This thesis uses S&P500 index options data to investigate the empirical pricing and hedging performance of the HAR-RV option pricing model relative to the EGARCH option pricing model.
Our results show that the HAR-RV model outperforms the EGARCH model in terms of one-day and five-days SPX call option pricing performance for all moneyness, except for OTM call options.
When considering put options, the HAR-RV model exhibits smaller pricing errors for all moneyness categories in terms of the three evaluating criteria. Comparing one-day performance with five-day pricing performance, both in call and put options, it can obviously find that five-days pricing errors are bigger than one-day, consistent with the intuition that using more previous data to forecast pricing will exhibit bigger pricing bias.
Furthermore, in hedging performance, this study shows that the HAR-RV option pricing model performs better than the EGARCH option pricing model only in the case of call options for all moneyness, while worse for put options. Although the EAGRCH option pricing model performs better than the HAR-RV model in put options, it is easy to discover that the difference of hedging errors between the HAR-RV model and the EGARCH model is close to zero, except for ITM put options.
Form the above results, it implies that the HAR-RV option pricing model is superior both for call and put options in one-day and five-days pricing performance and in call option hedging performance. The EGARCH option pricing model does not show significant superiority in hedging performance for put options.
Various issues could be examined by future studies. Since this study ignores the effects of the transaction cost, the dividend, and the borrowing-lending cost, in the future, it can add these factors for the improvement of evaluating pricing and hedging performance. Additionally, in financial markets, American options are heavily traded contracts, further researches can extend to evaluate American options pricing and hedging performance by applying the HAR-RV volatility model.
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Table 2.1
Summary Statistics of SPX in Call Options
Moneyness Full sample <0.94 0.94－0.97 0.97－1.00 1.00-1.03 1.03－1.06 >1.06
Average bid-offer option price 39.53 1.41 3.91 17.62 40.11 74.57 198.45 Minimum option price 0.40 0.40 0.40 1.23 17.00 51.80 91.50 Maximum option price 909.20 9.40 25.40 52.10 71.50 112.40 909.20 Total number of observations 2913 180 787 893 547 212 294 Note. This table summarizes the SPX call option data for the period from Apr. 2, 2007 to June 29, 2007. Moneyness is defined as S K , where S is the index level andK is the strike price.
Table 2.2
Summary Statistics of SPX in Put Options
Moneyness Full sample <0.94 0.94－0.97 0.97－1.00 1.00-1.03 1.03－1.06 >1.06
Average bid-offer option price 14.19 162.32 63.17 31.38 16.04 8.09 2.96 Minimum option price 0.40 92.10 44.70 13.30 4.40 1.85 0.40 Maximum option price 263.30 263.30 91.50 59.20 43.50 24.95 19.75 Total number of observations 4098 40 129 627 804 689 1809 Note. This table summarizes the SPX put option data for the period from Apr. 2, 2007 to June 29, 2007. Moneyness is defined asS K, where S is the index level andK is the strike price.
Table 4.1
Summary Statistic of the Parameter Estimators of the EGARCH(1,1) Option Pricing Model
Parameter Mean Standard Deviation min max
ω -0.1497 0.0090 -0.1719 -0.1357
ϕ 0.9885 0.0007 0.9865 0.9896
θ -0.0859 0.0013 -0.0894 -0.0832
γ 0.0469 0.0030 0.0430 0.0539
Note. The EGARCH(1,1) model parameters are estimated from daily S&P500 index return for the period from Jan. 2, 2002 to June 29, 2007.
Table 4.2
Summary Statistic of the Parameter Estimators of the HAR-RV Option Pricing Model
Parameter Mean Standard Deviation min max
β0 7.34E-06 7.28E-08 7.21E-06 7.60E-06
( )d
β 0.1078 0.0007 0.1066 0.1092
( )W
β 0.6518 0.0009 0.6492 0.6531
( )M
β 0.1518 0.0004 0.1513 0.1535
Note. The HAR-RV model parameters are estimated from daily S&P500 index return for the period from Jan. 2, 2002 to June 29, 2007.
Table 5.1
One-day Out-of-sample Valuation Error of the HAR-RV and EGARCH model for Call Options
MBIAS ($) MAE ($) RMAE (%) Moneyness HAR EGARCH HAR EGARCH HAR EGARCH
All Mean 1.2183 -6.6982 2.341 6.7065 0.5251 0.432
Stdev 2.6455 5.7242 1.7325 5.7144 0.9155 0.29
<0.94 Mean 2.1044 -1.3544 2.2114 1.3544 2.0548 0.9544
Stdev 1.6035 1.3777 1.4517 1.3777 1.4896 0.0459
0.94-0.97 Mean 2.7476 -2.8532 2.7885 2.8561 1.1862 0.6713
Stdev 1.9222 3.0866 1.8623 3.0839 1.0654 0.2202
0.97-1.00 Mean 1.7603 -8.0779 2.5215 8.0804 0.214 0.4223
Stdev 2.4863 6.2236 1.7083 6.2204 0.2213 0.1716
1.00-1.03 Mean -0.742 -10.3007 1.8722 10.3007 0.048 0.2532
Stdev 2.3505 5.3174 1.6016 5.3174 0.0399 0.0994
1.03-1.06 Mean -1.7368 -10.0362 1.9885 10.0362 0.027 0.1336
Stdev 1.9249 4.8689 1.6623 4.8689 0.0222 0.0552
>1.06 Mean 0.7135 -6.9621 1.8003 7.0292 0.0103 0.0493 Stdev 2.2099 3.9798 1.4637 3.8597 0.0086 0.0336 Note: MBIAS, MAE, and RMAE are the mean value of the valuation error in dollars, the mean absolute valuation error in dollars, and the mean value of percentage absolute error, respectively.
The parameters implied by the S&P 500 index data in our sample before the being priced date are used to calculate the forecasted call option prices. The valuation error is then calculated by comparing the observed and forecasted prices. Moneyness is defined as S K, where is the S&P 500 index level and is the strike price. The HAR-RV model follows
S
( ) ( ) ( ) ( ) ( ) X ( )
1 0
d d W M
d W
t d t t
M
RV+ =β +β RV +β RV +β RVt The EGARCH model, under risk-neutralized probability measure Q, has the following for ln
(
St St−1)
= −r 0.5σt2+σt tz* ; ln( )
σt2 =ω( )
2(
*)
*ln t 1 zt 1 zt 1 2
ϕ σ θ λ γ λ
+ − + − − +
⎡⎣
− − − π⎤⎦
, where λis restricted to zero.Table 5.2
One-day Out-of-sample Valuation Error of the HAR-RV and EGARCH model for Put Options
MBIAS ($) MAE ($) RMAE (%) Moneyness HAR EGARCH HAR EGARCH HAR EGARCH
All Mean -3.1351 -3.4014 3.2432 3.9542 0.5926 0.6062
Stdev 2.4449 3.8678 2.2995 3.3003 0.376 0.35
<0.94 Mean -3.0835 7.2592 3.2796 7.8156 0.0211 0.051
Stdev 2.3018 4.1135 2.0047 2.884 0.0149 0.0214
0.94-0.97 Mean -0.9555 2.9827 1.8819 4.4101 0.03 0.0703
Stdev 2.8748 3.9734 2.3696 2.2701 0.0361 0.0336
0.97-1.00 Mean -2.1006 -2.8639 2.5954 3.9577 0.0803 0.1187
Stdev 3.1105 5.0251 2.7108 4.2164 0.069 0.105
1.00-1.03 Mean -4.0164 -5.356 4.0233 5.3848 0.2755 0.3247
Stdev 2.3631 3.8934 2.3514 3.8534 0.1527 0.1582
1.03-1.06 Mean -4.7286 -5.3064 4.7286 5.3064 0.6393 0.6519
Stdev 1.836 3.0249 1.836 3.0249 0.1654 0.117
>1.06 Mean -2.6515 -2.6843 2.6515 2.6843 0.9462 0.9333 Stdev 1.8642 2.0835 1.8642 2.0835 0.0794 0.0671 Note: MBIAS, MAE, and RMAE are the mean value of the valuation error in dollars, the mean absolute valuation error in dollars, and the mean value of percentage absolute error, respectively.
The parameters implied by the S&P 500 index data in our sample before the being priced date are used to calculate the forecasted put option prices. The valuation error is then calculated by comparing the observed and forecasted prices. Moneyness is defined as S K, where is the S&P 500 index level and is the strike price. The HAR-RV model follows
S
( ) ( ) ( ) ( ) ( ) X ( )
1 0
d d W M
d W
t d t t
M
RV+ =β +β RV +β RV +β RVt The EGARCH model, under risk-neutralized probability measure Q, has the following for ln
(
St St−1)
= −r 0.5σt2+σt tz* ; ln( )
σt2 =ω( )
2(
*)
*ln t 1 zt 1 zt 1 2
ϕ σ θ λ γ λ
+ − + − − +
⎡⎣
− − − π⎤⎦
, where λis restricted to zero.Table 5.3
Five-days Out-of-sample Valuation Error of the HAR-RV and EGARCH model for Call Options
MBIAS ($) MAE ($) RMAE (%) Moneyness HAR EGARCH HAR EGARCH HAR EGARCH
All Mean 1.0466 -6.8791 2.3188 6.8807 0.4982 0.4439
Stdev 2.7269 5.9339 1.7757 5.9321 0.9252 0.2943
<0.94 Mean 2.0702 -1.4232 2.174 1.4232 1.9715 0.9593
Stdev 1.6905 1.4567 1.5541 1.4567 1.67 0.0432
0.94-0.97 Mean 2.5711 -2.964 2.6148 2.9659 1.0886 0.6896
Stdev 2.0394 3.1614 1.983 3.1596 1.0898 0.2127
0.97-1.00 Mean 1.54 -8.4067 2.516 8.4083 0.2074 0.4337
Stdev 2.6742 6.5829 1.7855 6.5808 0.221 0.1755
1.00-1.03 Mean -1.0039 -10.4699 2.0695 10.4699 0.0535 0.2582
Stdev 2.4468 5.4714 1.6449 5.4714 0.043 0.1023
1.03-1.06 Mean -1.7938 -10.1156 2.0456 10.1156 0.0279 0.1345
Stdev 1.8875 5.0194 1.6098 5.0194 0.022 0.0561
>1.06 Mean 0.7048 -7.2085 1.6825 7.2143 0.0096 0.0508 Stdev 2.0332 4.1893 1.3386 4.1792 0.0074 0.0353 Note: MBIAS, MAE, and RMAE are the mean value of the valuation error in dollars, the mean absolute valuation error in dollars, and the mean value of percentage absolute error, respectively.
The parameters implied by the S&P 500 index data in our sample before the being priced date are used to calculate the forecasted call option prices. The valuation error is then calculated by comparing the observed and forecasted prices. Moneyness is defined as S K, where is the S&P 500 index level and is the strike price. The HAR-RV model follows
S
( ) ( ) ( ) ( ) ( ) X ( )
1 0
d d W M
d W
t d t t
M
RV+ =β +β RV +β RV +β RVt The EGARCH model, under risk-neutralized probability measure Q, has the following for ln
(
St St−1)
= −r 0.5σt2+σt tz* ; ln( )
σt2 =ω( )
2(
*)
*ln t 1 zt 1 zt 1 2
ϕ σ θ λ γ λ
+ − + − − +
⎡⎣
− − − π⎤⎦
, where λis restricted to zero.Table 5.4
Five-days Out-of-sample Valuation Error of the HAR-RV and EGARCH model for Put Options
MBIAS ($) MAE ($) RMAE (%) Moneyness HAR EGARCH HAR EGARCH HAR EGARCH
All Mean -3.2914 -3.374 3.3924 3.9616 0.597 0.5992
Stdev 2.6031 4.0096 2.4699 3.4301 0.3746 0.3529
<0.94 Mean -3.2119 7.5248 3.3169 7.7094 0.0229 0.0522
Stdev 2.1032 3.2179 1.9288 2.7331 0.0158 0.0201
0.94-0.97 Mean -1.1772 2.9189 1.9551 4.4724 0.031 0.0709
Stdev 2.7517 4.17 2.2621 2.4129 0.035 0.035
0.97-1.00 Mean -2.4959 -2.8244 2.9464 4.0092 0.0892 0.1168
Stdev 3.311 5.1399 2.9167 4.279 0.0744 0.1015
1.00-1.03 Mean -4.3455 -5.3496 4.3619 5.3783 0.2915 0.3178
Stdev 2.6307 4.1553 2.6034 4.118 0.1556 0.161
1.03-1.06 Mean -4.8635 -5.2823 4.8635 5.2823 0.654 0.6455
Stdev 2.0774 3.2576 2.0774 3.2576 0.164 0.1207
>1.06 Mean -2.6773 -2.6946 2.6773 2.6946 0.9488 0.9319 Stdev 1.9442 2.1542 1.9442 2.1542 0.0781 0.0704 Note. MBIAS, MAE, and RMAE are the mean value of the valuation error in dollars, the mean absolute valuation error in dollars, and the mean value of percentage absolute error, respectively.
The parameters implied by the S&P 500 index data in our sample before the being priced date are used to calculate the forecasted put option prices. The valuation error is then calculated by comparing the observed and forecasted prices. Moneyness is defined as S K, where is the S&P 500 index level and is the strike price. The HAR-RV model follows
S
( ) ( ) ( ) ( ) ( ) X ( )
1 0
d d W M
d W
t d t t
M
RV+ =β +β RV +β RV +β RVt The EGARCH model, under risk-neutralized probability measure Q, has the following for ln
(
St St−1)
= −r 0.5σt2+σt tz* ; ln( )
σt2 =ω( )
2(
*)
*ln t 1 zt 1 zt 1 2
ϕ σ θ λ γ λ
+ − + − − +
⎡⎣
− − − π⎤⎦
, where λis restricted to zero.TBALE 6.1
Dynamic Delta Hedging Errors for SPX Call Options
1-Day Hedging 5-Day Hedging
Moneyness HAR EGARCH HAR EGARCH
Panel A: Absolute hedging errors
all 2.5354 2.8879 6.5231 7.3214
<0.94 0.0743 0.0813 0.2748 0.2887
0.94-0.97 0.4627 0.4825 0.9666 1.0118
0.97-1.00 2.3603 2.5071 5.1477 5.6212
1.00-1.03 4.2935 4.5678 18.6331 19.7269
1.03-1.06 7.4655 8.4452 22.7341 27.0333
>1.06 5.4789 6.7991 19.8412 23.7867
Panel B: Mean hedging errors
all -2.5186 -2.8788 -6.5115 -7.3104
<0.94 -0.0515 -0.0606 -0.1916 -0.2090
0.94-0.97 -0.4472 -0.4676 -0.9647 -1.0101
0.97-1.00 -2.3541 -2.5021 -5.1477 -5.6212
1.00-1.03 -4.2935 -4.5678 -18.6331 -19.7269
1.03-1.06 -7.4655 -8.4452 -22.7341 -27.0333
>1.06 -5.4315 -6.7949 -19.8412 -23.7867
Notes. This table presents the mean value of absolute hedging error ($), the mean value of hedging error ($) of a dynamic delta hedging strategies established daily or five days for the sample period. Delta is calculated daily using parameters implied by the index data period before the date of being priced call option. Moneyness is defined as S K, where Sis the S&P 500 index level and
Kis the strike price. The best model is the one with a near zero hedging error.
TBALE 6.2
Dynamic Delta Hedging Errors for SPX Put Options
1-Day Hedging 5-Day Hedging
Moneyness HAR EGARCH HAR EGARCH
Panel A: Absolute hedging errors
all 1.4344 1.3735 4.2896 3.9560
<0.94 7.4298 6.6413 43.3687 37.0871
0.94-0.97 4.2492 4.0576 13.5614 13.0808
0.97-1.00 1.6903 1.6333 3.5925 3.4118
1.00-1.03 1.4490 1.4244 1.1197 1.0955
1.03-1.06 0.6115 0.6022 1.8794 1.8272
>1.06 0.3133 0.3127 0.9276 0.9198
Panel B: Mean hedging errors
all -1.1724 -1.1155 -4.2896 -3.9560
<0.94 -7.4298 -6.6413 -43.3687 -37.0871
0.94-0.97 -4.2492 -4.0576 -13.5614 -13.0808
0.97-1.00 -1.3714 -1.3197 -3.5925 -3.4118
1.00-1.03 -0.0179 -0.0148 -1.1197 -1.0955
1.03-1.06 -0.6115 -0.6022 -1.8794 -1.8272
>1.06 -0.3133 -0.3127 -0.9276 -0.9198
Notes. This table presents the mean value of absolute hedging error ($), the mean value of hedging error ($) of a dynamic delta hedging strategies established daily or five days for the sample period. Delta is calculated daily using parameters implied by the index data period before the date of being priced put option. Moneyness is defined as S K, where is the S&P 500 index level and
S
Kis the strike price. The best model is the one with a near zero hedging error.
1-day valuation error for call option
<0.94 0.94-0.97 0.97-1.00 1.00-1.03 1.03-1.06 >1.06
moneyness
<0.94 0.94-0.97 0.97-1.00 1.00-1.03 1.03-1.06 >1.06 mone yne ss
MAE HAR
EGARCH
1-day valuation error for call option
0.00
<0.94 0.94-0.97 0.97-1.00 1.00-1.03 1.03-1.06 >1.06 moneyness
RMAE HAR
EGARCH
FIGURE 5.1 One-day forecasting errors for SPX Call Options
These figures show the 1-day forecasting errors of the HAR-RV model and the EGARCH model for SPX Call Options. The parameters implied by the S&P 500 index data in our sample before the being priced date are used to calculate the forecasted prices. The valuation error is then calculated by comparing the observed and forecasted prices. Moneyness is defined as
S K, where Sis the S&P 500 index level and X is the strike price.
1-day valuation error for put option
<0.94 0.94-0.97 0.97-1.00 1.00-1.03 1.03-1.06 >1.06
moneyness
MBIAS
HAR EGARCH
1-day valuation error for put option
0
<0.94 0.94-0.97 0.97-1.00 1.00-1.03 1.03-1.06 >1.06 moneyness
MAE HAR
EGARCH
1-day valuation error for put option
0.00
<0.94 0.94-0.97 0.97-1.00 1.00-1.03 1.03-1.06 >1.06 moneyness
RMAE HAR
EGARCH
FIGURE 5.2 One-day forecasting errors for SPX Put Options
These figures show the 1-day forecasting errors of the HAR-RV model and the EGARCH model for SPX Put Options. The parameters implied by the S&P 500 index data in our sample before the being priced date are used to calculate the forecasted prices. The valuation error is then calculated by comparing the observed and forecasted prices. Moneyness is defined as
S K, where Sis the S&P 500 index level and X is the strike price.
5-day valuation error for call option
<0.94 0.94-0.97 0.97-1.00 1.00-1.03 1.03-1.06 >1.06
moneyness
MBIAS HAR
EGARCH
5-day valuation error for call option
0
<0.94 0.94-0.97 0.97-1.00 1.00-1.03 1.03-1.06 >1.06 moneyness
MAE HAR
EGARCH
5-day valuation error for call option
0.00
<0.94 0.94-0.97 0.97-1.00 1.00-1.03 1.03-1.06 >1.06 moneyness
RMAE HAR
EGARCH
FIGURE 5.3 Five-days forecasting errors for SPX Call Options
These figures show the 5-day forecasting errors of the HAR-RV model and the EGARCH model for SPX Call Options. The parameters implied by the S&P 500 index data in our sample before the being priced date are used to calculate the forecasted prices. The valuation error is then calculated by comparing the observed and forecasted prices. Moneyness is defined as
S K, where Sis the S&P 500 index level and X is the strike price.
5-day valuation error for put option
<0.94 0.94-0.97 0.97-1.00 1.00-1.03 1.03-1.06 >1.06
moneyness
<0.94 0.94-0.97 0.97-1.00 1.00-1.03 1.03-1.06 >1.06 mone yne ss
MAE HAR
EGARCH
5-day valuation error for put option
0.00
<0.94 0.94-0.97 0.97-1.00 1.00-1.03 1.03-1.06 >1.06 moneyness
RMAE HAR
EGARCH
FIGURE 5.4 Five-days forecasting errors for SPX Put Options
These figures show the 5-day forecasting errors of the HAR-RV model and the EGARCH model for SPX Put Options. The parameters implied by the S&P 500 index data in our sample before the being priced date are used to calculate the forecasted prices. The valuation error is then calculated by comparing the observed and forecasted prices. Moneyness is defined as
S K, where Sis the S&P 500 index level and X is the strike price.
1-day Absolute hedging error (call)
FIGURE 6.1 Dynamic Delta Hedging Errors for SPX Call Options
These figures show the mean value of absolute error and the mean value of error of dynamic delta hedging strategies with 1-day and 5-day rebalancing periods for call options. The best model is the one that produces the smallest errors.
1-day absolute hedging errors (put)
FIGURE 6.2 Dynamic Delta Hedging Errors for SPX Put Options
These figures show the mean value of absolute error and the mean value of error of dynamic delta hedging strategies with 1-day and 5-day rebalancing periods for put options. The best model is the one that produces the smallest errors.