11 votes
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Ho and lee derivation for short rates model

For any $s \geq t$, note that \begin{align*} r_s = r_t + \sigma\int_t^s dW_u + \int_t^s \theta_u du. \end{align*} Then, \begin{align*} \int_t^T r_s ds &= (T-t)r_t + \sigma\int_t^T\int_t^s dW_u ds +...
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9 votes
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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 ...
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8 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). ...
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  • 2,059
7 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 ...
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  • 13.8k
6 votes
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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
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6 votes
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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 ...
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  • 4,208
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, \...
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5 votes
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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)=-\...
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4 votes
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Forward rates formulae

Note that $\frac{F(0,s,T)}{F(0,t,T)} = \frac{T-t}{T-s}\frac{B(0,s)-B(0,T)}{B(0,t)-B(0,T)}$ and $\frac{F(s,s,T)}{F(s,t,T)} = \frac{T-t}{T-s}\frac{B(s,s)-B(s,T)}{B(s,t)-B(s,T)}$. Multiplying the ...
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4 votes

Basic LIBOR curve question

Libor rates include credit risk. It is riskier to make a 6m loan than two 3m loan. So the 6M Libor curve is not the same as the 3M one. Their difference is the basis spread. When using a short rate ...
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  • 3,826
4 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 ...
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  • 4,961
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 ...
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  • 4,963
3 votes
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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)^{\...
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  • 2,406
3 votes

Basic LIBOR curve question

You wrote Given this, what does the value of 1M LIBOR curve at 1Y point represent? It is a real number X such that: The following deal can be agreed today in the swap market: You will pay me the ...
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  • 9,077
3 votes
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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(\...
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3 votes
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CIR model: is the short rate really non-central $\chi^2$ distributed?

To answer this I sum up a paragraph of "Interest rate models - An Introduction" by A.Cairns: For $i=1,\ldots,d$ consider the OU-processes $$ dX^i_t = -\frac 12 \alpha X^i_t dt + \sqrt{\alpha} dW^i_t. $...
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3 votes
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CallableFloatingRateBond in QuantLib: just a matter of multiple inheritance?

No, I don't think the raw solution you sketch is going to work. First and foremost, by extracting the cash flows from the bond you're discarding the dynamics of their rate under the Hull/White model ...
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3 votes

Timesteps in Vasicek model

Yes you can! Any SDE that has an analytic solution can be simulated exactly. The vasicek model has dynamics $dr=a(b-r)dt+\sigma dW_t$. By Ito's lemma, $$d\left(e^{at}r\right)=e^{at}\left(a(b-r)dt+\...
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  • 1,319
3 votes

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 ...
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  • 2,362
3 votes

How to price zero coupon bonds with short term rates model?

If you do not know anything about the dynamics of you short-rate $r_t$, then there is no way to express the price of the zero coupon bond better than what your already have: $ P(t,T) = \mathbb{E}^Q\...
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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 \...
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  • 711
3 votes
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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,...
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  • 316
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}...
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3 votes
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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 ...
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  • 1,933
3 votes
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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 ...
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  • 2,110
3 votes
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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) ...
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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)} ...
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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). ...
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  • 711
3 votes
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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 ...
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3 votes
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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(...
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