DEGREE PROJECT IN MATHEMATICS, SECOND CYCLE, 30 CREDITS STOCKHOLM, SWEDEN 2019 The Swap Market Model with Local Stochastic Volatility MOHAMMED BENMAKHLOUF ANDALOUSSI KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES The Swap Market Model with Local Stochastic Volatility MOHAMMED BENMAKHLOUF ANDALOUSSI Degree Projects in Mathematical Statistics (ECTS, 30 credits) Degree Programme in Engineering Physics (CTFYS, 300 credits) KTH Royal Institute of Technology year 2019 Supervisors at Kidbrooke Advisory: Edvard Sj¨ogren,Ludvig H¨allman Supervisor at KTH: Fredrik Viklund Examiner at KTH: Fredrik Viklund TRITA-SCI-GRU 2019:048 MAT-E 2019:18 Royal Institute of Technology School of Engineering Sciences KTH SCI SE-100 44 Stockholm, Sweden URL: www.kth.se/sci Abstract Modeling volatility is an intricate part of all financial models and the pricing of derivative contracts. And while local volatility has gained popularity in equity and FX models, it remained neglected in interest rates models. In this thesis, using spot starting swaps, the goal is to build a swap market model with non-parametric local volatility functions and stochastic volatility scaling factors. The local stochastic volatility formula is calibrated through a particle algorithm to match the market's swaption volatility smile. Nu- merical experiments are conducted for different currencies to compute the local stochastic volatility at different expiry dates, swap tenors and strike values. The results of the simulation show the high quality calibration of the algorithm and the efficiency of local stochastic volatility in interest rate smile building. 2 3 Sammanfattning Att modellera volatilitet ¨aren invecklad del av alla finansiella modeller och priss¨attningav derivatkontrakt. Medan lokala volatilitet har f˚attstor popu- laritet p˚aaktie- och FX-modeller, har de f¨orblivitf¨orsummadei r¨antemodeller. I detta examensarbete ¨arm˚aletatt bygga en r¨anteswapmarknads-modell med en icke-parametrisk lokal volatilitetsfunktion och en stokastisk volatilitets skalningsfaktor. Den lokala stokastiska volatiliteten kalibreras genom en partikelalgoritm f¨oratt matcha marknadens swaptions- volatilitetsleenden. F¨orolika valutor utf¨orsnumeriska experiment f¨oratt ber¨aknaden lokal stokastiska volatilitet f¨orolika utg˚angsdatum,swap-tenorer och l¨osenr¨antor. Resultaten av simuleringen visar att kalibreringen med den presenterade algo- ritmen ¨arav h¨ogkvalitet och effektiviteten i anv¨andandetav lokal stokastisk volatilitet vid ˚aterskapning av r¨anteleende. 4 5 Acknowledgements The writing of this thesis was possible through the advice and support of many people. First, I would like to thank my supervisor at the Royal In- stitute of Technology, Prof. Fredrik Viklund for his invaluable insight and his continuous guidance throughout the different stages of this thesis. I also would like to acknowledge my colleagues at Kidbrooke Advisory. In particu- lar, my two supervisors, Edvard Sj¨ogrenand Ludvig H¨allmanfor introducing me to the subject of the thesis, as well as their invaluable inputs, technical support and their insight into the financial industry. As well as Zaliia Gin- dullina, Mika Lindahl and Filip M¨orkfor their valuable advice, animated discussions and companionship. Mohammed Benmakhlouf Andaloussi Stockholm, April 2019 6 7 Nomenclature Abbreviations ARR Alternate Reference Rate ATM At-The-Money option, i.e. the strike price is equal to the current spot price of the underlying security. 1 bp 1 Basis Point, equal to 0.01% IBOR Inter-bank Offered Rate (LIBOR, EURIBOR ...) ICE Inter Continental Exchange IRS Interest Rate Swap ISDA International Swaps and Derivatives Association LIBOR London Inter-bank Offered Rate LMM LIBOR Market Model OIS Overnight Index Swap OTC Over The Counter RFR Risk-Free Rate SMM Swap Market Model ZCB Zero Coupon Bond xY x Years (1Y, 2Y, ...) xm x months (3m, 6m, ...) Mathematical notations 11x the dirac delta function at point x δi The time in year fractions between Ti−1 and Ti, measured with an appropriate day count convention i Pt The discount factor at time t with maturity Ti j;k At The Annuity factor at time t over the interval [Tj;Tk] j;k St The Swap rate at t starting at Tj and ending at Tk 8 Bt The value of the Bank account process at time t i;i+α∗ C j (Ti;K) Price (theoretical) of a payer swaption with strike K, on a ∗ spot starting swap with expiry date Ti and tenor αj . ∗ i;i+αj CMkt (Ti;K) Market price of a payer swaption with strike K, on a spot ∗ starting swap with expiry date Ti and tenor αj . EQ (...) The Expected value under the risk neutral measure Q ET (...) The Expected value under the forward measure QT EAi;j (...) The Expected value under the measure associated with the annuity factor Ai;j Ft The Filtration at time t, which contains all market informa- tion up to time t Π(t; X) the value at t of a contingent claim X (x)+ The positive part function of x :(x)+ = max(x; 0) gi;j(K) The local volatility function at strike K ct The stochastic volatility factor at time t nsim The number of simulation paths in the Monte-Carlo method 9 Contents 1 Introduction 15 1.1 Background............................ 16 1.2 Thesis objective.......................... 17 1.3 Limitations............................ 18 1.4 Disposition............................ 18 2 Mathematical Background 20 2.1 Interest rates, discount factors and ZCBs............ 20 2.1.1 Discount factors, ZCBs.................. 21 2.1.2 Forward rates....................... 22 2.1.3 The forward measure................... 22 2.2 Interest Rate Swaps........................ 23 2.3 Swaptions............................. 26 2.3.1 Definition, characteristics and contract terms...... 26 2.3.2 Payoff and pricing.................... 28 2.3.3 Bermudan swaptions................... 32 2.4 Discussion : Current changes in the interest rates market... 33 2.4.1 End of the IBOR era................... 33 2.4.2 OIS vs LIBOR discounting................ 36 3 The Swap Market Model Theory 39 3.1 The SMM with local stochastic volatility............ 39 3.1.1 The SMM setting..................... 40 3.1.2 Dynamics of a rolling maturity swap.......... 41 3.1.3 The drift term....................... 42 3.1.4 Volatility parameterisation................ 43 3.2 The Local Volatility function................... 44 3.3 The swap rate discount curve.................. 47 10 3.4 Swaption pricing under the Normal model........... 49 3.5 Relation to LMM and Libor rates................ 52 4 Methodology 54 4.1 Swap-Swap rates correlation................... 54 4.1.1 Historical Correlation................... 55 4.1.2 Smooth Correlation Matrix............... 57 4.2 Stochastic Volatility Scaling................... 59 4.3 Volatility interpolation/extrapolation.............. 61 4.4 A time series approximation................... 62 4.5 The SABR Model for normal volatility............. 64 4.6 The SMM Algorithm....................... 66 5 Results and Analysis 68 5.1 Algorithm Convergence...................... 68 5.1.1 Monte-Carlo simulation................. 68 5.1.2 Convergence and Confidence Intervals.......... 70 5.2 USD Simulation.......................... 75 5.2.1 Model parameters..................... 75 5.2.2 Local stochastic volatility surfaces............ 78 5.2.3 Results........................... 80 5.3 AUD simulation.......................... 83 5.3.1 Model parameters..................... 83 5.3.2 Local stochastic volatility surfaces............ 85 5.3.3 Results........................... 87 5.4 Bermudan Swaptions pricing................... 90 6 Discussion and Conclusion 92 6.1 Discussion............................. 92 6.2 Conclusion............................. 94 6.3 Further Work........................... 94 Bibliography 95 Appendices 99 11 List of Figures 2.1 the 3m US Libor - OIS spread during the period 2001-2017.. 26 2.2 Cash Flows structure for a swaption on the underlying asset Si;i+α ................................ 27 2.3 USD ATM Swaption prices.................... 30 2.4 Implied normal volatility of USD ATM swaptions....... 31 2.5 Implied lognormal volatility of USD ATM swaptions..... 31 2.6 Example of a Swap Spread for USD on 19/11/2018...... 38 3.1 Probability distribution function of a standard normal distri- bution plot (a) and a standard log normal distribution plot (b)................................. 42 3.2 Example of a Swap Curve for USD on 19/11/2018....... 48 3.3 ZCB discount curve for USD data on 19/11/2018, at T0=0 plot (a) and at T1=3m plot (b)................. 49 4.1 Historical swap rates (in %) for maturity 2Y, 5Y and 10Y over the period 2010-2018....................... 56 4.2 Swap rate correlation matrix generated through equation (4.1.7) with c = 0.22 and ρ1 = 0.17................... 58 4.3 Example of volatility interpolation and extrapolation results based on key tenor f1Y; 2Y; 5Y; 10Y; 15Y g ........... 62 4.4 Finite difference approximation of the first term of the time series sum for Ti = 9m and α = 10Y .............. 64 5.1 USD Swap Curve on 05/01/2017................ 69 3m;3m+5Y 5.2 Monte Carlo simulation for USD data of St for t 2 [0; 3m] (211 paths simulated).................. 70 1Y 10Y 5.3 Convergence test for the mean of S1Y as a function of the number of simulation paths................... 71 12 1Y 10Y 5.4 The absolute error of the convergence test for S1Y ..... 72 1Y 10Y 5.5 Convergence test, S1Y , with the 95% confidence interval.. 73 5.6 Convergence test for the model output g6m5Y (K = 3%) as a function of the number of simulation paths........... 74 5.7 The absolute error of the