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Instantaneous correlation and terminal correlation - 6.29

I am trying to derive the $(6.29)$ equation from brigo and Mercurio book. We have $dF_2(t)= \sigma_{2,\beta(t)}F_2(t)dW_2(t)$ $dF_3(t)= \frac{\delta_3\sigma_{3,\beta(t)}^2F_3^2(t)}{1+\...
JohnGalt's user avatar
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I am looking for help to derive this formula from Brigo & Mercurio

I think my use of numeraire change is incorrect and for sure my understanding is incomplete. $\frac {dQ2}{dQ1}=\frac {Pt(0,T2)P0(0,T1)}{P0(0,T2)Pt(0,T1)} = \frac {1+DeltaF_1(t)}{1+DeltaF_2(t)}$ Then ...
JohnGalt's user avatar
2 votes
1 answer
178 views

Intuition of drift in Libor market model

I studied the classical Libor market model, where the dynamics of rate $F_k$ from time $T_{k-1}$ to $T_k$ are given by $$ dF_k(t)/F_k(t) = \sigma_k(t) dZ_k(t) $$ under the forward measures $Q^k$ (...
Richi Wa's user avatar
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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)...
KiNest's user avatar
  • 21
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1 answer
192 views

Effect on Forward Swap Rate from a parallel shift in forward curve

Can anything be said on how a parallel shift in the forward curve affects the forward swap curve? To be more concise, say we have a model estimate of the implied vol for the 2Y-10Y point (2Y ...
J Muscat's user avatar
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0 answers
113 views

LMM (Libor market Model) Correlation Calibration

I use a LMM model from a well known vendor, using a SOFR swap curve and SOFR swaptions. The calibration set include many/all of the ATM swaptions from 1m-1y to 30y-30y and I get a very good fit for ...
Philippe Hatstadt's user avatar
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89 views

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 ...
Stann98's user avatar
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1 vote
1 answer
123 views

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 ...
Stann98's user avatar
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1 answer
310 views

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 ...
Hasek's user avatar
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93 views

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 ...
blah_crusader's user avatar
2 votes
1 answer
309 views

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)?
user avatar
1 vote
1 answer
157 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 ...
Hasek's user avatar
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1 vote
1 answer
276 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: ...
Hasek's user avatar
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7 votes
1 answer
256 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) +...
BEQuant's user avatar
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0 answers
336 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 ...
Hasek's user avatar
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1 answer
88 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 ...
Oscar's user avatar
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6 votes
0 answers
218 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 ...
BEQuant's user avatar
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2 votes
0 answers
910 views

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 ...
equanimity's user avatar
2 votes
1 answer
343 views

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^...
JoeBass's user avatar
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2 votes
1 answer
265 views

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}}{\...
pallo's user avatar
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3 votes
1 answer
196 views

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)...
JoeBass's user avatar
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2 votes
0 answers
95 views

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}...
JoeBass's user avatar
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5 votes
1 answer
278 views

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 ...
Jan Stuller's user avatar
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1 answer
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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 ...
iamlearning's user avatar
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74 views

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 ...
Matteo Campagnoli's user avatar
2 votes
1 answer
685 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 ...
Jan Stuller's user avatar
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2 votes
1 answer
308 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 (...
Jan Stuller's user avatar
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7 votes
1 answer
270 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? ...
davidhigh's user avatar
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3 votes
0 answers
238 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 + \...
Xman's user avatar
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1 vote
2 answers
195 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 ...
user357269's user avatar
0 votes
1 answer
242 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/...
Yassine Q.'s user avatar
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1 answer
449 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/...
foreignvol's user avatar
3 votes
1 answer
610 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: $$ ...
user25844's user avatar
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1 vote
0 answers
570 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 ...
Olórin's user avatar
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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!!
ababoua's user avatar
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4 votes
0 answers
165 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 ...
Tinkerbell's user avatar
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 ...
Tinkerbell's user avatar
0 votes
1 answer
119 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 ...
Tinkerbell's user avatar
1 vote
0 answers
670 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: ...
Tinkerbell's user avatar
2 votes
1 answer
300 views

Numerical Optimizer Matlab Calibration LMM

I am trying to mimimize the following function in order to calibrate the Libor Market Model $$\sum_{i=1}^{n} \left(\sigma_i^{market}-\sigma_i^{Reb}\left(a,b,c,d,\beta\right)/\sqrt{T_i}\right)^2,$$ ...
Tinkerbell's user avatar
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 ...
Quant2015's user avatar
-1 votes
1 answer
347 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 ...
salisboss's user avatar
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0 votes
1 answer
206 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 ...
William Hedén's user avatar
7 votes
2 answers
2k 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 ...
Adam's user avatar
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1 vote
0 answers
120 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 ...
lkjldfkjhljk's user avatar
2 votes
1 answer
631 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 ...
lkjldfkjhljk's user avatar
1 vote
0 answers
355 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 ...
lkjldfkjhljk's user avatar
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0 answers
980 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 ...
KaapstadKwant's user avatar
4 votes
2 answers
545 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....
athos's user avatar
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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. ...
olaker's user avatar
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