## The risk transition in LIBOR transition

This post continue on the "transformers" series related to LIBOR discontinuation and follows our Quant perspective on IBOR fallback. In this episode, we discuss the transition or transformation through time of the risk of a fixed portfolio. The explanation is done using the graph in Figure 1. If the meaning and content of the graph are obvious to you, then there is no need to read further; if this is not the case, you may want to spend a little bit of time reading.

Figure 1: A cryptic graph to be explained later.

### Single period swap

We start with the simplest portfolio, composed of a single swap on a single period. The date of the analysis is 30-Aug-2018 as in the previous episodes. The swap has a start date in 12 months and a 3-month tenor on USD-LIBOR-3M. The notional is 100m.

We look at the risk through the glasses of PV01. We compute the market quotes bucketed PV01 with respect to each tenors and then sum them by curve (OIS and LIBOR3M). This gives us two numbers for each date. Like mentioned in the previous episodes, those numbers have to be taken with a pinch of salt as they are obtained by adding sensitivities to different market realities (market quotes from different instruments with different conventions). They are enough for the qualitative analysis we perform, but may not be perfect for all purposes.

We first look at the trade risk in absence of LIBOR discontinuation. The risk is composed of the risk to the LIBOR fixing for roughly 2,500 USD/bp (100m/10,000/4) and a very small discounting amount from the fact that the swap is not ATM. The Y axis of the graph represent the PV01 in K USD/bp for the LIBOR and the OIS curves. The X axis is the date on which the risk is computed. To avoid complicating the picture, we have used the rate as of the first date and computed the implied forward curves for each day in the following year. The risk are computed with those forward curves. If we had used the actual market curves for each day, there would be on top of the changes described here some small ups and downs due to market fluctuations.

Figure 2: Risk transition for a one period swap in absence of discontinuation.

We now introduce the Announcement Date and the Discontinuation Date. We suppose that the announcement is 30-Dec-2018 and the actual discontinuation is 28-Feb-2018. Those dates do not affect our previous risk graph but we reported the dates for visualization facility.

Now we introduce a fallback option, starting with the OIS Benchmark option. The reason to start with that one, even if this is not in the ISDA consultation, is that this is the one the closest to the actual LIBOR in term of risk profile.

The big change happens on the announcement date. The only fixing in our swap is after the discontinuation date, it is then replaced by a fixing to the OIS benchmark. In term of risk, the OIS benchmark is on the Discounting/Overnight/OIS curve. On that date, the risk jumps from the LIBOR curve (dashed light blue) to the OIS curve (dashed dark blue). Then nothing spectacular happens to the risk up to the fixing date. On that date the risk decrease dramatically when the rate is known, leaving only a residual small OIS risk (coming from the difference between the fixing and the fixed rate of the trade) which disappears completely at maturity.

Note also that it is possible that the OIS fixing and LIBOR fixing dates will be slightly different because of non-good business days. For example USD-LIBOR is fixing according to the London calendar but SOFR according to the US Government Securities calendar (and obviously this is not yet defined for the OIS Benchmark financial fiction we use here).

Figure 3. Risk transition for a one period swap. OIS Benchmark option added.

Once the profile of one option is understood we can add the other three. The LIBOR profile will be the same for all options. It goes from something before the announcement date, and that something is the same for all options, to nothing. We do not repeat that part to avoid overloading the graph.

The other options included are Spot Overnight, Compounding Setting in Advance and Compounding Setting in Arrears. For the Spot Overnight, the fixing is also on one date, so the profile is very similar around the fixing date to the OIS Benchmark. The total PV01 risk between the announcement date and the fixing date is quite similar to the previous one. As discussed in a previous episode, this is not true when looking at the tenors/buckets level. For the Compounding Setting in Advance, the risk start to decrease three months before the actual fixing date. The fixing is obtained by compounding the rates over the three-month period preceding the fixing. So each day that is passing a small piece of the rate is know and there is no risk anymore on it. Each day the risk is decreasing slightly. Finally for the Compounding Setting in Arrears, the risk is roughly constant up to the start of the theoretical deposit underlying the fixing and slightly decrease up to the maturity date of the same fixing. This is a translated version of the previous description.
Figure 4. Risk transition for a one period swap. Legacy and all fallback options.

### Multi-periods swap

We now change the underlying instrument to a two-year swap starting in three months. The announcement date is 30-Dec-2018 and the discontinuation date is 30-Sep-2019. The swap has 8 3-month periods. The announcement date is in the first period and the discontinuation date is in the fourth period.

The profile in the absence of discontinuation is a standard profile with a small discounting risk and a LIBOR risk that steps down at each fixing date (yellow).

When we introduce the fallback, on the announcement date, the LIBOR risk of all the fixing after the discontinuation date (4) are transferred to the OIS curve. The light blue dashed line drops on the announcement date by the equivalent of 4 fixings risk and the OIS risk jumps in the opposite direction. The 3 fixings that are between the announcement date and the discontinuation date are not affected by the fallback, this is why they is still three quarterly drops on the LIBOR light blue line.

Figure 5: The risk profile for the OIS and LIBOR curves for a two-year swap. Legacy swap and all fallback options.

The three options propose a slightly different profile around the fixings. The OIS Benchmark and Spot Overnight are similar to the original LIBOR with risk drops. The Compounding options have the risk that linearly decrease (actually not completely linearly, but by a discrete drop of the same amount each day); they differ on when this decrease starts: on the fixing date or one original index tenor before.

No risk is lost in transition, most risks are transformed. In the end all risks die.

Don't fallback, step forward!

## Tuesday, 30 October 2018

### Course: Interest Rate Modelling in the Multi-curve Framework: Collateral and Regulatory Requirements - next delivery

Marc Henrard will present the course

Interest Rate Modelling in the Multi-curve Framework: Collateral and Regulatory Requirements

in New York on 25-26 March 2019.

The course details can be seen on the London Financial Studies web site at https://www.londonfs.com/programmes/interest-rate-modeling-new-york/Overview/

## Monday, 29 October 2018

### LIBOR Fallback transformers - magnified view on risk

In this new post, following questions by readers, we come back to the delta ladder associated to the different fallback options. In the LIBOR Fallback transformers - PV and Delta post, we have presented the risk associated to the first coupon of the forward swap. The coupon starting in 6 months and ending in 9 months. In the case of Overnight Spot fallback option, from the delta ladder, it looks like the the sensitivity of a 3-month instrument on the "wrong" period, while from the description it should be a 1-day risk starting at the fixing. Actually both descriptions are correct, but the delta ladder one is looked at with the wrong glasses.

The delta ladder using tenors with 3-month gaps is a good way to look at the risk when the risk is on 3-month benchmarks. When we look at single overnight fixings, like in the case of the Spot Overnight fallback option, you don't see clearly through those glasses. To see to which extend the glasses are deforming the view of the risk, we have recreated the delta ladder with weekly tenors up to 40 weeks (~ 9 months). Obviously there is no liquid market of weekly swaps up to 40 weeks, but we have created synthetic quotes for them and calibrated (synthetic) curves with those nodes and computed the sensitivities at each of them (by Algorithmic Differentiation, indeed)

If we look at the OIS Benchmark risk, nothing is changing, there is some risk at 26 weeks (6 months) and some risk at 39 weeks (9 months). If we look at the Overnight Spot option, the picture is very different, there is very large sensitivities with opposite signs at 25 and 26 weeks (note that due to the T+2 convention for OIS, the starting date of the LIBOR period is not the same as the starting date of the ON period). Nothing like a 3 months risk. There is clearly a very short term risk around that date. There is also a very small discounting risk at the payment date (39 weeks), but it is so small that you would be excused if you had not noticed it. Finally we look at the Compounding Setting in Advance case. We see large risks at 12 weeks and 26 weeks, corresponding to the period where the rate is compounded and a very small risk at the 39 weeks for the discounting of the payment.

OIS Benchmark Spot Overnight Compounding in Advance

Note that the total (parallel)  delta is almost the same for all options.

Looking at the reports, you will say: "Nobody is producing those kind of reports with weekly (or worst daily) risk". And I agree with you that nobody is doing something like that today. But if the Overnight Spot option is selected by ISDA, tomorrow everybody will be forced to do something similar. Risk reports with daily nodes will be required, specially in periods where the central banks are likely to hike (or cut) rates.

The same issue can be seen for the large test portfolio used previously.  Instead of showing it with a risk report as above, we are showing it by plotting the daily risk sensitivities. The graph below represent the LIBOR sensitivities at each fixing date. Each line represent the change of value of the portfolio for a change by one basis point of the forward rate fixing on that date. This is a risk report with daily precision. We displayed only a little bit more than 4 years and not the 50 years of sensitivities in the test portfolio. Those LIBOR-3M sensitivities are transformed by the fallback with Spot Overnight option to overnight sensitivities. Where there was before some averaging on a 3-month period, there is now a spike on a unique day. No risk compensation by overlapping periods is achieved anymore.

Don't fallback, step forward!

### LIBOR Fallback transformers - timing adjustments

As for previous blogs, we do it first for a single swap and then for our large test portfolio. We look at the timing adjustment for the Spot Overnight option.

For the adjustment we have selected the one-factor Hull-White model and the corresponding adjustment as described in our Quant Perspective on IBOR fallback Proposals.

In this case, we have selected a 10Y swap starting in 6 months. The coupon is 2.5% and the adjustment spread is 25bps. For the model parameters, we have taken round figures with the mean reversion at 2% and the (normal) volatility at 100bps.

The impact on the swap par rate is 1.2bps. This may look like somehow smaller than what is displayed in Figure 4 of the quant perspective, but in the paper the adjustment was for a single payment, while here it is for the full swap. The adjustments are somehow averaged between all the payments on the 10Y tenor.

If if lengthen the swap's maturity to 30Y, the impact is 2.4bps.

At the level of the delta, the difference between the adjusted and non-adjusted figures are very small.

If we look at the impact on a full portfolio, this can be important. For a test portfolio with 1000 swaps (slightly different from the installment 2), the PVs without and with timing adjustment are

The large difference comes from large positions in the very long term part of the curve for the LIBOR curve.

Don't fallback, step forward!

## Sunday, 28 October 2018

### LIBOR fallback transformers - forward discontinuation

The procedure for LIBOR discontinuation refers to two dates: the announcement date and the discontinuation date. The announcement date is the date when one of the triggers is made public, e.g. the benchmark administrator announces that he will cease to provide the IBOR permanently. The discontinuation date is the date when the benchmarks discontinuation is effective. It is expected that there will be some time between the two dates. One potential scenario would be that by mid-2021, the majority of LIBOR contributors, that have agreed with the FCA to contribute up to end-2021, have the opinion that they do not trade enough interbank term deposit anymore to contribute efficiently to the process; the lack of contributor prompt the administrator to announce the discontinuation, effective 1st January 2022.

In that period between the announcement date et the discontinuation date, the existing swaps (or other IBOR derivatives) are becoming some kind of hybrids. The payments related to IBOR with fixing before the discontinuation date are still IBOR coupons while the ones with fixing after the discontinuation date are becoming some type of overnight-indexed derivatives. The exact type depends on the fallback option.

We have run this type of scenario on different books. We start with the simplest case of one swap. We use the same swap as the one we have used in the first episode (Fallback transformers - Present value and delta). The delta for the OIS benchmark fallback type is provided below.

Bucketed delta for a discontinuation date 1-Jan-2022 as viewed from 29-Aug-2018. OIS benchmark fallback option.

We have selected the OIS benchmark type to start, as for this option the fixing for LIBOR and for the adjusted RFR are on the same period. The risk is transferred from LIBOR to overnight, but is quite similar in term of tenors. We see what we would expect: the swap behave as a LIBOR swap up to end 2021 (last LIBOR fixing for this swap: 2-Dec-2021; payment date of last LIBOR: 4-Mar-2022). From there on the swap has the risk of an OIS. For all practical purposed, this is an LIBOR swap up to early 2022 combined with a forward starting OIS starting in early 2022.

For other options, some interesting behavior can be observed. For example for the Compounded Setting in Advance, the risk for the period from December 2021 to March 2022 appears twice, both in LIBOR and in overnight. The reason is that the fallback procedure fixes the payment for Mar 2022 to Jun 2022 on the compounded rate for a period before the coupon start accrual. The period Dec 2021 to Mar 2022 is used in the last LIBOR coupon and in the first overnight coupon.

Bucketed delta for a discontinuation date 1-Jan-2022 as viewed from 29-Aug-2018. Compounding setting in advance fallback option.

If we look at a portfolio level, the picture is not very different. We have a LIBOR exposure up to beginning of 2022 and only OIS exposure after that. We used that same portfolio as in the previous episode (LIBOR Fallback transformers - portfolio valuation)

Bucketed delta for a discontinuation date 1-Jan-2022 as viewed from 29-Aug-2018. Large portfolio example with OIS Benchmark option.

Don't fallback, step forward!

## Tuesday, 23 October 2018

### LIBOR fallback consultation - our answer

The ISDA Consultation on Certain Aspects of Fallbacks for Derivatives Referencing GBP LIBOR, CHF LIBOR, JPY LIBOR, TIBOR, Euroyen TIBOR and BBSW closed yesterday.

We have send our answer to the consultation last week. The document of our answer can be found below.

## Monday, 22 October 2018

### LIBOR Fallback transformers - portfolio valuation

In this transformer's blog installment, we look at the fallback options impact on a portfolio level.

The portfolio we used for the examples is a portfolio of 1,000 USD swaps semi-randomly generated with different maturities, rate levels and conventions. Some trades are Fixed v LIBOR 3M, some OIS and some basis Overnight V LIBOR 3M.

### Present value

In the previous blog, we have seen that at a single trade level, the difference between the different options are not very large. On a portfolio level the result may be different. For the portfolio we have tested, we obtained the following results:

We have used a fixed spread of 25 bps. What we want to analyse is not only the difference between the legacy portfolio and the portfolio after fallback, we know that it will depend strongly on the spread selected, but also the difference between the different options.

The first test was run with curves calibrated with linear interpolation on the zero-rates. We see that, as expected, the Compounded Setting in Arrears and the OIS Benchmarks options gives almost the same value. Other options give relatively large differences, up to 160m on this large portfolio.

### Computation time

A natural question, looking at all those options and the present value and sensitivity computations, is to know how much computation power is required to obtain those results. The answer is probably less than what most would expect. In the first test for the 1000 trades portfolio above and for the legacy description plus the 4 options analysed, the total computation time for the PV and all the delta ladders was around than 2 seconds (to be exact, 250ms to load the portfolio from a csv file and 1900ms to generate the 5 versions of the swaps and compute PV and delta for all versions; time computed on the author's laptop; time for other interpolation mechanisms in next section).

### Interpolation

As the options change the way the rate is computed, we can expect the interpolation mechanism to have an impact. In the quant perspective, we have described how the interpolation mechanism has an impact on the forward spread, there is a lot of literature on the impact of interpolation on forward rates, and here we check the impact of the fallback option on the PV (through the forward rates) and see how much difference there is between the options depending on the interpolation mechanism. The interpolation schemes we have used are linear on the zero-rates, monotone cubic spline on the discount factors, natural cubic spline on the zero-rates and log-linear on the discount factors.

For the three options OIS Benchmark, Compounded in Arrears and compounded in Advance, there is a significant difference with LIBOR and between them, but the difference does not change significantly with the interpolation mechanism. This is not really surprising as each of them has an averaging mechanism on the coupons and the portfolio has a decent diversification in term of maturities and payment schedules. All yield curves are calibrated to the market and this is sufficient to have a relatively stable impact. In the case of the Spot Overnight option, the story is different. In this case, the rate selected for the fallback is a single day rate taken outside of the LIBOR period. The averaging and diversification does not work as well. The impact of the fallback can vary from almost nothing to more than 100m. It would certainly be possible to create portfolios where those impacts are even larger, just by selecting coupon fixing dates at special dates.

Don't fallback, step forward!

## Sunday, 21 October 2018

### LIBOR Fallback transformers - PV and Delta

The trade we look at is a plain vanilla payer USD Fixed v LIBOR 3M starting in 6M and with a tenor of 5Y. The coupon is 1% and the notional 100m. The trade is largely in-the-money (the forward rate is 2.95%). As SOFR-indexed instruments are not liquid yet, we have used EFFR curves as a proxy.

### Delta ladder (Bucketed PV01 - Key rate duration)

We first look at the delta risk (par rate sensitivities) implied by the different options (without convexity adjustment at this stage). The delta ladder is as expected, with most of the risk on the LIBOR curve, short the rate on the 6M and long the rates at 5Y6M (between 5Y and 7Y with most of it on the 5Y). As the trade is largely in-the-money, there is also a non-negligible delta on the OIS curve.

The total for each column has to be taken with pinch of salt. The individual numbers are not with respect to the same instruments and the conventions are not the same for all of them.

The first option we look at is the OIS benchmark. This is not part of the ISDA list (part of our quant perspective proposal). We start with it because this is the closest to the IBOR trade is term of dates and structure. The adjusted RFR is also a term rate and the term has, by construction, exactly the same dates as the theoretical deposit underlying the LIBOR.

The delta is entirely moved from the LIBOR curve to the OIS curve. On the tenor side, it is very similar, short on the 6M and long between 5Y and 7Y.

 Delta ladder for the option OIS Benchmark
The second option is the ISDA proposed Compounded Setting In Arrears. As discussed in details in our perspective, this option will either break the coherency of a single fixing and/or not be achievable in practice. Nevertheless we include it here as we deal only with vanilla swaps where it is achievable. For the vanilla swaps, if we ignore the detail of non-good business days, the valuation and risk are the same as for the OIS benchmark option as proved in the Section 5.2 of our perspective. Not surprising, the risk in this case is almost the same as in the previous one.

Delta ladder for the option Compounding Setting In Arrears

The next option we look at is the ISDA proposed Spot Overnight. This is the replacement of a term benchmark with an overnight benchmark with the same fixing date but a very different underlying period (and economic reality). From a delta ladder perspective, we see that most of the risk as been translated by three months backward.

Delta ladder for the option Spot Overnight

If we take the first period, with a LIBOR risk from 2BD+6M to 2BD+9M, it is now an overnight risk from 2BD+6M-2BD to 2BD+6M-1BD. A delta appears on the 3M because the forward rate of the first fixing is now impacted by the 3M rate and the 6M rate. If we were to look at the sensitivity of a 6Mx3M swap (one period LIBOR-3M swap starting in 6M), the risk from the forward is negative on 3M, positive on 6M and there is a little bit of discounting on the 9M.

What the delta ladder does not show is that most of the sensitivity is concentrated on few dates, one every 3 months. If the delta ladder had daily points we would see spikes at those dates. If such an option was selected, the delta ladder as presented here would become almost obsolete for risk management and other views of the risk would be required.

The last option we look at is the ISDA proposed Compounding Setting in Advance. The rate is for a period of the same tenor as the LIBOR and the rate is overnight rates compounded, but the period is the period before the fixing instead of the period after in the LIBOR case. The impact on the delta is that the sensitivities related to the forward rates are move backward by three-months in the ladder.

Note that the total delta is not very different from one option to the other.

Up to now, we have described only the risk (delta) for a given fixed adjustment spread of 0. What is happening when the spread is modified? In the next figure, we have displayed the impact of modifying the spread from 0 to 50 bps. The result is rather unspectacular. The change of present value is linear in the spread and the different options generate rather similar present values.

Profile of present value for different levels of spread.

At the level of a single swap the present values do not look very different and the value transfer is coming more from the spread choice than from the adjusted RFR option choice. When we will look at a large portfolio level in the next blog, we will see that those small differences add-up and can lead to significant impacts.

Don't fallback, step forward!

### LIBOR Fallback transformers!

In our quant perspective on IBOR fallback, we discuss the quantitative finance aspect of the fallback and its different options, even adding some options to the ones proposed by ISDA.

Once the theoretical side is done, we can move to the practical side. What would happen, not to a single payment and its detailed dates and convexity adjustment, but to a large portfolio? Portfolios a composed of hundred, thousand or even tens of thousand of swaps with many offsets. How do we estimate the impacts of the fallback when the trades' term sheet is transformed by it? Is there a quant solution to quickly analyze the fallback impacts?

This is not anymore a simple single curve to multi-curve problem. What we witnessed in 2008 by the move from single curve (LIBOR discounting) to multi-curve (OIS discounting) is very small with respect to fallback. In the multi-curve issue, it was simply the valuation estimate that was changed on a trade that remained unchanged. Here we have the trade itself - and obviously its valuation estimate - that changes.

Moreover, most of the ISDA proposed fallbacks lead to non-standard trades of which you probably don't have a single instance in you current portfolios - this is the case for option 1, 2 and 4. You have to transform the trades, in a different way for each option, and apply valuation method on each of them. Not a standard task for quants and risk managers. Your trades may be stored in data bases or systems that do not even support those new term sheets.  In theory, this is not very difficult, in practice this could be a lot of work especially that only one of the fallback options will be actually implemented, so 3/4 of the transformation work will just be thrown away.

As we have been working on the fallback issue since more than 6 months, we have developed a lot of  ideas, formulas, code and insight into the problem. In particular we have created a

Fallback Transformer.

The idea is that you pass your portfolio of legacy swaps, they are transformed into different version associated to the fallback options and for each option valuation and risks are computed in a flexible way. Depending of the hypothesis on the spread adjustment, the pricing mechanism and the curves, you can easily have 10 versions of the fallback's impact. It will not tell you exactly how to solve all your issues, but it will allow you to be

We will describe some extracts of the transformer results in forthcoming blogs.

The first part (to be published later today) describes the present value and (delta) risk of a single trade for the different options. The second part describes the valuation impact on a large portfolio and the computation performance. In the third part, we we look at the convexity adjustments.

Don't fallback, step forward!

## Saturday, 6 October 2018

### Risk-based overnight-linked futures - innovative design

Some years ago I proposed a innovative design for risk-based swap futures.

I worked with ASX to adapt it to the AUD BBSW swap market and a product based on that design has been traded on the exchange since 2016 (even if it has not attracted a lot of trading activity).

With the new importance of overnight benchmarks, the ETD market has to find a product that could be used for price discovery and risk management of the full term structure of interest rates.

I have detailed a version of the generic futures design to match the overnight conventions and OTC markets. The product has been presented to the main interest rate futures exchanges in Europe and the USA. We will see if it takes a new life.

All the technical details are now available in the form of a muRisQ Advisory Market Infrastructure Analysis. The document can be downloaded from SSRN at https://ssrn.com/abstract=3238640.

In the graph below I have displayed the convexity adjustments between the ETD futures and the OTC swaps that can be obtained with this design (in red) and the ones with the current design of overnight-linked futures. See the paper for the exact description of the graph.