## Friday, 21 December 2018

### Forward Initial Margin and multiple layers of AD

In the previous blog, I presented the application of AD to the question of Initial Margin (or Capital) attribution between desks in risk-weight based measures. In this installment, I incorporate this feature into a Monte Carlo forward IM computation mechanism. The Monte Carlo forward IM is one of the approach to compute Margin Value Adjustment (MVA). The full MVA also requires the introduction of the cost of funding (the IM) and the discouting; the funding will be the focus of the next installment.

The steps to obtain the forward IM in a Monte Carlo approach for interest rates in risk-weight based measures are the following:
1. Calibrate a multi-curve framework and a dynamic model
2. Evolve the curves to a sample of future dates using random scenarios
3. For each date and scenario, compute the sensitivities (market quotes deltas or bucketed PV01) of the portfolio
4. Computes the IM for each counterparty based on the sensitivities and apply the sub-portfolio attribution (see last blog)
The next steps, to obtain the MVA will be dealt with in the next blog.

### Calibration

The calibration of a multi-curve framework from market quotes is a standard procedure. I refer to my book on the multi-cure framework, Chapter 5, for the details. Note that Algorithmic Differentiation (AD) is already important at this stage. The calibration procedure is often done using root-finding algorithm of the Newton type. This requires the computation of the gradient of the market quotes function. This is done efficiently with AD. A multi-curve dynamic model is required and similarly it needs to be calibrated to the market.

For this blog, I'm using an hybrid multi-curve model, as described in the recent working paper Hybrid Model: A Dynamic Multi-Curve Framework. This is a relatively simple model than can be calibrated to the term structure of volatilities and includes a stochastic basis between LIBOR rates and OIS rates. This feature will be important when discussing cost of funding in the next instalment.

### Evolution

To evolve the curves, we use very standard techniques. The model describes the curves (OIS discount factors and LIBOR processes) at a future date in an explicit way based on Gaussian distributions. It is easy to obtain the value of discount factors and LIBOR processes at those forward dates.

### Sensitivities

The next step, obtaining the sensitivities of the portfolio with respect to the market quotes in the forward scenario, is probably less standard in derivative pricing. Most of derivative valuation is based on values and cash flows, not on risk measures. The technical requirements are not be very different: we have a model evolved curve and we want to compute a result that depends on those curves. But there is an extra catch, what we need for the risk-weight based measure is the sensitivity with respect to the methodology select market quotes, not arbitrary model parameters. Those market quotes are ot provided directly by the model. Fortunately, even if this is no something that we do explicitly in many places, this is something we do implicitly. With a variation of AD techniques this can be implemented efficiently. This can be obtained by a mixture of the chapters 5 and 6 ("Derivatives to non-inputs and non-derivatives to inputs" and "Calibration") from my Algorithmic Differentiation in Finance Explained book.

Once the sensitivities are computed for each trade at each date ad for each scenario, the measure and his attribution by portfolio is simply applying the techniques described in the previous blog.

### Figures

A picture is worth a thousand words. So let's put the above ideas in pictures.

First we take only one Monte Carlo scenario and look at the attribution. I have selected a small portfolio with one counterparty containing 30 swaps split between 6 sub-portfolios. The attribution is done on the different sub-portfolios.

The total IM is represented in red. The attribution is done using the Euler method described in the previous blog. With this attribution method, the offsets between positions are taken into account. This explain why some desks (Desk 2 and Desk 6) have negative attributions.

Figure 1: Forward IM attribution between desks.

What we can also see is that the relation between the different attributions change through time. Today, desk 1 is the largest, while through time, desk 2 is becoming the largest and even the only meaningful contributor after 8 years. This emphasises that a MVA attribution based only on today's spot IM attribution would not provide a faire representation of the contributions. The attribution along the path is really required.

Once we have done the attribution on one path, we can look at how the forward IM behaves along the different paths. In this case, we have kept the IM methodology unchanged through the life of the portfolios. In practice, the model parameters are reviewed on a regular basis (at least annually in case of SIMM). We should introduce change of methodology along the paths. This is not done here and may be the subject of another blog at a latter stage.

An example with 7 paths is proposed in the graph below. We use only a small number of paths to avoid overloading the pictures. The performance analysis will be done with more paths.

Figure 2: Forward IM along different paths

The least we can say is that the graph is underwhelming. This can be explain easily as our portfolio contains only vanilla swaps the present values of which are almost linear in the underlying market quotes. As the IM methodology is sensitivity (first order derivatives) based, the IM numbers do not change significantly from one path to another. This does not mean that multiple paths are unnecessary for MVA, as we will discuss in the next blog.

Obviously a financial institution will have more than one counterparty. The next figure reproduce the example of the forward computation with three different counterparties.

Figure 3: Forward IM for different counterparties

### Performance

What is the performance of such an implementation and where are the bottlenecks?

We have ran the above approach on a portfolio of 90 swaps split between 3 counterparties and 6 sub-portfolios. The horizon is 11 years with semi-annual dates and 101 paths. The total computation time (1) was 18s. The split is:
• Calibration in 340 ms.
• Path random variables in 11 ms.
• Paths fixings in 348 ms.
• IMs in 17430 ms.

The first line is the original calibration of the curves from market data stored in CSV files. The second line is the loading of the portfolio from a csv file. The model is a two-factor model based on Gaussian distribution, generating the underlying random variable was 11 ms. As the trades we want to value age through the different dates, for each path we need to generate a full time series of fixing consistently with the model used; not only on the path date but on all intermediary dates as a swap can have a fixing at any date. As we have OIS in the portfolio, in practice really each single date will be required. That is also relatively quick (348 ms).

As we expect, the bulk of the time is spent computing the sensitivities and combining them in the IM. One of the time consuming task is computing the market quote Jacobian matrices required to obtain the sensitivities to market quotes, even if the model does not provide the market quotes directly. In our example we use two curves (OIS and LIBOR) with 12 nodes. Computing the Jacobian is similar to computing 24 swaps parameter sensitivities, each with respect to 24 nodes and inverting a matrix. The inversion itself is almost irrelevant in term of computation time. We are left with the parameter sensitivities. In our implementation, this is done by AD and it takes around 6 PV time while a finite difference would take 25 PV times. A gain of a factor 4 for this task.

Then there are the sensitivities of the 90 swaps in the portfolio. The computation time was around twice the Jacobian computation time. The swap in the portfolio are not all long term, so the ratio with the Jacobian is the right size. Like for the previous element, the gain here is probably a factor 4 thanks for AD. This emphasises that for computational efficiency reason, it is better to run the simulation for all counterparties in one run. One of the time consuming task, the Jacobian computation, is common to all counterparties. Note that the representation of the swap here is the full representation with all the conventions, holidays and idiosyncratic details.

We come finally to the object of the previous blog, which was attribution. The computation of the IM itself, the marginal of each exposure and the attribution to 6 sub-portfolios took around 10 times less computation time than the Jacobian. The use of AD here has probably brought a gain of a factor 2 or 3, but this is almost inconsequential as the IM computation time from the sensitivities is dwarfed by the computation time of the sensitivities.

### Conclusions

On the performance side, the computation of forward IM using a risk-weight based methodology through Monte Carlo approach is feasible in reasonable time. The AD implementation brings real benefits. More curves are involved, more benefits it will bring. The measure computation from sensitivity itself is relatively fast and improvement to that computation are almost invisible in the final computation time.

On the business side, doing the attribution at each forward date is important to attribute the MVA correctly. A simple attribution based on the spot IM would provide unreliable results.

In forthcoming blogs we will look at the cost of funding, the change of the IM methodology parameters through time and the computation of marginal MVA.

(1) Time computed on the author laptop (MacBook Pro 13' , 3.1 GHz Intel Core i5). Personal Java code on a single thread.

## Wednesday, 12 December 2018

### Initial margin and double AD

The use of Algorithmic Differentiation (AD) in finance as become more popular in the last 5 to 10 years. AD can be described as "the art of calculating the differentiation of functions with a computer". An introduction to AD in finance can be found in my book with the same title.

The efficient computation of derivatives has been traditionally used in finance to compute the "greeks" associated to financial instruments and in particular the deltas or bucketed PV01.

Recent regulations have pushed in the direction of more computation of cost of capital for market risk (FRTB) and Initial Margin for uncleared trades (Uncleared Margin Regulation - UMR). The method used by most financial institutions already under the UMR is the ISDA proposed SIMM™ approach. The approach is very similar to FRTB capital computation with some small twists. The base idea of both is to compute a VaR-like number based on conventional risk weights and correlations. This is equivalent to a delta-normal VaR computation in the RiskMetrics style but with variance-covariance matrix in a stylized format with prescribed values. I will use the generic term of risk-weight based measure for those capital or IM methodologies.

The "delta" part of those methodologies is relying of the computation of PV01. This is where AD has been traditionally used in finance. This is the first layer of AD related to risk-weight based measure methodologies. As this is relatively standard, I will not focus on this aspect in this blog.

### Marginal measure

A second topic for which Algorithmic Differentiation can bring significant improvements is the topic of marginal risk measure and measure attribution. The marginal measure is the increase in the measure coming from adding a small sensitivity (or trade) to the existing portfolio. This is the derivative of the measure with respect to an increase in the sensitivity/exposure. This marginal measure can be computed at the single sensitivity level or at the trade level or at any combination of trades level. In the rest of the blog, I will consider the marginal measure at the most atomic level of our problem, the level of a single sensitivity. Obviously if the marginal measure is available at the lowest level, the marginals can be combined to obtained the marginals at any level above that. From a computational perspective, the lowest level of marginals is the most expensive and if we can solve it cheaply, then we can solve any other combination cheaply also.

In general a measure (Capital or IM) attribution between sub-portfolio is a way to divide in an additive way the total measure of a portfolio between different sub-portfolios.

The Euler attribution is based on the Euler's homogeneous function theorem. The theorem provides an equality for positively homogeneous functions. The standard approaches to capital, IM and VaR are in most cases positively homogeneous. This is the case for FRTB, SIMM (below the concentration risk threshold) and Delta-Normal VaR.

What are we trying to do with attribution? We start with a portfolio made of sub-portfolios. We have k sub-portfolios denoted Pi and the total portfolio is P:
P = Σi=1,...,k Pi = Σi=1,...,k 1 x Pi

We want to split the measure for the total portfolio in an additive way between the different sub-portfolios.  We cannot use directly the measure of each sub-portfolio as the measure itself is not additive.

The following equality, called Euler's homogeneous function formula, is satisfied for positively homogeneous functions
f(x) = Σi=1,...,k xi Di f(x)

We have a function which represents the measures μ on portfolios
f(X) = f((Xi)i=1,...,k) = μ(Σi=1,...,k Xi x Pi)

The measure applied to the total portfolio is
μ(P) = f(1,1,...,1)

Euler's theorem suggests an attribution based on
μ(P) = Σi=1,...,k 1 x Di f(1,1,...,1)

One of the reason this attribution is used is that it takes into account the offsets between sub-portfolios.

If you have the derivatives of the function f with respect to each individual sensitivity in the sub-portfolios, obtaining Di f(1,1,...,1) is simply the question of adding numbers for the sensitivities in the sub-portfolio.

### Performance example

What is the performance in practice of this method combined with AD? For this I have used a simple portfolio with 20 sub-portfolios and 500 exposures each. This is a total of 10,000 exposures. The measure selected is an IM computed using the SIMM methodology.

If we compute(1) a single IM for the portfolio (10,000 exposures), the computation time is 3.4 ms. If we were to compute the marginal IM of each exposure by finite difference, it would multiply the computation time by 10,000 (34,000 ms). If we were to compute the marginal IM for each sub-portfolio by finite difference it would multiply the computation time by 21 (714 ms).

What do we obtain by Algorithmic Differentiation? For the above portfolio, the time required for the measure, all the 10,000 marginal IM and the 20 sub-portfolios attribution is 10.3 ms. Obtaining all those 10,000+ figures multiplies the computation time only by 3. This is in line with the theory (on the good side of the range). This is more than 3,000 time faster than by finite difference!

Savings from full marginal IM : 3,000 times shorter computation time
Savings from full IM attribution: 7 times shorter computation time

### Conclusion

Using AD at two levels for risk-weight and correlation based risk measures improve significantly the computation time for marginal measures and attribution.

In a forthcoming blog, we will combine that with other uses of AD in MVA computations. We will add a third layer of AD. But that will probably be after the Christmas period.

(1) We have run all computations described 100 times in a loop and the figures reported are the averages by IM computation. If we run it only once, the times are too small. All times reported measured on the author's laptop running personal Java code.

Material similar to the one described in this blog was presented at the WBS xVA conference in March 2017 and at a Thalesians seminar in April 2017, that seminar that led The Wall Street Journal to use my picture (incorrectly to my opinion) in the article "The Quants Run Wall Street Now".

## Saturday, 8 December 2018

### Copenhagen Risk conference and workshop - 23-24 January 2019

Marc Henrard will present a seminar at the conference

CFA Society Denmark Risk Conference

which will take place on Wednesday 23 January 2019. The agenda of the conference can be found on the organizer web site:

On the next day, Thursday 24 January 2019, he will present the workshop

The future of LIBOR: Quantitative perspective on benchmarks, overnight, fallback and regulation.

The agenda of the workshop and registration details can be found on the organizer web site:

Marc will be in Copenhagen from 22 to 24 January. Don't hesitate to reach out if you want to meet during that time.

## Wednesday, 5 December 2018

### Event

Marc Henrard will attend the conference

Annual Forecast Event

which will take place at The Hotel Brussels on Monday 10 December 2018. The agenda of the conference can be found on the organizer web site:

Don't hesitate to reach out to Marc at the conference.

## Thursday, 29 November 2018

### Course "The future of LIBOR: Quantitative perspective on benchmarks, overnight, fallback and regulation"

Following the request by several clients, we have developed a training/workshop around the new benchmarks and the LIBOR fallback. A typical agenda of the course is presented below.

• Cash-collateral discounting.
• The standard collateral results and their exact application.
• What is hidden behind OIS discounting (and when it cannot be used)?
• Impact of new benchmarks on valuation
• EU Benchmark regulation
• The``alternative'' benchmarks:
• Progress in different jurisdictions
• SOFR, reformed SONIA, ESTER, SARON, TONAR.
• Secured v unsecured choice.
• Curve calibration
• SOFR and EFFR: two overnight rates in one currency!
• Status in different currencies. Cleared OTC products, liquidity. The different consultations in progress and what to expect from them.
• Fallback options
• ISDA consultation
• The different options for the "adjusted rate"
• Quantitative impacts: convexity adjustments and risk
• Risk management of transition.
• Risk impacts
• Potential impacts on systems
• What a risk solution would look like?
• Multi-curve: double or quit?
• Interest rate modelling
• New products associated to new benchmarks
• Futures on overnight benchmarks
• Deliverable swap futures
Detailed lecture notes for participants.

The training is usually proposed as a one-day program.

Don't fallback, step forward!

Other course proposal available on our Training Page.

## Saturday, 17 November 2018

### Event "Financial Regulation and Stability after Brexit"

Marc Henrard will attend the conference

Financial Regulation and Stability after Brexit

which will take place at the Palais d'Egmont (Brussels) on 21 November 2018. The agenda of the conference can be found on the organizer web site:

Don't hesitate to reach out to Marc at the conference regarding the quantitative impacts of regulation in the financial markets.

## 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.

## Friday, 31 August 2018

### Course: Interest Rate Modelling in the Multi-curve Framework: Collateral and Regulatory Requirements (2)

Marc Henrard will present the course

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

in New York on 1-2 October 2018.

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

## Tuesday, 7 August 2018

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

Marc Henrard will present the course

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

in London on 24-25 September 2018.

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

## Sunday, 5 August 2018

### Market infrastructure: A quant perspective on IBOR fallback proposals

A new note in the series Market infrastructure developments analysis is now available on SSRN. The note, titled

A quant perspective on IBOR fallback proposals

is written by Marc Henrard

#### Abstract

With the increased expectation of some IBORs discontinuation and the increasing regulatory requirements related to benchmarks, a more robust fallback provision for benchmark-linked derivatives is becoming paramount for the interest rate market. Several options for such a fallback have been proposed. This note describes and analyses some of those options. The focus is on the quantitative finance impacts. None of the options that have been proposed fits all of the criteria for a good fallback provision. It appears that some of the options that have gained traction failed even the achievability criterion. The note concludes with the author's personal preference.

The note is available on SSRN: http://ssrn.com/abstract=3226183

## Saturday, 30 June 2018

### IBOR Global Benchmark Transition Report

ISDA has published an IBOR Global Benchmark Transition Report. The report is available at : https://www.isda.org/2018/06/25/ibor-global-benchmark-transition-report/

The survey on which the report is based indicates that there is a general market awareness regarding the transition and that market participants are starting to prepare. Nevertheless, less than a quarter of the survey participants have started concrete steps in that direction.

Don't hesitate to contact us regarding IBOR and benchmark transition. We have worked intensively on related issue for several years. We provide to our clients a rapid access to the foundations on which to build their added value.

Our services include expertise and tools on valuation issues and senior staffing for our clients to initiate their program from a quantitative point of view. We also have a long experience in executive education and technical workshops.

## Thursday, 12 April 2018

### Working paper: Variation margin in presence of trade cash flows

A new working paper by Marc Henrard is now available on SSRN. The title of the paper is

Variation margin in presence of trade cash flows

With the generalisation of Variation Margin (VM) collateral, the derivative world is not driven anymore by discrete cash flows but by continuous dividend. Due to practical constraints, the VM is paid with a one day delay. This delay reduces significantly the effectiveness of the margin process as credit risk exposure reduction around the trade cash flow payments. This note presents an efficient and simple approach to bring back the effectiveness of the VM process even around trade flows dates. The approach is based on the usage of a forward valuation in the VM computation process.

The paper is available on SSRN: https://ssrn.com/abstract=3154329

## Sunday, 1 April 2018

### New working paper: overnight futures

A new working paper by Marc Henrard is now available on SSRN. The title of the paper is