# merton

`Merton` jump diffusion model

Since R2020a

## Description

The `merton` function creates a `merton` object, which derives from the `gbm` object.

The `merton` model, based on the Merton76 model, allows you to simulate sample paths of `NVars` state variables driven by `NBrowns` Brownian motion sources of risk and `NJumps` compound Poisson processes representing the arrivals of important events over `NPeriods` consecutive observation periods. The simulation approximates continuous-time `merton` stochastic processes.

You can simulate any vector-valued `merton` process of the form

`$d{X}_{t}=B\left(t,{X}_{t}\right){X}_{t}dt+D\left(t,{X}_{t}\right)V\left(t,{x}_{t}\right)d{W}_{t}+Y\left(t,{X}_{t},{N}_{t}\right){X}_{t}d{N}_{t}$`

Here:

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

• B(t,Xt) is an `NVars`-by-`NVars` matrix of generalized expected instantaneous rates of return.

• `D(t,Xt)` is an `NVars`-by-`NVars` diagonal matrix in which each element along the main diagonal is the corresponding element of the state vector.

• `V(t,Xt)` is an `NVars`-by-`NVars` matrix of instantaneous volatility rates.

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

• `Y(t,Xt,Nt)` is an `NVars`-by-`NJumps` matrix-valued jump size function.

• dNt is an `NJumps`-by-`1` counting process vector.

## Creation

### Syntax

``Merton = merton(Return,Sigma,JumpFreq,JumpMean,JumpVol)``
``Merton = merton(___,Name,Value)``

### Description

example

````Merton = merton(Return,Sigma,JumpFreq,JumpMean,JumpVol)` creates a default `merton` object. 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. NoteYou 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

````Merton = merton(___,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 `merton` 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 rates of asset return, denoted as B(t,Xt), specified as an array, a deterministic function of time, or a deterministic function of time and state.

If you specify `Return` as an array, it must be an `NVars`-by-`NVars` matrix representing the expected (mean) instantaneous rate of return.

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

If you specify `Return` as a deterministic function of time and state, it must return an `NVars`-by-`NVars` matrix when you call it with two inputs:

• A real-valued scalar observation time t

• An `NVars`-by-`1` state vector Xt

Data Types: `double` | `function_handle`

Instantaneous volatility rates, denoted as V(t,Xt), specified as an array, a deterministic function of time, or a deterministic function of time and state.

If you specify `Sigma` as an array, it must be an `NVars`-by-`NBrowns` matrix of instantaneous volatility rates or a deterministic function of time. In this case, each row of `Sigma` corresponds to a particular state variable. Each column corresponds to a particular Brownian source of uncertainty, and associates the magnitude of the exposure of state variables with sources of uncertainty.

If you specify `Sigma` as a deterministic function of time, when you call `Sigma` with a real-valued scalar time `t` as its only input, it must return an `NVars`-by-`NBrowns` matrix.

If you specify `Sigma` as a deterministic function of time and state, it must return an `NVars`-by-`NBrowns` matrix when you call it with two inputs:

• A real-valued scalar observation time t

• An `NVars`-by-`1` state vector Xt

Note

Although `merton` enforces no restrictions for `Sigma`, volatilities are usually nonnegative.

Data Types: `double` | `function_handle`

Instantaneous jump frequencies representing the intensities (the mean number of jumps per unit time) of the 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 an `NJumps`-by-`1` vector.

If you specify `JumpFreq` as a deterministic function of time, when you call `JumpFreq` with a real-valued scalar time `t` as its only input, `JumpFreq` must produce an `NJumps`-by-`1` vector.

If you specify `JumpFreq` as a deterministic function of time and state, it must return an `NVars`-by-`NBrowns` matrix when you call it with two inputs:

• A real-valued scalar observation time t

• An `NVars`-by-`1` 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 × `JumpVol`2) 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 an `NVars`-by-`NJumps` matrix.

If you specify `JumpMean` as a deterministic function of time, when you cal `JumpMean` with a real-valued scalar time `t` as its only input, it must return an `NVars`-by-`NJumps` matrix.

If you specify `JumpMean` as a deterministic function of time and state, it must return an `NVars`-by-`NJumps` matrix when you call it with two inputs:

• A real-valued scalar observation time t

• An `NVars`-by-`1` 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 an `NVars`-by-`NJumps` matrix.

If you specify `JumpVol` as a deterministic function of time, when you call `JumpVol` with a real-valued scalar time `t` as its only input, it must return an `NVars`-by-`NJumps` matrix.

If you specify `JumpVol` as a deterministic function of time and state, it must return an `NVars`-by-`NJumps` matrix when you call it with two inputs:

• A real-valued scalar observation time t

• An `NVars`-by-`1` 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, `merton` applies the same initial value to all state variables on all trials.

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

If `StartState` is a matrix, `merton` 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 an `NBrowns`-by-`NBrowns` positive semidefinite matrix, or as a deterministic function Ct that accepts the current time t and returns an `NBrowns`-by-`NBrowns` 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, you can specify a dynamic correlation structure.

Data Types: `double`

This property is read-only.

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 using the (t, Xt) interface.

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

The displayed parameters for a `drift` object are:

• `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 `A``B` 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`

This property is read-only.

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 `Merton` jump diffusion sample paths by Euler approximation `simBySolution` Simulate approximate solution of diagonal-drift `Merton` jump diffusion process `simByMilstein` Simulate `Merton` sample paths by Milstein approximation `simulate` Simulate multivariate stochastic differential equations (SDEs) for `SDE`, `BM`, `GBM`, `CEV`, `CIR`, `HWV`, `Heston`, `SDEDDO`, `SDELD`, `SDEMRD`, `Merton`, or `Bates` models

## Examples

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Merton jump diffusion models allow you to simulate sample paths of `NVARS` state variables driven by `NBROWNS` Brownian motion sources of risk and `NJumps` compound Poisson processes representing the arrivals of important events over `NPeriods` consecutive observation periods. The simulation approximates continuous-time merton stochastic processes.

Create a `merton` object.

```AssetPrice = 80; Return = 0.03; Sigma = 0.16; JumpMean = 0.02; JumpVol = 0.08; JumpFreq = 2; mertonObj = merton(Return,Sigma,JumpFreq,JumpMean,JumpVol,... 'startstat',AssetPrice)```
```mertonObj = Class MERTON: Merton Jump Diffusion ---------------------------------------- Dimensions: State = 1, Brownian = 1 ---------------------------------------- StartTime: 0 StartState: 80 Correlation: 1 Drift: drift rate function F(t,X(t)) Diffusion: diffusion rate function G(t,X(t)) Simulation: simulation method/function simByEuler Sigma: 0.16 Return: 0.03 JumpFreq: 2 JumpMean: 0.02 JumpVol: 0.08 ```

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## Algorithms

The Merton jump diffusion model (Merton 1976) is an extension of the Black-Scholes model, and models sudden asset price movements (both up and down) by adding the jump diffusion parameters with the Poisson process Pt.

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

`$\begin{array}{l}d{S}_{t}=\left(\gamma -q-{\lambda }_{p}{\mu }_{j}\right){S}_{t}dt+{\sigma }_{M}{S}_{t}d{W}_{t}+J{S}_{t}d{P}_{t}\\ \text{prob}\left(d{P}_{t}=1\right)={\lambda }_{p}dt\end{array}$`

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).

σM is the volatility of the asset price (σM> 0).

Under this formulation, extreme events are explicitly included in the stochastic differential equation as randomly occurring discontinuous jumps in the diffusion trajectory. Therefore, the disparity between observed tail behavior of log returns and that of Brownian motion is mitigated by the inclusion of a jump mechanism.

## 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.

## Version History

Introduced in R2020a

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