simByTransition

Simulate CIR sample paths with transition density

Syntax

[Paths,Times] = simByTransition(MDL,NPeriods)

[Paths,Times] = simByTransition(___,Name,Value)

Description

[Paths,Times] = simByTransition(MDL,NPeriods) simulates NTrials sample paths of NVars independent state variables driven by the Cox-Ingersoll-Ross (CIR) process sources of risk over NPeriods consecutive observation periods. simByTransition approximates a continuous-time CIR model using an approximation of the transition density function.

[Paths,Times] = simByTransition(___,Name,Value) specifies options using one or more name-value pair arguments in addition to the input arguments in the previous syntax.

You can perform quasi-Monte Carlo simulations using the name-value arguments for MonteCarloMethod, QuasiSequence, and BrownianMotionMethod. For more information, see Quasi-Monte Carlo Simulation.

Examples

Simulate Future Term Structures Using a CIR Model

Using the short rate, simulate the rate dynamics and term structures in the future using a CIR model. The CIR model is expressed as

$$dr(t)=\alpha (b-r(t))dt+\sigma \sqrt{r(t)}dW(t)$$

The exponential affine form of the bond price is

$$B(t,T)=e^{-A(t,T)r(t)+C(t,T)}$$

where

$$A(t,T)=\frac{2(e^{\gamma (T-t)}-1)}{(\gamma +\alpha )(e^{\gamma (T-t)}-1)+2\gamma }$$

$$B(t,T)=\frac{2\alpha b}{{\sigma }^2}\log \left(\frac{2\gamma e^{(\alpha +\gamma )(T-t)/2}}{(\gamma +\alpha )(e^{\gamma (T-t)}-1)+2\gamma }\right)$$

and

$$\gamma =\sqrt{{\alpha }^2+2{\sigma }^2}$$

Define the parameters for the cir object.

alpha = .1;
b = .05;
sigma = .05;
r0 = .04;

Define the function for bond prices.

gamma = sqrt(alpha^2 + 2*sigma^2);
A_func = @(t, T) ...
    2*(exp(gamma*(T-t))-1)/((alpha+gamma)*(exp(gamma*(T-t))-1)+2*gamma);
C_func = @(t, T) ...
    (2*alpha*b/sigma^2)*log(2*gamma*exp((alpha+gamma)*(T-t)/2)/((alpha+gamma)*(exp(gamma*(T-t))-1)+2*gamma));
P_func = @(t,T,r_t) exp(-A_func(t,T)*r_t+C_func(t,T));

Create a cir object.

obj = cir(alpha,b,sigma,'StartState',r0)

obj = 
   Class CIR: Cox-Ingersoll-Ross
   ----------------------------------------
     Dimensions: State = 1, Brownian = 1
   ----------------------------------------
      StartTime: 0
     StartState: 0.04
    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: 0.05
          Speed: 0.1

Define the simulation parameters.

nTrials = 100;
nPeriods = 5;   % Simulate future short over the next five years
nSteps = 12;    % Set intermediate steps to improve the accuracy

Simulate the short rates. The returning path is a (NPeriods + 1)-by-NVars-by-NTrials three-dimensional time-series array. For this example, the size of the output is 6-by-1-by-100.

rng('default');    % Reproduce the same result
rPaths = simByTransition(obj,nPeriods,'nTrials',nTrials,'nSteps',nSteps);
size(rPaths)

ans = 1×3

     6     1   100

rPathsExp = mean(rPaths,3);

Determine the term structure over the next 30 years.

maturity = 30;
T = 1:maturity;
futuresTimes = 1:nPeriods+1;

% Preallocate simTermStruc
simTermStructure = zeros(nPeriods+1,30);
for i = futuresTimes
    for t = T
        bondPrice = P_func(i,i+t,rPathsExp(i));
        simTermStructure(i,t) = -log(bondPrice)/t;
    end
end
plot(simTermStructure')
legend('Current','1-year','2-year','3-year','4-year','5-year')
title('Projected Term Structure for Next 5 Years')
ylabel('Long Rate Maturity R(t,T)')
xlabel('Time')

Quasi-Monte Carlo Simulation Using a CIR Model

The Cox-Ingersoll-Ross (CIR) short rate class derives directly from SDE with mean-reverting drift (SDEMRD): $d{X}_t=S(t)[L(t)-X_t]dt+D(t,X_t^{\frac{1}{2}})V(t)dW$

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

Create a cir object to represent the model: $d{X}_t=0.2(0.1-X_t)dt+0.05X_t^{\frac{1}{2}}dW$.

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

cir_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

Define the quasi-Monte Carlo simulation using the optional name-value arguments for 'MonteCarloMethod','QuasiSequence', and 'BrownianMotionMethod'.

[paths,time] = simByTransition(cir_obj,10,'ntrials',4096,'montecarlomethod','quasi','quasisequence','sobol','BrownianMotionMethod','principal-components');

Input Arguments

MDL — Stochastic differential equation model

object

Stochastic differential equation model, specified as a cir object. For more information on creating a CIR object, see cir.

Data Types: object

NPeriods — Number of simulation periods

positive scalar integer

Number of simulation periods, specified as a positive scalar integer. The value of NPeriods determines the number of rows of the simulated output series.

Data Types: double

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: [Paths,Times] = simByTransition(CIR,NPeriods,'DeltaTime',dt)

NTrials — Simulated trials (sample paths)

1 (single path of correlated state variables) (default) | positive integer

Simulated trials (sample paths) of NPeriods observations each, specified as the comma-separated pair consisting of 'NTrials' and a positive scalar integer.

Data Types: double

DeltaTime — Positive time increments between observations

1 (default) | scalar | column vector

Positive time increments between observations, specified as the comma-separated pair consisting of 'DeltaTime' and a scalar or a NPeriods-by-1 column vector.

DeltaTime represents the familiar dt found in stochastic differential equations, and determines the times at which the simulated paths of the output state variables are reported.

Data Types: double

NSteps — Number of intermediate time steps

1 (indicating no intermediate evaluation) (default) | positive integer

Number of intermediate time steps within each time increment dt (defined as DeltaTime), specified as the comma-separated pair consisting of 'NSteps' and a positive scalar integer.

The simByTransition function partitions each time increment dt into NSteps subintervals of length dt/NSteps, and refines the simulation by evaluating the simulated state vector at NSteps − 1 intermediate points. Although simByTransition does not report the output state vector at these intermediate points, the refinement improves accuracy by enabling the simulation to more closely approximate the underlying continuous-time process.

Data Types: double

StorePaths — Flag for storage and return method

True (default) | logical with values True or False

Flag for storage and return method that indicates how the output array Paths is stored and returned, specified as the comma-separated pair consisting of 'StorePaths' and a scalar logical flag with a value of True or False.

• If StorePaths is True (the default value) or is unspecified, then simByTransition returns Paths as a three-dimensional time series array.

• If StorePaths is False (logical 0), then simByTransition returns the Paths output array as an empty matrix.

Data Types: logical

MonteCarloMethod — Monte Carlo method to simulate stochastic processes

"standard" (default) | string with values "standard", "quasi", or "randomized-quasi" | character vector with values 'standard', 'quasi', or 'randomized-quasi'

Monte Carlo method to simulate stochastic processes, specified as the comma-separated pair consisting of 'MonteCarloMethod' and a string or character vector with one of the following values:

• "standard" — Monte Carlo using pseudo random numbers that has a convergence rate of O(N^-½).

• "quasi" — Quasi-Monte Carlo rate of convergence is faster than standard Monte Carlo with errors approaching size of O(N^-1).

• "randomized-quasi" — Quasi-random sequences, also called low-discrepancy sequences, are deterministic uniformly distributed sequences which are specifically designed to place sample points as uniformly as possible.

Data Types: string | char

QuasiSequence — Low discrepancy sequence to drive stochastic processes

"sobol" (default) | string with value "sobol" | character vector with value 'sobol'

Low discrepancy sequence to drive the stochastic processes, specified as the comma-separated pair consisting of 'QuasiSequence' and a string or character vector with the following value:

• "sobol" — Quasi-random low-discrepancy sequences that use a base of two to form successively finer uniform partitions of the unit interval and then reorder the coordinates in each dimension

Note

If MonteCarloMethod option is not specified or specified as"standard", QuasiSequence is ignored.

Data Types: string | char

BrownianMotionMethod — Brownian motion construction method

"standard" (default) | string with value "brownian-bridge" or "principal-components" | character vector with value 'brownian-bridge' or 'principal-components'

Brownian motion construction method, specified as the comma-separated pair consisting of 'BrownianMotionMethod' and a string or character vector with one of the following values:

• "standard" — The Brownian motion path is found by taking the cumulative sum of the Gaussian variates.

• "brownian-bridge" — The last step of the Brownian motion path is calculated first, followed by any order between steps until all steps have been determined.

• "principal-components" — The Brownian motion path is calculated by minimizing the approximation error.

The starting point for a Monte Carlo simulation is the construction of a Brownian motion sample path (or Wiener path). Such paths are built from a set of independent Gaussian variates, using either standard discretization, Brownian-bridge construction, or principal components construction.

Both standard discretization and Brownian-bridge construction share the same variance and, therefore, the same resulting convergence when used with the MonteCarloMethod using pseudo random numbers. However, the performance differs between the two when the MonteCarloMethod option "quasi" is introduced, with faster convergence seen for "brownian-bridge" construction option and the fastest convergence when using the "principal-components" construction option.

Data Types: string | char

Processes — Sequence of end-of-period processes or state vector adjustments

simByEuler makes no adjustments and performs no processing (default) | function | cell array of functions

Sequence of end-of-period processes or state vector adjustments, specified as the comma-separated pair consisting of 'Processes' and a function or cell array of functions of the form

$$X_t=P(t,X_t)$$

.

simByTransition applies processing functions at the end of each observation period. The processing functions accept the current observation time t and the current state vector X_t, and return a state vector that may adjust the input state.

If you specify more than one processing function, simByTransition invokes the functions in the order in which they appear in the cell array.

Data Types: cell | function

Output Arguments

Paths — Simulated paths of correlated state variables

array

Simulated paths of correlated state variables, returned as an (NPeriods + 1)-by-NVars-by-NTrials three-dimensional time series array.

For a given trial, each row of Paths is the transpose of the state vector X_t at time t. When the input flag StorePaths = False, simByTransition returns Paths as an empty matrix.

Times — Observation times associated with simulated paths

column vector

Observation times associated with the simulated paths, returned as an (NPeriods + 1)-by-1 column vector. Each element of Times is associated with the corresponding row of Paths.

More About

Transition Density

Transition density describes the probability distribution of a state variable transitioning from one state to another over a given time interval.

Transition Density Simulation

The SDE has no solution such that r(t) = f(r(0),⋯).

In other words, the equation is not explicitly solvable. However, the transition density for the process is known.

The exact simulation for the distribution of r(t_1 ),⋯,r(t_n) is that of the process at times t_1,⋯,t_n for the same value of r(0). The transition density for this process is known and is expressed as

$$\begin{array}{l}r(t)=\frac{{\sigma }^2(1-e^{-\alpha (t-u)}}{4\alpha }{x}_d^2\left(\frac{4\alpha e^{-\alpha (t-u)}}{{\sigma }^2(1-e^{-\alpha (t-u)})}r(u)\right),t>u\\ \text{where}\\ d\equiv \frac{4b\alpha }{{\sigma }^2}\end{array}$$

Algorithms

Use the simByTransition function to simulate any vector-valued CIR process of the form

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

where

• X_t is an NVars-by-1 state vector of process variables.

• S is an NVars-by-NVars matrix of mean reversion speeds (the rate of mean reversion).

• L is an NVars-by-1 vector of mean reversion levels (long-run mean or level).

• D is an NVars-by-NVars diagonal matrix, where each element along the main diagonal is the square root of the corresponding element of the state vector.

• V is an NVars-by-NBrowns instantaneous volatility rate matrix.

• dW_t is an NBrowns-by-1 Brownian motion vector.

References

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

Version History

Introduced in R2018b

R2022b: Perform Brownian bridge and principal components construction

Perform Brownian bridge and principal components construction using the name-value argument BrownianMotionMethod.

R2022a: Perform Quasi-Monte Carlo simulation

Perform Quasi-Monte Carlo simulation using the name-value arguments MonteCarloMethod and QuasiSequence.