IMA-FRTB – The Computational Challenge
A Benchmark Study of Fast-Pricing Solutions
Ignacio Ruiz1, Emilio Viudez2, Mariano Zeron3, Ruben Moral4
March 2017 Version 1.1
1 Founder & CEO, iRuiz Technologies. [email protected] 2 Head of Client Solutions, iRuiz Technologies. [email protected] 3 Head of R&D, iRuiz Technologies. [email protected] 4 Partner, Management Solutions. [email protected]
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Executive Summary The Fundamental Review of the Trading Book (FRTB) has created a number of important challenges in banks, especially for an advanced IMA-FRTB implementation. They include new demands for systems in the data management, pricing models and numerical approximating pricing techniques. In this paper, we explore the performance in the IMA-FRTB context of the currently most popular numerical techniques (Taylor expansions via sensitivities and linear-interpolation Grids) and we propose a novel technique via Smart Mocax Objects. Those Mocax Objects can be simply understood by the reader as “Smart” Grids. We perform a comparative study of all those numerical techniques using a sample portfolio, made of IR swaps, IR Bermudan Swaptions and Exotic Barrier Options, as illustrative examples of all the key challenges IMA-FRTB brings forward from a pricing standpoint: analytical, tree-based and finite differences pricers, with linear, non-linear and extreme non-linear behaviour in each trade type. In this paper, we show how the listed numerical techniques are considerably faster than a full-revaluation approach, with approximately similar orders of magnitude in computational time across each other. However, relevant differences appear when comparing how each of them performs in the critical P&L Attribution Test (PLAT). We see that Taylor series approximations perform well only for linear products (as somewhat expected), but as soon as non-linearities appear not even second order sensitivities are good enough. Linear interpolations in a multi-dimensional space performed better than sensitivity-based pricing under PLAT, but as soon as non-linearities become slightly strong, the technique also struggles under the demanding test. Finally, we see how the Smart Mocax Object is able to deal with PLAT in spite of the difficult pricing functions in the portfolio. The fundamental reason is that Mocax creates a numerical approximated pricing function that converges exponentially to the original function, which means that (i) it is able to achieve a much higher precision than linear interpolation for the same grid size or (ii) it is able to provide the same precision than linear interpolation with a considerable smaller grid. As a result the risk PnL of the portfolio passes the demanding P&L Attribution Test at an affordable computational cost only with the Mocax approach.
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This exhibit shows the PLAT Variance ratio performance over time under each of the numerical techniques:
In a second stage, we assumed that we had a few Non-Modellable Risk Factors (NMRF) and studied the performance of the numerical techniques in this framework. We observed again how the Smart Mocax Objects were optimal due to their ultra-high level of accuracy.
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Contents Executive Summary ...................................................................................................................................... 2 Standard vs. Internal Models Trade-off ....................................................................................................... 5
IMA options .............................................................................................................................................. 5 The computational challenge ................................................................................................................... 7
Summary of the Results ............................................................................................................................... 8 The sample portfolio ................................................................................................................................ 8 Calculations .............................................................................................................................................. 8 Results ...................................................................................................................................................... 9 Conclusions ............................................................................................................................................ 11
The Algorithmic Toolkit Used ..................................................................................................................... 12 Algorithmic alternatives to full-revaluation ........................................................................................... 12 Taylor expansions ................................................................................................................................... 12 Linear-interpolation Grids ...................................................................................................................... 12 “Smart” Grids via Mocax Algorithms ...................................................................................................... 13
Detailed Results ......................................................................................................................................... 14 Determining the period of stress ........................................................................................................... 14
Non-Modellable Risk Factors ..................................................................................................................... 19 Conclusions ................................................................................................................................................ 22 Appendix .................................................................................................................................................... 24
Practical Implementation of Linear and “Smart” Grids – Dimensionality Reduction ............................. 24 Number of revaluations for IMCC .......................................................................................................... 27 Details of the Mocax method ................................................................................................................. 28 The Mocax approach for FRTB ............................................................................................................... 34
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Standard vs. Internal Models Trade-off Basel’s Fundamental Review of the Trading Book (FRTB) was finalised in 2016 and is planned to go live in 2019. This set of regulations bring several new challenges that require organisational, operational and infrastructure changes. One of the key questions the industry is facing is should a bank stay with the Standard Approach (SA) or should it invest in the Internal Model Approach (IMA)? The SA is substantially more capital expensive that IMA. ISDA estimates, based on data from 21 major banks, indicate that SA capital will be between 2 and 6.2 times higher than IMA capital;5 the positive economic business case for IMA is clear. SA capital is based on trade sensitivities; once those are computed the capital calculation is given by a set of deterministic rules. However, the Internal Models Capital Charge (IMCC) is based on a number of Expected Shortfall calculations that must be calibrated to a period of stress. IMA requires two key tests: the back-testing and the P&L Attribution Test (PLAT). Back-testing deals with the quality of the generation of risk scenarios and valuation in the Expected Shortfall engine. The industry is not too worried about it because it has been doing back-testing for many years and FRTB does not introduce any substantial change. However, the PLAT is completely new. The spirit of it is that regulators want to make sure that banks are measuring risk with the same tools with which they value its portfolio of trades. To monitor this, FRTB requires banks to compare their official daily P&L (the “hypothetical P&L”, H-PnL) with the P&L given by the risk systems (the “risk-theoretical P&L”, R-PnL). For this, it has designed two PLAT metrics.
1. Mean Ratio: the mean of the Unexplained P&L (U-PnL), defined as the difference between H-PnL and R-PnL, divided by the standard deviation of H-PnL
2. Variance Ratio: the variance of the U-PnL divided by the variance of H-PnL These ratios must be based on a 12-month period and computed monthly. The Mean Ratio must be in the range between -10% and +10%, while the Variance Ratio must be smaller than 20%. These ratios must be computed per trading desk. If a desk reports more than four exceptions over the past 12 months, that desk FRTB capital must be based on the SA until it passes the test for a 12-month period. The way these tests have been created makes the IMA waiver extremely sensitive to the alignment of the official pricing vs. the risk-systems pricing; that was precisely the intention.6 IMA options The previous analysis yields two clear outcomes. Firstly, capital is so scarce and expensive in current times that the IMA decision will become a key driver in the profitability of the derivatives business from 2020 onwards. Secondly, the only way to have significant confidence that an IMA bank today will remain
5 ISDA study. See “The P&L attribution mess”, Duncan Wood, risk.net, 2nd August 2016. 6 There are speculations that the PLAT may be modified to overcome some of the unintended pitfalls it encounters; e.g. that it incentivizes non-fully hedged trading strategies. There is nothing solidly stablished in that regard at the time of writing this paper, so we will work based on the current PLAT.
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IMA during the foreseeable future is if its risk FRTB and official front office pricing systems are identical, or very close. All banks must make a balancing act between the first and second point. To be nicely profitable a bank needs to be IMA. However, aligning front office and risk pricing is going to be hard and expensive. There are three major sources of discrepancy: pricing models, data and the numerical methods applied. In this paper, we focus on the latter, i.e. we assume that the pricing models and data used by the front office and risk are identical and we investigate the potential numerical methods that could be applied to compute IMCC by the bank’s IMA-FRTB market risk system and meet the PLAT. Industry estimates indicate that an IMA-FRTB engine needs to be ready to reprice its entire portfolio many thousands of times,7 daily. Banks now face the following five options:
1. Full-revaluation: The obvious blunt way of solving the problem is using full revaluation in the IMA-FRTB engine. Indeed, if we could use full revaluation instantly, wouldn’t we use it in all our risk computations? However, full revaluation creates a most severe computational challenge given the lack of speed of the front-office pricers in the context of risk calculations.
2. Delta pricing: Traditionally banks have solved this type of problem by using a sensitivity based Taylor approximation for pricing in the market risk engine. This Taylor expansion often stops at the first derivative, hence it may fail the PLAT due to lack of accuracy as soon as non-linearities enter the game.
3. Delta-gamma pricing: As a step up from the Delta pricing, some risk engines incorporate second order sensitivities to compute an approximated pricing in the risk engine. Second order effects may make the pricing approximation more reliable, but as soon as we have medium or strong non-linear products, that approximation fails PLAT (as we are going to see).
4. Standard Interpolation Grids: An alternative to Taylor approximations consists in creating a grid of prices in a given range and interpolate or extrapolate from them in order to obtain the desired price. Standard interpolation methods are linear or spline methods. These methods tend to have the problem that, in order to ensure a decent precision in the price, the grid needs to be very fine to the extent that, often, it is not practical to build a grid that delivers prices with the desired resolution.
5. “Smart” Grids via Mocax Technology: An alternative to the above is using novel “smart” algorithmic based grids that do not have the outstanding computational needs that standard full revaluation has. We propose a solution in this space, using the Algorithmic Pricing Acceleration (APA) method, and compare it to the other alternatives.
7 The maximum number of Partial Expected Shortfall is sixty-three, each Expected Shortfall calculation requires around 250 revaluations. This is in addition to the search of the stress period, that must be done monthly, but also promptly if the market conditions change.
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The computational challenge Let’s say a bank has one million positions. Let’s say that IMA-FRTB requires “only” 10,000 revaluations per position and let’s assume the average derivative pricing time is 0.01 seconds. In that case the bank needs 28,000 computing hours for a IMA-FRTB calculation. If the bank wants to be able to do this calculation in the hour range, it is going to need a grid of around 25,000 CPUs. A grid of this size is going to cost several millions of dollars, annually. The problem becomes even worse when we face Non-Modellable Risk Factors (NMRF), as in many cases the bank has to find the period of stress and compute an Expected Shortfall for each NMRF. Given the scale of the problem, every bank is faced with a set of most important questions: does any of the fast-pricing possibilities solve the problem? Is the Delta pricing or the Delta-gamma pricing enough to ensure PLAT beyond reasonable doubt? Do we need Grids? Are Standard Grids good enough, or do we need Smart Grids? In this paper, we are going to analyse the impact on IMA-FRTB of those five approaches for a sample portfolio. We provide information of how the PLAT requirements would be met by each of them, highlighting their strengths and potential weaknesses. Then we assume some of the risk factors are non-modellable and run FRTB calculations for them.
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Summary of the Results The sample portfolio The following sample portfolio was built for this case study:
• Vanilla IR Swaps: It is a very linear product, the most common derivative and can be valued using fast analytical pricing techniques.
• IR Bermudan Swaptions: It is a common non-linear product. Its pricing is often based on slow Monte Carlo simulations or trees. So, it is computationally expensive to obtain their risk via full-revaluation.
• Exotic Barrier Options: These are very non-linear products that are very slow to price. So, they are also a big challenge for IMA-FRTB.
Portfolio Type Trade Type Asset Class Num. Trades Pricer Linear Vanilla Swap Interest Rate 20 Analytical Non-linear Bermudan Swaptions Interest Rate 20 Tree (Hull-White) Non-linear Exotic Barrier Option Equity 20 Finite differences
The risk in the portfolio of IR Swaps was computed by modelling its yield curve as per the IMA-FRTB requisites. The risk of the portfolios of Options was computed modelling all the yield curve, swaptions volatilities, equity prices and equity volatility respectively. We used a historical simulation with 10-day overlapping moves. We assume that all risk factors are modellable under the FRTB framework in the first part of the study, and then we considered the impact of a few of them being non-modellable. The notionals and maturities of the portfolio were randomly selected in a realistic fashion, with maturities up to twenty years. Calculations In order to complete IMA-FRTB computation in our simplified portfolio of 60 trades, we needed to do 0.4 million revaluations: 367,220 to find the period of stress, 35,280 to calculate the capital charge8, 15,120 for the backtesting, 60 for the PLAT. We did not compute a price twice for the same scenario: when a price calculation could be reused across different set of computations, they were stored and re-used.
8 Includes the full set and reduce set of risk factors.
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The following table shows the number of revaluations that were needed:
Calculation Number of revaluations Find period of stress 367,220 IMCC 35,280 Backtesting 1-day VaR 15,120 PLAT 60 Total: 417,680
Results In order to assess the different IMA computational alternatives, we compared the brute-force full revaluation approach using the open-source QuantLib library with the following four numerical methods: (i) Smart Grids via Mocax, (ii) Linear-interpolation Grids (Grid), (iii) expansion of the first order derivatives (Taylor 1) and (iv) expansion of the first and second order derivatives (Taylor 2).
In particular, we wanted to find answers to the following two key questions. Firstly, how computationally intense is each approach compared to full revaluation? Secondly, how do each of the four alternatives perform in the PLAT? This is key because, regardless of how fast a technique can be, it is of no use for IMA-FRTB if it does not pass the P&L Attribution Test. In terms of computational time, doing the full IMA-FRTB calculation via brute-force took around 13 hours using the QuantLib pricing library. All other accelerating techniques had first a preparation calculation that we refer to as “building” step. In this step the Greeks, Grids or Mocax Objects are constructed. That step is followed by a second step where the IMA calculation is derived as such, using the objects created in the building step. When the run time of both steps are considered together, all fast techniques were much faster than the “brute-force” approach, and all of them in the similar order
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of magnitude in velocity; the overall computational time under the numerical techniques was always dominated by the building phase. A summary of the results is shown in the following table:
Building IMA evaluation Speed gain over brute-
force Brute-force n/a Around 13 hours n/a Smart Grids (Mocax) 38 minutes 0.6 seconds 21x Linear-interpolation Grids
38 minutes 0.15 seconds 21x
Taylor 19 10 minutes 0.019 seconds 75x Taylor 210 11 minutes 0.021 seconds 70x
The important differences showed up when passing each of the techniques through the PLAT. In those tests, both Taylor expansion methods really struggled as soon as there was any relevant non-linearity in the pricing functions. Linear-interpolation grids performed generally better than Taylor, but they also could not cope with non-linearities well enough. The only method that showed solid PLAT results was the Smart Grids via Mocax. In the following graphs, we show the results of the PLAT, both the Mean and the Variance ratio. We performed a somewhat PLAT “backtest” by computing both its ratios on a daily rolling basis since 2007, keeping the sample portfolio constant.
9 Based on a 50-point yield curve and symmetric sensitivities. 10 Second order sensitivities on the underlying value (i.e. interest rates or equity price) were only computed for the Exotic Barrier Options, given the high linearity of all trades with respect to interest-rate gamma. Based on a 50-point yield curve and symmetric sensitivities.
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Conclusions As expected, from the pure computational effort standpoint, all four techniques outperform substantially the computational needs of the full-revaluation approach. Regarding the P&L Attribution Test, the Smart Grids via Mocax outperformed all other techniques. The reason will be seen in detail in subsequent sections, but it can be anticipated that it is because Mocax is the only technique that can deal with the non-linearity of the pricing functions sufficiently well at an affordable computational cost.
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The Algorithmic Toolkit Used The goal of this piece of research is to investigate how different accelerating techniques perform in the context of the strong IMA-FRTB computational requirements. In particular, we want to understand how their strengths and weakness compare to brute-force full-revaluation. In this section first we are going to describe the different techniques under consideration. Then we are going to see how each technique performs when undertaking different IMA tasks. Algorithmic alternatives to full-revaluation We considered these alternatives to full-revaluation: Taylor expansions, Standard Grids and Smart Grids via Mocax. For Standard Grids we used the industry-standard linear interpolation method. Taylor expansions Market risk engines have been using a sensitivity-based Taylor expansion approach for a long time. There are two versions that are the most popular:
• Delta pricing: The Taylor expansion is truncated at the first order derivatives
𝑃(𝑟, 𝑆, 𝜎) ≈ 𝑃(𝑟*+++⃗ , 𝑆*, 𝜎*) +-𝑑𝑣012
3
245
∙ 𝛿𝑟2 + ∆9 ∙ 𝛿𝑆 + 𝜗 ∙ 𝛿𝜎
where the vector �⃗� denotes a yield curve, 𝑆 is an underlying price if the derivative is other than the Interest Rate class (e.g. equity), 𝜎 the implied volatility of the option, 𝑑𝑣012 is the sensitivity of the price to the 𝑖-th component of the yield curve, ∆9 the price sensitivity respect to the stock price and 𝜗 the price sensitivity respect to the implied volatility. 11
• Delta-gamma pricing: in this case, we extend the Taylor expansion to the second order derivatives of the main risk factors (interest rates Gamma for IRS and stock price Gamma for equity options) without the cross-terms
𝑃(�⃗�, 𝑆, 𝜎) ≈ 𝑃(𝑟*+++⃗ , 𝑆*, 𝜎*) +-𝑑𝑣012
3
245
∙ 𝛿𝑟2 + ∆9 ∙ 𝛿𝑆 + 𝜗 ∙ 𝛿𝜎 +12-Γ2
3
245
∙ 𝛿𝑟2> +
12Γ9 ∙ 𝛿𝑆> +
12𝑉 ∙ 𝛿𝜎>
Linear-interpolation Grids Often, when market risk engines need to handle non-linear products, a grid method with linear interpolation is used. In this sample portfolio, the price of each derivative is a function of a multi-dimensional space (e.g. a whole yield curve for IRS, or stock price, volatility and interest rate for 11 In the case of Equity derivatives, typically the sensitivity to only one point in the yield curve is needed.
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options), so we implemented a multidimensional lattice method. This had to be done after a dimensionality reduction via Principal Components Analysis (PCA) was performed, as otherwise the dimensionality of the interpolation problem makes it unfeasible. We used five and four principal components respectively in the case of IR swaps and Bermudan Swaptions. The PLAT was performed considering the PCA approximation too; more details are shown in the Appendix. “Smart” Grids via Mocax Algorithms Mocax is a set of algorithms that create a replica of a pricing function that is both super-fast and super-accurate. In some ways, it can be understood as an intelligent “Smart” Grids method. Once built, the computational speed of Mocax compared to standard grids (e.g. linear or spline interpolation) is of the same order of magnitude; i.e. ultra-fast. However, standard grids have the limitation that they tend to converge to the original function with polynomial speed at best, while the algorithmic solution that underpins Mocax converges exponentially, orders of magnitude better. This property will be central in the P&L Attribution Test. Expanding on that, often, with Standard Grids, we need to increase the granularity of a grid a lot in order to increase the resolution of the underlying interpolation method so that it becomes useful. Therefore, when we try to reproduce a function with relevant non-linearities via standard interpolation grids, the granularity of the grid easily increases to the point that the excessive building time defeats its practicality. However, the intelligence in the Mocax technology makes its pricing function replicating objects to be exponentially convergent to the original functions as we increase the number of points in the building phase. As a result, we can achieve very high accuracy with relatively few calls to the original pricer. Consequently, Mocax Objects should outperform Standard Grids in two related ways:
1. For the same granularity (i.e. number of calls to the original function in the building phase), Mocax Objects are nearly always orders of magnitude more precise than the standard interpolation methods. Therefore, they will pass the PLAT where standard interpolations fail.
2. For a given level of accuracy, Mocax Objects can be built with significant smaller number of calls to the original function. Therefore, the building phase of the object (i.e. the standard interpolation or Mocax Object) can be reduced significantly with the Mocax technology.
These features make Mocax a strong candidate for a successful IMA-FRTB implementation, as the spirit of the FRTB document is to align very strongly the Front Office and Risk pricing functionalities. In addition, when we build a standard interpolation object, we are going to call the original functions “N” number of times, hoping that we achieve the required level of precision. However, the Mocax technology includes a way to construct a Smart Grid with the desired error as an input, which adds an important function to the practicalities of the problem at hand.
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In order to do a fair comparison, in this study we built the Standard Grids and Mocax Objects with the same number of anchor points (i.e. points where the original pricing function is called).
Detailed Results Determining the period of stress One of the calculations that needs to be done regularly is to determine the period of stress. This computation is required monthly at least, but the bank must be ready to promptly compute it “whenever there are material changes in the risk factors of the portfolio” (Jan’16 FRTB document, p. 54). In order to find the period of stress we need to compute the change in value of our portfolio in every rolling and overlapping 10-day period since 2007, and then compute a rolling Expected Shortfall. Given that this is a computationally expensive exercise, we may need to use one of the fast methods in order to compute a rolling Expected Shortfall calculation. That rolling calculation of our portfolio is shown in the following exhibit for each numerical method.
The true benchmark results are those obtained via brute-force revaluation. The reader can see how both Taylor fast computations provide results quite away from the true ones. As a matter of fact, they find the wrong period of stress. On the other hand, both Standard Grids and Mocax Objects deliver good results. P&L Attribution However, the difference between linear interpolation Grids and Mocax is going to be highlighted in the P&L Attribution Test; the critical test that fast pricing techniques need to pass.
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We showed the results previously in the Summary section, but let’s see them here again for completeness.
The P&L Attribution Test shows clearly how Taylor expansion pricing deals very poorly with non-linear products, even when second order adjustments are considered. Therefore, this type of technique should only be used for very linear products, if at all. One may hope that standard Grids would deal correctly with non-linearities. However, that is not the case. Indeed, that effect becomes quite relevant in 2012, when the ratios of the Grid pricing technique run away from 0%. That is the case because the strikes of the options portfolio were randomly chosen but not too far from the ATM strike today. That strike was very out of the money in the period 2007-2012, so the options portfolio was quite linear. However, when the underlying approached today’s strike (in 2012), non-linearities kicked in and the Grid method was not able to deal with them. On the other
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hand, the ultra-high precision of the Smart Mocax Objects hardly felt the difference. That happened because the “Smart” grids built by the Algorithmic Pricer Acceleration can capture the intricacies of the pricing functions with orders of magnitude better precision than the standard interpolation methods. In order to assess the performance of Standard Grids vs. Smart Mocax Object, we run a numerical test in which we increased the size of the grid for both methods, and compared the worst P&L Attribution Test ratios in each of them. The results are shown in the following graphs:
The reader can see how Mocax Objects outperform Standard Grids by orders of magnitude. In the main implementation that we did, each Object and Grid was built with around 300 exploration points. If we want Standard Grids to deliver the same PLAT ratios than using Mocax Objects (with 300 exploration points), the Grids will need to explore the original function around 8,000 times. That would make the Standard Grid method unfeasible in practice.12 The need for low cost accuracy becomes especially critical when the portfolio under consideration is well hedged. In such cases, the Unexplained PnL is amplified by a small denominator in the PLAT ratios. Only a good convergence to the original pricers can guarantee a successful PLAT. This high-precision feature can be depicted also in the next set of exhibits. The top left graph is the price of a Bermudan Swaption, in the context of Non-Modellable Risk Factors (NMRF, to be discussed in detailed later), in which one point in the yield is non-modellable (horizontal axis in the graph). The pricing function is very linear with that point of the yield curve, so all pricing methods are good. On the top right graph we see the price of the Bermudan Swaption, also in the context of Non-Modellable Risk Factors, in which the NMRF is a point in the volatility surface. We can appreciate how the price is clearly non-linear, and so the linear-interpolation grid starts to show at the bottom of the true pricing function, while Mocax stays perfectly on top of it. Finally, the bottom graph shows a 1-dimensional slice of the 3-dimensional pricing function (underlying, volatility, interest rate) of the portfolio of Equity Barrier Options; that portfolio has several strikes and barriers and, hence, the pricing function of the entire portfolio shows extreme non-linearity. The linear-interpolation grids clearly struggle to follow the true pricing function, while the Mocax Object follows perfectly.
12 The Appendix shows the same graphs in logarithmic scale, which illustrates clearly the power of the exponential converge of the Mocax methods.
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Capital Charge Under IMA, the Internal Model Capital Charge (IMCC) is given by
𝑰𝑴𝑪𝑪 = 𝝆 ∙ E𝑰𝑴𝑪𝑪(𝑪)F + (𝟏 − 𝝆) ∙ E∑ 𝑰𝑴𝑪𝑪(𝑪𝒊)𝑹𝒊4𝟏 F,
the unconstrained IMCC is
𝐼𝑀𝐶𝐶(𝐶) = OE𝐸𝑆Q(𝑃)F> +-R𝐸𝑆Q(𝑃, 𝑗) ∙ T
E𝐿𝐻W − 𝐿𝐻WX5F𝑇 Z
>
W[>
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and the constrained IMCC is
𝐼𝑀𝐶𝐶(𝐶2) = T\𝐸𝑆Q,2(𝑃)]>+ ∑ ^𝐸𝑆Q,2(𝑃, 𝑗) ∙ _
E`abX`abcdFQ
e>
W[> .
The 𝐼𝑀𝐶𝐶(𝐶2)’s are partial Expected Shortfalls where the i-th risk category is allowed to move while holding the other categories constant. This table summarises which trades need to be shocked in each risk class 𝐼𝑀𝐶𝐶(𝐶2).
On top of the risk-class ES partials, we also need to do partials mapped to the liquidity horizon of each risk type. This table shows the liquidity horizon partials that needed to be done to our sample portfolio.
This meant that a total of 2,623 revaluations were needed for each IR Swap, 5,246 for each Bermudan Swaption and 10,492 for each Equity barrier option. A table with details of these numbers is shown in the Appendix. We did not repeat a valuation when we could re-used a previous one.
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If we tried to do all partials via full revaluation, the number of times we need to call the slow original pricer would rocket, as we may need up to sixty-three expected shortfall calculations in a general portfolio.13 To do the partials, if we use Taylor expansions, all we need to do is set the sensitivities of risk factors that are not simulated at zero; if we use Standard Grids or Mocax, all we need to do is take orthogonal slices in the grids that keep the non-bumped risk factors constant. In either case, we can compute the partials at ultra-fast speed with all the methods. However, as shown in the previous section, Mocax’s “smart” grids are the only ones that are able to deliver price values that are systematically close to the real one. Hence, it is the only method that seems to be able to provide an ultra-fast and accurate enough IMCC computation.
Non-Modellable Risk Factors The computational challenge of NMRFs is, actually, ideal for Mocax’s Smart Grids. Let’s see why. In many NMRF cases, we need to compute the period of stress for each NMRF by computing a rolling Expected Shortfall. The capital associated from the NMRFs is driven by each NMRF’s SES, the maximum Expected Shortfall along its rolling series. If we try to compute the SES for each NMRF via brute-force revaluation, we need to price the portfolio affected by those MRMRs around 3,000 times per each NMRF. The computational demand for this is destined to explode. From the fast-pricing alternatives
• Taylor: we need to have the sensitivities for each of the NMRF. However, even if we have them, we have seen how this method seriously struggles with non-linearities, hence it is clearly suboptimal.
• Standard Grids: in this case we need to build a 1-dimensional pricing Grid and, then, use it to reprice the portfolio in each of the 3,000 scenarios. We have seen how those standard grids converge very slowly towards the true price, hence as soon as we have non-linearities we will need to build a grid with a lot of granularity. Additionally, once a grid is built, there is no way to know in advance what is the error we are going to get, hence all we can do is “hope for the best”.
• Mocax: The “Smart” Grids of the Mocax method outperform Standard Grids because
13 Due to the combination of Partial Expected Shortfall calculations. The original Jan’16 FRTB document required those calculations daily. The Q&A FRTB document from Jan’17 states that the collection of 𝐼𝑀𝐶𝐶(𝐶2) can be calculated weekly. However, a bank choosing to do this calculation weekly must make sure that “[it] does not lead to systematic underestimation of risks relative to daily calculation” and that “Banks need to be in a position to switch to daily calculation upon supervisory direction”. So, in practice and reality, it seems that banks should to be ready to do the partial 𝐼𝑀𝐶𝐶(𝐶2) computations on a daily basis.
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i. It converges exponentially to the real pricing functions; hence we will be able to achieve a high degree of accuracy, even when the pricing functions show strong non-linearities, with significantly smaller grids compared to any other typical method. We could visually see that in the 3-graph exhibit already shown above.
ii. Most importantly in a practical industrialised solution, the Mocax technology has an algorithm that conservatively estimates the error the Smart Grid in the Mocax Object will have. Thanks to that, we can build that Smart Grid with that error as an input, and in this way we can ensure that we will obtain a good quality approximated price without the need of manual intervention.
For our testing, we assumed that we had two points in the yield curve (point 23 and 42 in our 50-point yield curve) and one point in the volatility surface that are non-modellable. That affected all the IR swaps and swaptions
The NMRF calculations will require a total of 0.26 million evaluations. The following table shows the number of evaluations needed per trade type and NMRF.
For illustrative clarity, we show again the two graphs with the MNRF 1-D pricing functions
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The following two tables show the details of the building and evaluation face of the 1-D Mocax “Smart” Grids objects. Those numbers show the extraordinary precision of the Mocax methods: only six calls to the original pricer was enough to give a precision (maximum error) in the fifth significant figure (10-5) for the IR Bermudan Swaption, in spite of its strong convexity. In spite of that, the acceleration produced by the Mocax Object is extraordinary, 3,000,000 time faster than the original Bermudan pricer.
As a result, we were able to compute all the SES for the NMRFs 437 times faster than via brute force and with total security that the calculation is ultra-precise.
Finally, the following exhibit shows how Mocax was the only method able to find the NMRF period of stress correctly, in a few milliseconds. When the NMRF was a point in the yield curve, Taylor did not do a good job, but both Standard Grids and Mocax gave the correct result. However, when the NMRF was a point in the volatility surface, non-linearities kicked in and only Mocax was able to give the correct result.
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Conclusions We have seen that the IMA method for FRTB decreases capital by a factor of between 2 and 6.2 compared to the Standard Approach. An IMA implementation comes with a number of technology challenges. One of them is that it needs to use the same (or very similar) data and models. That comes with the problem that the official “front office” pricing models tend to be too slow for a practical and successful IMA implementation. In this paper, we review known alternative techniques like Taylor series approximation and Grid methods like linear interpolations, as well as a new approach that creates “Smart” Grids based on the Algorithmic Pricing Acceleration method within Mocax Intelligence. We created a sample portfolio that contains all major difficulties an IMA implementations is going to find from a pricing computation standpoint: linear, non-linear and extreme non-linear products, as well as a different range of computational requirements: analytical, tree and finite difference based pricing. We have used QuantLib pricers as the equivalent of Front Office official pricers, and we have benchmarked different IMA fast computation techniques. We have seen that:
1. All the alternative techniques to full-revaluation require much less computational power than full-revaluation. The computational time was dominated in all cases by the building phase.
2. One of the key tests, the P&L attribution test, is very unstable both under the Taylor and Standard Grid methods, but very stable under the newly proposed method with Smart Grids.
We have seen how the typical fast techniques that would not have such computational demands (Taylor expansion and Standard Grids) are clearly unstable in the P&L Attribution Test as soon as pricing functions have non-linearities, hence endangering the approval of an IMA waiver. Also, most importantly, even if we pass the P&L Attribution Test today, the volatility of the Test under those
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methods is so unstable that there is a considerable chance that we lose the IMA waiver when market moves are considerable, precisely when we least want to have it at risk. In this paper, we propose an alternative technique via the “Smart” Grids produced by the innovative technology in Mocax Intelligence. This novel technique does not require the outstanding computing power that full revaluation demands while ensuring ultra-fast pricing with ultra-high precision, which ensures stable P&L Attribution Test ratios and, hence, an ongoing IMA waiver approval. Finally, we applied the fast methods in the context of Non-Modellable Risk Factors. We show how Taylor and Standard Grids deliver incorrect results under very typical conditions, and how Mocax algorithms can do the job in a split of a second with absolute precision. We are aware that these results cannot be 100% conclusive in every single portfolio in the world, as this study is based on a relatively small sample portfolio. However, we have picked the portfolio so that it has all the major problems that FRTB imposes, so we expect that similar results should be achieved in real bank portfolios.
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Appendix Practical Implementation of Linear and “Smart” Grids – Dimensionality Reduction When applying any Grid method to a pricing function, either a Standard method like Linear or Spline interpolation, or a Smart method like Mocax, the first thing that needs to be done is to reduce the dimensionality of the problem to that of the true mathematical dimensionality. In general terms, a pricing function is a mapping from ℝg to ℝ. N comes from all the points in yield curves, volatility surfaces, etc.
N can easily be in the order of hundreds. However, as we know, the real dimensionality of the pricing function is much lower than that. For example, in the case of yield curves, even though we may have fifty points in it, only a few degrees of freedom (e.g. first few Principal Components) take the vast majority of the dynamics of the curve. Therefore, in risk calculations, we should only care about those relevant degrees of freedom. When we build Grids as a fast pricing method, first we need to explore the function f in the ℝg space. The size of that space grows exponentially with N, so it is important that we make use of the “relevant degrees of freedom” information; if we don’t do that, building the grid becomes unnecessarily unfeasible given the size of the grid for large N. Using the example of yield curves again, it does not make sense to explore the pricing function f in the region where, for example, 10-year swap rates are around 0.1%, 11-year rates are 5.0% and 12-year rates are again 0.1%. In mathematical terms, that means that we need to reduce the dimensionality of the problem so that we can apply the Grid method efficiently to a reduced set of variables (n) and, hence, optimise its performance. The lower the n the better the fast pricing Grid will be.
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There are many ways in which we can do that Dimensionality Reduction. Each of them can be the most appropriate depending on the problem at stake. Dimensionality reduction techniques include,
• Principal Components Analysis This is an obvious candidate for yield curves. Indeed, it is the technique we use in this FRTB case study. It must be noted that this technique always induces an error that compounds with the error given by the Grids; the higher the number of Principal Components, the lower the error. The P&L Attribution Test must be done considering the compound error; i.e., the official Risk-Theoretical price must be computed without any PCA dimensionality reduction.
• Pricer Parameter Mapping Let’s say that we are computing the risk of a Bermudan Swaptions in a risk engine, and that the pricer of the Swaption is based in a Monte Carlo simulation using a 1-factor Hull & White model, 𝑑𝑟h = (𝜃 − 𝛼 ∙ 𝒓𝒕)𝑑𝑡 + 𝜎𝑑𝑊𝒕. In each risk scenario where we want to price the swaption, we need to calibrate 𝑟*, 𝜃, 𝛼 and 𝜎. The price of that Swaption is unique for each combination of those four numbers. Hence, even if we have hundreds of points in the yield curve and volatility surface, this is really a 4-dimensional pricing problem.
Furthermore, in many cases like CVA or IMM-CCR, often the parameters 𝜃 and 𝛼 are left constant in the risk simulation, so in those cases this is only a 2-dimensional problem. If 𝜎 is also left constant (often the case), then we have a 1-dimensional pricing problem. We have used here a 1-factor Hull & White model only as an illustrative example. The same idea applies to any other pricing diffusion models like SABR, etc. Please note that, in this Pricer Parameter Mapping technique, the dimensionality reduction that takes place is exact, so the only source of error in the final price is given by the Grid Object. As said, that error is minimal in the case of Mocax as its methods are exponentially convergent.
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• Curve Parametrisation An alternative technique includes parametrization of curves. Some popular ones include Nelson-Siegel for yield curves or level-skew-convexity for volatility smiles.
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Number of revaluations for IMCC
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Details of the Mocax method Mocax is a set of novel mathematical algorithms that have been optimally implemented in a software library. It has three key algorithms: Algorithmic Pricer Acceleration (APA), Algorithmic Greeks Acceleration (AGA) and Portfolio Pricing Compression (PPC). APA & AGA When a pricing function is passed through Mocax, the algorithm explores the function on a number of points and creates a replica of the original pricing function that is both ultra-accurate and ultra-fast to evaluate. This is the APA method. A second algorithm is able to create a replica of the derivatives of the original pricing function that is also ultra-accurate and ultra-fast. This can be done without having to explore the original function again. This is the AGA method. The following exhibit illustrates that process. The top right panel shows the Mocax value 2D surface of an IR Swap in which the yield curve is determined by a 1-factor interest rate model “r” (e.g. 1-factor Hull-White, first principal component) and time to maturity “T”. It must be noted that the Mocax algorithms work for any dimensionality in the inputs of the pricing function. However, as explained in the section of the Appendix on Dimensionality Reduction, the higher the dimensionality the more complex (hence suboptimal) the algorithmic exploration and valuation is going to be.
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Another feature of the algorithm is that it can handle discontinuities. In derivative pricing, the discontinuities exist in payment days in the time dimension and on strikes or barriers in the risk-factor dimension. Indeed, the dents that can be observed in the IRS value surface in the graph above are precisely the drop in the price when the IRS has a payment day. The performance of the APA algorithm is based on two key properties:
• Exponential convergence – the accuracy of the Mocax pricing function increases exponentially with the number of points used in the exploration phase. Using the example of a Black-Scholes option pricer in one dimension, the algorithm can achieve a precision of 10-5 in most relevant cases with only ten calls to the original pricer in the function exploration phase. If a precision of 10-15 is needed, that can be achieved with only a few more points. This property has very positive consequences in the context of risk calculations and FRTB, as here we need an accurate replica of a pricing function so that the PLAT can be passed, but in an efficient manner so that it is computationally feasible.
To illustrate the power of the exponential convergence, the reader can see in the following exhibit the worst value of the rolling PLAT Variance Ratio that we obtained in our test case, compared to the linear-interpolation method used in the Standard Grids.
The horizontal axis is the number of exploration “anchor” points in the building phase. The vertical axis shows the Variance Ratio in log scale. It can be seen how, as we increase the number of anchor points, the Mocax method delivers increasingly better results in an exponential fashion, while the standard linear-interpolation method tends to flatten out. This implies that, if we need our numerical technique to perform better under the PLAT, a relatively small number of extra anchor points will be enough if we use Mocax, while we will need an orders-of-magnitude larger number of extra anchor points using Standard Grids.
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• Acceleration independence – we have said that the Mocax version of a pricer is ultra-fast. However, that speed acceleration is independent of the complexity of the original pricing function.14 This is best explained with a real example: we passed different QuantLib pricers through Mocax, in one dimension for illustrative reasons. First we passed an analytical pricer (Black-Scholes option). The QuantLib pricer would take 13 microseconds to provide a result, while the Mocax version of it would take 127 nanoseconds. Then we passed a Monte Carlo pricer that was taking 23 milliseconds in QuantLib, but it only took its Mocax version 96 nanoseconds to give a price. Finally, we passed a very slow tree-based pricer that was taking 0.23 seconds in QuantLib; its Mocax version needed 127 nanoseconds to provide an answer.15
14 By “complexity” we mean the amount of time the code takes to run, as opposed to the dimensionality of the state space. 15 Tests run in C++ in an i7 intel processor, all computations in one core (without parallelisation). Maximum error of the Mocax pricing function was 10-5.
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This example illustrates a key property of the algorithms: once the original pricing function exploration has been done, the time it takes the Mocax version of the pricer to run is in the nanosecond range (which makes it even faster than the already-considered ultra-fast Black-Scholes pricing function) and this time is independent of the speed of the original function. The APA method reduces the computation of one-dimensional pricing functions to around 100 CPU cycles, regardless of how long it took to evaluate the original function.
PPC The third algorithm, the Portfolio Pricer Compression (PPC) method, is based on the additive property of the APA method. Following the exhibit bellow for illustrative purposes, let’s say that we have a portfolio with many trades, some IR Swaps, some IR Swaptions and some Equity Options. Each trade has a pricing function with constant maturity date, notional, strikes (Trade001, Trade002, etc., in the exhibit). If we pass each of those pricing functions through Mocax, the algorithm is going to build an ultra-fast and ultra-accurate version of those pricing functions for each trade (mocax001, mocax002, etc., in the exhibit). The APA algorithm is additive in a way that we can combine all Mocax functions into a single function for those functions that share the same state space (i.e. the inputs to the function). In that way, we can combine all IR Mocax functions into a single IR sub-portfolio pricing function and all equity mocax pricers into one single equity pricer.
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One of the key features of this APA additive property is that the computational effort needed by the combined pricer is the same as that of each single pricer. Hence, if we have 1,000 trades that we combine into one, we gain a 1,000 times acceleration. Serialization The algorithms that lie inside Mocax have been implemented in a shared library so that any person can integrate them into an existing risk engine. The shared library contains a set of classes in an object-oriented environment. In order to explore a function and build a Mocax version of a pricing function, the user has to call the constructor of the Mocax class. The constructor then creates an object of the Mocax class that can execute all relevant algorithms through its methods. Those objects can then be serialized and stored so that they can be re-loaded and used whenever needed. As a result, the optimised pricing functionality offered by the created Mocax Objects can be used at different moments in time by the same risk engine, and shared across several risk engines. This type of set up makes the building phase of the computation with Mocax a shared overhead across many calculations.
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This feature can help substantially in the current environment, as regulators do not look kindly on financial institutions using different pricing functionality across different calculations.
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The Mocax approach for FRTB In this section, we are going to see how we can use all the Mocax machinery for our benefit in the FRTB working case. Step 1: Optimisation of the state space First, we need to work out the dimensionality of the problems at stake. It must be noted that this Step 1 is not unique to Mocax, it also needs to be done for Standard Grids. We have three trade types, each with the following dimensions:
The risk of all the Equity options was perfectly described by a three-dimensional state space. However, the IR swaps needed some care. The yield curves used as input to the pricer had 50 tenor points, so a blunt risk implementation would see a 50-dimensional state space. However, we have already seen (see Appendix on Dimensionality Reduction) how in reality we only have a few degrees of freedom in the ℝp* state space of the original pricer. We looked for those relevant degrees of freedom by applying a PCA technique and increase the number of principal components taken until we saw that we were complying with the FRTB requirements driven by the PLAT. We saw that three principal components could be enough, but we decided to go up to five to be on the safe side. The same analysis was done for the Bermudan Swaptions, and we found that a 4+1=5 (4 for the yield curve and one for the volatility surface) was optimal for the FRTB requirements. Step 2: Building After doing the dimensionality reduction for the IR Swaps and Swaptions, we created 5-dimensional pricing objects for each swap and swaption, and 3-dimensional objects for each barrier option. The following table shows the computational times of the building phase.
It must be noted the low number of calls to the original pricer that was needed to achieve a high level of accuracy in the pricing. They only needed to be called around 300 times to achieve an accuracy between
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10-2 and 10-4. It must be noted that most of the error was coming from the needed dimensionality reduction via PCA. These methods will be useful as long as we need to call the original pricer fewer times compared to the full-revaluation brute-force approach, and that by itself will be driven by the efficiency of the construction method given a level or desired error. As said, Mocax optimises that. For examples, the Barrier Options only needed to be called 336 times while, later in the computation, we will need 11,753 revaluations. The overhead relative computational cost was only 2.9% of the full-revaluation cost. Step 3: Evaluation Once the Mocax Objects were built we could value each trade with the following computational effort.
It must be noted that APA achieves significant accelerations even in the case of already fast analytical vanilla pricing. Then, on top of the single trade APA acceleration, we applied PPC. If, say, we have 1,000 swaps, PCC delivers x1,000 acceleration, on top of the already existing APA performance gain. Partial ES calculations FRTB requires a number of partial Expected Shortfall calculations in which some of the risk factors are kept constant in all scenarios. The Mocax Objects are ideal for these calculations as we can “slice” the state space conveniently and reduce the dimensionality of the Mocax Object, hence achieving further acceleration. All this can be done on the firstly built Mocax Object, without having to call the original pricer again. This pricing function slicing is illustrated in the following exhibit: if we have a 2D value surface and we need a 1D slice of it (the black line), we can do so quite easily with the Mocax Objects.
