# Tag Info

### Two papers - two different solutions of the Ornstein-Uhlenbeck process

Note that the Ito integral of a deterministic integrand $f: \mathbb{R}_+ \rightarrow \mathbb{R}$ is normally distributed \begin{equation} \int_0^t f(u) \mathrm{d}W_u \sim \mathcal{N} \left( 0, \...

### What is the purpose of short rate models?

Short rate models were first used in the 1970s and 1980s to try to fit and explain the term structure of interest rates - they went beyond simple parametric shapes (polynomials and exponential forms). ...

### Why isn't the Vasicek model arbitrage-free?

Short rate models are broadly divided into equilibrium models and no-arbitrage models. The models from Vasicek, Dothan and Cox, Ingersoll and Ross are examples of equilibrium short rate models. The ...
Accepted

### Vasicek short rate: Risk-neutral measure into real-world measure

Vasnicek by itself does not specify what form the change of measure should be and how you should parameterise the market price of risk. A very natural parameterisation is affine in the factor, i.e., ...
Accepted

Accepted

### How to find the transition distribution functions of these two processes?

Here is a derivation for the Ornstein-Uhlenbeck process. Solution to the SDE $$dX_t = \theta(\mu-X_t) dt + \sigma dW_t$$ subject to the initial condition $X_0=x$ has the form X_t= \mu + (x - \mu)e^{-...

### Help evaluating covariance integral when deriving vasiceks model

that symbol means "the min of". So for example, if: $s<t$, then $s$ ^ t = s. If you look in any book for the Covariance of a BM, you will see that same symbol and how to work with it. Cheers.
Accepted

### Vasicek model problem

For starters, the short rate model you mention in equation (1) is Cox-Ingersoll-Ross while the bond price in equations (2)-(4) correspond to the Vacisek model. So there is a problem somewhere, I would ...
$\theta(t) - a(t) r(t)$ is the risk neutral drift. The Hull & White models posits the dynamics $dr(t) = (\theta(t) - a(t) r(t)) dt + \sigma dW(t)$ under the risk neutral measure $P$ and then ...
It should be time dependent and set to the spot forward rate $= -\frac{\partial}{\partial t} \ln(\text{discount}(t))$ when simulating in continuous time. When discretizing the simulation use the ...