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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

Xt=P(t,Xt)

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

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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.

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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 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

dXt=B(t,Xt)Xtdt+D(t,Xt)V(t,xt)dWt+Y(t,Xt,Nt)XtdNt

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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