Day 13 : Vasicek Model (Ornstein-Uhlenbeck continuation)#

TheĀ Vasicek ModelĀ is one of the earliest and most influential mathematical frameworks for modeling interest rates. Developed by Czech mathematician Oldřich VaÅ”Ć­Äek in 1977, the model is a cornerstone of financial mathematics, especially in the valuation of bonds and interest rate derivatives.

The Vasicek Model is a one-factor short-rate model as it describes interest rate movements as driven by only one source of market risk whose dynamics follow anĀ Ornstein-Uhlenbeck process, known for its mean-reverting properties, which ensure that interest rates do not drift infinitely upward or downward.

Definition#

Under the Vasicek model, the dynamics of theĀ instantaneous interest rateĀ are defined by the following theĀ stochastic differential equation:

\[dX\_t = \theta(\mu - X\_t) dt + \sigma dW\_t, \qquad \geq 0,\]

with initial condition \(X\_0 =x\_0\in\mathbb{R}\), and where \(W\) is a standard Brownian motion, and the three parameters are constants:

- \(\theta>0\) : speed or mean reversion coefficient

- \(\mu \in \mathbb{R}\) : long term mean

- \(\sigma>0\) : volatility

In order to find the solution to this SDE, let us set the function \(f(t,x) = x e^{\theta t}\). Then, Ito’s formula implies

\[X\_te^{\theta t} = x\_0 +\int\_0^t X\_s \theta e^{\theta s}ds + \int\_0^t e^{\theta s}dX\_s\]
\[ = x\_0 + \int\_0^t \left\[ \theta X\_s e^{\theta s} +\theta e^{\theta s}(\mu - X\_s)\right\] ds + \int\_0^t e^{\theta s}\sigma dW\_s\]
\[ = x\_0 + \int\_0^t \left\[ \theta e^{\theta s}\mu\right\] ds + \int\_0^t e^{\theta s}\sigma dW\_s\]
\[ = x\_0 + \mu(e^{\theta t} - 1) + \int\_0^t e^{\theta s}\sigma dW\_s.\]

Thus

\[X\_t = x\_0e^{-\theta t} + \mu(1- e^{-\theta t}) + \sigma \int\_0^t e^{-\theta (t-s)}dW\_s., \]

This expression implies that for each \(t>0\), the random variable \(X\_t\) follows a normal distribution –since it can be expressed as the sum of a deterministic part and the integral of a deterministic function with respect to the Brownian motion.

šŸ”” Random Facts šŸ””#

  • Vasicek introduced the model in his 1977 paperĀ ā€œAn Equilibrium Characterization of the Term Structureā€ published in theĀ Journal of Financial Economics. His work was motivated by the need for a rigorous framework to explain the dynamics of interest rates and the term structure of bond yields.

  • Interest rates in the Vasicek Model naturally drift back to the long-term meanĀ \(\mu\)Ā over time, at a speed governed by the parameterĀ \(\theta\). This aligns with economic observations where extreme interest rate levels are unsustainable and mean reversion is observed.

  • The Vasicek Model is anĀ affine model, meaning bond prices and yields can be expressed as exponential-affine functions of time and the short rate. This property facilitates analytical pricing of zero-coupon bonds.

  • The model has closed-form solutions for bond prices, making it computationally efficient and widely adopted in practice.

More to Read šŸ“š#

P.s. If you are curious about probability distributions visit the Advent Calendar 2023 ✨