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The Variance Gamma (VG) Model with Long Range Dependence

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The Variance Gamma (VG) Model with Long Range Dependence View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Sydney eScholarship The Variance Gamma (VG) Model with Long Range Dependence Richard Finlay A thesis submitted in fulfilment of the requirements for the degree of Doctor of Philosophy School of Mathematics and Statistics University of Sydney February 2009 CONTENTS List of Figures ................................................................. v List of Tables .................................................................. vi Glossary ...................................................................... viii Abstract ...................................................................... x Acknowledgements ............................................................ xiv Chapter 1. Motivation, definitions and the model .............................. 1 1.1. Evidence against geometric Brownian motion . 2 1.1.1. Kurtosis. 2 1.1.2. Volatility and intermittency . 3 1.1.3. Dependence . 3 1.2. Self similarity and long range dependence. 8 1.3. The subordinator model . 10 1.3.1. The efficient markets hypothesis and the subordinator model.. 13 1.3.2. Volatility and the interpretation of fτtg .................................. 15 1.4. The VG and the t models . 16 1.4.1. The VG distribution . 16 1.4.2. The t distribution . 18 1.4.3. Comparison between the VG and t...................................... 20 1.4.4. The GH as a generalisation of the VG and t.............................. 21 1.5. Related Models. 22 1.5.1. Heston and the VG . 23 1.5.2. A correlated difference of gammas (DG) model . 23 1.6. Lévy processes . 25 1.7. Historical discussion . 26 Chapter 2. Activity time ...................................................... 31 2.1. The gamma case. 32 2.1.1. Construction of the gamma activity time process . 32 2.1.2. Convergence to a self similar limit . 37 2.1.3. Relaxing the assumptions on Z(s) ...................................... 42 2.1.4. Conclusion . 47 2.2. The inverse gamma case. 47 iii iv CONTENTS 2.2.1. Construction of the inverse gamma activity time process . 47 2.2.2. Conclusion . 52 2.3. Asymmetry and sign reversal in the asymptotic fTtg process . 52 2.4. Other possible activity time processes . 54 Chapter 3. Simulation, estimation and fit to data .............................. 58 3.1. Simulation and estimation techniques . 59 3.1.1. Estimation of the correlation structure. 62 3.2. Estimation results for the simulated data . 63 3.2.1. Estimation results for µ = θ = 0........................................ 64 3.2.2. Estimation results for µ, θ 6= 0.......................................... 71 3.3. The ECF method . 78 3.3.1. Estimation results for the ECF method . 79 3.3.2. Other characteristic function-based approaches . 82 3.4. Rejection of models. 84 3.5. Fitting to financial data . 87 3.6. Other approaches to simulation, estimation and model fit . 94 3.6.1. Simulation . 94 3.6.2. Estimation with WinBUGS . 96 3.6.3. Model fit . 98 Chapter 4. Option pricing .................................................... 101 4.1. Market completeness. 105 4.2. Established VG option pricing models . 107 4.3. Option pricing under the DG model . 109 4.4. Option pricing under a LRD VG model . 111 4.4.1. Some martingale constructions . 113 4.4.2. Fit to data . 116 Chapter 5. Conclusion ....................................................... 119 References .................................................................... 121 List of Figures 1.1 Histogram of S&P 500 index returns with fitted normal model........................ 4 1.2 Histogram of Microsoft Corporation’s share price returns with fitted normal model. ................................................................................................................ 4 1.3 30 day rolling annualised volatility (in per cent)............................................... 5 1.4 ACF of returns, absolute returns and squared returns for the S&P 500 index, Microsoft, and a simulated standard normal series. .......................................... 6 1.5 Smoothed Periodogram for S&P 500, (S&P 500)2 and |S&P 500| returns. ...... 7 1 −1 2.1 Boxplots of kH (Tk − k) and kH (Tk − k) for gamma and inverse gamma distributed increments........................................................................................ 54 3.1 Histogram of S&P 500 index returns with fitted models................................... 91 3.2 Histogram of Microsoft Corporation’s share price returns with fitted models.. 91 3.3 Histogram of the AUD/US exchange rate returns with fitted models. .............. 92 3.4 Histogram of WTI returns with fitted models.................................................... 92 v List of Tables 3.1 Descriptive statistics of VG simulation runs. .................................................... 64 3.2 Parameter estimation results of VG simulation runs. ........................................ 65 3.3 Descriptive statistics of t simulation runs.......................................................... 65 3.4 Parameter estimation results of t simulation runs.............................................. 66 3.5 MAD statistics for VG models with µ = θ = 0. ............................................... 67 3.6 Bias for VG models with µ = θ = 0. ................................................................ 67 3.7 Standard deviation for VG models with µ = θ = 0. ......................................... 68 3.8 MAD statistics for t models with µ = θ = 0..................................................... 68 3.9 Bias for t models with µ = θ = 0...................................................................... 69 3.10 Standard deviation for t models with µ = θ = 0............................................... 69 3.11 MAD of H^ estimates using the minimum χ2 method. ...................................... 69 3.12 Bias of H^ estimates using the minimum χ2 method. ........................................ 70 3.13 Standard deviation of H^ estimates using the minimum χ2 method. ................. 70 3.14 MAD of H^ estimates using the moment method............................................... 70 3.15 Bias of H^ estimates using the moment method................................................. 70 3.16 Standard deviation of H^ estimates using the moment method.......................... 71 3.17 Descriptive statistics of VG simulation runs. .................................................... 72 3.18 Parameter estimation results of VG simulation runs. ........................................ 72 3.19 Descriptive statistics of t simulation runs.......................................................... 72 3.20 Parameter estimation results of t simulation runs.............................................. 73 3.21 MAD statistics for VG models with θ small...................................................... 74 3.22 Bias of estimates from VG models with θ small. .............................................. 74 3.23 Standard deviation of estimates from VG models with θ small. ....................... 75 3.24 MAD statistics for t models with θ small.......................................................... 75 3.25 Bias of estimates from t models with θ small.................................................... 76 3.26 Standard deviation of estimates from t models with θ small............................. 76 3.27 MAD statistics for models with θ large............................................................. 76 3.28 Bias of estimates from models with θ large....................................................... 77 3.29 Standard deviation of estimates from models with θ large................................ 77 3.30 MAD statistics for symmetric VG models fitted via ECF. ................................ 79 3.31 Bias for symmetric VG models fitted via ECF. ................................................. 79 3.32 Standard deviation for symmetric VG models fitted via ECF. .......................... 79 3.33 MAD statistics for skew VG models fitted via ECF.......................................... 80 3.34 Bias for skew VG models fitted via ECF........................................................... 80 3.35 Standard deviation for skew VG models fitted via ECF. ................................... 80 vi LIST OF TABLES vii 3.36 MAD statistics for symmetric and skew VG models fitted via the Madan & Seneta ECF procedure. ...................................................................................... 83 3.37 Bias for symmetric and skew VG models fitted via the Madan & Seneta ECF procedure................................................................................................... 83 3.38 Standard deviation for symmetric and skew VG models fitted via the Madan & Seneta ECF procedure. .................................................................................. 84 3.39 Average χ2 values and 95th percentiles for correct models (CM), and per cent of incorrect models (IM) rejected, fitted via minimum χ2 and PMLE. ..... 86 2 2 3.40 Per cent of incorrect models rejected, using χ56 and χ57 cutoffs. ..................... 87 3.41 Descriptive statistics of data. ............................................................................. 88 3.42 Estimated parameters for data fitted via minimum χ2....................................... 88 3.43 Estimated parameters for data fitted via PMLE................................................. 88 3.44 H^ estimates for financial data............................................................................ 93 3.45 Estimated parameters for S&P 500 data fitted via WinBUGS and PMLE.......
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