# simByEuler

Euler simulation of stochastic differential equations (SDEs) for `SDE`, `BM`, `GBM`, `CEV`, `CIR`, `HWV`, `Heston`, `SDEDDO`, `SDELD`, or `SDEMRD` models

## Syntax

``[Paths,Times,Z] = simByEuler(MDL,NPeriods)``
``[Paths,Times,Z] = simByEuler(___,Name,Value)``

## Description

example

````[Paths,Times,Z] = simByEuler(MDL,NPeriods)` simulates `NTrials` sample paths of `NVars` correlated state variables driven by `NBrowns` Brownian motion sources of risk over `NPeriods` consecutive observation periods. `simByEuler` uses the Euler approach to approximate continuous-time stochastic processes.```

example

````[Paths,Times,Z] = simByEuler(___,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` and `QuasiSequence`. For more information, see Quasi-Monte Carlo Simulation.```

## Examples

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Load the Data and Specify the SDE Model

```load Data_GlobalIdx2 prices = [Dataset.TSX Dataset.CAC Dataset.DAX ... Dataset.NIK Dataset.FTSE Dataset.SP]; returns = tick2ret(prices); nVariables = size(returns,2); expReturn = mean(returns); sigma = std(returns); correlation = corrcoef(returns); t = 0; X = 100; X = X(ones(nVariables,1)); F = @(t,X) diag(expReturn)* X; G = @(t,X) diag(X) * diag(sigma); SDE = sde(F, G, 'Correlation', ... correlation, 'StartState', X);```

Simulate a Single Path Over a Year

```nPeriods = 249; % # of simulated observations dt = 1; % time increment = 1 day rng(142857,'twister') [S,T] = simByEuler(SDE, nPeriods, 'DeltaTime', dt);```

Simulate 10 trials and examine the SDE model

```rng(142857,'twister') [S,T] = simulate(SDE, nPeriods, 'DeltaTime', dt, 'nTrials', 10); whos S```
``` Name Size Bytes Class Attributes S 250x6x10 120000 double ```

Plot the paths

```plot(T, S(:,:,1)), xlabel('Trading Day'), ylabel('Price') title('First Path of Multi-Dimensional Market Model') legend({'Canada' 'France' 'Germany' 'Japan' 'UK' 'US'},... 'Location', 'Best')``` The Cox-Ingersoll-Ross (CIR) short rate class derives directly from SDE with mean-reverting drift (`SDEMRD`): $d{X}_{t}=S\left(t\right)\left[L\left(t\right)-{X}_{t}\right]dt+D\left(t,{X}_{t}^{\frac{1}{2}}\right)V\left(t\right)dW$

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

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

Simulate a single path over a year using `simByEuler`.

```nPeriods = 249; % # of simulated observations dt = 1; % time increment = 1 day rng(142857,'twister') [Paths,Times] = simByEuler(CIR,nPeriods,'Method','higham-mao','DeltaTime', dt)```
```Paths = 250×1 1.0000 0.8613 0.7245 0.6349 0.4741 0.3853 0.3374 0.2549 0.1859 0.1814 ⋮ ```
```Times = 250×1 0 1 2 3 4 5 6 7 8 9 ⋮ ```

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

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

`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'` and `'QuasiSequence'`.

`[paths,time,z] = simByEuler(cir_obj,10,'ntrials',4096,'method','basic','montecarlomethod','quasi','quasisequence','sobol');`

The Heston (`heston`) class derives directly from SDE from Drift and Diffusion (`sdeddo`). Each Heston model is a bivariate composite model, consisting of two coupled univariate models:

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

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

Create a `heston` object.

`heston_obj = heston (0.1, 0.2, 0.1, 0.05) % (Return, Speed, Level, Volatility)`
```heston_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 ```

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

`[paths,time,z] = simByEuler(heston_obj,10,'ntrials',4096,'montecarlomethod','quasi','quasisequence','sobol');`

Create a univariate `gbm` object to represent the model: $d{X}_{t}=0.25{X}_{t}dt+0.3{X}_{t}d{W}_{t}$.

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

`gbm` objects display the parameter `B` as the more familiar `Return`.

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

`[paths,time,z] = simByEuler(gbm_obj,10,'ntrials',4096,'montecarlomethod','quasi','quasisequence','sobol');`

## Input Arguments

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Stochastic differential equation model, specified as an `sde`, `bm`, `gbm`, `cev`, `cir`, `hwv`, `heston`, `sdeddo`, `sdeld`, or `sdemrd` object.

Data Types: `object`

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,Z] = simByEuler(SDE,NPeriods,'DeltaTime',dt)```

Method to handle negative values, specified as the comma-separated pair consisting of `'Method'` and a character vector or string with a supported value.

Note

The `Method` argument is only supported when using a `CIR` object. For more information on creating a `CIR` object, see `cir`.

Data Types: `char` | `string`

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`

Positive time increments between observations, specified as the comma-separated pair consisting of `'DeltaTimes'` 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`

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

The `simByEuler` 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 `simByEuler` does not report the output state vector at these intermediate points, the refinement improves accuracy by allowing the simulation to more closely approximate the underlying continuous-time process.

Data Types: `double`

Flag indicates whether `simByEuler` uses antithetic sampling to generate the Gaussian random variates that drive the Brownian motion vector (Wiener processes). This argument is specified as the comma-separated pair consisting of `'Antithetic'` and a scalar logical flag with a value of `True` or `False`.

When you specify `True`, `simByEuler` performs sampling such that all primary and antithetic paths are simulated and stored in successive matching pairs:

• Odd trials `(1,3,5,...)` correspond to the primary Gaussian paths.

• Even trials `(2,4,6,...)` are the matching antithetic paths of each pair derived by negating the Gaussian draws of the corresponding primary (odd) trial.

Note

If you specify an input noise process (see `Z`), `simByEuler` ignores the value of `Antithetic`.

Data Types: `logical`

Direct specification of the dependent random noise process used to generate the Brownian motion vector (Wiener process) that drives the simulation. This argument is specified as the comma-separated pair consisting of `'Z'` and a function or as an ```(NPeriods ⨉ NSteps)```-by-`NBrowns`-by-`NTrials` three-dimensional array of dependent random variates.

Note

If you specify `Z` as a function, it must return an `NBrowns`-by-`1` column vector, and you must call it with two inputs:

• A real-valued scalar observation time t.

• An `NVars`-by-`1` state vector Xt.

Data Types: `double` | `function`

Flag 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, `simByEuler` returns `Paths` as a three-dimensional time series array.

If `StorePaths` is `False` (logical `0`), `simByEuler` returns the `Paths` output array as an empty matrix.

Data Types: `logical`

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.

• `"quasi"` — Quasi-Monte Carlo using low-discrepancy sequences.

• `"randomized-quasi"` — Randomized quasi-Monte Carlo.

Note

If you specify an input noise process (see `Z`), `simByEuler` ignores the value of `MonteCarloMethod`.

Data Types: `string` | `char`

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

• `"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`

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

`${X}_{t}=P\left(t,{X}_{t}\right)$`

The `simByEuler` function runs processing functions at each interpolation time. They must accept the current interpolation time t, and the current state vector Xt, and return a state vector that may be an adjustment to the input state.

If you specify more than one processing function, `simByEuler` invokes the functions in the order in which they appear in the cell array. You can use this argument to specify boundary conditions, prevent negative prices, accumulate statistics, plot graphs, and more.

Data Types: `cell` | `function`

## Output Arguments

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Simulated paths of correlated state variables, returned as a ```(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 Xt at time t. When the input flag `StorePaths` = `False`, `simByEuler` returns `Paths` as an empty matrix.

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

Dependent random variates used to generate the Brownian motion vector (Wiener processes) that drive the simulation, returned as a ```(NPeriods ⨉ NSteps)```-by-`NBrowns`-by-`NTrials` three-dimensional time series array.

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### Antithetic Sampling

Simulation methods allow you to specify a popular variance reduction technique called antithetic sampling.

This technique attempts to replace one sequence of random observations with another of the same expected value, but smaller variance. In a typical Monte Carlo simulation, each sample path is independent and represents an independent trial. However, antithetic sampling generates sample paths in pairs. The first path of the pair is referred to as the primary path, and the second as the antithetic path. Any given pair is independent of any other pair, but the two paths within each pair are highly correlated. Antithetic sampling literature often recommends averaging the discounted payoffs of each pair, effectively halving the number of Monte Carlo trials.

This technique attempts to reduce variance by inducing negative dependence between paired input samples, ideally resulting in negative dependence between paired output samples. The greater the extent of negative dependence, the more effective antithetic sampling is.

## Algorithms

This function simulates any vector-valued SDE of the form

`$d{X}_{t}=F\left(t,{X}_{t}\right)dt+G\left(t,{X}_{t}\right)d{W}_{t}$`

where:

• X is an NVars-by-`1` state vector of process variables (for example, short rates or equity prices) to simulate.

• W is an NBrowns-by-`1` Brownian motion vector.

• F is an NVars-by-`1` vector-valued drift-rate function.

• G is an NVars-by-NBrowns matrix-valued diffusion-rate function.

`simByEuler` simulates `NTrials` sample paths of `NVars` correlated state variables driven by `NBrowns` Brownian motion sources of risk over `NPeriods` consecutive observation periods, using the Euler approach to approximate continuous-time stochastic processes.

• This simulation engine provides a discrete-time approximation of the underlying generalized continuous-time process. The simulation is derived directly from the stochastic differential equation of motion. Thus, the discrete-time process approaches the true continuous-time process only as `DeltaTime` approaches zero.

• The input argument `Z` allows you to directly specify the noise-generation process. This process takes precedence over the `Correlation` parameter of the `sde` object and the value of the `Antithetic` input flag. If you do not specify a value for `Z`, `simByEuler` generates correlated Gaussian variates, with or without antithetic sampling as requested.

• The end-of-period `Processes` argument allows you to terminate a given trial early. At the end of each time step, `simByEuler` tests the state vector Xt for an all-`NaN` condition. Thus, to signal an early termination of a given trial, all elements of the state vector Xt must be `NaN`. This test enables a user-defined `Processes` function to signal early termination of a trial, and offers significant performance benefits in some situations (for example, pricing down-and-out barrier options).

 Deelstra, G. and F. Delbaen. “Convergence of Discretized Stochastic (Interest Rate) Processes with Stochastic Drift Term.” Applied Stochastic Models and Data Analysis., 1998, vol. 14, no. 1, pp. 77–84.

 Higham, Desmond, and Xuerong Mao. “Convergence of Monte Carlo Simulations Involving the Mean-Reverting Square Root Process.” The Journal of Computational Finance, vol. 8, no. 3, 2005, pp. 35–61.

 Lord, Roger, et al. “A Comparison of Biased Simulation Schemes for Stochastic Volatility Models.” Quantitative Finance, vol. 10, no. 2, Feb. 2010, pp. 177–94