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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)...

5

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 ...

5

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

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. 3 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}} $$where a_i are our returns. We have to make an assumption here - that your underlying is described by \mathrm{d}S = \mu S \mathrm{d}t + \... 3 You are right. In the CIR++, \alpha parameter is absorbed into \phi. With the CIR++, \phi(t) will allow you to have to have negative rates. You will calibrate your \phi to fit the discount factors. The shifted idea is the one used to handle negative rates problem in caplet, swaption... 3 Yes you can! Any SDE that has an analytic solution can be simulated exactly. The vasicek model has dynamics dr=a(b-r)dt+\sigma dW_t. By Ito's lemma,$$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

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: $$dR_t = \sum_{i=1}^d (2 X^i_t dX^i_t) + d ... 3 This is indeed a standard result. You can convince yourself by noticing The bank account grows from 1 at t=\tau to E\left[\exp(\int_\tau^T r(u)du)|\mathscr{F}_\tau\right] at time T The price of a security paying X at time T discounted to t=\tau is then E\left[X \exp(-\int_\tau^T r(u)du)\right|\mathscr{F}_\tau] Hence the price of a credit risk-... 3 Yes, LIBOR rates can be simulated using short rate models. Or rather, Libor rates can be obtained from simulated short rate values. Usually, you have formulas giving you the zero-coupon bond price as a function of the short rate. For affine models for example, this would be of the form:$$P(t, T) = e^{A(t, T) - r(t)B(t,T)}$$(for example, for the one-factor ... 2 If you do not know anything about the dynamics of you short-rate r_t, then there is no way to express the price of the zero coupon bond better than what your already have:  P(t,T) = \mathbb{E}^Q\left[\left. \exp{\left(-\int_t^T r_s\, ds\right) } \right| \mathcal{F}_t \right]  You can use a model given in this page where you should be able to find close ... 2 First I must appreciate the @Richard's help that cause to solved this question. The Dothan model with this dynamic " dr_t=ar_tdt+\sigma r_tdW_t " is easily integrated r(t)=r(s)exp ( \mu (t-s)+\sigma (W_t-W_s)) Where \mu=a-\frac{\sigma^2}{2} so We have E^Q[B_t]=E^Q[exp(\int_0^t r(u)du)]\approx E^Q[e^{e^y}] Where y is Gaussian distributed so ... 2 Which is the paper/book you are reading? It should be noted there. But basically in a short-rate model you have a model for the short rate r_t you can calculate zero-coupon bond prices from it by P_T = E[\exp(-\int_{0}^T r_u du)] from these prices you can calculate the yield-to-maturity Y_T which fulfills$$ P_T =\exp( - Y_T T) thus Y_T = - \log(... 2 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(\left(\int_0^te^{\beta u}\sqrt{r_u}dW_u\right)^2 \right)\\ &=\sigma^2 e^{-2\beta t} E\left(\int_0^te^{2\beta u} r_u du \right)\\ &=\sigma^2 e^{-2\beta t}\... 2 Imagine you hold a zero coupon bond with a certain maturity T and the short rate follows a process like you specified. You might know deterministically what the cash bond pays this period, but you don't know how the interest rate itself is going to change. If the interest rate goes down, then the expectation of future rates goes down and the expected ... 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)} \...

2

If you have a simple instrument, short rate models capture all the key variance, but they impose structure on the shape of forward volatility curves (and, usually, forward tilt) that is often far from realistic. If you have instruments whose value is sensitive to what might happen with tilt or forward volatility, you need a multifactor model like HJM. ...

2

In practice, most derivatives traded on Fed Funds rates are linear(i.e. Forwards) rather than non-linear (options and exotics). As such, there has not been a strong case for precise modeling of the full distribution of a Fed Funds rate for a particular day. In contrast , there is a large market for derivatives on 3month USD Libor , which is less sensitive ...

2

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)\...

1

It comes down to what is meant by long rate. I will approach the problem as follow: Would a 5 year rate be called a long rate? How about 10 years rate then? How about 2 years rate? It won’t be easy to link all these rates, without some modelling of the rates between these tenors. And that’s exactly what the forward rates models do. Hope this helps!

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OK, so I think I have figured it out. I assumed that we need to use Ito's lemma here, however, it seems the authors mean to use Ito's isometry, which must be used to prove below equality $$\mathbb{E}_Q\Big(e^{-\sigma\int_t^T (T-u)dW_u} \mid r_t\Big)=e^{\frac{\sigma^2}{2}\int_t^T(T-u)^2 du}$$ We know that for normal-distibuted random variable $X$ (with mean \$...

1

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" parameters from historical data and compute the "Risk neutral" parameters from the yield curve. In such a case the HJM and Vasicek models are simply two way of ...

1

I have been working on, to generate vasicek model parameters as well. For what it's worth, your k seems large. However, what I do, is to fit my Vasicek parameters to real-quoted data. So, I have the USD treasury yields for 1y, 2y, 3y, 4y, 5y. I have the caplet volatilities for the same structure. In our set-up, I set k (mean-reversion) to be time-dependent; ...

1

Implied vol surfaces are just a convenient way to represent vanilla options (caps, European swaptions) prices in the context of simple models such as normal, log normal or shifted log normal. When using a more advanced model (BGM, etc.) with the goal of pricing non vanilla options the first step is to calibrate the advanced model vol parameters to vanilla ...

1

The model has effectively two free parameters, and therefore one cannot expect it to match bonds of different maturities. Typically this is how you 'get the parameters' by solving to match a given set of instruments. Of course you cannot match 100 instruments with 5 parameters, therefore you either solve in a least squares sense or increase the number of ...

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