A quant perspective on IBOR fallback consultation results,
is available on SSRN: http://ssrn.com/abstract=3308766
muRisQ Advisory offers quantitative finance advisory services. Its services focus on trainings and workshops, benchmark transitions, interest rate model development and validation, product design and risk management strategies.
Thursday, 10 January 2019
Updated Quant perspective on LIBOR fallback
The note on IBOR fallback in our series Market Infrastructure Analysis as been updated with the results of the ISDA consultation. The note, titled
is available on SSRN: http://ssrn.com/abstract=3308766
is available on SSRN: http://ssrn.com/abstract=3308766
Tuesday, 1 January 2019
MVA and cost of funding
In the previous blog on Margin Value Adjustment (MVA), we have shown that, for linear products, the forward Initial Margin (IM) along the different paths of Monte Carlo simulation may not be very different. This was visible in Figure 2 of the previous blog.
The IM level is only one half of the MVA, even if this is the one that is often the most computationally intensive. The other half is the cost of funding the IM. For illustration purposes, we use in this blog a cost of funding equal to the spread between OIS over a quarterly period and LIBOR. Obviously each institution need to had some idiosyncratic spread on top of that. But for illustration purposes, this simplified approach is enough.
When we have generated the paths on which we have computed the IM, we have used an interest rate model. Even if our portfolio contained only interest rate swaps linked to a unique IBOR rate, it is important to generate the paths with a model that take into account the stochastic spread between IBOR and OIS in a realistic way. The spread is used in the valuation/IM measurement, but more importantly in our case it is also used in the cost of funding computation. This is why the model used should match the current spread, take into account the spread dynamic (volatility) and have a realistic co-dependence (correlation) between IBOR and OIS curves.
For this blog, we have used a relatively simple model that fulfills those requirements: an hybrid Model for the Dynamic Multi-Curve Framework as described in a recent Model Development document. The model includes the spread stochasticity and the correlation between rate level and spread. For the examples we provide below, the model has been calibrated to USD cap/floor for the IBOR rates dynamic and to historical spreads and correlations behavior.
We first repeat the underwhelming graph which represent the IM level of our portfolio (with a small number of paths).
For the same paths, we have computed the quarterly cost of funding as the USD-LIBOR-3M/OIS-3M spread applied over the quarter to the IM. The cost are reproduced in the graph below. The colors for each path are the same in both graphs.
As appears clearly in that graph, the spread has more impact on the MVA than the precise IM level. In modelling term, it is very important to have a realistic multi-curve model with stochastic spread to which a good understanding of the dependence of the idiosyncratic institution specific funding spread should be added.
Other blogs on IM and MVA:
The IM level is only one half of the MVA, even if this is the one that is often the most computationally intensive. The other half is the cost of funding the IM. For illustration purposes, we use in this blog a cost of funding equal to the spread between OIS over a quarterly period and LIBOR. Obviously each institution need to had some idiosyncratic spread on top of that. But for illustration purposes, this simplified approach is enough.
When we have generated the paths on which we have computed the IM, we have used an interest rate model. Even if our portfolio contained only interest rate swaps linked to a unique IBOR rate, it is important to generate the paths with a model that take into account the stochastic spread between IBOR and OIS in a realistic way. The spread is used in the valuation/IM measurement, but more importantly in our case it is also used in the cost of funding computation. This is why the model used should match the current spread, take into account the spread dynamic (volatility) and have a realistic co-dependence (correlation) between IBOR and OIS curves.
For this blog, we have used a relatively simple model that fulfills those requirements: an hybrid Model for the Dynamic Multi-Curve Framework as described in a recent Model Development document. The model includes the spread stochasticity and the correlation between rate level and spread. For the examples we provide below, the model has been calibrated to USD cap/floor for the IBOR rates dynamic and to historical spreads and correlations behavior.
We first repeat the underwhelming graph which represent the IM level of our portfolio (with a small number of paths).
Figure 1: An example of forward IM paths for a swap portfolio.
For the same paths, we have computed the quarterly cost of funding as the USD-LIBOR-3M/OIS-3M spread applied over the quarter to the IM. The cost are reproduced in the graph below. The colors for each path are the same in both graphs.
Figure 2: An example of IM cost for a LIBOR-OIS related funding cost.
As appears clearly in that graph, the spread has more impact on the MVA than the precise IM level. In modelling term, it is very important to have a realistic multi-curve model with stochastic spread to which a good understanding of the dependence of the idiosyncratic institution specific funding spread should be added.
Other blogs on IM and MVA:
Wednesday, 26 December 2018
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:
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.
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.
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.
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.
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.
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.
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:
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.
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.
The steps to obtain the forward IM in a Monte Carlo approach for interest rates in risk-weight based measures are the following:
- Calibrate a multi-curve framework and a dynamic model
- Evolve the curves to a sample of future dates using random scenarios
- For each date and scenario, compute the sensitivities (market quotes deltas or bucketed PV01) of the portfolio
- Computes the IM for each counterparty based on the sensitivities and apply the sub-portfolio attribution (see last 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.
Attribution
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.
- Loaded trades in 88 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.
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.
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:
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
We have a function which represents the measures μ on portfolios
The measure applied to the total portfolio is
Euler's theorem suggests an attribution based on
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.
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".
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
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
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 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.
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
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.
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
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
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.
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.
- Fallback transformers - Introduction
- Fallback transformers - Present value and delta
- Fallback transformers - Portfolio valuation
- Fallback transformers - Forward discontinuation
- Fallback transformers - Convexity adjustments
- Fallback transformers - magnified view on risk
- Fallback transformers - Risk transition
Don't fallback, step forward!
Contact us for our LIBOR fallback quant solutions.
Subscribe to:
Posts (Atom)