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.
33
questions
1
vote
1
answer
74
views
Pricing & hedging vanilla interest rate options with SABR LMM
Are there any advantages of pricing and hedging plain vanilla interest rate options with more complex SABR LMM instead of simpler SABR model? Should one always go with the SABR LMM as a universal ...
- 535
1
vote
1
answer
75
views
SABR LMM vs no-arbitrage term structure of SABR parameters
There exists a LIBOR Market Model with stochastic volatility for pricing and hedging exotic (e.g. path-dependent) interest rate options with smile. However let us consider the following approach:
...
- 535
7
votes
1
answer
165
views
Why is the LMM with mixture dynamics (Brigo & Mercurio) inconsistent for the pricing of exotics?
I am reading about the LMM with lognormal-mixture dynamics. Consider the following dynamics for the forward rate $F_{i}(t)$ fixing at $T_{i-1}$ and paying at $T_i$:
\begin{align}
dF_{i}(t) = (F_i (t) +...
- 428
3
votes
0
answers
96
views
SABR LMM for RFR
Is there a research showing a way to use SABR LMM with new RFRs such as SOFR, i.e. pricing exotic path-dependent RFR derivatives with volatility smile and skew?
I'm aware that
Looking Forward to ...
- 535
0
votes
1
answer
46
views
Reconciling different specifications of drifts in the LMM
I've been going through the book "Fixed Income Securities" by Bruce Tuckman which gives the following definitions of the drift terms (after showing it for a specific example with 3 forward ...
- 788
6
votes
0
answers
149
views
SABR-LMM: best way to perform a MC simulation
I am working on a SABR-LMM model with the following system of SDEs under a numeraire $N$:
$$
\begin{align}
&\mathrm{d} F_i(t) = \sigma_i (t) (F_i(t) + s)^{\beta} \Big( \mu^f_i (t) \mathrm{d}t ...
- 428
0
votes
1
answer
59
views
Libor Market Model definitions in Options, Futures & Other Derivatives, Hull 9th Ed, p744
Re: Options, Futures & Other Derivatives, Hull 9th Ed, p744.
What does "m(t)" represent? I am struggling to understand the definition provided of:
"Index for the next reset date at ...
2
votes
1
answer
408
views
Pricing Swaption Analytically using Libor Market Model
I was asked the following question in a recent interview: "(i) Express a forward swap rate in terms of forward Libor rates. (ii) Apply Ito's lemma to this expression to derive the process for the ...
- 5,211
1
vote
1
answer
194
views
Intuition for consistent Derivative Prices under different Numeraires and Measures
This is essentially the Fundamental Theorem, however I am not asking for a thorough proof, I am more interested in the general intuition.
In words, it makes sense that whatever your unit of account (...
- 5,211
7
votes
1
answer
198
views
Benchmark a Libor Market Model implementation
Assume I have implemented a solution of the Libor Market model PDE in terms of the Finite Difference method. What is a good strategy for validating and benchmarking the results of this implementation?
...
- 308
3
votes
0
answers
133
views
Change of measure for BGM (LMM) Model
I've been checking the demos for BGM (LFM) forward rate model.
Here's a short reminder to help you follow:
Now, take the following
$$\frac{dL_j(t)}{L_j(t)} = \sigma_j. dW^j(t) = \mu_{ij} dt + \...
- 210
1
vote
2
answers
126
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 ...
- 475
0
votes
1
answer
181
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/...
- 136
0
votes
1
answer
343
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
1
answer
447
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:
$$ ...
- 353
1
vote
0
answers
495
views
The "I want to price swaptions" request
In a small buy-side structure I recently had the following request : "I want to trade swaptions, I need to price them".
After a quick discussion the need is to price vanilla options on fix vs float ...
- 1,162
0
votes
2
answers
1k
views
Accreting swaption
Is there any literature on the maths behind the computation of the price of an accreting swaption in the LMM model (no monte carlo, closed formula or close enough...)?
Thank you!!
- 177
4
votes
0
answers
147
views
Volatility Parametrization Libor Market Model - Underspecified Model?
Does the volatility parametrization that I have chosen give an underspecified model? Which volatility parametrization in the Libor Market Model would suit the best for the particular case described ...
- 333
1
vote
4
answers
1k
views
Exploding Libor Rates in Libor Market Model
I have implemented the Libor Market Model in Matlab. When I generate a number of paths, I notice that some of them explode. Does anybody have an idea what could cause this?
I already tried solving ...
- 333
0
votes
1
answer
108
views
Prove Volatility Parametrization of Libor Market Model is Bounded/Not Bounded
How can I prove that the function
$$\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]$$
is bounded/unbounded?
$\sigma_i\left(t\right)$ is the ...
- 333
1
vote
0
answers
575
views
Pricing back swaptions corresponding to underlying swaps of Bermudan Swaption in calibrated LMM
I do not know to which swaption volatility matrix I have to calibrate the LMM in order to price back correctly the swaptions corresponding to the underlying swaps of a Bermudan Swaption.
My problem: ...
- 333
2
votes
1
answer
267
views
Numerical Optimizer Matlab Calibration LMM
I am trying to mimimize the following function in order to calibrate the Libor Market Model
$$\sum_{i=1}^{n} \left(\sigma_i^{market}-\sigma_i^{Reb}\left(a,b,c,d,\beta\right)/\sqrt{T_i}\right)^2,$$
...
- 333
2
votes
1
answer
1k
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 ...
- 61
-1
votes
1
answer
302
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 ...
- 139
0
votes
1
answer
160
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
2
answers
1k
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 ...
- 463
1
vote
0
answers
110
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 γ ≤ β caplets can be
exercised, and they have to be automatically exercised when in the
...
- 156
2
votes
1
answer
564
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 ...
- 156
1
vote
0
answers
327
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 ...
- 156
0
votes
0
answers
843
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
...
- 144
4
votes
2
answers
502
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....
- 2,107
14
votes
2
answers
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. ...
- 4,910
15
votes
3
answers
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 ...
- 3,212