LIBOR Market Model

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Explains the LIBOR market model. Contains a step by step derivation of the drift under the forward and the spot measure, and also shows how the multi-dimensional LIBOR market model can be represented in terms of Variance-Covariance or Variance-Correlation matrices. Here is the outline of the content by timeline:
2:07/22:39: Illustrate what is being modelled in the LIBOR market model
4:00/22:39: How to define the Zero Coupon and Bank Account in the LIBOR framework
05:22/22:39: How to construct continuous process from discrete LIBORs
06:47/22:39: Link LIBOR to traded asset so that we can use the general valuation formula
07:43/22:39: Determine the dynamics of T period LIBOR under the T-forward measure
08:54/22:39: Determine the dynamics of other LIBORs under the T-forward measure
14:25/22:39: Determine the dynamics of LIBORs under the Spot measure
16:52/22:39: Explain multidimensional LIBOR, and how it can expressed in terms of Variance-Covariance (Variance-correlation) matrices
Note: For the purists, we use q(t) to represent both the index and the time value (e.g., 3 and T_3), but the context shall make it abundantly clear which one is meant. It is just that T_{q(t)} takes too much space.

Опубликовано:

22 июл 2024

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Комментарии : 30

@noebozinis5883 Год назад

Many thanks for the fantastic material! Since the video content has been extended to the covariance matrix of multiple observed rates, a good idea for future releases would be to expand towards swap rates and Constant Maturity Swap numeraire, convexity adjustment and pricing. Again, thank you very much, excellent work!

@quantpie 5 лет назад

For the purists, we use q(t) to represent both the index and the time value (e.g., 3 and T_3), but the context shall make it abundantly clear which one is meant. It is just that T_{q(t)} takes too much space.

@ghaithmhamdi4987 Год назад

Hello , thank you for the videos i have a question about the radon Nikodym derivative at 10:20 shouldn't the dQ(Tn) / dQ(Tn-1) be the inverse of the result that we have . if Tn is the probability measure with respect to the Pn+1 as numeraire and T(n-1) is the one with respect to the Pn i believe that the result is the inverse of what we have if i'm wrong please tell me how we are getting this result for dQ(T"n") / dQ(T"n-1")

@jonathankelly9106 19 дней назад

Hi, thanks so much for the great video. Can I ask, did you use any reference textbook for making this video?

@NeverBapet 2 месяца назад

could you or someone, elaborate how the radon-nikodym derivative can be extracted @10:16

@fabioniski3316 4 года назад

Great presentation, but isn't the current situation such that interest rates are low , and thus the black vols are not from log normal assumption on the forwards but on normal assumptions? Why dont you then use dL = sigma(t)dW instead of dL=L*sigma(t)dW ?

@quantpie 4 года назад

hello! This is the classic version of the model. To deal with negative rates, there are shifted versions of the model, which will shall cover at some point.

@saleelpai 4 года назад

Hi, excellent stuff. Could we have a video for the Swap Market Models too please?

@quantpie 4 года назад

thanks @Sal, added to the list!

@johnvanprague9540 4 года назад

This is an amazing video, thank you so much. I would like to ask a question: Although the zero-coupon bond is widely used in the industry for pricing purposes, it actually doesn't exists. For example, the Libor rates are not linked to Treasuries much at all, the two instruments live their own life. So even taking the short-dated Treasury securities as a proxy to the discount bond, it doesn't make sense to assume a no arbitrage between the Libor rates and the Treasuries. Isn't the assumption of using the zero-coupon bond as a function of the Libor rates quite an unrealistic tool?

@quantpie 4 года назад

@John Van Prague, thank you! Here the zero coupon bond is a mathematical construct in the following sense. In a multi-curve environment, you would have one curve for each rate - say instruments referencing 3 months GBP LIBOR will form an interest rate term structure (curve), whereas instruments referencing 6 months LIBOR will form a separate curve. Each Govies will have its own interest rate term structure. You can apply all the term structure concepts to each of the curves separately. Say you have the current LIBOR rate for 3 months deposits (covering period from today to 90 days), and a FRA referencing 3 months LIBOR covering the subsequent 3 months (91 days to 180 days), you can easily replicate a contract that pays 1 at time T=90 days and another contract that pays 1 at time T=180 using these instruments. In practice you will have FRAs/futures/swaps referencing 3 months LIBOR for more granular starting points, and you can always interpolate to make these continuous. So the zero coupon concept refereeing, 3 months LIBRO for example, can be viewed as a traded asset as you can always combine the LIBORs and FRA to replicate its payoff. Hence the zero coupon price is curve specific, and is not linked to the zero-coupon Govies. Hope this helps, but please do let me know if any of the points is not clear!

@johnvanprague9540 4 года назад

@@quantpie Thank you so much for that: that is the best answer I have seen. On quant stack exchange, the claim is that 3-m T-bills can be used as the zero-coupon bonds for USD 3-m Libor, which is very confusing. So I totally get the first part: each curve lives its own independent life. I also get that up to 90 days, the zero-coupon bond is effectively the Libor deposit, so we can lend 1-unit discounted at the 3-m Libor today and receive 1-unit in 3-months time with certainty (forgetting credit risk) - is that correct? From your answer, I understand that up to 90 days, the deposit market is liquid even in forward terms. How can I replicate the discount bond with FRAs though, going behind the 90 days? Ps: do we also effectively assume that we can close out the deposit at any point in time? I.e. that we can effectively trade the zero-coupon bonds (via deposits) once these have been "issued" every day, and not just at maturities? (i.e. I lend today at Libor for 3 months and I want to close it out in a week's time).

@quantpie 4 года назад

@@johnvanprague9540 Thank you! The FRA/futures are liquid up to 2/3 years maturities (here maturity being the start point of the underlying LIBOR deposit), and you then have swaps that can go all the way to 50/70 years. The derivatives (FRA/swaps) these days are collateralised and cleared, so the risk up to the start point of the deposit (same thing as 3m LIBOR here) will be minimal, and if the derivative is not collateralised, you can price the risk in. The deposit (that the LIBOR references)will be uncollateralised by its nature, hence the segmentation by the tenor type (1 month LIBOR, 3 months LIBOR etc). Re-treasury vs LIBOR, the rates generally do move together but they represent different things, so the moves will not be perfectly aligned, 1 basis point can make a huge P&L. And then the differences are so pronounced in the times of stress (remember 2007-2008 and the ensuing search for multiple benchmarks?) that they don't seem to represent the same thing. Hope this helps!

@johnvanprague9540 4 года назад

@@quantpie Apologies if this is a stupid question: whilst it is obvious how to replicate the 1-unit pay off via a Libor deposit, how can this be done via FRAs? Consider a 3*6 FRA (so maturing in 3 months, underlying 3-month Libor): you lock in today and receive the FRA: so in 3-months time, you receive fixed rate corresponding to the currently prevailing implied 3m-forward 3m Libor. You trade notional that will produce 1-unit pay-off when multiplied by the fixed rate of the FRA. Fast forward: in 3-months time, the FRA settles and you receive the fixed rate * notional (so you get 1-unit), but you need to pay the prevailing spot 3-m Libor against it. Even if you lend notional at the then prevailing 3-m spot Libor, you will only receive this in 3-months time. So you don't really get a 1-unit pay-off. Is there a different way of doing it?

@quantpie 4 года назад

No such thing as stupid question! It is great question! This paper, available on SSRN, covers these subtleties: Everything You Always Wanted to Know About Multiple Interest Rate Curve Bootstrapping but Were Afraid to Ask (ssrn.com/abstract=2219548)

@jc8414 3 года назад

Thank you very much for the video! I have a few questions if that's ok: 1. why is a part of RN Derivative around 10:24 to be P(0, Tn) / P(0, Tn+1) rather than P(S, Tn) / P(S, Tn+1)? 2. Around 16:45 you said the drift under spot measure is the same as the drift under shortest maturity zc which we already know - Where did you mention that before? Not sure if I know that already. Sorry my technical knowledge is very poor and the above questions may not even make sense. Apologise in advance!

@jc8414 3 года назад

Would be really useful if you can produce videos about implementations of these models! Theoretically I managed to understand the steps but not sure exactly how these models (Short Rate model + LLM) are implemented in real life and couldnt seemed to find any books/papers that describe the actual implementation...unless it involves very complex mathematical reasoning which really confuses me). Thank you very much!

@jc8414 3 года назад

Sorry for the vast amount of messages here - it's kind of the same story as learning maths theories and only really understand what they meant when you see a exam question and the solutions!

@claudem5527 2 года назад

there is a typo in 8'20 : Libor maturing in Tn is a QTn+1 (and not QTn) martingale.

@efectorami 3 года назад

Hi quantpie! Great video, as always! I have a few questions, hope it is okay! 1. You define `q(t)` as a mapping function being left continuous. Then, your formulas for `B(t)` and `P(t, Tn)` make use of `q(t)`. I believe your formulas are correct but, how do you explain the differences that arise with respect to Andersen L. and Piterbarg V. in the Interest Rate Modeling book? You can check that formulas for `B(t)` (eq. 14.8) and `P(t, Tn)` (eq. 14.2) are identical to the ones that you have shown here but Andersen uses the right continuous interpretation for `q(t)`. My insight on this topic is that Andersen formulas for `B(t)` and `P(t, Tn)` are "incorrect" and that the use of right continuous interpretation comes in handy for the drift in the spot measure in equation (14.7) and not in the money market and zero coupon bond. When dealing with the drift, the left continuous interpretation removes one term in the summation when `t` equals a mesh point T in the tenor structure of the Libor market model. 2. Is there something to say about the `\sigma_L(t, T_n)` expression? Because I have noticed that in many textbooks the drift depends on correlations and I believe this is coming from the product of `\sigma_L(t, T_n) * \sigma_L(t, T_k)`, right? 3. I don't know if I have understood correctly the multi dimensional Libor market model. Are you suggesting that, when you have to simulate say, 30 Libors, the common practice is to use less brownian motions? 4. Why is it called Libor market model and not Ibor market model? I don't know if this question makes sense but I would like to ask it anyways :) Thank you so much!

@quantpie 3 года назад

many thanks for the kind words and the great questions!! Answers in order: 1) will check and revert back. 2) yes that's right, essentially covariance, and when the Brownians are correlated, you can get the formula in terms of correlation. We shall do a sequel on vol and correlation, we had the video almost ready but never published it! 3) Yes that's correct, usually fewer than 10 suffice to capture the joint dynamics. In simples cases, you see under 5, even 3 are not uncommon, 7 are good! 4) Probably because Jamshidian's paper was titled LIBOR and swap market models, and his choice was probably due to the fact that LIBOR based derivatives had been more liquid/widespread. Hope this helps!

@efectorami 3 года назад

@@quantpie Thank you! 1) I will be waiting for your response. 2) I will be waiting for the video. 3) Excellent! 4) Maybe I am asking something that does not make sense again! But... what is the difference between Ibor and Libor? Are those the same but Ibor englobes all rates of this type and the Libor consideres only the London interbank rate? So when I read something about Ibor it can be applied not only to the Libor but to others? Thanks again!!

@quantpie 3 года назад

Please see the introductory section of this report: www.iosco.org/library/pubdocs/pdf/IOSCOPD526.pdf So yes IBOR is a collective term for LIBOR, EURIBOR, TIBOR, etc.

@efectorami 3 года назад

@@quantpie hey! I just wanted to ask you what do you think about this paper: papers.ssrn.com/sol3/papers.cfm?abstract_id=3330240 ? Since it is a really hot topic, maybe a video would we awesome?

@quantpie 3 года назад

@@efectorami thanks! yes this is a topic we need to cover! thanks for the suggestion!!