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

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 ...


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 ...


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

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\...


2

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 vol for most of the swaption surface (EUR) because of the negative forwards, you can either use shifted Black Vol or Normal Vol. In the example you are using ...


2

GSR stands for Gaussian Short Rate model. It describes the short rate $r(t)$ dynamics under the Risk Neutral measure as: $$ dr(t) = \kappa(t) \cdot (\theta(t) - r(t)) \cdot dt + \sigma(t) \cdot dW(t). $$ Please, note that this document describes the QuantLib implementation, which is also described in the Andersen and Piterbarg book: Interest Rate Modeling. I ...


2

For Q1, Indeed the ratio of 2 zero coupon bonds associated with the forward is an exact lognormal process (Just apply Ito's lemma to the ratio, as you already know the dynamics of the 0 coupon bonds. You can disregard the drift term as the forward rate is a martingale in the bond forward measure.). The forward rate is then obtained by just adding a scalar, ...


2

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 which compares two discount factors and shows their difference. red line : mean of simulated discount factors blue line : market discount factor green line : ...


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

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 or Floorlet), the Libor Market Model framework (just focusing on one particular forward Libor rate) would be more useful and the preferred way to price. Short ...


1

Just adding my two cents. Without taking the logarithm of the price, the Ito's Lemma should result in: $d p(t,T) = \left( \partial_t A(t,T) - \partial_t B(t,T) r + \frac{1}{2}\sigma^2B(t,T)^2 \right)p(t,T) dt - B(t,T) p(t,T) dr_t$ substituting now the partial derivatives and the differential $dr_t$, and simplifying the identical terms: $d p(t,T) = r_t p(t,T) ...


1

You can simply use Ito's lemma under the risk neutral measure $Q$.For the log-bond price $p(t,T)$ this gives $$dp(t,T)=(A_t(t,T)-B_t(t,T)r_t)dt-B(t,T)dr_t$$ $$=[A_t(t,T)-(B_t(t,T)+B(t,T)a)r_t]dt-B(t,T)\sigma dW_t$$ Here $A_t(t,T)$ and $B_t(t,T)$ are partial derivatives wrt $t$ and $W_t$ is Wiener process under $Q$.


1

You are on the right path but here $P(t,t+1)$ is not your overnight rate but a daily discount factor. To convert to a simply compounded daily rate, which would be your overnight rate, you would do something like this $$ R(t,t+1) = (1 - P(t,t+1))/(\tau P(t,t+1)) $$ Here $\tau$ is your daycount fraction for $1$ day As I mentioned in the comment $R(t,t+1)$ is ...


1

It is an annual rate, with a Actual/360 day count so the interest paid on an overnight loan is -0.56%/360.


1

I know its been a while but I would like to answer this question for all the people that arrives from now on. I hope that is okay. Let's divide the problem in two main parts. The first one is the computation of the zero coupon bond $P(t, T)$. In this case, you are using a short rate model given by the factor dynamics $dy(t)$ and the short rate dynamics $r(t)$...


1

quantlib does not seem able to handle negative rates The slides by Peter Caspers discuss 'QL_NEGATIVE_RATES" and related topics in a very accessible style.


1

A short rate model provides an analytical solution for the zero coupon bond $P(t, T)$, given by the following expectation: $$ P(t, T) = E_t^Q \left[ \exp \left( - \int_t^T r(s) ds \right) \right]. $$ For example, depending on notation, when $r(t)$ follows a short rate model, the previous equation yields to: $$ P(t, T) = \exp(A(t, T) - B(t, T) \cdot r(t)) $$ ...


1

The Hull-White short rate model (or any other short rate model) describes the short rate dynamics $dr(t)$ as well as provide the analytical solution of the zero coupon bond $P(t, T)$: $$ P(t, T) = E_t^Q \left[ \exp \left( - \int_t^T r(s) ds \right) \right] = \exp(A(t, T) - B(t, T) \cdot r(t)) $$ Depending on the notation you are using, the zero coupon bonds ...


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