Model l i ngand For ecast i ng Hi ghFr equency Fi nanci alDat a St avr osDegi annaki sandChr i st osFl or os Modelling and Forecasting High Frequency Financial Data This page intentionally left blank Modelling and Forecasting High Frequency Financial Data Stavros Degiannakis and Christos Floros © Stavros Degiannakis and Christos Floros 2015 All rights reserved No reproduction, copy or transmission of this publication may be made without written permission No portion of this publication may be reproduced, copied or transmitted save with written permission or in accordance with the provisions of the Copyright, Designs and Patents Act 1988, or under the terms of any licence permitting limited copying issued by the Copyright Licensing Agency, Saffron House, 6–10 Kirby Street, London EC1N 8TS Any person who does any unauthorized act in relation to this publication may be liable to criminal prosecution and civil claims for damages The authors have asserted their rights to be identified as the authors of this work in accordance with the Copyright, Designs and Patents Act 1988 First published 2015 by PALGRAVE MACMILLAN Palgrave Macmillan in the UK is an imprint of Macmillan Publishers Limited, registered in England, company number 785998, of Houndmills, Basingstoke, Hampshire RG21 6XS Palgrave Macmillan in the US is a division of St Martin’s Press LLC, 175 Fifth Avenue, New York, NY 10010 Palgrave Macmillan is the global academic imprint of the above companies and has companies and representatives throughout the world Palgrave® and Macmillan® are registered trademarks in the United States, the United Kingdom, Europe and other countries ISBN 978-1-349-56690-7 ISBN 978-1-137-39649-5 (eBook) DOI 10.1057/9781137396495 This book is printed on paper suitable for recycling and made from fully managed and sustained forest sources Logging, pulping and manufacturing processes are expected to conform to the environmental regulations of the country of origin A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data Degiannakis, Stavros, author Modelling and forecasting high frequency financial data / Stavros Degiannakis, Christos Floros pages cm Finance–Mathematical models Speculation–Mathematical models Technical analysis (Investment analysis)–Mathematical models I Floros, C (Christos), author II Title HG106.D44 2015 2015013168 332.01 5195–dc23 To Aggelos, Andriana and Rebecca Stavros Degiannakis To Ioanna, Vasilis-Spyridon, Konstantina-Artemis and Christina-Ioanna Christos Floros This page intentionally left blank Contents List of Figures xi List of Tables xiv Acknowledgments xvii List of Symbols and Operators xviii Introduction to High Frequency Financial Modelling The role of high frequency trading Modelling volatility Realized volatility Volatility forecasting using high frequency data Volatility evidence Market microstructure 10 11 14 14 15 Intraday Realized Volatility Measures The theoretical framework behind the realized volatility Theory of ultra-high frequency volatility modelling Equidistant price observations 3.1 Linear interpolation method 3.2 Previous tick method Methods of measuring realized volatility 4.1 Conditional – inter-day – Variance 4.2 Realized variance 4.3 Price range 4.4 Model-based duration 4.5 Multiple grids 4.6 Scaled realized range 4.7 Price jumps 4.8 Microstructure frictions 4.9 Autocorrelation of intraday returns 4.10 Interday adjustments Simulating the realized volatility Optimal sampling frequency 24 24 27 31 31 32 32 32 34 35 37 37 37 37 37 38 38 42 47 Methods of Volatility Estimation and Forecasting 58 Daily volatility models – review 58 vii viii Contents 59 59 60 60 60 61 61 61 62 62 63 63 64 64 64 67 67 70 70 70 72 73 Multiple Model Comparison and Hypothesis Framework Construction Statistical methods of comparing the forecasting ability of models 1.1 Diebold and Mariano test of equal forecast accuracy 1.2 Reality check for data snooping 1.3 Superior Predictive Ability test 1.4 SPEC model selection method Theoretical framework: distribution functions A framework to compare the predictive ability of two competing models A framework to compare the predictive ability of n competing models 4.1 Generic model 4.2 Regression model 4.3 Regression model with time varying conditional variance 4.4 Fractionally integrated ARMA model with time varying conditional variance Intraday realized volatility application 110 110 111 111 112 112 113 4 1.1 ARCH(q) model 1.2 GARCH(p, q) model 1.3 APARCH(p, q) model 1.4 FIGARCH(p, d, q) model 1.5 FIAPARCH(p, d, q) model 1.6 Other methods of interday volatility modelling Intraday volatility models: review 2.1 ARFIMA k, d , l model 2.2 ARFIMA k, d , l - GARCH p, q model 2.3 HAR-RV model 2.4 HAR-sqRV model 2.5 HAR-GARCH(p, q) model 2.6 Other methods of intraday volatility modelling Volatility forecasting 3.1 One-step-ahead volatility forecasting: Interday volatility models 3.2 Daily volatility models: program construction 3.3 One-step-ahead volatility forecasting: intraday volatility models 3.4 Intraday volatility models: program construction The construction of loss functions 4.1 Evaluation or loss functions 4.2 Information criteria 4.3 Loss functions depend on the aim of a specific application 115 119 119 121 121 122 123 ix Contents 128 128 130 133 Realized Volatility Forecasting: Applications Measuring realized volatility 1.1 Volatility signature plot 1.2 Interday adjustment of the realized volatility 1.3 Distributional properties of realized volatility Forecasting realized volatility Programs construction Realized volatility forecasts comparison: SPEC criterion Logarithmic realized volatility forecasts comparison: SPA and DM Tests 5.1 SPA test 5.2 DM test 161 161 162 165 174 176 178 190 Recent Methods: A Review Modelling jumps 1.1 Jump volatility measure and jump tests 1.2 Daily jump tests 1.3 Intraday jump tests 1.4 Using OxMetrics (Re@lized under G@RCH 6.1) The RealGARCH model 2.1 Realized GARCH forecasting 2.2 Leverage effect 2.3 Realized EGARCH Volatility forecasting with HAR-RV-J and HEAVY models 3.1 The HAR-RV-J model 3.2 The HEAVY model Financial risk measurements 4.1 The method 217 217 218 219 220 221 230 232 234 234 235 235 236 238 238 Intraday Hedge Ratios and Option Pricing Introduction to intraday hedge ratios Definition of hedge ratios 2.1 BEKK model 2.2 Asymmetric BEKK model 2.3 Constant Conditional Correlation (CCC) model 2.4 Dynamic Conditional Correlation (DCC) model 2.5 Estimation of the models Data Estimated hedge ratios 243 243 246 248 248 249 250 251 251 253 Simulate the SPEC criterion 6.1 ARMA(1,0) simulation 6.2 Repeat the simulation 6.3 Intraday simulated process 200 200 202 264 Modelling and Forecasting High Frequency Financial Data Even with all the above assumptions p(t) does not become a martingale To see that, take s < t to get: ⎛ E p(t) Fsσ = E ⎝ N (t) ⎞ e yi Fsσ ⎠ = i=1 N (s) ⎛ e yi E ⎝ ⎞ e yi Fsσ ⎠ i=N (s)+1 i=1 N (t) N (t) E(e yi ) = p(s) N (t) i=N (s)+1 e yi (7.33) i=N (s)+1 The price process is martingale if and only if the last product of expected prices is equal to one, i.e N (t) E(e yi ) = ⇔ E(e yi ) = 1, ∀i (7.34) i=N (s)+1 Even if this is the case, an equivalent martingale measure can be constructed This is done through the following steps: Step 1: Introduce a modified process for the logarithm of the price through N (t) q˜ (t) = N (t) (yi − log E(e yi )) = i=1 (yi − ly ), (7.35) i=1 where ly := log E(e yi ) Step 2: Introduce the corresponding modified price process through p˜ (t) = e q˜ (t) (7.36) One can readily see that E p˜ (t) Fsσ = p˜ (t) N (t) E(e yi −ly ) = p˜ (s), i=N (s)+1 which yields p˜ (t) to be a martingale (7.37) Intraday Hedge Ratios and Option Pricing 265 Option pricing 9.1 The approach of Merton Following the set up of previous section we get the following formula for the European call option price: E c (t) = e −λ(T−t) +∞ n=0 (λ(T − t))n E cn (p(0), K , μ, σ ), n! (7.38) where λ denotes the activity of the Poisson process for the trades that take place, K is the strike price, μ is the expected value and σ is the variance of the jumps of the logarithm of the price The jumps are assumed to be normally distributed In addition cnE is given by the equation (Merton, 1976; Scalas and Politi, 2013): cnE (p(0), K , μ, σ ) = N (d1, n )p(0) − N (d2, n )K , (7.39) where N stands for the normal cumulative distribution function and log(p(0) K ) + n(μ + σ 2) , √ σ n √ d2, n = d1, n − σ n d1, n = (7.40) 9.2 The approach of Scalas and Politi The payoff of a European call option at maturity is given by: c E (T) = max{p(T) − K ; 0}, (7.41) where K denotes as before the strike price The price of the option at any prior time instant is given by: c E (t) = e −r(T−t) Ep˜ c E (T) Ftσ = Ep˜ c E (T) Ftσ , (7.42) where r is the risk-free rate, which has been assumed to be equal to zero.8 To further proceed with the option valuation we consider two cases; time t is an epoch, and time t is not an epoch 266 Modelling and Forecasting High Frequency Financial Data 9.3 Time t is an epoch Without loss of generality it can be assumed that t = Then the value of the option is derived as: c E (0) = Ep˜ c E (T) F0σ +∞ = c E (τ )dFp˜ (T) (τ ) (7.43) Following our standard notation Fp˜ (T) (τ ) is nothing but the cumulative distribution function of the random variable p˜ (T) We set n e yi −log E(e ) yi p˜ n = (7.44) i=1 This is the product of n independent identically distributed random variables.9 Therefore, its cumulative distribution function is the n-fold Mellin convolution of Fy˜ (u), the common cumulative distribution function of y˜i = yi − logE(e yi ) (7.45) This yields Fp˜ n = Fy˜∗ M (u) n (7.46) As the number of trades in the interval [0, T] can be an arbitrary non-negative integer, one can see that +∞ Fp˜ (T) = P(N (T) = n)Fy˜∗ M (u), n (7.47) n=0 due to the mutual independence of the tick-by-tick logarithmic returns and the inter-trade durations If we were to we calculate the probabilities that appear in the sum above, we would be in business First observe that for n = the Mellin convolution is a cumulative distribution function such that Fy˜∗ M = 0 if u = if u > (7.48) Intraday Hedge Ratios and Option Pricing 267 When n = 0, then there are no trades, hence the price remains unchanged, i.e p(T) = p(0) = 1, whence c E (T) = c E (1) The probability P(N (T) = 0) decreases as T increases The contribution to the sum is P(N (T) = 0)c E (1) If n > 0, then we have: P(N (T) = n) = P ({Tn ≤ T} ∩ {Tn+1 > T}) = E I{Tn ≤T} I{Tn+1 >T} = E I{Tn ≤T} I{Jn+1 >T−Tn } +∞ T = T−τ T = dFJ∗ n (τ )dFJ (u) (1 − FJ (T − τ ))dFJ∗ n (τ ) (7.49) 9.4 Time t is not an epoch Without loss of generality we assume that p(t) = 1, whence p(t) acts as numeraire We define q(t, T) = q(T) − q(t) = log(p(T)/p(t)) = log p(T) The option price is then given by: c E (t) = Ep˜ c E (T) Ftσ +∞ = c E (τ )dFp˜nt (τ ), (7.50) where N (t) = nt is the known number of trades and the cumulative distribution t (τ ) is given by: function Fp˜n(T) t Fp˜n(T) (τ ) = +∞ P (N (T) − N (t) = n |N (t) = nt )Fy˜∗ M (τ ) n (7.51) n=0 The probability in the above sum is given by: T−t P (N (T) − N (t) = n |N (t) = nt ) = P (N (T) − N (t + τ ) = n − 1)dFJt, nt (τ ) (7.52) The probability in the integral above is calculated as in the case when time t is an epoch, i.e via equation (7.49), with T replaced by T − (t + τ ) FJt, nt (τ ) is the cumulative distribution function of the residual life-time at time t, conditioned by the fact that there were nt trades up to time t It is denoted with Jt, nt The residual 268 Modelling and Forecasting High Frequency Financial Data life-time is nothing but the time interval from t to the next renewal epoch TN (t)+1 Its distribution depends heavily on the knowledge of its previous history In our case the total number of trade until time t is known The right-hand side of the last equation contains the probability of having nt − trades between the renewal epoch t + τ and T We still need to compute the cumulative distribution function of the residual life-time To that we observe that: Jt, nt ≤ τ = Tnt +1 − t ≤ τ |N (t) = nt (7.53) The latter can be written as: Jt, nt ≤ τ = Tnt +1 − t ≤ τ Tnt ≤ t ∩ Tnt +1 > t (7.54) This yields: FJt, nt (τ ) = Pr(Jt, nt ≤ τ ) = P = P Tnt +1 − t ≤ τ Tnt ≤ t ∩ Tnt +1 > t Tnt +1 − t ≤ τ ∩ Tnt ≤ t ∩ Tnt +1 > t Pr Tnt ≤ t ∩ Tnt +1 > t (7.55) The denominator has already been calculated in the case where t was an epoch; see equation (7.49) To calculate the numerator we see that: Tnt +1 − t ≤ τ ∩ Tnt ≤ t ∩ Tnt +1 > t Tnt ≤ t ∩ t − Tnt < Jnt +1 ≤ t + τ − Tnt (7.56) This gives: P Tnt +1 − t ≤ τ ∩ Tnt ≤ t ∩ Tnt +1 > t Tnt ≤ t ∩ t − Tnt < Jnt +1 ≤ t + τ − Tnt =P = E I{Tnt ≤t } I{t−Tnt