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.

Euler attribution

The marginal measure is also closely linked one standard method of attribution, the attribution method called "Euler attribution".

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


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


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.
    • What about term rates?
    • 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"
    • The different options for the "adjustment spread"
    • Quantitative impacts: convexity adjustments and risk
    • Clearing house adoption
  • 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!

Contact us for our LIBOR fallback training and quant solutions.

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.

Thursday, 15 November 2018

LIBOR Fallback Transformers - Risk transition

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. 

  1. Fallback transformers - Introduction
  2. Fallback transformers - Present value and delta
  3. Fallback transformers - Portfolio valuation
  4. Fallback transformers - Forward discontinuation
  5. Fallback transformers - Convexity adjustments
  6. Fallback transformers - magnified view on risk
  7. Fallback transformers - Risk transition

Don't fallback, step forward!

Contact us for our LIBOR fallback quant solutions.

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

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.

  1. Fallback transformers - Introduction
  2. Fallback transformers - Present value and delta
  3. Fallback transformers - Portfolio valuation
  4. Fallback transformers - Forward discontinuation
  5. Fallback transformers - Convexity adjustments
  6. Fallback transformers - magnified view on risk

Don't fallback, step forward!

Contact us for our LIBOR fallback quant solutions.

LIBOR Fallback transformers - timing adjustments

In this transformer's blog installment, we look at the impact of timing adjustment.

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.

Discounting delta Adjusted delta

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 adjustments add to a total of 37m.

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

  1. Fallback transformers - Introduction
  2. Fallback transformers - Present value and delta
  3. Fallback transformers - Portfolio valuation
  4. Fallback transformers - Forward discontinuation
  5. Fallback transformers - Convexity adjustments
  6. Fallback transformers - magnified view on risk

Don't fallback, step forward!

Contact us for our LIBOR fallback quant solutions.

Sunday, 28 October 2018

LIBOR fallback transformers - forward discontinuation

In this transformer's blog installment, we look at forward starting 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.

  1. Fallback transformers - Introduction
  2. Fallback transformers - Present value and delta
  3. Fallback transformers - Portfolio valuation
  4. Fallback transformers - Forward discontinuation
  5. Fallback transformers - Convexity adjustments
  6. Fallback transformers - magnified view on risk

Don't fallback, step forward!

Contact us for our LIBOR fallback quant solutions.