Questions tagged [libor-market-model]
The libor-market-model tag has no usage guidance.
46
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How to simulate from instantaneously correlated Brownian motions?
Say I have obtained a distribution for different forward rates F_k such that:
$$
dF_k (t) = \sigma (t) * F_k (t) * dW_k(t)
$$
with
$$ dW_k(t) * dW_l(t) = \rho_{k,l} (t) dt.
$$
From this I want to ...
- 13
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1
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Is it possible to perform a VaR analysis based on the forwards obtained by the LMM?
I am in the process of building a LMM model and I ideally want to use this not only to price LIBOR swaps at the current time but also provide a price distribution in a future time. For example we have ...
- 13
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1
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Choosing a time step in Monte Carlo simulation of forward rates in LIBOR Market Model
Lets talk about the Monte Carlo simulation of forward rates in Euler discretization scheme under the $T_N$-forward measure, a so called terminal measure. Suppose that we have a number of time steps ...
- 636
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Why are LMM+ parameters becoming more unstable when using an inverted volatility term structure
I have an implementation of an LMM+ model (a shifted Libor Market Model with rebonato volatility function) and am seeing recently that the calibrated parameters are becoming more unstable over time; I ...
- 171
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Double exponential parametrization of a correlation matrix
I'm implementing a LIBOR Market Model with stochastic volatility following this book and ran into a problem trying to parametrize the forward-forward and volatility-volatility correlation matrices ...
- 636
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1
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144
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Relationship between simple Libor spot and forward rates
How is the simple forward rate L(0,T,T+1) calculated given the spot rate L(0,T)?
user62731
1
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1
answer
99
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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 ...
- 636
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122
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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:
...
- 636
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1
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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
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191
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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 ...
- 636
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51
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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 ...
- 842
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174
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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
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339
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Replacement for LIBOR Market Model (LMM)?
With the transition from LIBOR to SOFR, will the LIBOR Market Model be replaced by a new model? Perhaps this has already happened. If yes, what is this new model? If not, will the LIBOR Market ...
- 389
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1
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227
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Should the Libor Market Model using spot measure as numeraire simulate an arbitrage free forward curve?
I have been looking at the following resource:
Reference Paper
Using equation [4] for the discretized version of the forward libor rate:
$\tilde{L}^i_{T_{j+1}} = \tilde{L}^i_{T_{j}} exp[\sigma^i(\sum^...
- 123
2
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1
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Using converted lognormal volatilities for negative rates in a lognormal Libor Market Model (LMM)
There exist formulas to convert between normal and lognormal interest rate volatilities. In the most simple form the approximation for ATM volatilities would be $\sigma_{LogNorm}=\frac{\sigma_{Norm}}{\...
- 21
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122
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Can you shift a standard libor market model with regard to only at-the-money options?
Suppose I have an LMM defined using the spot measure as in Brigo and Mercurio:
$dF_k(t) = \sigma_k(t)F_k(t)\sum^k_{j=\beta(t)}\frac{\tau_j\rho_{j,k}\sigma_j(t)F_j{t}}{1+\tau_jF_k(t)}dt + \sigma_k(t)...
- 123
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0
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80
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LMM multifactor swaption calibration
Brigo and Mercurio give Rebonato's approximation for Black-like swaption volatility as
$(v^{LFM}_{\alpha,\beta})^2=\sum^\beta_{i,j=\alpha+1}\frac{w_i(0)w_j(0)F_i(0)F_j(0)p_{i,j}}{S_{\alpha,\beta}(0)^2}...
- 123
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Forward starting zero-coupon bonds
We trivially have that:
$$\frac{Z(t_0,t_1)}{Z(t_0,t_2)}=1+\tau L(t_0,t_1,t_2)$$
Where $L(t_0,t_1,t_2)$ is the forward Libor between $t_1$ and $t_2$, as of $t_0$.
Simply inverting this relationship ...
- 5,513
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1
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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 ...
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Where to find Caps/Floor historical data?
I'm trying to calibrate the Lognormal Forward Libor Model to market data, in order to calculate market implied volatilities. However, I'm having some troubles finding any Cap/Floor historic price. Not ...
2
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1
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471
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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 ...
- 5,513
1
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1
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223
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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 (...
- 5,513
7
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1
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218
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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?
...
- 328
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171
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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 + \...
- 237
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2
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168
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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 ...
- 485
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201
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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
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384
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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/...
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1
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498
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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:
$$ ...
- 355
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525
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The "I want to price swaptions" request
In a small buy-side structure I recently had the following request : "I want to trade swaptions, I need to price them".
After a quick discussion the need is to price vanilla options on fix vs float ...
- 1,170
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2
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1k
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Accreting swaption
Is there any literature on the maths behind the computation of the price of an accreting swaption in the LMM model (no monte carlo, closed formula or close enough...)?
Thank you!!
- 177
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155
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Volatility Parametrization Libor Market Model - Underspecified Model?
Does the volatility parametrization that I have chosen give an underspecified model? Which volatility parametrization in the Libor Market Model would suit the best for the particular case described ...
- 343
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4
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1k
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Exploding Libor Rates in Libor Market Model
I have implemented the Libor Market Model in Matlab. When I generate a number of paths, I notice that some of them explode. Does anybody have an idea what could cause this?
I already tried solving ...
- 343
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1
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114
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Prove Volatility Parametrization of Libor Market Model is Bounded/Not Bounded
How can I prove that the function
$$\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]$$
is bounded/unbounded?
$\sigma_i\left(t\right)$ is the ...
- 343
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613
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Pricing back swaptions corresponding to underlying swaps of Bermudan Swaption in calibrated LMM
I do not know to which swaption volatility matrix I have to calibrate the LMM in order to price back correctly the swaptions corresponding to the underlying swaps of a Bermudan Swaption.
My problem: ...
- 343
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1
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281
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Numerical Optimizer Matlab Calibration LMM
I am trying to mimimize the following function in order to calibrate the Libor Market Model
$$\sum_{i=1}^{n} \left(\sigma_i^{market}-\sigma_i^{Reb}\left(a,b,c,d,\beta\right)/\sqrt{T_i}\right)^2,$$
...
- 343
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1
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1k
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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
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319
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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
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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
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2
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2k
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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
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0
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114
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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
...
- 166
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1
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577
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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 ...
- 166
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0
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338
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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 ...
- 166
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888
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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
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529
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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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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. ...
- 4,980
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3
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2k
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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 ...
- 3,222