# simBySolution

Simulate approximate solution of diagonal-drift Merton jump diffusion process

## Description

example

[Paths,Times,Z,N] = simBySolution(MDL,NPeriods) simulates NNTrials sample paths of NVars correlated 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 jump diffusion process by an approximation of the closed-form solution.

example

[Paths,Times,Z,N] = simBySolution(___,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

collapse all

Simulate the approximate solution of diagonal-drift Merton process.

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

Use simBySolution to simulate NTrials sample paths of NVARS correlated 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 function approximates continuous-time Merton jump diffusion process by an approximation of the closed-form solution.

nPeriods = 100;
[Paths,Times,Z,N] = simBySolution(mertonObj, nPeriods,'nTrials', 3)
Paths =
Paths(:,:,1) =

1.0e+03 *

0.0800
0.0600
0.0504
0.0799
0.1333
0.1461
0.2302
0.2505
0.3881
0.4933
0.4547
0.4433
0.5294
0.6443
0.7665
0.6489
0.7220
0.7110
0.5815
0.5026
0.6523
0.7005
0.7053
0.4902
0.5401
0.4730
0.4242
0.5334
0.5821
0.6498
0.5982
0.5504
0.5290
0.5371
0.4789
0.4914
0.5019
0.3557
0.2950
0.3697
0.2906
0.2988
0.3081
0.3469
0.3146
0.3171
0.3588
0.3250
0.3035
0.2386
0.2533
0.2420
0.2315
0.2396
0.2143
0.2668
0.2115
0.1671
0.1784
0.1542
0.2046
0.1930
0.2011
0.2542
0.3010
0.3247
0.3900
0.4107
0.3949
0.4610
0.5725
0.5605
0.4541
0.5796
0.8199
0.5732
0.5856
0.7895
0.6883
0.6848
0.9059
1.0089
0.8429
0.9955
0.9683
0.8769
0.7120
0.7906
0.7630
1.2460
1.1703
1.2012
1.1109
1.1893
1.4346
1.4040
1.2365
1.0834
1.3315
0.8100
0.5558

Paths(:,:,2) =

80.0000
81.2944
71.3663
108.8305
111.4851
105.4563
160.2721
125.3288
158.3238
138.8899
157.9613
125.6819
149.8234
126.0374
182.5153
195.0861
273.1622
306.2727
301.3401
312.2173
298.2344
327.6944
288.9799
394.8951
551.4020
418.2258
404.1687
469.3555
606.4289
615.7066
526.6862
625.9683
474.4597
316.5110
407.9626
341.6552
475.0593
478.4058
545.3414
365.3404
513.2186
370.5371
444.0345
314.6991
257.4782
253.0259
237.6185
206.6325
334.5253
300.2284
328.9936
307.4059
248.7966
234.6355
183.9132
159.6084
169.1145
123.3256
148.1922
159.7083
104.0447
96.3935
92.4897
93.0576
116.3163
135.6249
120.6611
100.0253
109.7998
85.8078
81.5769
73.7983
65.9000
62.5120
62.9952
57.6044
54.2716
44.5617
42.2402
21.9133
18.0586
20.5171
22.5532
24.1654
26.8830
22.7864
34.5131
27.8362
27.7258
21.7367
20.8781
19.7174
14.9880
14.8903
19.3632
23.4230
27.7062
17.8347
16.8652
15.5675
15.5256

Paths(:,:,3) =

80.0000
79.6263
93.2979
63.1451
60.2213
54.2113
78.6114
96.6261
123.5584
126.5875
102.9870
83.2387
77.8567
79.3565
71.3876
80.5413
90.8709
77.5246
107.4194
114.4328
118.3999
148.0710
108.6207
110.0402
124.1150
104.5409
94.7576
98.9002
108.0691
130.7592
129.9744
119.9150
86.0303
96.9892
86.8928
106.8895
119.3219
197.7045
208.1930
197.1636
244.4438
166.4752
125.3896
128.9036
170.9818
140.2719
125.8948
87.0324
66.7637
48.4280
50.5766
49.7841
67.5690
62.8776
85.3896
67.9608
72.9804
59.0174
50.1132
45.2220
59.5469
58.4673
98.4790
90.0250
80.3092
86.9245
88.1303
95.4237
104.4456
99.1969
168.3980
146.8791
150.0052
129.7521
127.1402
113.3413
145.2281
153.1315
125.7882
111.9988
112.7732
118.9120
150.9166
120.0673
128.2727
185.9171
204.3474
194.5443
163.2626
183.9897
233.4125
318.9068
356.0077
380.4513
446.9518
484.9218
377.4244
470.3577
454.5734
297.0580
339.0796

Times = 101×1

0
1
2
3
4
5
6
7
8
9
⋮

Z =
Z(:,:,1) =

-2.2588
-1.3077
3.5784
3.0349
0.7147
1.4897
0.6715
1.6302
0.7269
-0.7873
-1.0689
1.4384
1.3703
-0.2414
-0.8649
0.6277
-0.8637
-1.1135
-0.7697
1.1174
0.5525
0.0859
-1.0616
0.7481
-0.7648
0.4882
1.4193
1.5877
0.8351
-1.1658
0.7223
0.1873
-0.4390
-0.8880
0.3035
0.7394
-2.1384
-1.0722
1.4367
-1.2078
1.3790
-0.2725
0.7015
-0.8236
0.2820
1.1275
0.0229
-0.2857
-1.1564
0.9642
-0.0348
-0.1332
-0.2248
-0.8479
1.6555
-0.8655
-1.3320
0.3335
-0.1303
0.8620
-0.8487
1.0391
0.6601
-0.2176
0.0513
0.4669
0.1832
0.3071
0.2614
-0.1461
-0.8757
-1.1742
1.5301
1.6035
-1.5062
0.2761
0.3919
-0.7411
0.0125
1.2424
0.3503
-1.5651
0.0983
-0.0308
-0.3728
-2.2584
1.0001
-0.2781
0.4716
0.6524
1.0061
-0.9444
0.0000
0.5946
0.9298
-0.6516
-0.0245
0.8617
-2.4863
-2.3193

Z(:,:,2) =

0.8622
-0.4336
2.7694
0.7254
-0.2050
1.4090
-1.2075
0.4889
-0.3034
0.8884
-0.8095
0.3252
-1.7115
0.3192
-0.0301
1.0933
0.0774
-0.0068
0.3714
-1.0891
1.1006
-1.4916
2.3505
-0.1924
-1.4023
-0.1774
0.2916
-0.8045
-0.2437
-1.1480
2.5855
-0.0825
-1.7947
0.1001
-0.6003
1.7119
-0.8396
0.9610
-1.9609
2.9080
-1.0582
1.0984
-2.0518
-1.5771
0.0335
0.3502
-0.2620
-0.8314
-0.5336
0.5201
-0.7982
-0.7145
-0.5890
-1.1201
0.3075
-0.1765
-2.3299
0.3914
0.1837
-1.3617
-0.3349
-1.1176
-0.0679
-0.3031
0.8261
-0.2097
-1.0298
0.1352
-0.9415
-0.5320
-0.4838
-0.1922
-0.2490
1.2347
-0.4446
-0.2612
-1.2507
-0.5078
-3.0292
-1.0667
-0.0290
-0.0845
0.0414
0.2323
-0.2365
2.2294
-1.6642
0.4227
-1.2128
0.3271
-0.6509
-1.3218
-0.0549
0.3502
0.2398
1.1921
-1.9488
0.0012
0.5812
0.0799

Z(:,:,3) =

0.3188
0.3426
-1.3499
-0.0631
-0.1241
1.4172
0.7172
1.0347
0.2939
-1.1471
-2.9443
-0.7549
-0.1022
0.3129
-0.1649
1.1093
-1.2141
1.5326
-0.2256
0.0326
1.5442
-0.7423
-0.6156
0.8886
-1.4224
-0.1961
0.1978
0.6966
0.2157
0.1049
-0.6669
-1.9330
0.8404
-0.5445
0.4900
-0.1941
1.3546
0.1240
-0.1977
0.8252
-0.4686
-0.2779
-0.3538
0.5080
-1.3337
-0.2991
-1.7502
-0.9792
-2.0026
-0.0200
1.0187
1.3514
-0.2938
2.5260
-1.2571
0.7914
-1.4491
0.4517
-0.4762
0.4550
0.5528
1.2607
-0.1952
0.0230
1.5270
0.6252
0.9492
0.5152
-0.1623
1.6821
-0.7120
-0.2741
-1.0642
-0.2296
-0.1559
0.4434
-0.9480
-0.3206
-0.4570
0.9337
0.1825
1.6039
-0.7342
0.4264
2.0237
0.3376
-0.5900
-1.6702
0.0662
1.0826
0.2571
0.9248
0.9111
1.2503
-0.6904
-1.6118
1.0205
-0.0708
-2.1924
-0.9485

N =
N(:,:,1) =

3
1
2
1
0
2
0
1
3
4
2
1
0
1
1
1
1
0
0
3
2
2
1
0
1
1
3
3
4
2
4
1
1
2
0
2
2
3
2
1
3
2
2
1
1
1
3
0
2
2
1
0
1
1
1
1
0
2
2
1
1
5
7
3
2
2
1
3
3
5
3
0
1
6
2
0
5
2
2
1
2
1
3
0
2
4
2
2
4
2
3
1
2
5
1
0
3
3
1
1

N(:,:,2) =

4
2
2
2
0
4
1
2
3
1
2
1
4
2
6
2
2
2
2
1
4
3
1
3
3
1
3
6
1
4
2
2
1
2
1
1
5
0
2
2
3
2
2
1
0
1
5
4
0
1
1
2
1
2
3
2
2
1
2
2
0
3
1
6
3
3
0
2
1
2
0
6
1
3
1
2
2
2
1
0
2
2
2
2
1
1
3
1
2
2
1
4
1
3
3
0
1
1
1
2

N(:,:,3) =

1
3
2
2
1
4
2
3
0
0
4
3
2
3
1
1
1
1
3
4
1
2
3
1
1
1
1
0
3
0
1
0
4
0
2
4
3
1
0
1
5
3
3
2
1
2
3
1
5
4
1
1
2
2
1
1
1
2
1
5
1
2
1
3
2
2
1
3
1
6
0
1
4
1
1
3
5
3
1
2
2
1
2
1
1
1
1
1
2
3
6
2
1
3
2
1
1
0
1
3

This example shows how to use simBySolution with a Merton model to perform a quasi-Monte Carlo simulation. Quasi-Monte Carlo simulation is a Monte Carlo simulation that uses quasi-random sequences instead pseudo random numbers.

Create a merton object.

AssetPrice = 80;
Return = 0.03;
Sigma = 0.16;
JumpMean = 0.02;
JumpVol = 0.08;
JumpFreq = 2;

Merton = merton(Return,Sigma,JumpFreq,JumpMean,JumpVol,'startstat',AssetPrice)
Merton =
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

Perform a quasi-Monte Carlo simulation by using simBySolution with the optional name-value arguments for 'MonteCarloMethod','QuasiSequence', and 'BrownianMotionMethod'.

[paths,time,z,n] = simBySolution(Merton, 10,'ntrials',4096,'montecarlomethod','quasi','QuasiSequence','sobol','BrownianMotionMethod','brownian-bridge');

## Input Arguments

collapse all

Merton model, specified as a merton object. You can create a merton object using merton.

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,N] = simBySolution(merton,NPeriods,'DeltaTimes',dt,'NNTrials',10)

Simulated NTrials (sample paths) of NPeriods observations each, specified as the comma-separated pair consisting of 'NNTrials' 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 an NPeriods-by-1 column vector.

DeltaTimes 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 DeltaTimes), specified as the comma-separated pair consisting of 'NSteps' and a positive scalar integer.

The simBySolution 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 simBySolution 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 to use antithetic sampling to generate the Gaussian random variates that drive the Brownian motion vector (Wiener processes), specified as the comma-separated pair consisting of 'Antithetic' and a scalar numeric or logical 1 (true) or 0 (false).

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

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

• Even NTrials (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), simBySolution ignores the value of Antithetic.

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 and N), simBySolution 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.

• If you specify an input noise process (see Z), simBySolution ignores the value of QuasiSequence.

Data Types: string | char

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.

Note

If an input noise process is specified using the Z input argument, BrownianMotionMethod is ignored.

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

Direct specification of the dependent random noise process for generating the Brownian motion vector (Wiener process) that drives the simulation, specified as the comma-separated pair consisting of 'Z' and a function or an (NPeriods * NSteps)-by-NBrowns-by-NNTrials three-dimensional array of dependent random variates.

The input argument Z allows you to directly specify the noise generation process. This process takes precedence over the Correlation parameter of the input merton object and the value of the Antithetic input flag.

Specifically, when Z is specified, Correlation is not explicitly used to generate the Gaussian variates that drive the Brownian motion. However, Correlation is still used in the expression that appears in the exponential term of the log[Xt] Euler scheme. Thus, you must specify Z as a correlated Gaussian noise process whose correlation structure is consistently captured by Correlation.

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

Dependent random counting process for generating the number of jumps, specified as the comma-separated pair consisting of 'N' and a function or an (NPeriodsNSteps) -by-NJumps-by-NNTrials three-dimensional array of dependent random variates. If you specify a function, N must return an NJumps-by-1 column vector, and you must call it with two inputs: a real-valued scalar observation time t followed by 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 numeric or logical 1 (true) or 0 (false).

If StorePaths is true (the default value) or is unspecified, simBySolution returns Paths as a three-dimensional time series array.

If StorePaths is false (logical 0), simBySolution returns Paths as an empty matrix.

Data Types: logical

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

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

simBySolution applies processing functions at the end of each observation period. These functions must accept the current observation time t and the current state vector Xt, and return a state vector that can be an adjustment to the input state.

The end-of-period Processes argument allows you to terminate a given trial early. At the end of each time step, simBySolution 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).

If you specify more than one processing function, simBySolution 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

collapse all

Simulated paths of correlated state variables, returned as an (NPeriods + 1)-by-NVars-by-NNTrials 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 StorePaths is set to false, simBySolution returns Paths as an empty matrix.

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.

Dependent random variates for generating the Brownian motion vector (Wiener processes) that drive the simulation, returned as a (NPeriods * NSteps)-by-NBrowns-by-NNTrials three-dimensional time-series array.

Dependent random variates for generating the jump counting process vector, returned as an (NPeriods ⨉ NSteps)-by-NJumps-by-NNTrials three-dimensional time-series array.

collapse all

### 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 that has the same expected value but a 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 other pairs, 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 NTrials.

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

The simBySolution function simulates the state vector Xt by an approximation of the closed-form solution of diagonal drift Merton jump diffusion models. Specifically, it applies a Euler approach to the transformed log[Xt] process (using Ito's formula). In general, this is not the exact solution to the Merton jump diffusion model because the probability distributions of the simulated and true state vectors are identical only for piecewise constant parameters.

This function simulates 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.

## 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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