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)...


7

Let $$Z_t = \exp(-X_t)$$ with $$X_t = \sigma(T-t)W_t+\sigma\int_0^tW_sds+\int_0^Tf(0,u)du+\int_0^t\int_s^T\alpha(s,u)du ds $$ and $W_t$ a standard Brownian motion, along with the usual assumptions. We can write $X_t=f(t,W_t)$ and apply Itô's lemma to get: $$ dX_t = \frac{\partial f}{\partial t}(t,W_t) dt + \frac{\partial f}{\partial W_t} (t,W_t) dW_t + \...


3

Based on this question, for the Hull-White model of the form \begin{align*} dr_t = (\theta(t)-a r_t) dt + \sigma dW_t, \end{align*} where $a$ and $\sigma$ are constants, $a(t)$ is a deterministic function, and $W_t$ is a standard Brownian motion, the price at time $t$ of a zero-coupon bond with maturity $T$ and unit face value is given by \begin{align*} P(t, ...


3

You almost get there. However, you ca not conclude that $\rho^2$ is a constant based on $(10)$. Note that, from your $(7)$ and $(8)$, \begin{align*} \frac{\rho(z_t)^2}{\beta} e^{\beta \tau} (e^{\beta \tau} - 1) = -h'(\tau)+e^{\beta \tau}h'(0). \end{align*} Taking derivative with respect to $\tau$ on both sides, we obtain that \begin{align*} \frac{\rho(...


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 $f(t, T)$ ...


2

This is my first ever answer so please bear with me. Apologies in advance for terrible formatting. Also fyi, you have some typos in your post that may be making things more confusing. The issue here is that $\int_{0}^{t}\int_{s}^{T}\sigma(s,u)dudW_s$ is itself a stochastic process and trying to take its partial derivative with respect to time (which you ...


2

We assume that \begin{align*} d \ln P(t,T) = \mu(t, T) dt - \sigma (t,T) dW(t). \end{align*} Then, \begin{align*} \ln P(t,T) = \ln P(0,T) + \int_0^t \mu(s, T) ds - \int_0^t \sigma (s,T) dW(s). \end{align*} Moreover, \begin{align*} f(t, T) &= -\frac{\partial\ln P(t,T)}{\partial T} \\ &= -\frac{\partial\ln P(0,T)}{\partial T} - \int_0^t \frac{\partial\...


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. ...


1

The problem should go away if you simulate $r_t$. Ho Lee should work for the function of the form you assumed: $P(0,T)=e^{-aT^2-bT}=e^{-(aT+b)T}$ The problem with your simulation is that the forward rate, as you correctly derived, is as follows: $f(0,T)=2aT+b$ So when you take the derivative to calculate $\theta$, you lose b. But remember the short rate ...


1

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

Your issue is that you misinterpreted the NA criterion, it reads: $$ a_t(x) \triangleq \sum_{i=1}^{\infty} \left(b^i_t(x) \int_0^x (b_t^i(u))^T du\right)e_i, $$ where $b^i_t$ denotes the $i^{th}$ column of the volatility matrix $b_t$, $^T$ the transpose and $e_i$ the $i^{th}$ standard basis vector in $\ell^1$. In other words the $i^th$ coordinate of the ...


1

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 ...


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