This thesis evaluates how the evolutionary algorithm CMA-ES (Covariance Matrix Adaption Evolution Strategy) can be used for optimizing the total initial margin for a network of banks trading bilateral OTC derivatives. The algorithm is a stochastic method for optimization of non-linear and, but not limited to, non-convex functions. The algorithm searches for an optimum by generating normally distributed samples and iteratively updating the mean and covariance matrix of the search distribution using the best candidate solutions in the sampled population. In this thesis, feasible solutions are represented by the null space obtained from the constraint of keeping all banks' market exposure unchanged throughout the optimization, and the generated samples for each iteration correspond to linear combinations of the base vectors spanning this null space. In particular, this thesis investigates how different representations of this null space affect the convergence speed of the algorithm. By applying the algorithm to problems of varying sizes, using several different null space representations coming from different matrix decomposition methods, it is found that as long as an orthonormal representation is used it does not matter which matrix decomposition method it comes from. This is found to be because, given any orthonormal null space representation, the algorithm will at start generate a rotationally invariant sample space in its search for the optimal solution, independent of the specific null space representation. If the representation is not orthogonal, the initial sample will in contrast be in the shape of an ellipsoid and thus biased in certain directions, which in general affects the performance negatively. A non-orthonormal representation can converge faster in specific optimization problems, if the direction of the solution is known in advance and the sample space is pointed towards that direction. However, the benefit of this aspect is limited in a realistic scenario and an orthonormal representation is recommended. Furthermore, as it is shown that different orthonormal representations perform equally, it is implied that other characteristics can be considered when deciding which matrix decomposition method to use; such as the importance of fast computation or desire for a sparse representation. / Denna avhandling utvärderar hur CMA-ES (Covariance Matrix Adaption Evolution Strategy) kan användas för att optimera en total "initial margin" för ett nätverk av banker som handlar bilaterala OTC derivat. Algoritmen är en stokastisk metod för optimering av icke-linjära och, men inte enbart, icke-konvexa funktioner. Algoritmen söker efter ett optimum genom att generera normalfördelade utfall och iterativt uppdatera medelvärdet och kovariansmatrisen för sök-fördelningen med hjälp av de bästa lösningarna i varje iteration. I detta arbete representeras tillåtna lösningar till problemet av nollrummet från bivillkoret att alla bankers marknadsexponering ska vara oförändrade genom optimeringen och de genererade utfallen består av slumpade linjärkombinationer av nollrummets basvektorer. I synnerhet undersöks hur olika representationer av nollrummet påverkar konvergenshastigheten för algoritmen. Algoritmen har applicerats med flera olika nollrumsrepresentationer, framtagna genom olika matrisfaktoriseringsmetoder, och det kan konstateras att så länge nollrummsrepresentationen är ortonormal är valet av faktoreringsmetod obetydlig. Detta då användande av orthornormala nollrumsrpresentationer i algoritmen leder till en initialt symmetrisk, rotationsmässigt invariant, sökning efter den optimala lösningen. Om representationen inte är ortogonal kommer det resulterande sökområdet i varje iteration att ha formen av en ellipsoid och sålunda viktas i vissa riktningar, vilket i allmänhet påverkar prestandan negativt. Emellertid kan en icke-ortonormal representation konvergera snabbare i specifika scenarier, givet att lösningens riktning är känd i förväg och sökområdet kan pekas mot den riktningen. Vidare, eftersom det har visats att olika ortonormala representationer konvergerar lika fort, innebär resultatet att andra egenskaper kan beaktas vid val av matrisfaktoriseringsmetod, såsom vikten av snabb beräkning eller önskan om en gles representation.

Within Central Counterparty Clearing, the Clearing House collects Initial Margin from its Clearing Members. The Initial Margin can be calculated in many ways, one of which is by applying the commonly used risk measure Value-at-Risk. However, Value-at-Risk has one major flaw, namely its inability to encapsulate Tail Risk. Due to this, there has for long been a desire to replace Value-at-Risk with Expected Shortfall, another risk measure that has shown to be much better suited to encapsulate Tail Risk. That said, Value-at-Risk is still used over Expected Shortfall, something which is mainly due to the fact that there is no consensus regarding how one should backtest Expected Shortfall. The goal of this thesis is to evaluate some of the most commonly proposed methods for backtesting Expected Shortfall. In doing this, several non-parametric backtests of Expected Shortfall are investigated using simulated data as well as market data from different types of securities. Moreover, this thesis aims to shed some light on the differences between Value-at-Risk and Expected Shortfall, highlighting why a change of risk measure is not as straightforward as one might believe. From the investigations of the thesis, several backtests are found to be sufficient for backtesting the Initial Margin with Expected Shortfall as the risk measure, the so called Minimally Biased Relative backtest showing the overall best performance of the looked at backtests. Further, the thesis visualizes how Value-at-Risk and Expected Shortfall are two risk measures that are inherently different in a real-world setting, emphasizing how one should be careful making conversions between the two based upon parametric assumptions.

Machine learning is a rapidly growing field within artificial intelligence that an increasing number of individuals and corporations are beginning to utilize. In recent times, the financial sector has also started to recognize the potential of these techniques and methods. Nasdaq Clearing is responsible for managing the clearing business for the clearinghouse's members, and the objective of this thesis has been to explore the possibilities of using machine learning to replicate a subpart of the SPAN model's margin call calculations, known as initial margin, in the commodities market. The purpose of replicating SPAN's initial margin calculations is to open up for possibilities to create transparency and understanding in how the input variables affect the output. In the long run, we hope to broaden the insights on how one can use machine learning within the margin call processes. Various machine learning algorithms, primarily focused on regression tasks but also a few classification ones, have been employed to replicate the initial margin size. The primary objective of the methodology was to determine the algorithm that demonstrated the best performance in obtaining values that were as close as possible to the actual initial margin values. The findings revealed that a model composed of a combination of classification and regression, with non-parametric algorithms such as Random Forest and KNN, performed the best in both cases. Our conclusion is that the developed model possesses the ability to effectively compute the size of the initial margin and thus accomplishes its objective. / Maskininlärning är ett snabbt växande område inom artificiell intelligens som allt fler individer och företag börjar använda. Finanssektorn har nu också börjat undersöka hur dessa tekniker och metoder kan skapa värde. Nasdaq Clearing hanterar clearingverksamheten för clearinghusets medlemmar och syftet med denna uppsats har varit att undersöka möjligheterna att använda maskininlärning för att replikera en del av SPAN-modellens beräkningar av marginkravet som kallas Initial Marginal. Syftet med att replikera SPANs initiala marginberäkningar är att öppna upp för möjligheter att skapa transparens och förståelse för hur inputvariablernapåverkar outputen. På sikt hoppas vi kunna bredda insikterna hur maskininlärningslösningar skulle kunna användas inom "Margin Call"- processen. De metoder som användes för att replikera storleken på Initial Margin var olika maskininlärningsalgoritmer, främst fokuserade på regressionsuppgifter men några klassificeringsalgoritmer användes också. Fokus i metoden var att hitta vilken algoritm som presterade bäst, det vill säga den algoritm som predikterade närmst de faktiska värdena för Initial Margin. Resultatet visade sig vara en modell som kombinerade klassificering och regression, där icke-parametriska algoritmer såsom Random Forest och KNN var de som presterade bäst i båda fallen. Vår slutsats är att den utvecklade modellen har en god förmåga att beräkna storleken på Initial Margin och därmed uppfyller den sitt syfte.

Trading contracts with future commitments requires posting a collateral, called initial margin requirement, to cover associated risks. Differences in estimating those risks and varying risk appetites can however lead to identical contracts having different initial margin requirements at different market places. This creates a potential for minimizing those requirements by reallocating contracts. The task of minimizing the requirement is identified as a black-box optimization problem with constraints. The aim of this project was to investigate that optimization problem, how it can best be tackled, and comparing different techniques for doing so. Based on the results and obstacles encountered along the way, some guidelines are then outlined to provide assistance for whomever is interested in solving this or similar problems. The project consisted both of a literature study to examine existing knowledge within the subject of optimization, and an implementation phase to empirically test how well that knowledge can be put to use in this case. During the latter various algorithms were tested in a number of different scenarios. Focus was put on practical aspects that could be important in a real situation, such as how much they could decrease the initial margin requirement, execution time, and ease of implementation. As part of the literature study, three algorithms were found which were evaluated further: simulated annealing, differential evolution, and particle swarm optimization. They all work without prior knowledge of the function to be optimized, and are thus suitable for black-box optimization. Results from the implementation part showed largely similar performance between all three algorithms, indicating that other aspects such as ease of implementation or parallelization potential can be more important to consider when choosing which one to use. They were all well able to optimize different portfolios in a number of different cases. However, in more complex situations they required much more time to do so, showing a potential need to speed up the process.

The introduction of initial margin requirements for non-cleared OTC derivatives has made it possible to optimize initial margin when considering a network of trading participants. Applying CMA-ES, this thesis has explored a new method to handle the nonlinear constraints present in the initial margin optimization problem. The idea behind the method and the research question in this thesis are centered around leveraging data created during optimization. Specifically, by creating a linear approximation of the feasible region using support vector machines and in turn applying a repair strategy based on projection. The hypothesis was that by repairing solutions an increase in convergence speed should follow. In order to answer the research question, a reference method was at first created. Here CMA-ES along with feasibility rules was used, referred to as CMA-FS. The proposed method of optimization data leveraging (ODL) was then appended to CMA-FS, referred to as CMA-ODL. Both algorithms were then applied to a single initial margin optimization problem 100 times each with different random seeds used for sampling in the optimization algorithms. The results showed that CMA-ODL converged significantly faster than CMA-FS, without affecting final objective values significantly negatively. Convergence was measured in terms of iterations and not computational time. On average a 5% increase in convergence speed was achieved with CMA-ODL. No significant difference was found between CMA-FS and CMA-ODL in terms of the percentage of infeasible solutions generated. A reason behind the lack of a reduction in violations can be due to how ODL is implemented with the CMA-ES algorithm. Specifically, ODL will lead to a greater number of feasible solutions being available during recombination in CMA-ES. Although, due to the projection, the solutions after projection are not completely reflective of the actual parameters used for that generation. The projection should also bias the algorithm towards the boundary of the feasible region. Still, the performative difference in terms of convergence speed was significant. In conclusion, the proposed boundary constraint handling method increased performance, but it is not known whether the method has any major practical applicability, due to the restriction to only considering the number of iterations and not the computational time. / Införandet av initial margin för non-cleared OTC derivatives har gjort det möjligt att optimera initial margin när ett flertal marknadsdeltagare tas till hänsyn. Denna uppsats har applicerat CMA-ES och specifikt undersökt en ny metod för hantering av de icke-linjära bivillkoren som uppstår när initial margin optimeras. Idén bakom metoden och forskningsfrågan i rapporten bygger på att utnyttja data som generas vid optimering. Detta görs specifikt genom att den icke-linjära tillåtna regionen approximeras linjärt med support vector machines. Därefter används en reparationsstrategi bestående av projicering för att reparera otillåtna lösningar. Hypotesen i uppsatsen var att genom att reparera lösningar så skulle konvergenshastigheten öka. För att svara på forskningsfrågan så togs en referensmetod fram, där CMA-ES och feasibility rules användes för att hantera icke-linjära bivillkor. Denna version av CMA-ES kallades CMA-FS. Sedan integrerades den nya metoden med CMA-FS, denna version kallades för CMA-ODL. Därefter så applicerades båda algoritmer 100 gånger på ett initial margin optimeringsproblem, där olika seeds användes för generering av lösningar i algoritmerna. Resultaten visade att CMA-ODL konvergerade signifikant snabbare än CMA-FS utan att påverka optimeringsresultatet negativt. Med CMA-ODL så ökade konvergenshastigheten med ungefär 5%. Konvergens mättes genom antal iterationer och inte beräkningstid. Ingen signifikant skillnad mellan CMA-ODL och CMA-FS observerades när de jämfördes med avseende på mängden icke-tillåtna lösningar genererade. En anledning varför ingen skillnad observerades är hur den nya metoden var integrerad med CMA-ES algoritmen. Den tilltänkta metoden leder till att fler tillåtna lösningar är tillgängliga när CMA-ES ska bilda nästa generation men eftersom lösningar projiceras så kommer dom inte att reflektera dom parametrar som användes för att faktiskt generera dom. Projiceringen leder också till att fler lösningar på randen av det tillåtna området kommer att genereras. Sammanfattningsvis så observerades fortfarande en signifikant ökning i konvergenshastighet för CMA-ODL men det är oklart om algoritmen är praktiskt användbar p.g.a. restriktionen att enbart betrakta antalet iterationer och inte total beräkningstid.