Questions tagged [lmm]
The LIBOR market model is a financial model of interest rates. It is used for pricing interest rate derivatives. The quantities that are modeled are a set of forward rates (also called forward LIBORs), which have the advantage of being directly observable in the market, and whose volatilities are naturally linked to traded contracts.
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1answer
189 views
Pricing Swaption Analytically using Libor Market Model
I was asked the following question in a recent interview: "(i) Express a forward swap rate in terms of forward Libor rates. (ii) Apply Ito's lemma to this expression to derive the process for the ...
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1answer
128 views
Intuition for consistent Derivative Prices under different Numeraires and Measures
This is essentially the Fundamental Theorem, however I am not asking for a thorough proof, I am more interested in the general intuition.
In words, it makes sense that whatever your unit of account (...
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115 views
Benchmark a Libor Market Model implementation
Assume I have implemented a solution of the Libor Market model PDE in terms of the Finite Difference method. What is a good strategy for validating and benchmarking the results of this implementation?
...
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0answers
84 views
Displaced diffusion LMM
In the standard LMM a rate $L_i(t)=L(t,T_{i-1},T_i)$ has under the $T_n$-forward measure ($n>i$) the dynamics
\begin{equation}
d{L_i}(t) = - {\sigma _i}(t){L_i}(t)\sum\limits_{j = i + 1}^n {\frac{{...
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0answers
77 views
Change of measure for BGM (LMM) Model
I've been checking the demos for BGM (LFM) forward rate model.
Here's a short reminder to help you follow:
Now, take the following
$$\frac{dL_j(t)}{L_j(t)} = \sigma_j. dW^j(t) = \mu_{ij} dt + \...
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2answers
100 views
LIBOR Market Model - tenors?
In the LIBOR market model, we have a bunch of forward rates $L_j$ on $[T_j, T_{j+1}]$ for some collection on $j$.
My question is, is it the delivery dates or the time to maturities that are fixed?
So ...
0
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1answer
148 views
Libor Market Model Implementation
I'm trying to implement an LMM-MultiCurve for caplet pricing following the analytical formula mentioned in this article (pg 20):
https://www.researchgate.net/publication/...
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1answer
220 views
Calibrate SABR-LMM using only data from Bloomberg?
I'm exploring the SABR-LMM model. In particular, have been trying to study the effect of the parameters and their time evolution.
However, the data seems to be a major issue here. Prices for caps/...
3
votes
1answer
296 views
Libor Market Model (LMM) under risk neutral measure
I would like to establish the equations of forward libors under risk neutral measure. Here is how I do it, and what I get :
Under the $P_{T_j} $ measure, forward Libor $L_j$ is martingale. Thus:
$$ ...
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0answers
351 views
The “I want to price swaptions” request
In a small buy-side structure I recently had the following request : "I want to trade swaptions, I need to price them".
After a quick discussion the need is to price vanilla options on fix vs float ...
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2answers
801 views
Accreting swaption
Is there any literature on the maths behind the computation of the price of an accreting swaption in the LMM model (no monte carlo, closed formula or close enough...)?
Thank you!!
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138 views
Volatility Parametrization Libor Market Model - Underspecified Model?
Does the volatility parametrization that I have chosen give an underspecified model? Which volatility parametrization in the Libor Market Model would suit the best for the particular case described ...
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4answers
988 views
Exploding Libor Rates in Libor Market Model
I have implemented the Libor Market Model in Matlab. When I generate a number of paths, I notice that some of them explode. Does anybody have an idea what could cause this?
I already tried solving ...
0
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1answer
101 views
Prove Volatility Parametrization of Libor Market Model is Bounded/Not Bounded
How can I prove that the function
$$\sigma_i\left(t\right) = k_i\left[\left(a+b\left(T_i-t\right)\right)e^{-c\left(T_i-t\right)}+d\right]$$
is bounded/unbounded?
$\sigma_i\left(t\right)$ is the ...
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0answers
473 views
Pricing back swaptions corresponding to underlying swaps of Bermudan Swaption in calibrated LMM
I do not know to which swaption volatility matrix I have to calibrate the LMM in order to price back correctly the swaptions corresponding to the underlying swaps of a Bermudan Swaption.
My problem: ...
2
votes
1answer
250 views
Numerical Optimizer Matlab Calibration LMM
I am trying to mimimize the following function in order to calibrate the Libor Market Model
$$\sum_{i=1}^{n} \left(\sigma_i^{market}-\sigma_i^{Reb}\left(a,b,c,d,\beta\right)/\sqrt{T_i}\right)^2,$$
...
2
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1answer
822 views
instantaneous forward rates vs forward LIBOR rates
HJM describes the behavior of instantaneous forward rates while BGM describes the behavior of forward Libor rates. From concept perspective, I understand forward libor rate are like forward Libor rate ...
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1answer
251 views
Finding Discount Bond Matrix in LMM Model C++
I am working on a 1 Factor Libor Market Model (LMM) in C++ and I working my implementation of the formula to find my Discount Bond matrix via the following formula:
In the case of my model alpha is ...
0
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1answer
134 views
Test Log-Normality for LIBOR forward rates under the Libor Market Model
As far as I understand, under the Libor Market Model the forward rates are assumed to have a log-normal distribution. Given that I have constructed my LMM model and now have a matrix of:
k different ...
7
votes
2answers
931 views
Practical implementation of Libor Market Model
I am trying to implement a project about the BGM model, suggested in the book "The Concepts and Practice of mathematical finance" by Mark Joshi.
My question is related to the forward volatility ...
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0answers
105 views
Price 3m libor autocap with LMM calibrated on 1y swaption data
I need to calculate a price of an autocap contract which is
An autocap is similar to a cap, but at most Ī³ ā¤ Ī² caplets can be
exercised, and they have to be automatically exercised when in the
...
2
votes
1answer
503 views
LMM. Calibration to swaptions by Brigo and Morini. Volatility of swaption that matures at T=0
I'm reading Brigo D., Mercurio F. Interest Rate Models - Theory and Practice (Springer, 2006)(ISBN 3540221492) and also a source article on LMM cascade calibration to swaptions by Brigo and Morini.
I ...
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0answers
311 views
Incompatibility of Lognormal Forward Model (LMM\BGM) and Lognormal Swap Model
In his paper On the distributional distance between the Libor and the Swap market models (and also in his book about IR modeling) D.Brigo says:
10, 11, 12 are defined in the end of message. Do I ...
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0answers
758 views
Calibration Problem in the LMM-Skew (Shifted Diffusion) Model
I have implemented the LIBOR market model (LMM) and I am quite satisfied with the results. I have now added a skew to the model as described in 10.1 of Brigo/Mercurio. That is, I have replaced the SDE
...
4
votes
2answers
478 views
question on Leif Andersen's “Interest Rate Modeling, vol 2 Term Structure Models”
I'm reading Leif Andersen's "Interest Rate Modeling, vol 2 Term Structure Models" and met a problem on Chapter 14 LM Dynamics and Measures, $\S$ 14.2.5 Stochastic Volatility, Lemma 14.2.6, on page 602....
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2answers
1k views
When is the LIBOR market model Markovian?
The question is inspired by a short passage on the LMM in Mark Joshi's book.
The LMM cannot be truly Markovian in the underlying Brownian motions due to the presence of state-dependent drifts. ...
15
votes
3answers
2k views
Rate interpolation in Libor Market Model
Libor Market Model (LMM) models the interest rate market by simulating a set of simply compounded, non-overlapping Libor rates which reset and mature on predefined dates. How do I obtain from them a ...
