Model OUV

Finance

Model OUV (Ornstein, Uhlenbeck, Vasicek) is used to calculate the options on rate.

Mathematics

Named after Leonard Solomon Ornstein and George Eugene Uhlenbeck and which is also known under the name of Mean-reverting process, the process $r$ is a Ornstein-Uhlenbeck process (OU), if its stochastic differential equation (EDS) is form:

$dr\_t\; =\; -\; \backslash \; theta\; (r\_t-\; \backslash \; driven)\; \backslash ,\; dt\; +\; \backslash \; sigma\; \backslash ,\; dW\_t,\; \backslash ,$

where θ , μ and σ are deterministic parameters and W _T indicate process Wiener.

Solution

This horsemanship is solved by variation of parameters. To apply Itō' S lemma to the function $f\; (r\_t,\; T)\; =\; r\_t\; e^\; \{\backslash \; theta\; T\}$ to obtain

$df\; (r\_t,\; T)\; =\; \backslash \; theta\; r\_t\; e^\; \{\backslash \; theta\; T\}\; \backslash ,\; dt\; +\; e^\; \{\backslash \; theta\; T\}\; \backslash ,\; dr\_t\; \backslash ,$

$=\; e^\; \{\backslash \; theta\; T\}\; \backslash \; theta\; \backslash \; driven\; \backslash ,\; dt\; +\; \backslash \; sigma\; e^\; \{\backslash \; theta\; T\}\; \backslash ,\; dW\_t.\; \backslash ,$

To integrate of 0 into T one obtains

$r\_t\; e^\; \{\backslash \; theta\; T\}\; =\; r\_0\; +\; \backslash \; int\_0^t\; e^\; \{\backslash \; theta\; S\}\; \backslash \; theta\; \backslash \; driven\; \backslash ,\; ds\; +\; \backslash \; int\_0^t\; \backslash \; sigma\; e^\; \{\backslash \; theta\; S\}\; \backslash ,\; dW\_s\; \backslash ,$

of what we see

$r\_t\; =\; r\_0\; e^\; \{-\; \backslash \; theta\; T\}\; +\; \backslash \; driven\; (1-e^\; \{-\; \backslash \; theta\; T\})\; +\; \backslash \; int\_0^t\; \backslash \; sigma\; e^\; \{\backslash \; theta\; (St)\}\backslash ,\; dW\_s.\; \backslash ,$

Thus, the first moment (mathematical) is given by (while assuming that $r\_0$ is a constant),

$E\; (r\_t)\; =\; r\_0\; e^\; \{-\; \backslash \; theta\; T\}\; +\; \backslash \; driven\; (1-e^\; \{-\; \backslash \; theta\; T\}).$

$s\; \backslash \; wedge\; T\; =\; \backslash \; min\; (S,\; T)$ One can use Itō isometry to calculate covariance

$\backslash \; operatorname\; \{cov\}\; (r\_s,\; r\_t)\; =\; E\; -\; E\; [r\_s)\; (r\_t\; -\; E)]$

$=\; E\; \backslash \; sigma\; e^\; \{\backslash \; theta\; (custom)\}\backslash ,\; dW\_u\; \backslash \; int\_0^t\; \backslash \; sigma\; e^\; \{\backslash \; theta\; (v-t)\}\backslash ,\; dW\_v$

$=\; \backslash \; sigma^2e^\; \{-\; \backslash \; theta\; (s+t)\}E\; e^\; \{\backslash \; theta\; U\}\; \backslash ,\; dW\_u\; \backslash \; int\_0^t\; e^\; \{\backslash \; theta\; v\}\; \backslash ,\; dW\_v$

$=\; \backslash \; frac\; \{\backslash \; sigma^2\}\; \{2\; \backslash \; theta\}\; \backslash ,\; e^\; \{-\; \backslash \; theta\; (s+t)\}(e^\; \{2\; \backslash \; theta\; (S\; \backslash \; wedge\; T)\}-\; 1).\; \backslash ,$

It is also possible (and often convenient) to represent $r\_t$ (without reserve) as a measurement transformed of time the Wiener process:

$r\_t=\; \backslash \; mu+\; \{\backslash \; sigma\; \backslash \; over\; \backslash \; sqrt\; \{2\; \backslash \; theta\}\}\; W\; (e^\; \{2\; \backslash \; theta\; T\})\; e^\; \{-\; \backslash \; theta\; T\}$

or condition (given $r\_0$) like

$r\_t=r\_0\; e^\; \{-\; \backslash \; theta\; T\}\; +\; \backslash \; driven\; (1-e^\; \{-\; \backslash \; theta\; T\})\; +$

{\ sigma \ over \ sqrt {2 \ theta}} W (e^ {2 \ theta T} - 1) e^ {- \ theta T}.

The process Ornstein-Uhlenbeck (an example of the Gaussian process with variance dependant) admits a stationary process with probable distribution, in opposition to the process Wiener.

The Temps integral of the process can be used to generate noise with has 1 F to be able spectrum.

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