Stochastic Processes Overview

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Finance Package Commands For Stochastic Processes

Overview

The Financial Modeling package supports a wide range of stochastic processes used in Financial Engineering. This includes processes for modeling equity prices, mean-reverting processes, pure jump processes, jump diffusions as well as multi-variate Ito processes. In addition, the package provides tools for building more complicated processes from simple ones. Below is the list of supported processes:

BrownianMotion	-	uni- or multi-variate Brownian motion
CEVProcess	-	constant elasticity of variance (CEV) process
DeterministicProcess	-	deterministic time-dependent process
DynamicPortfolio	-	a portfolio of stochastic processes that can be dynamically rebalanced
MarkovChain	-	a finite state Markov chain
GammaProcess	-	gamma process
GaussMarkovProcess	-	Gaussian process for modeling short-term interest rate introduced by Hull and White
GeometricBrownianMotion	-	geometric Brownian motion
HestonProcess	-	stochastic volatility process introduced by Heston
ItoDiffusion	-	general uni- or multi-variate Ito process defined by drift and diffusion
MertonJumpDiffusion	-	jump-diffusion process introduced by Merton
RegimeSwitchingProcess	-	stochastic process with multiple regimes
OrnsteinUhlenbeckProcess	-	Ornstein-Uhlenbeck process
PoissonProcess	-	Poisson process including the doubly-stochastic Poisson (Cox) process
SquareRootDiffusion	-	square-root diffusion
SVJJProcess	-	stochastic volatility with jumps (SVJJ) process
WienerProcess	-	uni- and multi-variate Wiener process including subordinated Wiener process

Here are some related commands.

ExpectedValue	-	compute a Monte Carlo estimate of the specified expression
Drift	-	compute the drift component of an Ito process
Diffusion	-	compute the diffusion component of an Ito process
PathPlot	-	plot sample path(s) of the specified stochastic process
SamplePath	-	generate a sample path for the specified stochastic process
SampleValues	-	generate sample values for the specified expression involving stochastic variables
ValueAtRisk	-	estimate value-at-risk for the specified expression

Basic commands

>	$restart &colon; with (Finance) &colon;$

For the first example consider the standard Wiener process.

>	$W := WienerProcess ()$

$W :=_W$

(2.1)

The previous command created a new Maple variable representing the standard Wiener process. This is an ordinary Maple variable with additional attributes containing some information about the underlying stochastic process. This variable is typically accessed in the form $W (t)$ , where $t$ is the time of interest. For example, the following command can be used to generate some replications of the sample path for the underlying stochastic process.

>	$S := SamplePath (W (t), t = 0 .. 3, timesteps = 50, replications = 20)$

>	$PathPlot (S, timegrid = 0 .. 3, color = red .. blue, thickness = 3, axes = BOXED, tickmarks = [10, 10], gridlines)$

You can now generate a larger sample and use tools from the Statistics package to analyze the generated data.

>	$T := 1.0$

$T := 1.0$

(2.2)

>	$S := SamplePath (W (t), t = 0 .. T, timesteps = 100, replications = 10^{4})$

>	$A := S_{1 .. 10^{4}, 50}$

>	$Statistics :- Mean (A)$

$- 0.004119959338$

(2.3)

>	$Statistics :- StandardDeviation (S_{1 .. 10^{4}, 50}) \approx \sqrt{T \cdot \frac{50}{100}} &semi;$

$(0.6983974595) \approx (0.7071067812)$

(2.4)

>	$Statistics :- StandardDeviation (S_{1 .. 10^{4}, 75}) \approx \sqrt{T \cdot \frac{75}{100}} &semi;$

$(0.8626421722) \approx (0.8660254038)$

(2.5)

You can also simulate any expression involving our stochastic variable.

>	$ExpectedValue (\exp (W (3)), timesteps = 100, replications = 10^{4})$

$[value = 4.321931125, standarderror = 0.1820788275]$

(2.6)

>	$R := t \to {&ExponentialE;}^{\int_{0}^{t} 0.01 W (u) &DifferentialD; u} &colon;$

>	$ExpectedValue (R (3), timesteps = 100, replications = 10^{3}) &semi;$

$[value = 1.001642347, standarderror = 0.0009304064761]$

(2.7)

Simple expressions involving multiple stochastic parameters can be handled the same way.

>	$W_{1} ≔ WienerProcess () &colon;$

>	$W_{2} ≔ WienerProcess () &colon;$

>	$ExpectedValue (\max ({&ExponentialE;}^{W_{1} (3)}, {&ExponentialE;}^{W_{2} (3)}), timesteps = 1000, replications = 10^{4}) &semi;$

$[value = 8.322545370, standarderror = 0.3466860178]$

(2.8)

>	$ExpectedValue (\min ({&ExponentialE;}^{W_{1} (3)}, {&ExponentialE;}^{W_{2} (3)}), timesteps = 1000, replications = 10^{4}) &semi;$

$[value = 0.9821223967, standarderror = 0.02041880265]$

(2.9)

Alternatively, you can implement any kind of path function using Maple procedures. This function will be called on every replication of the sample path. In the case of a single stochastic factor the path function will be passed a one-dimensional array containing the sample path. In the case of multiple stochastic factors, the path function will be passed a two-dimensional array where each row contains a sample path for the corresponding stochastic factor.

Consider, for example, a path function that returns $W_{1} (3) &period;$ In the case when the path function is given explicitly, you must know the number of time steps in advance. You will use $100$ time steps in this example, which means that the sample path will contain $101$ points.

>	$PathFunction := A \to {&ExponentialE;}^{A_{101}} &colon;$

Now you can compute the expected value of this path function.

>	$ExpectedValue (PathFunction, W_{1}, 0 .. 3, timesteps = 100, replications = 10^{4})$

$[value = 4.682401575, standarderror = 0.2271049960]$

(2.10)

Compare this to:

>	$ExpectedValue ({&ExponentialE;}^{W_{1} (3)}, timesteps = 100, replications = 10^{4})$

$[value = 4.412244372, standarderror = 0.1617301713]$

(2.11)

Consider an example involving two stochastic factors (the same as above).

>	$PathFunction := A \to \max ({&ExponentialE;}^{A_{1, 1001}}, {&ExponentialE;}^{A_{2, 1001}}) &colon;$

>	$ExpectedValue (PathFunction, ⟨W_{1}, W_{2}⟩, 0 .. 3, timesteps = 1000, replications = 10^{4})$

$[value = 7.727181378, standarderror = 0.2169239333]$

(2.12)

Here is a procedure that computes the expected value of the absorbing Wiener process.

>	$PathFunction := proc (A) local i &semi; for i from 2 to 11 do if A [i] <= Float (0, 0) then return Float (0, 0) elif Float (10, - 1) <= A [i] then return Float (10, - 1) end if end do &semi; return A [11] end proc &colon;$

>	$ExpectedValue (PathFunction, W_{1}, 0 .. 3, timesteps = 10, replications = 10^{5})$

$[value = 0.2356212806, standarderror = 0.001339482953]$

(2.13)

Ito Processes

An Ito process is a stochastic process $X (t)$ governed by the stochastic differential equation (SDE)

$dX (t) = μ (X (t), t) \cdot dt + σ (X (t), t) \cdot dW (t),$

where $μ (X (t), t)$ is the drift parameter and $σ (X (t), t)$ is the volatility parameter. The coefficients of the above SDE can be extracted using the Drift and Diffusion commands.

>	$restart &semi; with (Finance) &colon;$

>	$X := GeometricBrownianMotion (100, a, 0.5, t) &colon;$

>	$Drift (X (t)) &semi;$

$a_X (t)$

(3.1)

>	$Diffusion (X (t)) &semi;$

$0.5_X (t)$

(3.2)

Here is a slightly more complicated example.

>	$Y := BrownianMotion (100, a, b, t) &colon;$

>	$Drift ({&ExponentialE;}^{Y (t)})$

$a {&ExponentialE;}^{_X0 (t)} + \frac{1}{2} b^{2} {&ExponentialE;}^{_X0 (t)}$

(3.3)

>	$Diffusion ({&ExponentialE;}^{Y (t)})$

$b {&ExponentialE;}^{_X0 (t)}$

(3.4)

>	$Drift ({({&ExponentialE;}^{Y (t)})}^{2})$

$2 a {({&ExponentialE;}^{_X0 (t)})}^{2} + 2 b^{2} {({&ExponentialE;}^{_X0 (t)})}^{2}$

(3.5)

>	$Drift (\exp (X (t) \cdot Y (t)))$

$a_X (t)_X0 (t) {&ExponentialE;}^{_X (t)_X0 (t)} + a_X (t) {&ExponentialE;}^{_X (t)_X0 (t)} + 0.1250000000 {_X (t)}^{2} {_X0 (t)}^{2} {&ExponentialE;}^{_X (t)_X0 (t)} + \frac{1}{2} b^{2} {_X (t)}^{2} {&ExponentialE;}^{_X (t)_X0 (t)}$

(3.6)

Here is an example involving a multi-dimensional process.

>	$Σ ≔ ⟨⟨1.0, 0.5⟩ \| ⟨0.5, 1.0⟩⟩ &semi;$

>	$W := WienerProcess (Sigma) &colon;$

>	$Drift (\exp (W (t) [1] + W (t) [2])) &semi;$

$2.250000000 {&ExponentialE;}^{{_W (t)}_{1} + {_W (t)}_{2}}$

(3.7)

Here is an Ito process defined explicitly.

>	$X ≔ ItoProcess (1, x \cdot t, 0.05 \cdot x \cdot t, x, t)$

$X :=_X1$

(3.8)

>	$Drift (X (t)) &semi;$

$t_X1 (t)$

(3.9)

>	$Diffusion (X (t)) &semi;$

$0.05 t_X1 (t)$

(3.10)

>	$A := SamplePath (X (t), t = 0 .. 3, timesteps = 100, replications = 10) &semi;$

>	$PathPlot (A, timegrid = 0 .. 3, tickmarks = ([5, 5]), axes = BOXED, gridlines = true, thickness = 2)$

	See Also
	Term Structures worksheet, Cash Flow Analysis worksheet, Day Counters worksheet, Lattice Methods worksheet, Calendars worksheet

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