# 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 \int_0^t f(u) \mathrm{d}W_u \sim \mathcal{N} \left( 0, \...
• 6,044

### 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). ...
• 2,167

### 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 ...
• 16k
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., ...
• 1,078
Accepted

### Problem with pricing a call option using the Monte Carlo Vasicek model

To make sure that I understand the problem: you are trying to price a call option expiring at time 0.5, which will exercise into a unit notional zero-coupon bond with a maturity of 1.0 at a strike (...
• 845
Accepted

• 741
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### Difference between the Basel IRB and the Vasicek formula

The paper continues "The quantity p(Y) provides the loan default probability under the given scenario." But the default probability is 0.001, not 0.999 as in the IRB version. So G(0.999) = -G(1 - 0....
• 66

### How to price a stock under Q and stochastic interest rates?

The derivation in Appendix A of the paper Valuation of Equity-Indexed Annuities under Stochastic Interest Rates that you mentioned is Wrong: the Girsanov transformation is applied to an $n$-...
• 21.1k
Accepted

### LIBOR rates from Vasicek/Hull-White model?

In practice, you can calibrate to either 1 month libor or 3 month libor, but not both. That's because there's a basis swap between 1 month libor and 3 month libor that can't be explained by your ...
• 17.2k
Accepted

### Affine Structure Resolution for the Vasicek model

We begin with the equation $1+B_t(t,T)-kB(t,T) = 0 \quad(1)$ \begin{align} (1) & \iff e^{-kt}+e^{-kt}B_t(t,T)+(-k)e^{-kt}B(t,T) = 0 \\ & \iff e^{-kt}+ \frac{\partial}{\partial t}\left(e^{-kt}B(...
• 1,008
Accepted

### Why can a two-factor interest rate model not be used to value a coupon bearing bond as the sum of options on ZCBs

On a conceptual level an option on a coupon bonds is an option on a sum of the coupons (and principal), and we are comparing it to the sum of the options on coupons. In a one-factor model all coupons/...
• 940

### Change of numeraire to the forward measure in the Vasicek model

Let's assume you are working with 1-dimensional Brownian motion, the instantaneous correlation matrix $\rho$ drops to 1. $C$ and $C'$ both are 1. Now, referring to Proposition 2.3.1, in particular ...
• 31
Accepted

• 5,080
Accepted

### Hull-White Extension of Vasicek Model

$\theta(t) - a(t) r(t)$ 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 ...
• 5,672

### Term structure used in Geometric Brownian Motions under Risk Neutral Measure?

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 ...
• 5,672
### If short rates $r(t)$ do not determine the bond prices $P(t, T)$, then what is the basis for short rate models?
Let $r(s)$ be the process of a short rate. Then, by risk neutral pricing, $$P(t,T) = \mathbb{E}^\mathbb{Q}\left[ \exp\left( -\int_t^T r(s)\mathrm{d}s\right) \Bigg| \mathcal{F}_t\right].$$ Thus, the ...