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7 votes
Accepted

Baxter & Rennie HJM: differentiating Ito integral

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. ...
Quantuple's user avatar
  • 14.7k
5 votes

Instantaneous forward rate within the HJM framework

This is known as the classical Leibniz rule. The link sends to Wikipedia, where you can find a proof. It allows to differentiate under the integral sign. A general statement of the formula is: $$\text{...
Daneel Olivaw's user avatar
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, \...
Gordon's user avatar
  • 21.2k
5 votes

Zero-coupon bond price volatility with one factor Hull White interest rate model

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 ...
Gordon's user avatar
  • 21.2k
4 votes
Accepted

HJM framework problem - showing that HJM drift condition implies that $b(z)=b+βz$ and $(ρ)^2=α$

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^{\...
Gordon's user avatar
  • 21.2k
4 votes
Accepted

HJM in infinite dimensions

Keeping it simple, you know HJM SDE gives the dynamics of an instantaneous forward referencing a fixed maturity T, $f\left(t, T\right)$, but there is a continuum of such maturities - the whole forward ...
Magic is in the chain's user avatar
3 votes
Accepted

Difference HJM Framework versus Short rate model

Most principal component analyses (PCAs) on historical data of yield curves find that typically a yield curve moves parallel flips from normal to inverse (or vice versa) twists (changes its ...
Kurt G.'s user avatar
  • 2,033
3 votes
Accepted

Modelling limitations and understanding of long term goverment bonds

OK, your framework on this is right. Long-term yields embed "expectations" about future short rates. That is: what do I think I'll get if I sat for T years with cash in the bank for swaps, or the same ...
demully's user avatar
  • 5,071
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)} ...
Gordon's user avatar
  • 21.2k
2 votes

Understanding the HJM drift condition's dimensions

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 $i^{th}$ ...
ABIM's user avatar
  • 373
2 votes
Accepted

HJM framework and expectations hypothesis, updated

In the real world, many people believe that the yield of a government bond consists of an expectations part plus a term premium. However financial models such as HJM are built in a “risk -neutral” ...
dm63's user avatar
  • 17.2k
2 votes

Instantaneous forward rate within the HJM framework

It is just an application of the Leibniz integral rule, written in differential form. Please see here: https://en.m.wikipedia.org/wiki/Leibniz_integral_rule Capital T is constant, t is changing, so ...
Magic is in the chain's user avatar
2 votes
Accepted

HJM model Baxter Rennie: differentiating the discounted asset price using Ito

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 ...
Slade's user avatar
  • 656
2 votes
Accepted

Getting $df(t,T)$ when given $d\ln P(t,T)$ and $f(t,T)=-\frac{\partial}{\partial T} \ln P(t,T)$

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{...
Gordon's user avatar
  • 21.2k
2 votes
Accepted

Non-recombining lattice in non-markovian models

I think I might have found the solution to my own question. The Markov property as stated above has no direct relation with the recombination of the approximating lattice. However, if we consider the &...
Matteo Campagnoli's user avatar
2 votes

HJM or Short rates model?

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 ...
Brian B's user avatar
  • 14.9k
1 vote

Can volatility assume negative values under multi-factor HJM framework?

At first, I also could not find a single source that formally restricts $\sigma_i(t,T)$. However, the formally very precise article [1] explicitly states that the $\sigma_i(t,T)$ are assumed to be non-...
Hans-Peter Schrei's user avatar
1 vote
Accepted

HJM drift condition problem: Show that the HJM drift condition implies $b(t) \equiv b, \rho^{2}(t) \equiv a$

Fixing again some typos of yours, we know that in HJM under the risk-neutral measure $$ f(t, T)=f(0, T)+\int_0^t\left(\sigma(s, T) \int_s^T \sigma(s, u) \,du\right)\,ds+\int_0^t \sigma(s,T)\,dW_t^* $$...
Kurt G.'s user avatar
  • 2,033
1 vote
Accepted

Inverting the Black formula for Cap price to find Black implied volatility

Notice that cap prices are given as %, therefore your vector of cap prices should be CapPrices = [0.002, 0.008, 0.012, 0.016] while you have ...
emot's user avatar
  • 876
1 vote

No-arbitrage in term-structure models

In the typical HJM model, during a single time step forward rates evolve according to their individual volatilities and according to pairwise correlations which can be specified. That arrangement is ...
dm63's user avatar
  • 17.2k
1 vote

Ho Lee model in Baxter&Rennie

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}...
siwy9's user avatar
  • 63
1 vote

Ho and lee derivation for short rates model

It seems this specific passage of the Ho-Lee short rate model has left many readers puzzled, so the authors themselves have expanded on this derivation with a pdf add-on that can be found at the book ...
Giogre's user avatar
  • 366

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