A practical implementation of XVA in the new normal by Christopher Kairinos Submitted in partial fulfillment of the requirements for the degree Magister Scientiae to the Department of Mathematics and Applied Mathematics in the Faculty of Natural and Agricultural Sciences University of Pretoria Pretoria Supervisor: Prof. E Mar´e June 2017 Declaration I, Christopher Kairinos, declare that this dissertation, which I hereby submit for the degree Master of Science at the University of Pretoria, is my own work and has not previously been submitted by me for a degree at this or any other tertiary institution. Christopher Kairinos June 2017 Abstract The Great Financial Crisis (GFC) of 2008 left many financial institutions dev- astated. Despite the practice of advanced risk management at the time, society witnessed the collapse of the \too big to fail" institutions. Gaping holes within the existing risk framework lurked, which both regulators and practitioners failed to detect. This dissertation discusses the symptoms of the crisis that were over- looked and explores the financial engineering implemented post-2008 to avoid the next crisis. The author considers the work of Hull, White, Gregory, Brigo, Kenyon, Green, Morini, Pallavicini, Piterbarg, Burgard, Kjaer, Elouerkhaoui, and Castagna. A literature review is provided for each of the mentioned names to highlight each author's contribution to the field of Total Value Adjustment (XVA) pricing. An in-depth analysis on the funding invariance principle suggested by Elouerkhaoui is provided followed by a model implementation. The core aim of this dissertation is to review XVA valuations from a practitioners perspective using the framework provided by Elouerkhaoui. A secondary aim of the dissertation is to briefly explore the work of Aboura and Maillard on the Cornish-Fisher Transformation (CF). The CF is considered as a parsimonious approach in estimating non-normal distribu- tions, therefore an interesting alternative to price XVA using Monte Carlo (MC) simulation. Acknowledgments I would like to thank my supervisor Prof. Eben Mar´efor the patience he has shown and the invaluable advice he has given me throughout my course. To my friends and family, thank you for providing me with the motivation to complete what I started. To Berenice, your encouragement gave me the will I needed to cross the finish line, I cannot thank you enough. \To strive, to seek, to find, and not to yield." Lord Alfred Tennyson Contents List of Figures vii List of Tables x Acronyms xii Notation xiv I Introduction and Context 1 1 Structure of the Dissertation 2 2 Introduction 10 3 The Law of One Price 15 II Literature Review 24 4 Hull and White 25 5 Pricing by Hedging - Piterbarg, Kjaer, and Burgard 30 6 Pricing by Expectation - Brigo, Morini, and Pallavicini 38 7 Including MVA and KVA in the Semi-Replication Framework - Kenyon and Green 44 8 The Funding Invariance Principle with XVA - Elouerkhaoui 49 v III Mathematical Preliminaries 56 9 The Funding Invariance Principle 57 10 From FVA to KVA 70 11 The Poisson Process 78 IV Pricing Model Implementation 80 12 Pricing Commodities without XVA 81 13 Accounting for a Non-Normal World 87 14 Adding CVA, DVA, COLVA, and FVA to the Batch 90 15 Finishing off with MVA and KVA 103 16 Conclusion 110 V Appendices 112 A Model Calibration Code and Results 113 A.1 Estimating State Variables and Parameters using the Kalman Filter and Maximum Likelihood Estimation . 113 A.2 Implying Parameters from Current Market Data by Minimising Squared Errors . 120 A.3 Kalman Filter Best Fit Results . 123 A.4 Implied Parameter Best Fit Results . 124 B Model Pseudocode 126 B.1 MC and QMC code for GBM, OU, and SL processes . 126 B.2 Generating XVA Numbers for Collateralised and Uncollateralised trades132 Bibliography 146 vi List of Figures 3.1 The spread between three month LIBOR and OIS during the 2008 crisis. Source Bloomberg, April 1 2016 . 17 3.2 The history of counterparty credit risk in financial institution [Ruiz 2015], page 127 . 18 3.3 A simplistic view of the considerations required to price a derivative pre-2008 accurately. 22 3.4 A complicated overview of the many considerations required to accu- rately price a derivative post-2008 accurately. 23 12.1 Comparison of MC vs QMC techniques when pricing a ICE Brent call option using T = 0.5 and the same parameters as listed in Table 12.1. 86 13.1 Surface A implies k=0.57 and s=0.21; surface B implies k=4.2 and s=0.96 . 88 A.1 The difference between the estimated and actual time series of ICE Brent crude oil for each model, 24 February 2006 - 20 January 2016. 123 A.2 The cumulative sum of the squared errors for each model produced via the Kalman filter parameter estimate, 24 February 2006 - 20 January 2016. 123 A.3 The estimated GBM ICE Brent curve vs actual. 124 A.4 The estimated OU ICE Brent curve vs actual. 124 A.5 The estimated SL ICE Brent curve vs actual. 125 vii List of Tables 12.1 Parameters used in the MC Simulation. 84 12.2 Summary of European call prices for three different processes using the MC Simulation. 84 13.1 Summary of European call prices using MC and QMC techniques across different strikes and changes in k and s where indicated. Pa- rameters from Table 12.1 were used with T=0.5. 89 14.1 Summary of European call prices across different strikes. A nor- mal distribution was assumed for pricing with RC = 0:3; RB = 0:4; CDSB = 400 bps; CDSC = 700 bps; rf = 0:05; rc = 0:01 and rv = 0:1. Bank B sells to counterparty C. 92 14.2 Summary of European call prices across different strikes using the CF transformation with bank B selling to counterparty C. 93 14.3 Summary of European call prices across different strikes using a nor- mal distribution with bank B buying from counterparty C. 94 14.4 Summary of European call prices across different strikes using the CF transformation with bank B buying from counterparty C, s=0.97 and k=4.2. 95 14.5 Summary of European put prices across different strikes using a nor- mal distribution with bank B selling to counterparty C. 95 14.6 Summary of European put prices across different strikes using the CF transformation with bank B selling to counterparty C, s=0.97 and k=4.2. 96 14.7 Summary of European put prices across different strikes using a nor- mal distribution with bank B buying from counterparty C. 97 viii 14.8 Summary of European put prices across different strikes using the CF transformation with bank B buying from counterparty C, s=0.97 and k=4.2. 97 14.9 Summary of XVAs to apply when pricing uncollateralised options. 98 14.10Brent crude oil forward price with XVAs calculated using a normal distribution. Bank B sells to counterparty C. 98 14.11Brent crude oil forward price with XVAs calculated using the CF transformation with bank B sellng to counterparty C, s=0.97 and k=4.2. 98 14.12Brent crude oil forward price with XVAs calculated using a normal distribution. Bank B buys from counterparty C. 99 14.13Brent crude oil forward price with XVAs calculated using the CF transformation with bank B buying from counterparty C, s=0.97 and k=4.2. 99 14.14Collateralised Brent crude oil forward price with XVAs calculated using a normal distribution. Bank B sells to counterparty C. 100 14.15Collateralised Brent crude oil forward price with XVAs calculated using the CF transformation with bank B selling to counterparty C, s=0.97 and k=4.2. 100 14.16Collateralised Brent crude oil forward price with XVAs calculated using a normal distribution. Bank B buys from counterparty C. 100 14.17Collateralised Brent crude oil forward price with XVAs calculated using the CF transformation with bank B buying from counterparty C, s=0.97 and k=4.2. 100 14.18Collateralised Brent crude oil forward price with XVAs calculated using a normal distribution. Bank B buys from counterparty C, col- lateral payment lag is set to 10 days. 101 14.19Collateralised Brent crude oil forward price with XVAs calculated using the CF transformation with bank B buying from counterparty C, s=0.97 and k=4.2, collateral payment lag is set to 10 days. 101 14.20Collateralised Brent crude oil forward price with XVAs calculated using a normal distribution. Bank B buys from counterparty C, col- lateral payment lag is set to 100 days. 101 ix 14.21Collateralised Brent crude oil forward price with XVAs calculated using the CF transformation with bank B buying from counterparty C, s=0.97 and k=4.2, collateral payment lag is set to 100 days. 101 15.1 Model parameters calibrated to real-world data from 2007 to 2008 to obtain 99th percentile Stressed VAR (SVAR). 104 15.2 Model parameters calibrated to real-world data from 2015 to 2016 to obtain 99th percentile Value-at-Risk (VAR). 104 15.3 Unhedged and uncollateralised forward price, bank B buys from coun- terparty C. A normal distribution was assumed for pricing with RC = 0:3; RB = 0:4; CDSB = 400 bps; CDSC = 700 bps; rf = 0:05; rk = 0:12; rc = 0:01 and rv = 0:1. 105 15.4 Hedged forward price with IM, bank B buys from counterparty C. A normal distribution was assumed for pricing with the same parame- ters as Table 15.3. 106 15.5 Unhedged and uncollateralised forward price, bank B buys from coun- terparty C.