# Tag Info

### 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). ...
Accepted

### How to get set the theta function in the Hull-White model to replicate the current yield curve

Concerning your first question, this depends on what curve, currency, etc. you are interested in. The general method for constructing yield curves is called bootstrapping which allows you to derive ...

### 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

### Differences between main classes of interest pricing derivatives models

I am not sure if you can classify it like that. Mind you, I never wrote a book. I'll write what I know below and you can decide if the classification makes sense or not. 1 ) STIR: as the term ...
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 (...
Accepted

### QuantLib Gsr model

the model is described in Andersen, Piterbarg: Interest Rate Modeling. The formulas that are acutally implemented are derived here https://ssrn.com/abstract=2246013 Best Peter

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

I might get down-voted for this, but in my opinion, short-rate models are not very useful for any practical pricing problems in today's finance. Even for simple vanilla rate derivatives (i.e. Caplet ...

### Ho Lee model in Baxter&Rennie

Here we provide another answer using Ito's calculus. It appears involved, but it also has its own interest. Given the short rate dynamics \begin{align*} dr_t = \nu(r_t, t) dt + \rho(r_t, t) dW_t, \...
Accepted

Accepted

Accepted

### Variance of the Cox-Ingersoll-Ross short rate

The independence assumption is not needed. In fact, based on Ito's isometry and the Fubini theorem, \begin{align*} Var(r_t) &= E\left((r_t-E(r_t))^2 \right)\\ &=\sigma^2 e^{-2\beta t} E\left(\...

### Extensions of CIR

You are right. In the CIR++, $\alpha$ parameter is absorbed into $\phi$. With the CIR++, $\phi(t)$ will allow you to have to have negative rates. You will calibrate your $\phi$ to fit the discount ...
Accepted

Accepted

### Short rate models

The main thing we want is the $P(t,T)$ function. In the short rate model, we model the system as an instantaneous short rate variable which evolves stochastically. Different models assign different ...
Accepted

### Deriving interest rate term structure in a short rate model

This is indeed a standard result. You can convince yourself by noticing The bank account grows from 1 at $t=\tau$ to $E\left[\exp(\int_\tau^T r(u)du)|\mathscr{F}_\tau\right]$ at time $T$ The price of ...
Accepted

### Negative Libor Simulation

Yes, LIBOR rates can be simulated using short rate models. Or rather, Libor rates can be obtained from simulated short rate values. Usually, you have formulas giving you the zero-coupon bond price as ...
Accepted

### How to determine components of Affine Term Structure for an Ohrnstein-Uhlenbeck process?

Let $\mathrm{d}r_t=\mu(t,r_t)\mathrm{d}t+\sigma(t,r_t)\mathrm{d}W_t$ be a model for the short rate under the risk-neutral measure $\mathbb{Q}$. Starting from the bond PDE \begin{align*} P_t + \mu(t,r) ...

### Hull-White model: match between HJM framework and short model formulation

Note that \begin{align*} f(t, T) = f(0, T) + \int_0^t\alpha(u,T)du+\int_0^t\sigma e^{-a(T-u)}dW_u, \end{align*} where, based on this question, \begin{align*} f(0, T) = \int_0^T \theta(u) e^{-a(T-u)} ...

### Current discount rate of Hull White One-Factor Monte Carlo Simulation

The average of simulated discount factors from the Hull-White model and market discount factor are the same in theory but very similar in the simulation due to numerical error. I draw one figure ...

### What is gsr model for short term interest rate

GSR stands for Gaussian Short Rate model. It describes the short rate $r(t)$ dynamics under the Risk Neutral measure as:  dr(t) = \kappa(t) \cdot (\theta(t) - r(t)) \cdot dt + \sigma(t) \cdot dW(t). ...
Accepted

### QuantLib - Calibrating Hull White one-factor on negative interest rates

When building a SwaptionHelper, you have to tell QuantLib what kind of volatility you are inputting. There are three options: Black Vol, Shifted Black Vol and Normal Vol. Since you don't have black ...
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(...
Accepted

### Calibration of Heston model with stochastic short rate

If we take your model literally (with the correction that I suggested as a comment), then there exists no (semi-)closed form, IMHO, that you can use for asset pricing. What you could do is then to ...
In more standard notation the instantaneous forward rate is written as $f(t,T)$, that is, the continuously compounded interest rate seen at $t$ for the infinitesimal interest period $[T,T+dt]\,.$ ...