A central hypothesis of the Black-Scholes model is that the return on the underlying asset distributed log-normally with constant volatility. However, it has been widely recognized that financial asset return processes possess heavy-tailed marginal distributions and volatility clustering. These features are interpreted as the evidence of the stochastic volatility of financial assets, and estimating the term structure of volatility has become an important issue in finance engineering.
We introduced the GARCH option pricing model of Duan (1995), using the LRNVR change measure to price options by Monte Carlo simulation. We then evaluate the empirical performance of different option pricing models on TAIEX options. We had considered the improved and constant volatility (non-update) Black-Scholes models and the update, non-update GARCH option pricing models. We then compare their pricing performance according to the absolute and percentage pricing errors.
Under Duan’s model setting, we compute the option prices according to the information set of the underlying asset, say, stock index; while for the Black-Scholes the information set of index options. There are a lot of authors utilizing information from the option data, for examples, Heston and Nandi (2000), Brigo and Mercurio (2001). They estimated their model by non-linear least square method, and the performance could depend on the parameter dimension.
References
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Appendix A. Tables
Table 1: Summary Statistics for TAIEX Call Options (2003/9/1-2003/12/10)*
Table 1. Moneyness (S/K)
Stat. DOTM OTM ATM ITM DITM Over all Average 18.35 44.16 118.15 252.45 501.58 210.52 Std. Dev. 10.90 23.93 43.71 45.35 162.85 197.61 Market Price
Number 52 121 160 113 138 584
*The summary statistics of TAIEX call option near closing prices are reported for each moneyness category.
Moneyness is defined as S/K, where S denotes the closing value of the TAIEX and K denotes the exercise price of the option. The sample period is from September 1, 2003 to December 10, 2003 with a total of 584 call options.
Table 2: The market option prices and the estimated prices of the alternative models+
Table 2 Moneyness (S/K)
Model DOTM OTM ATM ITM DITM Over all
Market Price 18.35 44.16 118.15 252.45 501.58 210.52 Non-update BS 29.02 62.57 136.16 260.63 489.41 218.93 Update BS 29.34 57.76 130.36 255.40 488.15 215.06 Non-update GARCH 29.75 60.62 130.43 259.54 490.28 217.02 Update GARCH 12.72 38.28 102.10 240.08 485.80 198.28
+The GARCH option pricing model is computed via 50,000 Monte Carlo simulation runs. The only difference between the non-update and update GARCH model is in the parameters being obtained. The implied volatilities of the update Black-Scholes model were obtained from the previous day, while the non-update model was obtained from the available sample.
Table 3A: The option prices with various initial conditional variance ratios in the GARCH option pricing model (Non-Update case)
Table 3A Moneyness (S/K)
0 σ
h DOTM OTM ATM ITM DITM Over all
0.8 29.91 60.59 130.14 258.61 511.08 235.11 1 29.75 60.62 130.43 258.65 511.20 235.22 1.2 29.93 60.89 131.02 258.48 511.20 235.41
Table 3B: The option prices with various initial conditional variance ratios in the GARCH option pricing model (Update case)
Table 3B Moneyness (S/K)
0 σ
h DOTM OTM ATM ITM DITM Over all
0.8 12.68 37.94 101.87 239.90 507.62 217.38 1 12.72 38.28 102.10 239.76 507.51 217.45 1.2 12.62 38.23 102.25 240.50 507.44 217.59
Table 4A: The Pricing Error of Alternative Option Pricing Models
Table 4A. Moneyness (S/K)
Moneyness DOTM OTM ATM ITM DITM Overall
S/K <0.95 (0.95, 0.98) (0.98, 1.02) (1.02, 1.05) >1.05 BS 10.68 18.42 18.02 8.18 -12.17 8.41 update BS 10.99 13.60 12.22 2.95 -13.43 4.54 Non-update GARCH 11.40 16.46 12.28 7.09 -11.30 6.49 Update GARCH -5.63 -5.88 -16.05 -12.37 -15.78 -12.24
Table 4B: The Percentage Pricing Error of Alternative Option Pricing Models
Table 4B. Moneyness (S/K)
Moneyness DOTM OTM ATM ITM DITM Overall
S/K <0.95 (0.95, 0.98) (0.98, 1.02) (1.02, 1.05) >1.05
BS 0.68 0.55 0.20 0.04 -0.02 0.23
update BS 0.74 0.39 0.12 0.02 -0.02 0.18 Non-update GARCH 0.57 0.30 0.10 0.03 -0.02 0.14 Update GARCH -0.43 -0.26 -0.18 -0.05 -0.03 -0.16
Table 5A: The Absolute Pricing Error of the Alternative Option Pricing Models
Table 5A. Moneyness (S/K)
Moneyness DOTM OTM ATM ITM DITM Overall
S/K <0.95 (0.95, 0.98) (0.98, 1.02) (1.02, 1.05) >1.05 BS 10.74 18.80 20.00 15.76 17.69 17.56 update BS 11.97 16.19 16.55 13.87 16.71 15.59 Non-update GARCH 12.30 20.50 22.22 21.81 21.63 20.76 Update GARCH 9.55 19.54 29.14 25.47 23.11 23.27
Table 5B: The Percentage Absolute Pricing Error of Alternative Option Pricing Models
Table 5B. Moneyness (S/K)
Moneyness DOTM OTM ATM ITM DITM Overall
S/K <0.95 (0.95, 0.98) (0.98, 1.02) (1.02, 1.05) >1.05
BS 0.69 0.56 0.22 0.07 0.04 0.26
update BS 0.79 0.47 0.16 0.06 0.03 0.23 Non-update GARCH 0.66 0.47 0.20 0.09 0.05 0.24 Update GARCH 0.58 0.52 0.29 0.10 0.05 0.27
Appendix B. Figures
Figure 1: The volatility smile of the TAIEX options in September, 2, 2003.
Volatility Skew
0 0.1 0.2 0.3 0.4 0.5
5000 5100 5200 5300 5400 5500 5600 5700 5800 5900 Strike Price
Implied Volatility
Figure 2: The Daily Closing price of TAIEX (2000/1/1 to 2003/12/10)
TAIEX index
3000 4000 5000 6000 7000 8000 9000 10000 11000
2000/1/4
2000/7/17
2001/1/29
2001/8/22
2002/3/28
2002/10/18
2003/5/19
2003/12/8
Figure 3: The Daily Log Return series of the TAIEX (2000/1/1 to 2003/12/10) Log Return Series of TAIEX index
-0.06 -0.04 -0.02 0 0.02 0.04 0.06
2000/1/4 2000/9/4 2001/5/18 2002/2/6 2002/11/1 2003/7/25
Figure 4: Maximum Likelihood Estimations of update GARCH (1, 1) process*
10 30 50 70
Day 0.00e0
5.00e-6 1.00e-5 1.50e-5 2.00e-5
8.50e-1 8.70e-1 8.90e-1 9.10e-1 9.30e-1
6.00e-2 6.50e-2 7.00e-2 7.50e-2
-1.00e-3 4.00e-3 9.00e-3 1.40e-2
alpha0 by Day
beta1 by Day
alpha1 by Day
lam bda by Day
*The parameters of the GARCH model were obtained by constrained optimization. Since we use the
“rolling the GARCH model” method, the parameter estimations changed daily. All the estimation sets satisfy the stationary GARCH conditions, i.e. α1+β1 <1. The estimates of the risk-premium parameter,λ, are all very close to zero.
Figure 5A: The Pricing Error of Alternative Option Pricing Models
DOTM OTM ATM ITM DITM
BS
update BS Non-update GARCH
Update GARCH
Figure 5B: The Percentage Pricing Error of Alternative Option Pricing Models
5B. Percentage Error
DOTM OTM ATM ITM DITM
BS
update BS
Non-update GARCH
Update GARCH
Figure 6A: The Absolute Pricing Error of Alternative Option Pricing Models.
DOTM OTM ATM ITM DITM
BS
update BS Non-update GARCH
Update GARCH
Figure 6B: The Absolute Percentage Pricing Error of Alternative Option Pricing Models.
6B. Absolute Percentage Error
0.00
DOTM OTM ATM ITM DITM
BS
update BS
Non-update GARCH
Update GARCH