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Questions tagged [heath-jarrow-morton]

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1 answer
113 views

What does volatility process mean and how is it different from volatility?

I have been reading the paper "Bridging P-Q Modeling Divide with Factor HJM Modeling Framework" by Lyashenko and Goncharov (2022). On Equation 5 of page 4 of the paper, I came across the ...
1 vote
0 answers
54 views

Interest rate models history

I am familiar with some interest rate models, such as the Vasicek, CIR. I also have an understanding of the basic formalization of other models such as Ho-Lee, Hull-White, HJM, Libor market model (LMM)...
2 votes
1 answer
138 views

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

I could find any reference restricting the sign of the volatilities in the multi-factor HJM framework. Can someone please confirm if $\sigma_i(t,T)$ can assume negative values for some $i,t$ and $T$? $...
5 votes
0 answers
58 views

Separability of Stochastic Volatility Model

After having read the article of Trolle & Schwartz regarding their general stochastic volatility term structure model (https://papers.ssrn.com/sol3/papers.cfm?abstract_id=966364), it is not clear ...
0 votes
0 answers
42 views

Instantaneous forward rate function to use in HJM framework

HJM framework uses the instantaneous forward rate $f(t,T)$ in the resulting dynamics and pricing formulas (like in Hull-White or Ho-Lee model). But clearly market does not have an $f(t,T)$ formula, so ...
0 votes
0 answers
180 views

How to convert the parameters of multi-factors cheyette model (quasi-Gaussian model) from tenors to factors?

The book "Interest Rate Modeling" by Andersen and Piterbarg is an extermely fascinating book on interest rate derivatives. Recently, I have encoutered some issues while reading this book. ...
0 votes
0 answers
129 views

Backtesting One-Factor HJM model with selling European Receiver Swaption

I am attempting back test the performance of a model - namely the Musiela equation used to model instantaneous forward rates with constant time to maturity: $$r(t,x)=r(0,x)+\int_0^t\left(\frac{\...
2 votes
1 answer
384 views

Ito's lemma for special case

Assume a HJM framework with the same Brownian motion driving the dynamics for every tenor. $$ df(t,T) = \alpha(t, T)dt + \sigma(t,T) dw_t \,, $$ with $\alpha(t, T) = \sigma(t,T)\int_t^T \sigma(t,s)ds$....
1 vote
1 answer
768 views

Difference HJM Framework versus Short rate model

Recently I study some interest rate models. When I moved on to forward rate models, I see this documents https://en.wikipedia.org/wiki/Heath-Jarrow-Morton-_framework It said "HJM-type models ...
9 votes
2 answers
2k views

Ho and lee derivation for short rates model

A silly question that is bugging me. I am working my way through Baxter and Rennie (again) and I am getting my wires crossed on the short rate models in particular the straight forward Ho and Lee ...
2 votes
0 answers
183 views

In the Black-Scholes model with stochastic interest rates, what are the 3 assets used to compute measures?

Suppose I have a model with 2 primary assets, a stock $S$ and a short rate. The stock will be driven by a Brownian motion $W_1$. The short rate will be random and will be driven by a Brownian motion $...
2 votes
1 answer
247 views

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

I need your help with understanding and solving the HJM framework. I am hoping I can get some help as I feel so lost with HJM and learning online because of the pandemic is adding more stress. Anyway ...
1 vote
0 answers
463 views

Calibrate 1-factor Gaussian HJM model on forward rates and ATM caps prices

I'm trying to solve the following problem as a part of the Interest Rate Models course The algorithm that I'm following is derive simple rates from the given forward rates via $L(0, T_i) = \frac{(1+\...
1 vote
1 answer
334 views

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

I'm solving the following problem as a part of Interest Rate Models class on Coursera I'm having a hard time using nonlinear root solver to invert the Black formula for Cap price in order to obtain a ...
4 votes
1 answer
4k views

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

I have been trying to understand the H&W model expression for zero coupon bond price volatilities: $\nu_B(t_0,t_M)=-\frac{\nu_r}{m}(1-e^{-m\tau_{0,M}})$, where $\nu_B(t_0,t_M)$ is zero coupon ...
2 votes
1 answer
109 views

Non-recombining lattice in non-markovian models

Brigo&Mercurio Interest Rate Models - Theory and Practice, 2nd edition, when treating not markovian HJM models, says the following "the approximating lattice will not be recombining and the ...
2 votes
2 answers
208 views

Heath–Jarrow–Morton under real-world measure

In HJM model (framework), the drift of the forward is determined by its diffusion coefficient: $$ \mu(t,s) = \sigma(t,s)\int_t^s \sigma(t,v)^Tdv $$ My understanding, is that the change of measure ...
0 votes
1 answer
256 views

Modelling limitations and understanding of long term goverment bonds

Been trying to understand the yield curve for a while now. This is what I collected so far, There is a relation between short rates and long rates that goes via the forward rate, and so by the ...
2 votes
1 answer
2k views

Hull-White model: match between HJM framework and short model formulation

I need to show that the Hull-White model $$dr=(\theta(t)-ar)dt+\sigma dW^Q$$ corresponds to the Heath-Jarrow-Morton formulation $$df(t,T)=\alpha(t,T)dt+\sigma e^{-a(T-t)}dW^Q.$$ I obtained the drift ...
0 votes
1 answer
179 views

HJM framework and expectations hypothesis, updated

Is there a way one can decompose the yield of say a government bond with respect the the HJM framework? (into say an expectations component and a term premium component). As far as I can see the HJM ...
6 votes
1 answer
318 views

No-arbitrage in term-structure models

I am a bit confused about what the implication of "no-arbitrage" in popular term struchture models (such as affine term struchtre models or HJM models) are? Is it solely a restriction on the cross-...
1 vote
0 answers
290 views

Calibrate an HJM model in a multicurve setup

I am a mathematician and I'm working on my thesis on Financial Mathematics. I studied this model HJM in a multicurve setup: $$ \begin{cases} df(t,T)=a(t,T)dt+\sigma(t,T)dW_t & (\mbox{rik-free})\...
1 vote
0 answers
157 views

HJM Model proofs

I am looking for a source that possibly has the proofs for the material in the first paper on the HJM model Heath, David, et al. “Bond Pricing and the Term Structure of Interest Rates: A New ...
2 votes
2 answers
383 views

Instantaneous forward rate within the HJM framework

within the HJM framework, the dynamics of the instantaneous forward rate are defined by: $$f_t(T)=f_0(T) + \int_0^t\alpha_s(T)ds+\int_0^t\sigma_s(T)dW_s$$ or in differential form: $$df_t(T)=\alpha_t(...
1 vote
0 answers
75 views

Arbitrage pricing models

I have been reading Wu's Interest rate modeling and in his chapter on the HJM model he says that With arbitrage pricing models, the prices of the basic instruments are treated as model inputs ...
3 votes
1 answer
256 views

HJM in infinite dimensions

I recently started reading Filipovic's Consistency problems for HJM interest rate models and came across the Musiela reparametrization $$r_t(x)=f(t,x+t)$$ so the forward curve can be thought of as a ...
2 votes
1 answer
614 views

Ho-Lee short rate model under the Heath-Jarrow-Morton framework

Under the Heath-Jarrow-Morton (HJM) framework the dynamics of the Ho-Lee short rate model are defined as following: $$dr(t)=\theta(t)dt+\sigma dW^{\mathbb{Q}}(t)$$ with $\mathbb{Q}$ the risk-neutral ...
2 votes
2 answers
517 views

Ho Lee model in Baxter&Rennie

I am currentyl reading Baxter&Rennie and I have a difficulty with understanding a derivation of formula for one function, $g(x,t,T)$ (this can be found on page 152 in the book). I know that there ...
3 votes
1 answer
201 views

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

From Baxter and Rennie Page 145: $Z(t,T) = exp(\int_{0}^{t}\Sigma(s,T)dW_s - \int_{0}^{T}f(o,u)du - \int_{0}^{t}\int_{s}^{T}\alpha(s,u)duds)$ where $\Sigma(t,T) = \int_{t}^{T}\sigma(t,u)du$ How ...
7 votes
1 answer
737 views

Baxter & Rennie HJM: differentiating Ito integral

From Baxter and Rennie, page 138: $$f(t,T)=\sigma W_t+f(0,T)+\int_0^t\alpha(s,T)ds$$ $$Z_t=\exp-\bigg(\sigma(T-t)W_t+\sigma\int_0^tW_sds+\int_0^Tf(0,u)du+\int_0^t\int_s^T\alpha(s,u)ds\bigg)$$ $$dZ_t=...
2 votes
1 answer
106 views

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

Let the HJM dynamics of $\ln P(t,T)$ (log of bond prices) given by (In the risk neutral measure ) : $$d \ln P(t,T) = \mathcal{O}( dt) - \sigma_P (t,T) dW(t)$$ Knowing that $f(t,T)=-\frac{\partial}{\...
2 votes
0 answers
402 views

Markovianity of the short rate process in the HJM framework

In Andersen and Piterbarg (2010), the authors study the short rate process under a HJM framework and derive the following expression (Section 4.4.3): $$ r(t)=f(t,t)=f(0,t)+\int_0^t\sigma_f(u,t)^\...
2 votes
1 answer
611 views

HJM or Short rates model?

When market practitioners do prefer HJM models to short rates models when it comes to pricing derivatives (other than swaptions and caps, let say light exotics to exotics) ? To be more specific, ...
1 vote
0 answers
134 views

Extension of HJM to multiple factors

The HJM model calibrates the entire forward curve using the existing yield curve data and this results in the following expression for its instantaneous forward rate- $$df(t,T)=\sigma(t,T)\int_0^T\...
1 vote
0 answers
148 views

HJM model, existence of arbitrage:

The Setup: Suppose I know the yield curve of a Bond satisfies: f (0, t) = 0.04 for t ≥ 0 and f (ω, 1, t) = 0.06, t ≥ 1, ω = ω 1 , 0.02, t ≥ 1, ω = ω 2 , where Ω = {ω 1 , ω 2 } with P[ω i ] > 0, i = 1,...
0 votes
0 answers
122 views

Vol specifications under Heath Jarrow Morton framework

What are some of the common forward vol specifications under HJM framework used in the industry. I guess most common would be 2 and 3 factor models, but any pointers to more details would be very ...
2 votes
0 answers
954 views

Stochastic Leibniz rule

We have the following single-factor HJM model $$d_tf(t,T)=\sigma(t,T)dW_t+\alpha(t,T)dt$$ $$f(t,T)=f(0,T)+\int_0^t\sigma(s,T)dW_s+\int_0^t\alpha(s,T)ds$$ The discounted T bond is then \begin{align} Z(...
1 vote
3 answers
942 views

Understanding the HJM drift condition's dimensions

In an HJM model the forward rate dynamics follow $$ df_t(T) =a_t(f_t(T))dt+b_t(f_t(T))dW_t $$ where $W_t$ is a $d$-dimensional brownian motion, $b_t$ takes values in $\mathbb{R}^{d\times d}$ and $a_t$ ...
2 votes
1 answer
508 views

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

Hi I am looking for some general clarification to Heath–Jarrow–Morton framework. I am analyzing a problem where the forward rate is modeled as $$ f(t,T)=e^{\beta(T-t)} Z_t+h(T-t) \tag{1}$$ for some ...
1 vote
0 answers
361 views

Bond pricing with HJM simulation

I'm using Glasserman 3.16 and 3.17 algorithm to price bonds. The algorithms evaluates the forward rates and the discount factor $B(0,t_j)$. My question is: How can I price bonds in a future time? I ...
4 votes
1 answer
1k views

HJM simulation problem

I'm trying to simulate a 3-factor HJM model. I got the algorithms from Glasserman book. In my case, I have $3$ maturity:$ 0.25y, 0.5y, 0.75y$. So my time grid is: $t_0=0,t_1=0.25,t_2=0.5,t_3=0.75$. ...