DEGREE PROJECT IN MATHEMATICS, SECOND CYCLE, 30 CREDITS STOCKHOLM, SWEDEN 2016 Modelling of Stochastic Volatility using Partially Observed Markov Models HJALMAR HEIMBÜRGER KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES Modelling of Stochastic Volatility using Partially Observed Markov Models HJALMAR HEIMBÜRGER Master’s Thesis in Mathematical Statistics (30 ECTS credits) Master Programme in Applied and Computational Mathematics (120 credits) Royal Institute of Technology year 2016 Supervisor at Handelsbanken: Björn Löfdahl Supervisor at KTH: Jimmy Olsson Examiner: Jimmy Olsson TRITA-MAT-E 2016:62 ISRN-KTH/MAT/E--16/62-SE Royal Institute of Technology School of Engineering Sciences KTH SCI SE-100 44 Stockholm, Sweden URL: www.kth.se/sci Modelling of Stochastic Volatility using Partially Observed Markov Models Abstract In this thesis, calibration of stochastic volatility models that allow correla- tion between the volatility and the returns has been considered. To achieve this, the dynamics has been modelled as an extension of hidden Markov models, and a special case of partially observed Markov models. This the- sis shows that such models can be calibrated using sequential Monte Carlo methods, and that a model with correlation provide a better fit to the ob- served data. However, the results are not conclusive and more research is needed in order to confirm this for other data sets and models. i Modellering av Stokastisk Volatilitet genom Partiellt Observerbara Markovmodeller Sammanfattning Detta examensarbete behandlar kalibrering av stokastiska volatilitetsmod- eller som till˚aterkorrelation mellan volatiliteten och avkastningen. F¨or att uppn˚adetta beteende har dynamiken modellerats som ett specialfall av partiellt observerbara Markovmodeller som ¨aren utvidgning av dolda Markovmodeller (HMMer). I denna masteruppsats visas att dessa typer av modeller kan kalibreras med sekventiella Monte Carlo-metoder och att dessa modeller ger en b¨attreanpassning till observerad data. Resultaten ¨aremellertid inte entydiga och det ¨arn¨odv¨andigtutreda fr˚aganvidare f¨or andra modelltyper och andra datam¨angder. ii Acknowledgements I would like to thank my supervisor Jimmy Olsson for introducing me to the subject of computational statistics, valuable input and suggestions on everything concerning this thesis. Furthermore, I would like to thank my supervisor at Handelsbanken Bj¨ornL¨ofdahlfor the idea behind the model as well as guiding me through the process. Moreover, I would also like to thank Handelsbanken for providing me with the data necessary to perform the analysis. Lastly, I would like to thank my other half Carolina Eriksson for listening to my problems and always being there for me. Without you I would not have finished this thesis. iii Contents 1 Introduction1 1.1 Stochastic volatility................................1 1.2 Hidden Markov models...............................1 1.3 Partially observed Markov models........................3 1.4 Thesis objectives..................................3 1.5 Outline.......................................4 2 Background5 2.1 Hidden Markov models...............................5 2.2 Parameter estimation in HMMs | the Expectation-Maximisation algorithm.6 2.2.1 The EM algorithm.............................7 2.2.2 Numerical approximations.........................8 2.2.3 Gradient ascent EM............................8 2.2.4 Averaging..................................9 2.3 Sequential Monte Carlo methods.........................9 2.3.1 The bootstrap particle filter........................ 10 3 Sequential Monte Carlo Methods for Virtually Hidden Markov Models 12 3.1 Virtually hidden Markov models......................... 12 3.2 Filtering in virtually hidden Markov models................... 13 3.3 Smoothing in virtually hidden Markov models.................. 14 3.3.1 Fixed-lag smoothing............................ 14 3.3.2 The time-reversed process......................... 15 3.3.3 Forward-filtering, backward-smoothing.................. 17 3.3.4 Forward-only FFBSm........................... 17 3.3.5 Forward-filtering, backward-simulation.................. 18 3.3.6 The PaRIS algorithm........................... 19 3.4 Instrumental density design............................ 20 3.4.1 Designing the proposal density...................... 21 3.4.2 Measures of weight imbalance....................... 22 3.4.3 Instrumental kernel for backward index draws.............. 23 4 Evaluation of Models 26 4.1 Information criteria................................. 26 4.1.1 Akaike information criterion........................ 26 4.1.2 Bayesian information criterion...................... 27 iv 4.1.3 Bootstrap information criterion...................... 27 4.1.4 Posterior probabilities of model candidates............... 28 5 Stochastic Volatility Models and Implementation 29 5.1 Choice of proposal density............................. 30 5.2 The intermediate quantity............................. 31 5.3 Volatility index model............................... 33 5.4 Model evaluation.................................. 34 5.5 A few notes on implementation.......................... 35 5.5.1 Starting guesses for the EM algorithm.................. 35 5.5.2 Accept-reject algorithm.......................... 35 5.5.3 Working with logarithms......................... 36 6 Simulations and Results 37 6.1 Design of the PaRIS algorithm.......................... 37 6.1.1 Simulation time complexity........................ 38 6.2 Calibration of SV models............................. 40 6.2.1 Prefatory study............................... 40 6.2.2 Main study................................. 40 7 Discussion 50 7.1 Design of the PaRIS algorithm.......................... 50 7.1.1 Proposal kernel selection.......................... 50 7.1.2 Computational complexity for the PaRIS algorithm.......... 51 7.2 Stochastic volatility models............................ 51 7.2.1 Prefatory study............................... 51 7.2.2 Main study................................. 52 8 Conclusions and Future Work 54 8.1 Conclusions..................................... 54 8.2 Future work..................................... 54 A Extension of the Accept-Reject Sampling Algorithm 56 B Derivation of the Intermediate Quantity 58 B.1 E-step........................................ 58 B.2 M-step........................................ 60 B.2.1 Maximisation with respect to µ ...................... 60 B.2.2 Maximisation with respect to βζ ..................... 61 B.2.3 Maximisation with respect to Σ ..................... 61 B.2.4 Updating formula for the HMM version................. 62 B.3 Summary...................................... 62 v Nomenclature R Correlation matrix of the log-returns, i.e. the correlation corresponding to the covariance matrix in (5.4). Its elements are denoted [R]ij = rij. PN i Ωt i=1 !t. i i pAIC Akaike weights, wAIC / exp(−∆AICi=2), see (4.5). i i pBIC Posterior probability of model gi being the true model, pBIC / exp(−∆BICi=2) see (4.4). xs:t A vector. xs:t , (xs; xs+1; : : : ; xt), 8s ≤ t. PaRIS(·) One iteration of Algorithm 3.2. PF(·) One iteration of Algorithm 3.1. i N i N j Pr(f!tgi=1) The categorical distribution, i.e. if J ∼ Pr(f!tgi=1), then P(J = j) = !t =Ωt. φ0:t;θ The smoothing distribution. φ0:t;θ , φ0:tjt;θ. φs:s0jt(xs:s0 ) Density of Xs:s0 , conditionally on the observations, i.e. for any (measurable) s0+1−s R A ⊆ X , P(Xs:s0 2 Ajy1:t) = A φs:s0jt(xs:s0 ) dxs:s0 . φt;θ The filtering distribution. φt;θ , φt:tjt;θ. 0 0 0 gt(x; x ) g(x; x ; yt), i.e. the emission density associated with YtjXt−1 = x; Xt = x . q(x; ·) The transition density associated with Xt+1jXt = x. 1(·) The indicator function. E Expectation of a stochastic variable. V Variance of a stochastic variable. D(pkq) Kullback-Leibler Divergence between distributions p and q. kAk det(A). x ^ y min(x; y). (X; X ) The measurable space of the state process fXtg. (Y; Y) The measurable space of the observation process fYtg. vi m; n fk 2 N : m ≤ k ≤ n; (m; n) 2 N2g, i.e. the set of all non-negative integers J K between integers m and n. N∗ fn 2 N : n > 0g, i.e. the positive integers. R+ fx 2 R : x > 0g, i.e. the positive real numbers. Xn The Cartesian product space X × · · · × X. | {z } n times vii Chapter 1 Introduction 1.1 Stochastic volatility The pioneering work of Black and Scholes [3] and their derivation of the famous Black-Scholes formula was a major advancement for financial mathematics and pricing of European-style derivatives. However, as time passed, several of the assumptions have proved to be too crude. The observed data exhibit features that are impossible under the simplistic assumptions made. One of the most notable such assumption is that the volatility of a stock is constant over time. To address these shortcomings, the main approach has been to add randomness to the volatility in the model. These models have become known as stochastic volatility models.A few models that have been proposed are the Heston model [18], Bates model [1], and the model proposed by Hull and White [19]. A main concept of all the models listed is the existence of volatility clusters. Empirical data suggests that there are clusters when the volatility is high, and any plausible model should allow these clusters. Furthermore, the market and the volatility is often assumed to be negatively correlated [2]. The concept is intuitive, as the increase of risk should make it less favourable to invest. However, not all literature points to this fact, as it only seems to be present