10

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 + \int_t^T \int_t^s\theta_u du ds\\ &=(T-t)r_t + \sigma\int_t^T\int_u^T ds\, dW_u +\int_t^T\int_u^T\theta_u ds du\\ &=(T-t)r_t + \sigma\int_t^T (T-u)...


9

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 spot, zero-coupon rates from the known price of coupon-bearing instruments $-$ such as bonds or swaps. In general: You start picking short-term (typically less ...


6

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


6

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). They were not used for pricing as the fact that these short-rate models (Vasicek, CIR and Ho-Lee) had only two or three free parameters meant that they could ...


6

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 models from Ho-Lee, Hull-White and Black-Karasinski are no-arbitrage models. Take Vasicek and Hull-White as an example. The short rate processes are $\mathrm{d}...


6

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 indicates - short term like Eurodollar frequently modelled with Black or Bachelier (normal) model. HW1F is also a short rate model. 2 ) HJM is a framework (M is not ...


5

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)=-\frac{\frac{d}{du}P(t,u)}{P(t,u)}\\ =-\frac{\frac{d}{du}E^Q_t[e^{-\int_t^u r_s ds}]}{P(t,u)} =\frac{E^Q_t[e^{-\int_t^u r_s ds} r_u]}{P(t,u)} =E^Q_t\left[\frac{e^{-\...


4

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 numerator and denominator of the last expression with $B(0,s)$ and noting that $B(0,s)B(s,u)=B(0,u)$ (investing one Dollar for $s$ years and then for another $u-s$ ...


4

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 model, you are modelling one curve. As a first approximation, you can deduce the other curves by adding a deterministic basis spread.


4

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, \end{align*} we define the function \begin{align*} g(x, t, T) = -\ln E\left(e^{-\int_t^T r_s ds} \,\big|\, r_t = x\right). \end{align*} The forward rate $f(t, T)$ ...


3

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. $$ \left( \prod_{i=0}^n a_i \right)^{\frac{1}{n}} $$ where $a_i$ are our returns. We have to make an assumption here - that your underlying is described by $\mathrm{d}S = \mu S \mathrm{d}t + \...


3

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, $$d\left(e^{at}r\right)=e^{at}\left(a(b-r)dt+\sigma dW_t\right) +a e^{at} r dt$$ Simplifying, $$d\left(e^{at}r\right)=e^{at} ab +e^{at}\sigma dW_t$$ Integrating, $$e^{aT} r_T=r_0+b(e^{aT}-1)+\sigma \int_0 ...


3

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 factors. The shifted idea is the one used to handle negative rates problem in caplet, swaption...


3

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 you're using. You should both forecast and discount them on the tree; the way to do it correctly is implemented, e.g., in the DiscretizedSwap class (and ...


3

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 amount X (fixed in advance) one year from now, and in return I agree to pay you one year from now the amount Y equal to the 1 Month Libor Rate published at that ...


3

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 $$ dX^i_t = -\frac 12 \alpha X^i_t dt + \sqrt{\alpha} dW^i_t. $$ Looking at the squared radius $R_t = \sum_{i=1}^d (X^i_t)^2 $ (in $\mathbb{R}^d$) of this process we get by Ito: $$ dR_t = \sum_{i=1}^d (2 X^i_t dX^i_t) + d ...


3

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 $\mathscr{F}_t$ should also be considered. For this particular question, based on an answer to this question, \begin{align*} P(t, T) = e^{-(T-t)r_t - \int_t^T (T-u)\...


3

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 a security paying $X$ at time $T$ discounted to $t=\tau$ is then $E\left[X \exp(-\int_\tau^T r(u)du)\right|\mathscr{F}_\tau]$ Hence the price of a credit risk-...


3

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 a function of the short rate. For affine models for example, this would be of the form: $$P(t, T) = e^{A(t, T) - r(t)B(t,T)}$$ (for example, for the one-factor ...


3

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) P_r + \frac{1}{2}\sigma(t,r)^2P_{rr} -rP=0, \end{align*} subject to $P(T,T)=1$ whose general solution is $P(t,T)=\mathbb{E}^\mathbb{Q}\left[e^{-\int_t^T r_u\...


3

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)} du - \frac{\sigma^2}{2a^2}\big(e^{-a T} -1\big)^2 + e^{-a T} r_0. \end{align*} Note also that \begin{align*} \int_0^t\alpha(u,T)du &= \int_0^t\sigma(u,T)\...


3

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(t,T)\right) = 0 \\ & \iff \int_t^Te^{-ku}du+ \int_t^T\frac{\partial}{\partial u}\left(e^{-ku}B(t,T)\right)du = 0 \\ & \iff \int_t^Te^{-ku}du+ \int_t^T\...


3

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 main disadvantage of HJM, high-dimensional stochastic process as underlying, was overcome by Cheyette, back in 1994, by restricting the general HJM model to a ...


3

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 make the model a bit simpler or to simulate. Simulation This is the nasty part. Based on your model, you simulate a very large number of the discount factor(s) and ...


2

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\left[\left. \exp{\left(-\int_t^T r_s\, ds\right) } \right| \mathcal{F}_t \right] $ You can use a model given in this page where you should be able to find close ...


2

I am not sure about this specific algorithmic implementation, but I am a bit confused by your indexes and suspect you might be as well (e.g. $M$ not defined, you're showing cases of $i$ looping when it seems you mean $j$). I think it would be useful to revisit the basics: Let $D_t \in (0,1]$ be the present value factor for a cash flow at time $t$. By ...


2

First I must appreciate the @Richard's help that cause to solved this question. The Dothan model with this dynamic " $dr_t=ar_tdt+\sigma r_tdW_t$ " is easily integrated $r(t)=r(s)exp ( \mu (t-s)+\sigma (W_t-W_s))$ Where $\mu=a-\frac{\sigma^2}{2}$ so We have $E^Q[B_t]=E^Q[exp(\int_0^t r(u)du)]\approx E^Q[e^{e^y}]$ Where $y$ is Gaussian distributed so ...


2

Which is the paper/book you are reading? It should be noted there. But basically in a short-rate model you have a model for the short rate $r_t$ you can calculate zero-coupon bond prices from it by $P_T = E[\exp(-\int_{0}^T r_u du)]$ from these prices you can calculate the yield-to-maturity $Y_T$ which fulfills $$ P_T =\exp( - Y_T T) $$ thus $Y_T = - \log(...


2

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: $$ P(t, T) = E_t^Q \left[ \exp \left( - \int_t^T r(s) ds \right) \right]. $$ Otherwise, when pricing interest rate derivatives using Monte Carlo simulations, you would have to perform Monte Carlo ...


2

Regarding your first question: the equation for $\theta(t)$ is obtained from the consistency condition $$ \forall T, \;\; E\left[e^{-\int_0^T r(t) dt} \right] = P^M(0,T) $$ after a somewhat involved calculation using the integrated version of the SDE for $r$ $$ r(t)=e^{-\kappa t}r(0) + \int_0^t e^{-\kappa (t-u)} \theta(u) du + \int_0^t e^{-\kappa (t-u)} \...


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