This thesis proposes a model combining the GARCH option valuation and innovations for the asset returns following the NIG distribution. The model is called NIG-GARCH and will include the GARCH option pricing model of Duan (1995) as a limiting case. In addition, we also consider the “leverage effects” in the model, called NIG-NGARCH, which can explain the asymmetric effect of stock volatility. There are two results in our empirical studies. First, empirical results on index shows that our models can adequately fit the daily returns on S&P 500 index because the distribution assumption of normal inverse Gaussian has fatter tails than the Gaussian and can be used to explain excess kurtosis. Second, empirical results on call option shows that in out-of-sample performance our model is superior to the GARCH models of Duan (1995) and Heston and Nandi (2000) for at-the-money calls and does not perform too badly for in-the-money or out-of-the-money calls. Besides, we also find that although nonanalytical solution GARCH models take a lot of time in computing, they can fit the observed implied volatility of call option data much better than the closed-form models.
References
Andersen, J. (2001), “On the Normal Inverse Gaussian Stochastic Volatility Model,”
Journal of Business and Economic Statistics, 19, 44-54.
Bakshi, G., C. Cao and Z. Chen (1997), “Empirical Performance of Alternative Option Pricing Models,” Journal of Finance, 52, 2003-2049.
Barndorff-Nielsen, O. E. (1997), “Normal Inverse Gaussian Distributions and Stochastic Volatility Modeling,” Scandinavian Journal of Statistics, 24, 1-13.
Barndorff-Nielsen, O. E. (1998), “Processes of Normal Inverse Gaussian Type,” Finance and Stochastics, 2, 41–68.
Barndorff-Nielsen, O. E. and N. Shephard (2001), “Non-Gaussian
Ornstein-Uhlenbeck-based Models and Some of their Uses in Financial Economics,”
Journal of the Royal Statistical Society, Series B, 63, 167–241.
Barndorff-Nielsen, O. E. and Shephard, N. (2003), “Integrated OU Processes and non-Gaussian OU-based Stochastic Volatility,” Scandinavian Journal of Statistics, 30, 277-295.
Black, F. and Scholes, M. (1973), “The Pricing of Options and Corporate Liabilities,”
Journal of Political Economy, 81, 637-654.
Bollerslev, T. (1986), “Generalized Autoregressive Conditional Heteroskedasticity, ” Journal of Econometrics, 31, 307-327.
Bollerslev T. (1987), “A Conditionally Heteroskedastic Time Series Model for
Speculative Prices and Rates of Return,” Review of Economics and Statistics, 69, 542–547.
Bollerslev, T. and Forsberg, L. (2002), “Bridging the Gap Between the Distribution of Realized (ECU) Volatility and ARCH Modelling (of the Euro): The GARCH-NIG Model,” Journal of Applied Econometrics, 17, 535-548.
Cox, J.C., Ross, S.A., and Rubinstein, M. (1979), “Option Pricing: a Simplied Approach,” Journal of Financial Economics, 7(3), 229-263.
Christoffersen, P., S. Heston and K. Jacobs (2003), “Option Valuation with Conditional Skewness,” Manuscript, McGill University.
Christoffersen, P. and K. Jacobs (2004a), “The Importance of the Loss Function in Option Valuation,” Journal of Financial Economics, 72, 291-318.
Christoffersen, P. and K. Jacobs (2004b), “Which GARCH model for Option Valuation?”
Management Science, 50, 1204-1221.
Duan, J. C. (1995), “The GARCH Option Pricing Model,” Mathematical Finance, 5, 13-32.
Duan, J. C., G. Gauthier and J. Simonato (1999), “An Analytical Approximation for the GARCH Option Pricing Model,” Journal of Computational Financl, 2, 75-116.
Dumas, B., J. Fleming, and R. Whaley (1998), “Implied Volatility Functions: Empirical Tests,” Journal of Finance, 53, 2059-2106.
Engle, R. (1982), “Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of UK Inflation,” Econometrica, 50, 987-1008.
Engle, R. and V. Ng (1993), “Measuring and Testing the Impact of News on Volatility,”
Journal of Finance, 48, 1749-1778.
Heston, S. L. (1993), “A Closed Form Solution for Options with Stochastic Volatility, with Applications to Bond and Currency Options,” Review of Financial Studies, 6, 327-343.
Heston, S. L. and Nandi, S., (2000), “A Closed-Form GARCH Option Valuation Model,”
Review of Financial Studies, 13, 585-625
Hull, J. and A. White (1987), “The Pricing of Options on Assets with Stochastic Volatility,” Journal of Finance, 42, 281-300.
Nelson, D.B. (1991), “Conditional Heteroskedasticity in Asset Returns: A New Approach,” Econometrica, 59, 347-370.
Pan, J. (2002), “The Jump-Risk Premia Implicit in Options: Evidence from an Integrated Time-Series Study,” Journal of Financial Economics, 63, 3-50.
Scott, L.O. (1987), “Option Pricing when the Variance Changes Randomly: Theory, Estimation and an Application,” Journal of Financial and Quantitative Analysis, 22, 419-438.
Appendix A. Tables
Table 1 Maximum likelihood estimates and Model characteristics
ω 1.10E-06 3.80E-06 6.30E-07 3.50E-06 5.30E-13 6.10E-13 α 0.0194 0.0233 0.0218 0.0307 1.10E-06 7.00E-07
persistence 0.9859 0.9505 0.992 0.9545 0.9854 0.9906 Annualized
Volatility 0.1388 0.1395 0.1412 0.1388 0.1395 0.1374 JB test 59.63 Maximum likelihood estimates of all the GARCH (1,1) models using the daily returns on S&P 500 index from July 1,1988 to June 28,1991. The sample size of observations is equal to 756. JB test = Jarque-Bera test, KS test = Kolmogorov-Smirnov (KS) test, LB-Qz (p) = Ljung-Box Q test on autocorrelation for p lagged standardized innovations, LB-Qz2 (q) = Ljung-Box Q test on GARCH effects for q lagged squared standardized innovations. The parenthesis gives the p-value of all the tests.
Table 2 Basic description of Call Options data Panel A. Average Call Option Prices
moneyness DTM<20 20<DTM<50 50<DTM<80 All S / K<0.95 0.91 1.34 2.98 2.07 0.95<S / K<0.99 1.60 3.42 6.21 3.87 0.99<S / K<1.01 4.82 8.41 12.09 8.42 1.01<S / K<1.05 12.96 16.16 19.53 16.17
1.05<S / K 25.47 27.65 30.70 27.87
All 11.79 13.25 15.49 13.51
Panel B. Number of Call Options
moneyness DTM<20 20<DTM<50 50<DTM<80 All
<0.95 2 84 70 156
[0.95,0.99) 210 712 339 1261
[0.99,1.01) 164 354 163 681
[1.01,1.05) 299 658 286 1243
>1.05 216 542 225 983
All 891 2350 1083 4324
The table shows the basic description of call option contracts written on the S&P 500 index for a total 235 trading days from January 8, 1991 to December 31, 1991. Panel A describes the average call option prices and Panel B describes the number of call options across days to maturity (DTM) and moneyness (S/K) where S/K is defined as the S&P500 index value over the option strike price.
Table 3 Out-of-Sample Performance with loss function RMSE Panel A. NIG-NGARCH Model RMSE
moneyness DTM<20 20<DTM<50 50<DTM<80 All S / K<0.95 0.6704 0.7767 1.1945 0.9853 0.95<S / K<0.99 0.5439 1.1522 1.5337 1.1963 0.99<S / K<1.01 0.6531 1.1815 1.3820 1.1338 1.01<S / K<1.05 0.5240 0.8985 1.1383 0.8897 1.05<S / K 0.4200 0.7021 0.8749 0.6970 All 0.5331 0.9809 1.2717 0.9964
Panel B. NGARCH Model RMSE
moneyness DTM<20 20<DTM<50 50<DTM<80 All S / K<0.95 0.6802 0.7394 1.2058 0.9761 0.95<S / K<0.99 0.5216 1.1345 1.5252 1.1821 0.99<S / K<1.01 0.6651 1.2011 1.3819 1.1461 1.01<S / K<1.05 0.5319 0.9508 1.1289 0.9164 1.05<S / K 0.4147 0.7126 0.8727 0.7015 All 0.5322 0.9928 1.2666 1.0010
Panel C. Heston-Nandi Model RMSE
moneyness DTM<20 20<DTM<50 50<DTM<80 All S / K<0.95 0.4360 1.0603 1.6100 1.3308 0.95<S / K<0.99 0.5957 1.3006 1.7922 1.3703 0.99<S / K<1.01 0.6687 1.1686 1.4306 1.1434 1.01<S / K<1.05 0.5132 0.8197 1.0462 0.8190 1.05<S / K 0.4327 0.6599 0.7379 0.6371 All 0.5479 1.0198 1.3723 1.0500 The table shows the root mean square error (RMSE) of call option across days to maturity (DTM) and moneyness (S/K). Panel A, B, C stand for the models of NIG-NGARCH, NGARCH and HN-GARCH, respectively.
Table 4 Out-of-Sample Performance with loss function MAPE Panel A. NIG-NGARCH Model MAPE
moneyness DTM<20 20<DTM<50 50<DTM<80 All S / K<0.95 0.0067 0.6192 0.4377 0.5391
moneyness DTM<20 20<DTM<50 50<DTM<80 All S / K<0.95 0.0068 0.5887 0.4426 0.5250
Panel C. Heston-Nandi Model MAPE
moneyness DTM<20 20<DTM<50 50<DTM<80 All S / K<0.95 0.0042 0.8587 0.5909 0.7333 The table shows the mean absolute percentage error (MAPE) of call option contracts across days to maturity (DTM) and moneyness (S/K). Panel A, B, C stand for the models of NIG-NGARCH, NGARCH and HN-GARCH, respectively.
Table 5 Out-of-Sample Performance with loss function log-IVMAE Panel A. NIG-NGARCH Model log-IVMAE
moneyness DTM<20 20<DTM<50 50<DTM<80 All S / K<0.95 0.2362 0.1523 0.1522 0.1542
Panel B. NGARCH Model log-IVMAE
moneyness DTM<20 20<DTM<50 50<DTM<80 All S / K<0.95 0.2409 0.1460 0.1539 0.1519
Panel C. Heston-Nandi Model log-IVMAE
moneyness DTM<20 20<DTM<50 50<DTM<80 All S / K<0.95 0.1575 0.2026 0.2015 0.2015 The table shows the mean absolute error of log-implied volatility (log-IVMAE) of call option contracts across days to maturity (DTM) and moneyness (S/K). Panel A, B, C stand for the models of NIG-NGARCH, NGARCH and HN-GARCH, respectively.
Table 6 Four characteristics for IG and NIG distributions
Appendix B. Figures
Figure 1 QQ-plot of all the GARCH (1,1) models
The figure shows QQ-plot of the standardized innovations for daily returns on S&P 500 index from July 1, 1988 to June 28, 1991.
Figure 2 Implied volatility of data observed and model forecasted
The figure shows the implied volatility of data observed and model forecasted where obs is data observed implied volatility, NIGNG, NG and HN are model implied volatilities of NIG-NGARCH, NGARCH and HN-GARCH, respectively.
Figure 3a NIG densities with different σ
The figure shows NIG densities with β= 0,μ= 0,η= 3 andσ= 1, 2, 3 (from inner to outer).
Figure 3b NIG densities with different β
The figure shows NIG densities withσ= 1,μ= 0,η= 3 andβ= -1.6, -1, 0, 1, 1.6 (from left to right).
Figure 3c NIG densities with different μ
The figure shows NIG densities withσ= 1,β= 0,η= 3 andμ= -1, 0, 1 (from left to right).
Figure 3d NIG densities with different η
The figure shows NIG densities withσ= 1,β= 0,μ= 3 andη= 0.1, 0.5, 1, 2, 5 (from inner to outer).
Appendix C. Show that (3.1) is risk-neutral under measure Q