DEGREE PROJECT IN MATHEMATICS, SECOND CYCLE, 30 CREDITS STOCKHOLM, SWEDEN 2016 Modelling the Stochastic Correlation PENG CHEN KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES Modelling the Stochastic Correlation PENG CHEN Master’s Thesis in Financial Mathematics (30 ECTS credits) Master Programme in Applied and Computational Mathematics (120 credits) Royal Institute of Technology year 2016 Supervisor at KTH: Fredrik Armerin Examiner: Boualem Djehiche TRITA-MAT-E 2016:27 ISRN-KTH/MAT/E--16/27--SE Royal Institute of Technology School of Engineering Sciences KTH SCI SE-100 44 Stockholm, Sweden URL: www.kth.se/sci Modelling the Stochastic Correlation Abstract In this thesis, we mainly study the the correlation between stocks. The correlation between stocks has been receiving increasing atten- tion. Usually the correlation is considered to be a constant, although it is observed to be varying over time. In this thesis, we study the properties of correlations between Wiener processes and introduce a stochastic correlation model. Following the calibration methods by Zetocha, we implement the calibration for a new set of market data. i Modellering av stokastisk korrelationen Sammanfattning I det h¨ar examensarbetet fokuserar vi fr¨amst p˚aatt studera kor- relation mellan aktier. Korrelationen mellan aktier har f˚attallt st¨orre uppm¨arksamhet. Vanligtvis antas korrelation vara konstant, trots att empiriska studier antyder att den ¨ar tidsvarierande. I det h¨ar exa- mensarbetet studerar vi egenskaper hos korrelationen mellan Wiener- processer och inf¨or en stokastisk korrelationsmodel. Baserat p˚aka- libreringsmetoder av Zetocha implementerar vi kalibrering f¨or en ny upps¨attning av marknadsdata. ii Acknowledgements I would like to express my gratitude to my supervisor at KTH Fredrik Armerin for the guidance and comments on this thesis. Thanks for your kindness and encouragement throughout the thesis work. I woud like to thank my family for supporting my study in Sweden. It is a valuable experience. Peng Chen Stockholm, June 2016 . iii Contents 1 Introduction 1 2 Theoretical background 3 2.1 It^ointegral . 3 2.2 Variation and covariation of continuous martingales . 5 2.2.1 Martingale . 5 2.2.2 Variation and covariance . 6 2.2.3 Quadratic variation of Wiener process . 7 2.3 It^o'sformula . 8 2.3.1 Constructing correlated Wiener processes . 9 2.4 Stochastic differential equations . 11 2.5 The Girsanov theorem . 13 3 The Black-Scholes model 15 3.1 The one dimensional Black-Scholes model . 15 3.1.1 Historic volatility . 17 3.1.2 Implied volatility . 17 3.2 Correlation . 19 3.2.1 Correlation between two random variables . 19 3.2.2 Basket correlation and implied correlation . 20 3.3 Multidimensional models with correlations . 22 3.3.1 Constant volatility model . 23 3.4 Multi-asset derivatives . 24 3.4.1 Worst-of options . 25 3.4.2 Dispersion trading . 25 3.4.3 Variance and volatility swaps . 27 3.4.4 The correlation swap . 28 4 The stochastic correlation model 29 4.1 A general model . 29 4.2 A simplified multi-asset model . 29 4.3 Properties of the model . 31 4.3.1 Existence and uniqueness of solutions . 31 4.3.2 Moment evaluation . 31 4.3.3 The boundary condition . 33 4.3.4 Simulation . 33 4.3.5 Numerical experiments for the parameters . 34 5 Calibration 37 5.1 Methods for calibration . 37 5.2 Data description and the calibration . 39 5.2.1 Model analysis . 41 5.2.2 Summary . 46 v 1 Introduction There is a large number of stocks traded in financial markets. The prices of these stocks change every day in light of the news and the expected future performance of the equities. These changes are not independent. There are some co-movements between these stocks, which are called correlations. Firms that run some common business lines will have some correlations, and in general, these correlations do not remain the same over time. For instance, suppose A and B are two firms that have some common business lines. If firm A drop a certain business line, then the correlation between A and B will change. According to [8], the price of a stock reflects the expectation of future performance of the firm. Every news will affect this expectation for all stocks but with different extent. The effect may be large for some equities but negligible for others, thus the correlation will change according to the news. For simplicity, the correlation is usually modeled as a constant. In some cases, the constant correlation cannot explain the market well. In this thesis, we will study a stochastic correlation model. It is observed that the correlation between stocks rises in a bear market and falls in a bullish market. In order to verify this claim again, we explore the OMXS30 index to observe its correlation and return. Figure 1 is a scatter plot of the correlation and return for the index OMXS30. It shows that the correlation of the index OMXS30 is negatively correlated to its return. The red line is the linear regression with coefficients y^ = 0:194477 − 0:365424x; wherey ^ denotes the correlation and x is the log return. It indicates that the correlation and the log return are negatively correlated. There are many multi-asset equity derivatives traded in the market. The correlation market mainly includes the following types of contracts: • Worst-of options • Dispersion trading • Variance and volatility swaps • Correlation swaps We will study these multi-asset equity derivatives later after the introduction of some theoretical backgrounds. 1 three−month realized correlaiton three−month realized 0.05 0.10 0.15 0.20 0.25 0.30 0.35 −0.2 −0.1 0.0 0.1 0.2 three−month log return Figure 1: Scatter of three month realized correlation vs three month log-return 2 2 Theoretical background 2.1 It^ointegral Through this section, we will introduce some theoretical background of stochas- tic calculus. This section mainly refers to [9]. Definition 2.1. Let (Ω; F; P) be a probability space. A filtration (Ft)t≥0 on the probability space is an increasing family of σ-field that satisfies Fs ⊂ Ft ⊂ F; 8s ≤ t: A probability endowed with a filtration is called a filtered probability space. A stochastic process X on (Ω; F; P) is adapted to the filtration (Ft) if for each t ≥ 0, Xt is Ft-measurable. The natural filtration of the stochastic process X is the smallest σ-algebra with respect to which all the variables (Xs; s ≤ t) are measurable. We will X use (Ft )t≥0 to denote the natural filtration of X. We know that a stochastic process is always adapted to its natural filtration. Definition 2.2. (One-dimensional Wiener process) A stochastic process W is called a Wiener process or a Brownian motion if 1. W (0)=0 2. The process has independent increments, i.e. if s < t then W (t)−W (s) W is independent of Fs . 3. For s < t the random variable W (t)−W (s) has the normal distribution N[0; t − s], where t − s is the variance. 4. W has continuous trajectories. We will interchangeable to use the notation W (t) and Wt in this thesis. Definition 2.3. (Multidimensional Wiener process) An Rd-valued process 1 2 d Wt = (Wt ;Wt ; ··· ;Wt ) is called d-dimensional Wiener process, if its com- 1 2 d ponents Wt ;Wt ; ··· ;Wt are independent Wiener processes. Note that an n-dimensional Wiener process has n independent compo- nents. In this thesis, we will use term, correlated Wiener process, if the components are correlated. Definition 2.4. We say that a process g belongs to the class L2[a; b] if R b 2 • a E[g (s)]ds < 1 3 W • the process g is adapted to (Ft )-filtration. If g belongs to L2[0; t] for all t > 0, then we say g belongs to L2 or g 2 L2: Before we define the It^ointegral for general process, we firstly define the It^o integral for simple processes. Then the general It^ointegral can be defined as a limit. Definition 2.5. A process g is called a simple process if there exist a parti- tion Π : a = t0 < t1 < ··· < tn = b of [a; b] such that n−1 X g(t) = gtk 1[tk;tk+1)(t): (2.1) k=0 Then the It^ointegral of a simple process is defined by n−1 Z b X g(s)dWs = gtk [W (tk+1) − W (tk)]; (2.2) a k=0 where g is a random variable that is F W -measurable. tk tk Theorem 2.1. For any process g 2 L2[a; b], we can find a sequence of simple process gn such that Z b 2 E[(g(s) − gn(s)) ]ds ! 0: (2.3) a Then the It^ointegral for general process g 2 L2[a; b] can be defined by Z b Z b g(s)dW (s) = lim gn(s)dW (s): (2.4) a n!1 a From the definition above, we have following properties of the stochastic integral with respect to a Wiener process. Theorem 2.2. Let g; f 2 L2[a; b]. Then each It^ointegral is defined and has the following properties: 1. Linearity. If α and β are some constants then Z b Z b Z b (αf(s) + βg(s))dW (s) = α f(s)dW (s) + β g(s)dW (s) a a a 4 2. Zero mean property. Z b E g(s)dW (s) = 0: a 3. It^oisometry. "Z b 2# Z b E g(s)dW (s) = E[g2(s)]ds: a a 4.