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5 votes

Exploding Libor Rates in Libor Market Model

this is a well-known problem. One solution is to make volatility zero when rates exceed a certain high level. It's less problematic than it looks because any cash-flows generated will be divided by ...
Mark Joshi's user avatar
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4 votes
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Forward starting zero-coupon bonds

$Z(t_0,t_1,t_2)$ is the $t_1$-forward price of the ZC bond with maturity $t_2$, as of $t_0$. We have: $$ Z(t_0,t_1,t_2) = E_{t_0}^{t_1}[Z(t_1,t_2)]\not= Z(t_1,t_2).$$ With a not-trivially stochastic ...
ir7's user avatar
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3 votes
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Intuition of drift in Libor market model

Paying F(K-1) at a lag (K) is a delayed payment and involves a convexity adjustment that can be understood as a consequence of 2 parts - a "stochastic part" where if rates rise whenever F(K-...
Arshdeep's user avatar
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3 votes

Using converted lognormal volatilities for negative rates in a lognormal Libor Market Model (LMM)

I've not seen the abs function on the forward rate here before, the approximation comes from matching variances of a Black (Lognormal) and Bachelier (Normal) SDE. The Black SDE doesn't have this ...
BrownianBread's user avatar
3 votes

Accreting swaption

From a practitioner standpoint, we know the prices of non accreting swaptions. The price of the accreting swaption in any model calibrated to these non accreting swaptions, is heavily dependent on ...
dm63's user avatar
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3 votes
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Pricing Swaption Analytically using Libor Market Model

The present value of a Vanilla Swap (the word Vanilla is used since I am considering the simplest swap, i.e., notional equal to one, contiguous time intervals, constant rate, etc) is given by: \begin{...
rvignolo's user avatar
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2 votes
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Intuition for consistent Derivative Prices under different Numeraires and Measures

Your dynamics under the $T_{i-1}$-forward measure is wrong. Specifically, let $P_{i-1}$ and $P_i$ be, respectively, the $T_{i-1}$- and $T_i$-forward probability measures. Moreover, let $\Delta_i = ...
Gordon's user avatar
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2 votes
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Libor Market Model definitions in Options, Futures & Other Derivatives, Hull 9th Ed, p744

Regarding 1: $m(t)$ returns the index in a tenor structure formed by the "reset times" such that $t \leqslant t_{m(t)}$. So, for example, given a tenor structure formed by the reset times $...
rvignolo's user avatar
  • 741
2 votes

LIBOR Market Model - tenors?

Just to be precisely clear, your mathematical formulation will not necessarily capture the nuances of the physical dates that libor is valued between, due to holiday calendars and modification rules. ...
Attack68's user avatar
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2 votes
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SABR LMM vs no-arbitrage term structure of SABR parameters

I am guessing that the first model you are referring to is the one from Rebonato: Linking caplets and swaptions prices in the LMM-SABR model (2009)? If yes, then I would say that your approach is a ...
BEQuant's user avatar
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2 votes
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Can you shift a standard libor market model with regard to only at-the-money options?

The cash bond doesn't require a displacement, as such the denominator in your drift term should be $F_k(t)$ not $\bar{F}_k(t)$, i.e. $dF_k(t) = \sigma_k(t)\bar{F}_k(t)\sum^k_{j=\beta(t)}\frac{\tau_j\...
BrownianBread's user avatar
2 votes

Relationship between simple Libor spot and forward rates

USD Libor rates are quoted on a Act/360 basis. You can determine USD Libor forward rates by application of the following formula. $$ (1 + \text{SpotRate}(t) \times (\text{Act}(0,t)/360)) \times (1 + \...
AlRacoon's user avatar
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2 votes
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Libor Market Model (LMM) under risk neutral measure

We assume that, under the $T_j$-forward probability measure $P_{T_j}$, \begin{align*} \frac{dP(t, T_j)}{P(t, T_j)} = \mu_P(t, T_j) dt + \sigma_P(t, T_j) dW_t^{T_j}, \end{align*} where $\mu_P(t, T_j)$ ...
Gordon's user avatar
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1 vote
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Effect on Forward Swap Rate from a parallel shift in forward curve

This actually depends upon the definition of your parallel curve shift. I actually define the curve shift of a curve based on discount factors to be a shift in the overnight rates on that curve. ...
Attack68's user avatar
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1 vote
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Pricing & hedging vanilla interest rate options with SABR LMM

I can hardly think of any advantage of using the SABR LMM for a vanilla book. The SABR LMM should be anyway calibrated to the smile (usually marked in the SABR model)? If the calibration is perfect (...
jChoi's user avatar
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1 vote
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Why is the LMM with mixture dynamics (Brigo & Mercurio) inconsistent for the pricing of exotics?

My impression is that Piterbarg criticizes the mixture approach because it is not consistent with the stochastic volatility (SV) world. In the SV world, instanteneous volatility (or variance) ...
jChoi's user avatar
  • 1,164
1 vote

Reconciling different specifications of drifts in the LMM

Björk writes a few pages earlier: "We can also allow for correlation between the various Wiener processes but this will not affect the swaption prices. Such a correlation will however affect the ...
Kurt G.'s user avatar
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1 vote
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Should the Libor Market Model using spot measure as numeraire simulate an arbitrage free forward curve?

I think that under the spot (money market) measure, the ratio of bond prices to the money market account is a martingale. So consider a world where the continuously compounded rate for the first year ...
dm63's user avatar
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1 vote

Libor Market Model Implementation

For example a Caplet with Expiry of 3year with tenor = 0.5 has to be priced (following the analytical formula) with the LIBOR rate L(0,2.5,3). Am I getting it right ? Thats right. The caplet hast a ...
TomDecimus's user avatar
1 vote
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Calibrate SABR-LMM using only data from Bloomberg?

Bloomberg (and depending on the service one subscribes to) has ITM/OTM volatility quotes and price quotes for a wide-range of expiry/tenor pairs. The most liquid currencies are USD, and EUR. You ...
Kiann's user avatar
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1 vote
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Benchmark a Libor Market Model implementation

Answering versus your specific queries Against which analytical formula or numerical method can I compare my simulation results? Basic would be to benchmark the accuracy against European Swaptions ...
Kiann's user avatar
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1 vote
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LIBOR Market Model - tenors?

In the libor market Model, what are modeled are the forward rates. Hence you have to see the fixing date $T_j$ and the maturity date $T_{j+1}$ as dates and not durations. Both are fixed dates and are ...
Jiem's user avatar
  • 436
1 vote

Exploding Libor Rates in Libor Market Model

The explosion of the forward rates in the log-normal LMM simulated in the spot measure seems to be related to the explosion of the Eurodollar futures prices in this model which was studied in this ...
danp's user avatar
  • 116
1 vote

Exploding Libor Rates in Libor Market Model

The rates will explode in the current low rates environment my friend where empirically they are at a too low level to use a log-normal model if you want to preserve your log-normality please use a ...
Bond007's user avatar
  • 83
1 vote
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Prove Volatility Parametrization of Libor Market Model is Bounded/Not Bounded

If I have read the question correctly then I will assume that $a$, $b$, $c$, $d$, $T_i$, and $k_i$ are constants. If this is the case then the only term which we need to show is bounded is $$ \big(a +...
oliversm's user avatar
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