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Finance[GaussMarkovProcess] - create new Gauss-Markov short-rate process
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Calling Sequence
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GaussMarkovProcess(![r[0]](/support/helpjp/helpview.aspx?si=8912/file01632/math10.png) , g, h, sigma, t, opts)
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Parameters
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![r[0]](/support/helpjp/helpview.aspx?si=8912/file01632/math33.png)
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algebraic expression; initial value
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g
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algebraic expression, procedure, or yield term structure; the adjusted mean
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h
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algebraic expression, procedure, or yield term structure; speed of mean reversion
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sigma
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algebraic expression, procedure, or yield term structure; the non-negative volatility
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t
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(optional) name; time variable
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opts
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(optional) equation(s) of the form option = value where option is scheme; specify options for the GaussMarkovProcess command
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Description
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The GaussMarkovProcess command creates a linear Gauss-Markov stochastic process  , which is governed by the stochastic differential equation (SDE)
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This process can be used to model short-term interest rates. This model - introduced by Hull and White (1990) - contains many popular term structure models as special cases.
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Vasicek model:
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 ,  and  are constant.
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Ho-Lee model:
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 and  is constant.
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Hull-White model:
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 and  are constant.
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The parameter ![r[0]](/support/helpjp/helpview.aspx?si=8912/file01632/math182.png) defines the initial value of the underlying stochastic process.
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The parameter g is the adjusted mean. It can be either an algebraic expression, a procedure, or a yield term structure. The parameters h and sigma are the volatility parameters. In general, g, h, and sigma can be any algebraic expressions. However, if the process is to be simulated, these parameters must be assigned numeric values.
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Options
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scheme = unbiased or Euler -- This option specifies which discretization scheme should be used for simulating the process. By default the standard Euler scheme is used. When scheme is set to unbiased the transition density will be used to simulate a value  given  .
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Compatibility
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The Finance[GaussMarkovProcess] command was introduced in Maple 15.
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Examples
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![T := [2.0, 5.0, 10.0, 30.0]](/support/helpjp/helpview.aspx?si=8912/file01632/math256.png)
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![T := [2.0, 5.0, 10.0, 30.0]](/support/helpjp/helpview.aspx?si=8912/file01632/math259.png)
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![R := [0.4976e-1, 0.4970e-1, 0.5064e-1, 0.5143e-1]](/support/helpjp/helpview.aspx?si=8912/file01632/math263.png)
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![R := [0.4976e-1, 0.4970e-1, 0.5064e-1, 0.5143e-1]](/support/helpjp/helpview.aspx?si=8912/file01632/math266.png)
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![g := PIECEWISE([0.4980000000e-1-0.2000000000e-4*t, t < 5.0], [0.4876000000e-1+0.1880000000e-3*t, t < 10.0], [0.5024500000e-1+0.3950000000e-4*t, otherwise])](/support/helpjp/helpview.aspx?si=8912/file01632/math273.png)
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![rates := [0.2e-1, 0.1e-1, 0.4e-1, 0.6e-1, 0.7e-1]](/support/helpjp/helpview.aspx?si=8912/file01632/math293.png)
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![times := [0., .5, 1.0, 1.5, 2.0]](/support/helpjp/helpview.aspx?si=8912/file01632/math297.png)
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![rates := [0.2e-1, 0.1e-1, 0.4e-1, 0.6e-1, 0.7e-1]](/support/helpjp/helpview.aspx?si=8912/file01632/math324.png)
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![times := [0., .5, 1.0, 1.5, 2.0]](/support/helpjp/helpview.aspx?si=8912/file01632/math328.png)
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![[value = 0.8445415108e-1, standarderror = 0.1785016918e-2]](/support/helpjp/helpview.aspx?si=8912/file01632/math358.png)
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![RTABLE(18446790986426614598, float[8], Array, rectangular, C_order, [], 1, 1 .. 10000)](/support/helpjp/helpview.aspx?si=8912/file01632/math365.png)
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Note that the overhead of computing the transition densities does not depend on the number of replications.
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See Also
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Finance[BlackScholesProcess], Finance[BrownianMotion], Finance[Diffusion], Finance[Drift], Finance[ExpectedValue], Finance[GeometricBrownianMotion], Finance[ItoProcess], Finance[OrnsteinUhlenbeckProcess], Finance[SamplePath], Finance[SampleValues], Finance[SquareRootDiffusion], Finance[StochasticProcesses], Finance[WienerProcess]
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References
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Brigo, D., Mercurio, F., Interest Rate Models: Theory and Practice. New York: Springer-Verlag, 2001.
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Glasserman, P., Monte Carlo Methods in Financial Engineering. New York: Springer-Verlag, 2004.
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Ho, T.S.Y, and Loo, S.-B., Term Structure Movements and Pricing Interest Rate Contingent Claims, Journal of Finance, 41 (1986), pp. 1011-29.
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Hull, J., Options, Futures, and Other Derivatives, 5th. edition. Upper Saddle River, New Jersey: Prentice Hall, 2003.
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Hull, J., and White, A., Pricing Interest Rate Derivative Securities, Review of Financial Studies, 3 (1990), pp. 573-92.
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Vasicek, O.A., An Equilibrium Characterization of the Term Structure, Journal of Financial Economics, 5 (1977), pp. 177-88.
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