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Finance

  

HestonProcess

  

create new Heston process

 

Calling Sequence

Parameters

Description

Examples

References

Compatibility

Calling Sequence

HestonProcess(x0, v0, mu, theta, kappa, sigma, rho)

Parameters

x0

-

algebraic expression; initial value of the state variable

v0

-

algebraic expression; initial value of the variance

mu

-

algebraic expression; risk-neutral drift

theta

-

algebraic expression; long-run mean of the variance process

kappa

-

algebraic expression; speed of mean reversion of the variance process

sigma

-

algebraic expression; volatility of the variance process

rho

-

algebraic expression; instantaneous correlation between the return process and the volatility process

Description

• 

The HestonProcess command creates a new stochastic process St governed by the following stochastic differential equation (SDE)

dSt=μStdt+VtStdWt1

dVt=κθVtdt+σVtdWt2

  

where

– 

μ is the drift parameter

– 

θ is the long-run mean

– 

κ is the speed of mean reversion

– 

σ is the volatility of the variance process

  

and

– 

Wt is the two-dimensional Wiener process with instantaneous correlation ρ.

• 

This is a stochastic volatility process that was introduced by Heston in A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options.

• 

The parameter X0 defines the initial value of the underlying stochastic process.

• 

The parameter mu is the drift parameter of state process.

• 

The volatility of this process evolves as a SquareRootDiffusion.

• 

The parameter kappa is the speed of mean-reversion of the variance process. The parameter theta is the long-term running mean of the variance process. The parameter sigma is the volatility of the variance process. In general, kappa, theta, and sigma can be any algebraic expressions. However, if the process is to be simulated, these parameters must be assigned numeric values.

• 

The parameter rho is the instantaneous correlation between the state process and the volatility process.

Examples

withFinance:

SHestonProcessS0,V0,μ,θ,κ,σ,ρ:

DriftSt

μ_Xt1κt_Xt2

(1)

DiffusionSt

_Xt2ρ_Xt1_Xt2_Xt1ρ2+1_Xt2σ0

(2)

You can now simulate the Heston process.

S0700

S0:=700

(3)

V00.1

V0:=0.1

(4)

κ1.0

κ:=1.0

(5)

θ0.1

θ:=0.1

(6)

μ0.05

μ:=0.05

(7)

σ0.1

σ:=0.1

(8)

ρ0.5

ρ:=0.5

(9)

SHestonProcessS0,V0,μ,θ,κ,σ,ρ:

ASamplePathSt,t=0..3,timesteps=100,replications=5

A:= 1..5 x 1..2 x 1..101 ArrayData Type: float8Storage: rectangularOrder: C_order

(10)

These are sample paths for the state variables.

PathPlotA,timegrid=0..3,1,thickness=3,color=red..blue,axes=BOXED,gridlines=true

And these are the corresponding sample paths for the volatility.

PathPlotA,timegrid=0..3,2,thickness=3,color=red..blue,axes=BOXED,gridlines=true

References

  

Brigo, D., Mercurio, F., Interest Rate Models: Theory and Practice. New York: Springer-Verlag, 2001.

  

Gatheral, J., The Volatility Surface: A Practitioner's Guide, (with foreword by Nassim Taleb), Wiley, 2006.

  

Glasserman, P., Monte Carlo Methods in Financial Engineering. New York: Springer-Verlag, 2004.

  

Heston, Steven L., A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options, The review of Financial Studies, Volume 6, Issue 2, 327-343, 1993.

  

Hull, J., Options, Futures, and Other Derivatives, 5th. edition. Upper Saddle River, New Jersey: Prentice Hall, 2003.

Compatibility

• 

The Finance[HestonProcess] command was introduced in Maple 15.

• 

For more information on Maple 15 changes, see Updates in Maple 15.

See Also

Finance[BlackScholesProcess]

Finance[BrownianMotion]

Finance[Diffusion]

Finance[Drift]

Finance[ExpectedValue]

Finance[GeometricBrownianMotion]

Finance[ItoProcess]

Finance[SamplePath]

Finance[SampleValues]

Finance[SquareRootDiffusion]

Finance[StochasticProcesses]

Finance[WienerProcess]

 


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