structure.

Data Types: double

Drift-rate component of continuous-time stochastic differential equations (SDEs), specified as a drift object or function accessible by (t, Xt).

The drift rate specification supports the simulation of sample paths of NVars state variables driven by NBrowns Brownian motion sources of risk over NPeriods consecutive observation periods, approximating continuous-time stochastic processes.

Use the drift function to create drift objects of the form

$F\left(t,{X}_{t}\right)=A\left(t\right)+B\left(t\right){X}_{t}$

Here:

• A is an NVars-by-1 vector-valued function accessible by the (t, Xt) interface.

• B is an NVars-by-NVars matrix-valued function accessible by the (t, Xt) interface.

The displayed parameters for a drift object follow.

• Rate — Drift-rate function, F(t,Xt)

• A — Intercept term, A(t,Xt), of F(t,Xt)

• B — First-order term, B(t,Xt), of F(t,Xt)

A and B enable you to query the original inputs. The function stored in Rate fully encapsulates the combined effect of A and B.

Specifying AB as MATLAB double arrays clearly associates them with a linear drift rate parametric form. However, specifying either A or B as a function allows you to customize virtually any drift-rate specification.

### Note

You can express drift and diffusion objects in the most general form to emphasize the functional (t, Xt) interface. However, you can specify the components A and B as functions that adhere to the common (t, Xt) interface, or as MATLAB arrays of appropriate dimension.

Example: F = drift(0, 0.1) % Drift-rate function F(t,X)

Data Types: object

Diffusion-rate component of continuous-time stochastic differential equations (SDEs), specified as a drift object or function accessible by (t, Xt).

The diffusion-rate specification supports the simulation of sample paths of NVars state variables driven by NBrowns Brownian motion sources of risk over NPeriods consecutive observation periods for approximating continuous-time stochastic processes.

Use the diffusion function to create diffusion objects of the form

$G\left(t,{X}_{t}\right)=D\left(t,{X}_{t}^{\alpha \left(t\right)}\right)V\left(t\right)$

Here:

• D is an NVars-by-NVars diagonal matrix-valued function.

• Each diagonal element of D is the corresponding element of the state vector raised to the corresponding element of an exponent Alpha, which is an NVars-by-1 vector-valued function.

• V is an NVars-by-NBrowns matrix-valued volatility rate function Sigma.

• Alpha and Sigma are also accessible using the (t, Xt) interface.

The displayed parameters for a diffusion object are:

• Rate — Diffusion-rate function, G(t,Xt)

• Alpha — State vector exponent, which determines the format of D(t,Xt) of G(t,Xt)

• Sigma — Volatility rate, V(t,Xt), of G(t,Xt)

Alpha and Sigma enable you to query the original inputs. (The combined effect of the individual Alpha and Sigma parameters is fully encapsulated by the function stored in Rate.) The Rate functions are the calculation engines for the drift and diffusion objects, and are the only parameters required for simulation.

### Note

You can express drift and diffusion objects in the most general form to emphasize the functional (t, Xt) interface. However, you can specify the components A and B as functions that adhere to the common (t, Xt) interface, or as MATLAB arrays of appropriate dimension.

Example: G = diffusion(1, 0.3) % Diffusion-rate function G(t,X)

Data Types: object

User-defined simulation function or SDE simulation method, specified as a function or SDE simulation method.

Data Types: function_handle

## Object Functions

 simByEuler Simulate Bates sample paths by Euler approximation simByQuadExp Simulate Bates, Heston, and CIR sample paths by quadratic-exponential discretization scheme simulate Simulate multivariate stochastic differential equations (SDEs)

## Examples

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Bates models are bivariate composite models, composed of two coupled and dissimilar univariate models, each driven by a single Brownian motion source of risk and a single compound Poisson process representing the arrivals of important events over NPeriods consecutive observation periods. The simulation approximates continuous-time Bates stochastic volatility processes.

Create a bates object.

AssetPrice = 80;
Return = 0.03;
JumpMean = 0.02;
JumpVol = 0.08;
JumpFreq = 0.1;

V0 = 0.04;
Level = 0.05;
Speed = 1.0;
Volatility = 0.2;
Rho = -0.7;
StartState = [AssetPrice;V0];
Correlation = [1 Rho;Rho 1];

batesObj = bates(Return, Speed, Level, Volatility,...
JumpFreq, JumpMean, JumpVol,'startstate',StartState,...
'correlation',Correlation)
batesObj =
Class BATES: Bates Bivariate Stochastic Volatility
--------------------------------------------------
Dimensions: State = 2, Brownian = 2
--------------------------------------------------
StartTime: 0
StartState: 2x1 double array
Correlation: 2x2 double array
Drift: drift rate function F(t,X(t))
Diffusion: diffusion rate function G(t,X(t))
Simulation: simulation method/function simByEuler
Return: 0.03
Speed: 1
Level: 0.05
Volatility: 0.2
JumpFreq: 0.1
JumpMean: 0.02
JumpVol: 0.08

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## Algorithms

The Bates model (Bates 1996) is an extension of the Heston model and adds not only stochastic volatility, but also the jump diffusion parameters as in Merton (1976) were also added to model sudden asset price movements.

Under the risk-neutral measure the model is expressed as follows

$\begin{array}{l}d{S}_{t}=\left(\gamma -q-{\lambda }_{p}{\mu }_{j}\right){S}_{t}dt+\sqrt{{\upsilon }_{t}}{S}_{t}d{W}_{t}+J{S}_{t}d{P}_{t}\\ d{\upsilon }_{t}=\kappa \left(\theta -{\upsilon }_{t}\right)dt+{\sigma }_{\upsilon }\sqrt{{\upsilon }_{t}}d{W}_{t}^{\upsilon }\\ \text{E}\left[d{W}_{t}d{W}_{t}^{\upsilon }\right]=pdt\\ \text{prob}\left(d{P}_{t}=1\right)={\lambda }_{p}dt\end{array}$

Here:

ᵞ is the continuous risk-free rate.

q is the continuous dividend yield.

J is the random percentage jump size conditional on the jump occurring, where

$\mathrm{ln}\left(1+J\right)~N\left(\text{ln(1+}{u}_{j}\right)-\frac{{\delta }^{2}}{2},{\delta }^{2}$

(1+J) has a lognormal distribution:

$\frac{1}{\left(1+J\right)\delta \sqrt{2\pi }}\text{exp}\left\{\frac{-{\left[\mathrm{ln}\left(1+J\right)-\left(\text{ln(1+}{\mu }_{j}\right)-\frac{{\delta }^{2}}{2}\right]}^{2}}{2{\delta }^{2}}\right\}$

Here:

μj is the mean of Jj > -1).

ƛp is the annual frequency (intensity) of the Poisson process Ptp ≥ 0).

υ is the initial variance of the underlying asset (υ0 > 0).

θ is the long-term variance level (θ > 0).

κ is the mean reversion speed for the variance (κ > 0).

συ is the volatility of volatility (συ > 0).

p is the correlation between the Weiner processes Wt and Wtυ (-1 ≤ p ≤ 1).

The "Feller condition" ensures positive variance: (2κθ > συ2).

The stochastic volatility along with the jump help better model the asymmetric leptokurtic features, the volatility smile, and the large random fluctuations such as crashes and rallies.

## References

[1] Aït-Sahalia, Yacine. “Testing Continuous-Time Models of the Spot Interest Rate.” Review of Financial Studies 9, no. 2 ( Apr. 1996): 385–426.

[2] Aït-Sahalia, Yacine. “Transition Densities for Interest Rate and Other Nonlinear Diffusions.” The Journal of Finance 54, no. 4 (Aug. 1999): 1361–95.

[3] Glasserman, Paul. Monte Carlo Methods in Financial Engineering. New York: Springer-Verlag, 2004.

[4] Hull, John C. Options, Futures and Other Derivatives. 7th ed, Prentice Hall, 2009.

[5] Johnson, Norman Lloyd, Samuel Kotz, and Narayanaswamy Balakrishnan. Continuous Univariate Distributions. 2nd ed. Wiley Series in Probability and Mathematical Statistics. New York: Wiley, 1995.

[6] Shreve, Steven E. Stochastic Calculus for Finance. New York: Springer-Verlag, 2004.