9
votes
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 ...
- 8,219
9
votes
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). ...
- 2,197
8
votes
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 ...
- 16.3k
7
votes
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 ...
- 9,884
7
votes
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
6
votes
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
- 126
5
votes
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,
\...
- 21.3k
5
votes
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 ...
- 6,490
5
votes
Accepted
why $f(t,u) \neq E_t^Q [r(u)]$ when $r$ is random?
Your equations are flawed. Also there is no expectation if the process $\{r_s\}$ is deterministic.
The correct reasoning is, assuming $\{r_s\}$ is stochastic:
$$
f(t,u)=-\frac{d}{du}\ln P(t,u)=-\...
- 5,702
4
votes
Differences between main classes of interest pricing derivatives models
Just an addendum to the above answers and comments:
The main decision is whether to use single or multiple factor dynamics.
LMM models term forward rates. HJM models instantaneous forward rates.
The ...
- 5,173
3
votes
Accepted
Deriving interest rate term structure in a short rate model
This is indeed a standard result. You can convince yourself by noticing
The bank account grows from 1 at $t=\tau$ to $E\left[\exp(\int_\tau^T r(u)du)|\mathscr{F}_\tau\right]$ at time $T$
The price of ...
- 2,023
3
votes
Accepted
Short rate models
The main thing we want is the $P(t,T)$ function.
In the short rate model, we model the system as an instantaneous short rate variable which evolves stochastically. Different models assign different ...
- 3,719
3
votes
Bond dynamics in Ho Lee model
When taking the partial derivative $\frac{\partial}{\partial t}$ in a conditional expectation, not only the parameter $t$ within the expectation needs to be considered, the information set $\mathscr{F}...
- 21.3k
3
votes
Accepted
Vasicek model: joint simulation with discount factor
Although it's been a long time this question has been asked, I'd like to propose an answer in case someone was looking for the same thing.
First, I think there's a confusion between $P(t,T)$ and $DF(t,...
- 316
3
votes
What is the purpose of short rate models?
Long story short, the main reason of a short rate model is to provide an analytical solution for the zero coupon bond $P(t, T)$, given by the following expectation:
$$
P(t, T) = E_t^Q \left[ \exp \...
- 741
3
votes
Accepted
Why is logarithmic mean equal to the arithmetic expectation less one-half its variance?
So i'm kinda guessing what you really mean by the logarithmic mean - i'm guessing you mean the logarithmic average of returns - where you mean geometric average.
$$
\left( \prod_{i=0}^n a_i \right)^{\...
- 2,611
3
votes
Accepted
Variance of the Cox-Ingersoll-Ross short rate
The independence assumption is not needed. In fact, based on Ito's isometry and the Fubini theorem,
\begin{align*}
Var(r_t) &= E\left((r_t-E(r_t))^2 \right)\\
&=\sigma^2 e^{-2\beta t} E\left(\...
- 21.3k
3
votes
Accepted
Negative Libor Simulation
Yes, LIBOR rates can be simulated using short rate models. Or rather, Libor rates can be obtained from simulated short rate values.
Usually, you have formulas giving you the zero-coupon bond price as ...
- 2,280
3
votes
Accepted
How to determine components of Affine Term Structure for an Ohrnstein-Uhlenbeck process?
Let $\mathrm{d}r_t=\mu(t,r_t)\mathrm{d}t+\sigma(t,r_t)\mathrm{d}W_t$ be a model for the short rate under the risk-neutral measure $\mathbb{Q}$. Starting from the bond PDE
\begin{align*}
P_t + \mu(t,r) ...
- 16.3k
3
votes
Hull-White model: match between HJM framework and short model formulation
Note that
\begin{align*}
f(t, T) = f(0, T) + \int_0^t\alpha(u,T)du+\int_0^t\sigma e^{-a(T-u)}dW_u,
\end{align*}
where, based on this question,
\begin{align*}
f(0, T) = \int_0^T \theta(u) e^{-a(T-u)} ...
- 21.3k
3
votes
Current discount rate of Hull White One-Factor Monte Carlo Simulation
The average of simulated discount factors from the Hull-White model and market discount factor are the same in theory but very similar in the simulation due to numerical error.
I draw one figure ...
- 41
3
votes
What is gsr model for short term interest rate
GSR stands for Gaussian Short Rate model. It describes the short rate $r(t)$ dynamics under the Risk Neutral measure as:
$$
dr(t) = \kappa(t) \cdot (\theta(t) - r(t)) \cdot dt + \sigma(t) \cdot dW(t).
...
- 741
3
votes
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 ...
- 5,895
3
votes
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(...
- 1,043
3
votes
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 ...
- 7,140
3
votes
Why should future short rates tend towards the current term structure of interest rates?
It really depends for what purpose you are using the model. Let’s say you are using it for valuation of some instrument. If you want the fair market value, then a) is irrelevant and you would instead ...
- 17.9k
3
votes
what's the difference between instantaneous short rate and instantaneous forward rate?
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 $[T,T+dt]\,.$ ...
- 2,339
3
votes
Accepted
Difference HJM Framework versus Short rate model
Most principal component analyses (PCAs) on historical data of yield curves find that typically a yield curve
moves parallel
flips from normal to inverse (or vice versa)
twists (changes its ...
- 2,339
2
votes
Risk neutral measure of short rate model
For any given process for the short rate $\{r_t,, t >0\}$, the price at time $t$ of a zero-coupon bond with maturity $T$, where $t\le T$, is given by
\begin{align*}
P(t, T) = E\left(e^{-\int_t^T ...
- 21.3k
2
votes
Risk neutral measure of short rate model
The Vasicek and other short rate models are only "incomplete" until they are calibrated to market data. If rates actually followed Vasicek processes, it would be trivial to estimate the "Real world" ...
- 1,439
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