Over the last year, I have used this blog to discuss questions related to the multi-curve framework in a large sense. I have decided to expand the blog's scope to "quantitative finance" in general.

This will be reflected in the forthcoming blogs related to the mandatory bilateral margining and in the medium term with the announcement of a new book.

Stay tuned.

## 2015-12-06

## 2015-11-21

### Treasury / Swap spreads are negative. And what?

In recent weeks there were several news articles related to the negative treasury/Swap spread. This is in particular the case of a Bloomberg article and a Zero Hedge blog.

Some of the “information” in those notes are

Let’s be clear, to me, those claims are purely

What does a swap represent? As indicated in the first extract above - yes part of those quotes are correct - the swap rate is a fixed rate paid in exchange of floating ones. The floating rates are overnight or term Ibor rate indices. How are those indices computed? As the average of interbank rate over their term - actual rates for overnight and invented rates for Ibor. They represent some fixed amount that the market thinks is equivalent to periodic reinvestment with banks

On one side there is the risk of default of the state over 30 years and on the other side the risk of default of a basket of banks over 3 months - the standard in USD; see Appendix B.8 in the book - rolled over 30 years with the weak members of the basket eliminated every 3 months.

The question of the swap spread is: can the difference be negative? You may have a quantitative economic model that tells you that the difference should always be positive, but there is nothing “illogical” per se is negative spreads. How do you compare 30 years risk with 3 months repeated risk of something else that is changing every 3 months? There is no obvious comparison. And if your model tells you that the

And if you ask me my opinion about the U.S. government creditworthiness - I know you have not really asked and you probably don’t care about my credit risk opinion, but I give it anyway -, with a very high and increasing deficit, periodic fights about debt ceiling from a government that already morally defaulted on its debt when it abolished the USD/gold convertibility, 30 years is a very long time!

Some of the “information” in those notes are

Swap rates are what companies, investors and traders pay to exchange fixed interest payments for floating ones. That rate falling below Treasury yields […] is illogical in the eyes of most market observers, because it theoretically signals that traders view the credit of banks as superior to that of the U.S. government.

It’s hard to overstate how illogical it is when swap spreads are inverted. That’s because it suggests that governments are less creditworthy than the very financial institutions they bailed out during the credit crisis just seven years ago.

Let’s be clear, to me, those claims are purely

**wrong**, nothing is “*illogical*” in negative swap spread. But why do I rant about that in the “multi-curve framework” blog? Because it is exactly the same that the spread between swaps of same tenor illustrated in Figure 1.1 of the book. Remember, those spreads going up and down while they refer to the same swap tenor. If those swap rates were representing the banks creditworthiness over the term of the swap, there would be no spreads between swaps referencing different indices. The multi-curve framework foundation is a contradiction with the idea that swaps represent the creditworthiness of banks over the tenor of the swap. Those press articles indicate that the basis of the multi-curve framework are not understood by “*most market observers*”. That leaves a lot of potential sales for the book ;-)What does a swap represent? As indicated in the first extract above - yes part of those quotes are correct - the swap rate is a fixed rate paid in exchange of floating ones. The floating rates are overnight or term Ibor rate indices. How are those indices computed? As the average of interbank rate over their term - actual rates for overnight and invented rates for Ibor. They represent some fixed amount that the market thinks is equivalent to periodic reinvestment with banks

*with substitution*. The banks that loose too much in creditworthiness or default during one period are not included in the next period. The rate is somehow the creditworthiness of good banks with elimination of weakening ones on a regular basis. It is the fear of bank default without the actual defaults.On one side there is the risk of default of the state over 30 years and on the other side the risk of default of a basket of banks over 3 months - the standard in USD; see Appendix B.8 in the book - rolled over 30 years with the weak members of the basket eliminated every 3 months.

The question of the swap spread is: can the difference be negative? You may have a quantitative economic model that tells you that the difference should always be positive, but there is nothing “illogical” per se is negative spreads. How do you compare 30 years risk with 3 months repeated risk of something else that is changing every 3 months? There is no obvious comparison. And if your model tells you that the

*spread cannot “theoretically” be negative*, it just mean that*your model is wrong*, not that there is a problem with the economy.And if you ask me my opinion about the U.S. government creditworthiness - I know you have not really asked and you probably don’t care about my credit risk opinion, but I give it anyway -, with a very high and increasing deficit, periodic fights about debt ceiling from a government that already morally defaulted on its debt when it abolished the USD/gold convertibility, 30 years is a very long time!

## 2015-03-21

### Home library

It had been my dream for many year now!

A majestic library to store all my books. After a lot of design, planing and book carrying ... its here. More exactly it has been there for several months, but I had not reported on it.

Two-storey, seven meters wide, five meter high, an embedded desk and a glass platform...

I now have enough space to store all my books and the new one I'm buying. With a full shelve available to stored my multi-curve book (yet) unsold copies.

A majestic library to store all my books. After a lot of design, planing and book carrying ... its here. More exactly it has been there for several months, but I had not reported on it.

Two-storey, seven meters wide, five meter high, an embedded desk and a glass platform...

I now have enough space to store all my books and the new one I'm buying. With a full shelve available to stored my multi-curve book (yet) unsold copies.

**Two-storey**

**Overview**

**Glass platform**

**Desk on the left**

### Commenting on the Fundamental Review of the Trading Book

In December 2014, the Basel Committee on Banking Supervision (BCBS) issued a consultative paper on outstanding issues for its fundamental review of the trading book capital standards.

The paper indicated that the Committee welcomed comments on the proposal. Some of the subjects described in the consultative document are related to the multi-curve framework. I provided comments to the BCBS on some of the issues described in the document. In this blog, I describe how some of those comments are related to the book content.

The comments made to the consultative paper are in principle public. The comments may be published on the BIS website. Even if the deadline for the comments was Friday 20 February 2015, the pages related to the consultative document have not changed since and the comments made are not publicly available yet. The link to my comments can be found at the bottom of this blog. Don’t hesitate to comment of my comments!

The main items in my comments are:

In general my comments are in the following direction. The regulation should be

It is difficult to be prescriptive when the market is so diverse. Can the regulators be omniscient? Personally, I doubt it. By being very prescriptive, it seems they think they are. I would prefer a principle based approach than a rule based approach. That would mean that the people in charge of verifying the rules need enough knowledge to understand the principles and estimate if they are respected in diverse practical situations. Checking a list of fixed rules is easier than understanding fundamental principles and the functioning of a complex institution as a bank. But does it make the financial systems safer?

This description raises several questions: What instruments should be used? What is the conventions for those instruments. In some markets, the most liquid (and maybe unique) quote is a spread. For example in USD, the most liquid LIBOR6M instrument is the basis swap LIBOR3M v LIBOR6M. How is this instrument incorporated in the framework. Should we use a

A second type of question relates to the Chapter 3 of the book titled

The second comment on that same article 3.(c).20 is the formula requested. The standardized approach for market risk is based on first order sensitivities. The regulation goes to the point of indicating how that sensitivity should be computed. The formula that the regulation enforces is a

If you have implemented Algorithmic Differentiation in your libraries (see Appendix C.3 in the book), too bad for you, you can not use it. You have to implement a new approach using finite difference and multiply you computation time by 5 on your vanilla swap book and maybe by 10 for cross currency swaps. The required formula creates numerical instability in the sensitivity from you tree methods (see Pelsser and Vorst (1994)), never mind, this is the regulation.

The public comments made to the consultative paper are now available on the BIS website at: http://www.bis.org/bcbs/publ/comments/d305/overview.htm

The paper indicated that the Committee welcomed comments on the proposal. Some of the subjects described in the consultative document are related to the multi-curve framework. I provided comments to the BCBS on some of the issues described in the document. In this blog, I describe how some of those comments are related to the book content.

The comments made to the consultative paper are in principle public. The comments may be published on the BIS website. Even if the deadline for the comments was Friday 20 February 2015, the pages related to the consultative document have not changed since and the comments made are not publicly available yet. The link to my comments can be found at the bottom of this blog. Don’t hesitate to comment of my comments!

The main items in my comments are:

- Curves nodes or vertices.
- Computation method for PV01.
- Implied volatility

In general my comments are in the following direction. The regulation should be

*more explicit*on the*goal*of the procedure/regulation and*less prescriptive*on the means used to achieve that goal. The implementation validation by the regulators can check that the goals are effectively achieved. If one institution can achieve that goal in a*more efficient*way that the way imagined at the time of writing the regulation, this should be seen as a positive aspect of the implementation, not as a bypass of the regulation.It is difficult to be prescriptive when the market is so diverse. Can the regulators be omniscient? Personally, I doubt it. By being very prescriptive, it seems they think they are. I would prefer a principle based approach than a rule based approach. That would mean that the people in charge of verifying the rules need enough knowledge to understand the principles and estimate if they are respected in diverse practical situations. Checking a list of fixed rules is easier than understanding fundamental principles and the functioning of a complex institution as a bank. But does it make the financial systems safer?

## Interest rate curve vertices

In the document, Section 3.(a).11, the Committee describes the*tenors/vertices*that should be used for the different interest rate curves. There is one curve for each index (like USD LIBOR3M or EUR EONIA). The risk factors should be to the*market rates*as described in Section 3.(c).20. The ten tenors are 0.25Y, 0.50Y, 1Y, 2Y, 3Y, 5Y, 10Y, 15Y, 20Y, 30Y.This description raises several questions: What instruments should be used? What is the conventions for those instruments. In some markets, the most liquid (and maybe unique) quote is a spread. For example in USD, the most liquid LIBOR6M instrument is the basis swap LIBOR3M v LIBOR6M. How is this instrument incorporated in the framework. Should we use a

*risk weight*of the spread equal to the risk weight of a standard swap? Or should we convert the market data of the basis spread into a synthetic quote of the Fixed v LIBOR6M IRS? If the second method is used, can we call the synthetic number computed a*market rate*?A second type of question relates to the Chapter 3 of the book titled

*Variation on a theme*. What does it mean to use a ‘’market quote’’ for instruments with maturity smaller than the underlying index? There is no liquid market instrument with maturity smaller than the index they refer to. What would a 3 months instrument with a IBOR6M index be? In Section 3.1 of the book I describe the pseudo-discount factors approach to forward curve description. One of the important aspect of that description (Definition 3.1) is the fact that below the index maturity, the pseudo-discount factor curve is arbitrary. Asking a 3 months point on the IBOR6M curve is an arbitrary request on an arbitrary number.

Suggestions to the BCBSSuggestions to the BCBS

*Indicate that for curve related to Ibor indices, the first vertex of the curve is the tenor of the index. The curves related to longer tenors will have less vertices at the short term*

## Computation method for PV01

The first comment on the article 3.(c).20 is that there is a scaling factor missing somewhere. The figure described in the document as “PV01” is not the “present value of 1 basis point” but the “present value of 1”. This is a trivial point to correct, just add a scaling factor ( x 0.0001) somewhere. Even if this is just a typo, it is still surprising that a Committee supposed to proposed world-wide regulations impacting billions of capital had not the document proof-checked by “quants” who would have spotted it at first reading.The second comment on that same article 3.(c).20 is the formula requested. The standardized approach for market risk is based on first order sensitivities. The regulation goes to the point of indicating how that sensitivity should be computed. The formula that the regulation enforces is a

*forward finite difference*. To fulfill the regulation you have to compute each sensitivity for interest rate using a finite difference on the market rate approach! If you read it in the strict sense, each time you want to compute the capital related to a simple swap based on 2 curves, you have to calibrate your curves 21 times (once for the market rate, and once for each of the bumps on the 2x10 vertices) and compute the present value of your swap with those 21 curves sets.If you have implemented Algorithmic Differentiation in your libraries (see Appendix C.3 in the book), too bad for you, you can not use it. You have to implement a new approach using finite difference and multiply you computation time by 5 on your vanilla swap book and maybe by 10 for cross currency swaps. The required formula creates numerical instability in the sensitivity from you tree methods (see Pelsser and Vorst (1994)), never mind, this is the regulation.

**Suggestions to the BCBS***Add a scaling factor of one basis point (×0.0001) in the definition of sensitivity.**Rephrase point 20 to require the sensitivity to be computed as the partial derivative, in the mathematical sense, of the instrument value with respect to the relevant rate.**Leave the implementation and approximation choices, if any, on how to compute those numbers to the implementing institutions.*

### References

Pelsser, A. and Vorst, T. (1994) The binomial Model and the Greeks, The journal of Derivatives, 1(3), 45-49## Links

**Marc's comments on BCBS FRTB**

**Edit on 27 March 2015**

The public comments made to the consultative paper are now available on the BIS website at: http://www.bis.org/bcbs/publ/comments/d305/overview.htm

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