simBySolution
Simulate approximate solution of diagonal-drift Merton
jump
diffusion process
Since R2020a
Syntax
Description
[
simulates Paths
,Times
,Z
,N
] = simBySolution(MDL
,NPeriods
)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.
[
specifies options using one or more name-value pair arguments in addition to the
input arguments in the previous syntax. Paths
,Times
,Z
,N
] = simBySolution(___,Name,Value
)
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
Use simBySolution with merton
Object
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
Quasi-Monte Carlo Simulation Using Merton Model
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
MDL
— Merton model
merton
object
Merton model, specified as a merton
object. You can
create a merton
object using merton
.
Data Types: object
NPeriods
— Number of simulation periods
positive 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,Z,N] =
simBySolution(merton,NPeriods,'DeltaTime',dt,'NNTrials',10)
NNTrials
— Simulated NTrials (sample paths)
1
(single path of correlated state
variables) (default) | positive integer
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
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 an 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 within each time increment
1
(indicating no intermediate
evaluation) (default) | positive integer
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 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
Antithetic
— Flag to use antithetic sampling to generate the Gaussian random variates
false
(no antithetic
sampling) (default) | logical with values true
or
false
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
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"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
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.If you specify an input noise process (see
Z
),simBySolution
ignores the value ofQuasiSequence
.
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.
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
for the "brownian-bridge"
construction option and the
fastest convergence for the "principal-components"
construction option.
Data Types: string
| char
Z
— Direct specification of the dependent random noise process for generating Brownian motion vector
generates correlated Gaussian variates based on the
Correlation
member of the SDE
object (default) | function | three-dimensional array of dependent random variates
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
N
— Dependent random counting process for generating number of jumps
random numbers from Poisson distribution with
merton
object parameter
JumpFreq
(default) | three-dimensional array | 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
(NPeriods
⨉ NSteps
)
-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
StorePaths
— Flag that indicates how Paths
is stored and returned
true
(default) | logical with values true
or
false
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
istrue
(the default value) or is unspecified,simBySolution
returnsPaths
as a three-dimensional time series array.If
StorePaths
isfalse
(logical0
),simBySolution
returnsPaths
as an empty matrix.
Data Types: logical
Processes
— Sequence of end-of-period processes or state vector adjustments of the form
simBySolution
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
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
Paths
— Simulated paths of correlated state variables
array
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.
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
.
Z
— Dependent random variates for generating Brownian motion vector
array
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.
N
— Dependent random variates for generating the jump counting process vector
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.
More About
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
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 anNVars
-by-NVars
diagonal matrix in which each element along the main diagonal is the corresponding element of the state vector.V(t,Xt)
is anNVars
-by-NVars
matrix of instantaneous volatility rates.dWt is an
NBrowns
-by-1
Brownian motion vector.Y(t,Xt,Nt)
is anNVars
-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, Vol. 9, No. 2 ( Apr. 1996): 385–426.
[2] Aït-Sahalia, Yacine. “Transition Densities for Interest Rate and Other Nonlinear Diffusions.” The Journal of Finance, Vol. 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 R2020aR2022b: 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
.
See Also
Topics
- Simulating Equity Prices
- Simulating Interest Rates
- Stratified Sampling
- Price American Basket Options Using Standard Monte Carlo and Quasi-Monte Carlo Simulation
- Base SDE Models
- Drift and Diffusion Models
- Linear Drift Models
- Parametric Models
- SDEs
- SDE Models
- SDE Class Hierarchy
- Quasi-Monte Carlo Simulation
- Performance Considerations
MATLAB 명령
다음 MATLAB 명령에 해당하는 링크를 클릭했습니다.
명령을 실행하려면 MATLAB 명령 창에 입력하십시오. 웹 브라우저는 MATLAB 명령을 지원하지 않습니다.
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