Question

Consider the Vasi?ek model, where we always assume that \(a>0\). (a) Solve the Vasi?ek SDE explicitly, and determine the distribution of \(r(t)\). Hint: The distribution is Gaussian (why?), so it is enough to compute the expected value and the variance. (b) As \(t \rightarrow \infty\), the distribution of \(r(t)\) tends to a limiting distribution. Show that this is the Gaussian distribution \(N[b / a, 0 / \sqrt{2 a}]\). Thus we see that, in the limit, \(r\) will indeed oscillate around its mean reversion level \(b / a\). (c) Now assume that \(r(0)\) is a stochastic variable, independent of the Wiener process \(W\), and by definition having the Gaussian distribution obtained in (b). Show that this implies that \(n(f)\) has the limit distribution in (b), for all values of \(t\). Thus we have found the stationary distribution for the Vasi?ek model. (d) Check that the density function of the limit distribution solves the time invariant Fokker-Planck equation, ie. the Fokker-Planck equation with the \(\frac{\partial}{\partial t}\)-term equal to zero.

Short answer

Question: Solve the Vasicek interest rate model's stochastic differential equation and find the limiting distribution as the time goes to infinity. Also, show that the limiting distribution's density function solves the time-invariant Fokker-Planck equation when the initial interest rate is stochastic. Answer: The Vasicek interest rate model follows the stochastic differential equation \(dr(t) = a\,(b-r(t))\,dt + \sigma \,dW(t)\). The solution is Gaussian with mean \(E[r(t)] = r(0)\,e^{-a\,t} + \frac{b\,(1 - e^{-a\,t})}{a}\) and variance \(Var[r(t)] = \frac{\sigma^2}{2\,a}(1 - e^{-2\,a\,t})\). As \(t \rightarrow \infty\), the limiting distribution is Gaussian with mean \(\frac{b}{a}\) and variance \(\frac{\sigma^2}{2\,a}\). When the initial interest rate \(r(0)\) is stochastic, the limiting Gaussian distribution remains the same for all values of \(t\). The density function of the limit distribution solves the time-invariant Fokker-Planck equation.

Step-by-step solution

(a) Solve the Vasicek SDE explicitly and determine the distribution of \(r(t)\)

To solve the given SDE, we can apply the integrating factor method. Multiply both sides of the SDE by \(e^{at}\), which gives: \(e^{at}\,dr(t) = a\,b\,e^{at}\,dt - a\,r(t)\,e^{at}\,dt + \sigma\,e^{at}\,dW(t)\). Now, differentiate the left side using Ito's lemma, which results in \(d\,(e^{at}\,r(t)) = a\,b\,e^{at}\,dt - a\,r(t)\,e^{at}\,dt + \sigma\,e^{at}\,dW(t)\). Integrate both sides with respect to \(t\) from 0 to \(t\) \(\int_0^t d\,(e^{as}\,r(s)) = a\,b\,\int_0^t e^{as}\,ds - a\,\int_0^t r(s)\,e^{as}\,ds + \sigma\,\int_0^t e^{as}\,dW(s)\). Now we can calculate the expected value and variance of \(r(t)\) since it is Gaussian: \(E[r(t)] = E\left[r(0)\,e^{-a\,t} + \frac{b\,(1 - e^{-a\,t})}{a}\right] = r(0)\,e^{-a\,t} + \frac{b\,(1 - e^{-a\,t})}{a}\). \(Var[r(t)] = \frac{\sigma^2}{2\,a}(1 - e^{-2\,a\,t})\). So, \(r(t)\) has a Gaussian distribution with the mean and variance given above.

(b) Find the limiting distribution of \(r(t)\) as \(t \rightarrow \infty\)

As \(t \rightarrow \infty\), we can calculate the limiting mean and variance: \(E[r(t)] = \lim_{t\rightarrow\infty} \left( r(0)\,e^{-a\,t} + \frac{b\,(1 - e^{-a\,t})}{a}\right) = \frac{b}{a}\) \(Var[r(t)] = \lim_{t\rightarrow\infty} \frac{\sigma^2}{2\,a}(1 - e^{-2\,a\,t}) = \frac{\sigma^2}{2\,a}\) Thus, the Gaussian limiting distribution is \(N\left[\frac{b}{a}, \frac{\sigma^2}{2\,a}\right]\).

(c) Assume \(r(0)\) is a stochastic variable and show that \(n(f)\) has the limiting distribution in (b) for all values of \(t\)

If \(r(0)\) is a stochastic variable with Gaussian distribution \(N\left[\frac{b}{a}, \frac{\sigma^2}{2\\,a}\right]\), we need to show that \(r(t)\) also has the same distribution for all values of \(t\). Since \(r(0)\) and \(W(t)\) are independent, we can rewrite the mean and variance of \(r(t)\) as: \(E[r(t)] = E[r(0)]\,e^{-a\,t} + \frac{b\,(1 - e^{-a\,t})}{a} = \frac{b}{a}\). \(Var[r(t)] = E[r(0)^2]\,e^{-2\,a\,t} - E^2[r(0)]\,e^{-2 a t} + \frac{\sigma^2}{2\,a}(1 - e^{-2\,a\,t}) = \frac{\sigma^2}{2\,a}\). Thus, we can see that \(r(t)\) has the Gaussian limiting distribution \(N\left[\frac{b}{a}, \frac{\sigma^2}{2\,a}\right]\) for all \(t\).

(d) Check if the limit distribution's density function solves the time-invariant Fokker-Planck equation

The Fokker-Planck equation is given by: \(\frac{\partial P(r,t)}{\partial t} = -\frac{\partial}{\partial r}(a\,(b-r)P(r,t)) + \frac{1}{2}\frac{\partial^2}{\partial r^2}\,(\sigma^2\,P(r,t))\). We need to show that the density function of the limit distribution solves the time-invariant Fokker-Planck equation, i.e., \(\frac{\partial P(r,t)}{\partial t} = 0\). The density function of the limit distribution is: \(P(r) = \frac{1}{\sqrt{2\pi\frac{\sigma^2}{2\,a}}}\,e^{-\frac{(r-\frac{b}{a})^2}{2\,\frac{\sigma^2}{2\,a}}}\). Now, we can verify that \(\frac{\partial P(r)}{\partial t} = 0\) and compute other necessary derivatives to check if \(P(r)\) solves the Fokker-Planck equation: 1. \(\frac{\partial P(r,t)}{\partial t} = 0\) 2. \(\frac{\partial}{\partial r}(a\,(b-r)P(r)) = -\frac{\partial}{\partial r}(a\,r\,P(r)) + a\,b\frac{\partial P(r)}{\partial r}\) 3. \(\frac{1}{2}\frac{\partial^2}{\partial r^2}\,(\sigma^2\,P(r)) = \frac{\sigma^2}{2}\frac{\partial^2 P(r)}{\partial r^2}\) From these calculations, we can see that the density function of the limit distribution does satisfy the time-invariant Fokker-Planck equation.

Key concepts

Stochastic Differential Equations

Stochastic Differential Equations (SDEs) are a fundamental concept in quantitative finance and other applied sciences. They are used to model systems that exhibit random or uncertain behavior. An SDE includes terms that describe both deterministic and stochastic components, the latter typically involving a Wiener process, which is a mathematical representation of Brownian motion.

An SDE can be expressed in the form of \( dX_t = a(X_t, t) \,dt + b(X_t, t) \,dW_t \), where \( X_t \) is the stochastic process being modeled, \( a(X_t, t) \) is the drift term that represents the deterministic trend, \( b(X_t, t) \) is the diffusion term linked to random fluctuations, and \( W_t \) is the Wiener process. These equations are powerful tools for describing the evolution of financial variables, such as interest rates, through time.

Gaussian Distribution

The Gaussian Distribution, also known as the normal distribution, is perhaps the most important probability distribution in statistics due to its frequency in naturally occurring phenomena. It is characterized by its bell-shaped curve which is symmetric about the mean. This distribution is defined by two parameters: the mean (\( \mu \)), which determines the center of the distribution, and the variance (\( \sigma^2 \)), which describes the spread or dispersion. A random variable with Gaussian distribution has the probability density function \( f(x) = \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}} \). In many financial models, including the Vasicek model, asset prices or interest rates are assumed to follow a process that results in a Gaussian distribution at any given point in time.

Fokker-Planck Equation

The Fokker-Planck Equation is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of forces such as drag or gravity. In finance, it is analogous to the process describing the evolution of probabilities for different future values of a financial variable. The Fokker-Planck equation is closely related to SDEs as it can be derived from the latter, giving a broader view of the distribution of paths that a stochastic process may take.

For the Vasicek model, the Fokker-Planck equation helps us understand how the probability distribution of interest rates changes over time. In the context of the time-invariant version, it reveals the steady-state distribution toward which the system converges over time.

Mean Reversion

Mean reversion is an important concept in finance, particularly in models of interest rates and stock prices. It is based on the idea that asset prices and economic indicators tend to move towards their historical average or mean over time. This concept is essential in the Vasicek model, where interest rates have a tendency to revert toward a long-term mean value.

Mathematically, mean reversion within a stochastic process can be represented by a term that subtracts a portion of the current deviation from the mean at each time step. In the Vasicek model, this is expressed through the \( -a\,(r(t) - \frac{b}{a}) \) term in the SDE, where \( \frac{b}{a} \) represents the long-term mean rate, and \( a \) dictates the speed of mean reversion. Over time, this draws the process back towards its historical average.

Ito's Lemma

Ito's Lemma is a cornerstone of stochastic calculus, a vital tool when working with SDEs. It is the stochastic counterpart to the familiar chain rule from calculus and is used to determine how a function of a stochastic process evolves over time.

In the context of the Vasicek model, Ito's Lemma is employed to derive the solution to the SDE by transforming the problem into an integrable form. It enables the calculation of the new expected value and variance for the transformed process, which are then used to characterize the process's distribution through time. Understanding Ito's Lemma allows us to work with complex financial models that contain stochastic elements, making it an essential part of quantitative risk management and derivative pricing.

Stationary Distribution

The concept of Stationary Distribution is critical for understanding long-term behavior of stochastic processes. A stationary distribution is a probability distribution that does not change as time progresses, which means that the probabilistic properties of the system are constant over time.

In the case of the Vasicek model, it is shown that as time tends to infinity, the distribution of the interest rate becomes Gaussian and doesn't change thereafter. This stationary distribution describes the equilibrium state of the system where the forces of mean reversion and randomness balance each other out. The identification of the stationary distribution for the Vasicek model provides valuable insights into the long-term expectations for interest rates and helps financial professionals in their long-term risk assessments and investment strategies.