CIR process

The root diffusion process ( next to it is also in the German-speaking the English name square-root diffusion- use) is a stochastic process defined on a stochastic differential equation. Since this process class was studied by the Croatian- American mathematician William Feller for the first time in 1951, she has found numerous areas of application in various areas of financial mathematics.

Definition and parameters

Be real parameters and a Brownian motion given. A stochastic process is called root diffusion process with those parameters, if the stochastic differential equation

Solves. In most cases, is determined by an initial condition, so that from the above differential equation, a stochastic initial value problem.

One reason for the popularity of this process is surely the fact that the action of the individual parameters are easily interpretable (and hence well as " screws " can be used in the modeling ):

• Is the equilibrium level of the process ( engl. mean reversion level ). If so the drift term is positive and "pulls" the process up, otherwise it is negative and X tends downward. So drifts in anticipation always opposed.
• Are the strength of this regulatory function ( engl. mean reversion speed). For this is overridden and the course of X only by the diffusion term dependent. The larger, the more " rigid " X is bound to. In theory, would also be conceivable for negative values ​​, but then the process would have completely different properties: started above, he would face diverge, otherwise he would soon be negative - and thus fall out from their own domain.
• The parameter is the volatility of the process. It controls how strongly exposed to X the fluctuations of Brownian motion. For large values ​​of X will vary greatly over time, converges in X to exponentially.

Properties

Although above differential equation has no closed-form solution ( ie it is not possible to express the resulting process X as a function of t and W), nevertheless it can be said a lot about the characteristics of the process:

• For X is never negative, that it applies. Applies also the stability condition, then X is even almost certainly strictly positive. The reason is that for the diffusion term also tends to zero and hence the drift of the process again positive "pulls" upwards.
• Given a previous (u < t) value Xu, then Xt has a non- central chi-square distribution in which the non -centrality parameter of Xu depends.
• In the long term, the distribution of Xt converges independent of the starting point for a gamma distribution with mean and variance. This distribution is selected as the initial distribution, X has the same distribution at any time.
• The root diffusion process is a process Affiner

Application

Cox - Ingersoll- Ross model

In their seminal work, A theory of the term structure of interest rates, the American mathematician Cox, Ingersoll and Ross 1985 hit the root diffusion process the first time as a model for short-term interest rates. At the same time, they presented new approaches for the calculation of transition probabilities of the process. Your interest rate model was soon so that the root diffusion process soon nicknamed CIR process was one of the standard models on the market.

Heston model for stochastic volatility

In 1993, Stephen L. Heston gave the ' square-root diffusion ' new popularity when he developed a complex capital market model by a stochastic volatility, he expanded the model of Black and Scholes, who was still assumed to be constant in the latter. This was simulated by a root diffusion and was able for the first time are correlated on the Brownian motion with the market price: This made it possible, the natural phenomenon ( leverage ), after which falling prices increase the excitement ( volatility ) on the markets to capture mathematically. The Heston model is now considered the most important extension to the Black- Scholes and gave the root - diffusion finally a regular place in the textbooks on stochastic processes.

Stochastic Time difference

Another way to provide them with the help of the CIR process general Lévy processes ( a much broader class than that studied by Heston process class ) with stochastic volatility, based on an idea by Wolfgang Döblin (son of Alfred Döblin ) from the 30s: Doblin had then proposed to control the volatility of a process by the speed of time, in the process process will be controlled. Many decades later, it turned out that the root diffusion process is ideally suited to influence as an upstream "time machine" that speed. If, for example, such a Lévy process X and the above- defined process, we obtain through a process Y with stochastic time.

Simulation of the process

In practice, there are two ways to simulienen the CIR process, both of which have their advantages and disadvantages:

• One possibility is to utilize the known transition probability ( non-central chi-square distribution) and to draw at any given point to the next ( at any distance ) directly. This method provides an accurate ( in the sense of an unbiased ) simulation of the process, however, must be taken for each step to determine the chi-square distributed updates a normal and a gamma distributed random variable. The latter is, however, numerically -consuming to draw, what can appear to be unsuitable, for example, Monte Carlo simulations this technique.
• The second possibility - not exact, but is faster and suitable for finer grid - is the simulation by a simple Euler - Maruyama method ( in this way are the sample pictures were taken on this item ). Here one should note that the Euler update in every step with positive probability can be negative ( which is not the case with the real process ), which not only prepares numerical problems.

• Stochastic process