7.4 Model Performance
7.4.1 In Sample Pricing Performance
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parameters will generally lead to better in sample fitting than simple model. However, it may bring the poor results in the out-sample testing. Therefore, it is important to analyze the model which should perform better than other models in both in-sample and out-of-sample testing.
7.4.1 In Sample Pricing Performance
To examine in-sample cross-sectional pricing performance for each model, we compute in-sample pricing errors by absolute difference and relative difference between the market prices of S&P500 option and the model prices whose parameters are estimated by mini-mizing the sum of squared errors (SSE) between actual and theoretical implied volatility.
That is,
maxθt
SSEt= 1 mt
mt
X
j=1
IVt OBSt,j − Ot,jM odel2
, t = 1, ..., T, (7.8)
After the model parameters are calibrated by minimizing the the sum of squared errors of implied volatility, the in-sample average absolute pricing errors can be represented as follows:
εAt,in−sample = 1 mt
mt
X
j=1
|OBSt,j − OM odelt,j |, t = 1, ..., T, (7.9)
And the in-sample average relative pricing errors can be represented as follows:
εRt,in−sample = 1 mt
mt
X
j=1
Ot,jBS− Ot,jM odel OM odelt,j
!
, t = 1, ..., T, (7.10)
where mt is the number of option prices at time t in the market, IVt OBSt,j
is the Black-Scholes implied volatility of the j-th market-observed call option price OBSt,j = C (St,j, Kt,j, rt,j, τt,j) at time t, IVt OM odelt,j is the Black-Scholes implied volatility of the j-th call option price OM odelt,j = C (Θt|St,j, Kt,j, rt,j, τt,j) which is computed using the spe-cific model at time t, where the parameters St,j, Kt,j, rt,j, τt,j are the underlying S&P500 index, strike, days-to-expiration, riskless rate and Θt is the parameter vector containing the model risk-neutral parameter and risk premiums at time t.
Table 10 reports the in-sample absolute and relative pricing errors for all models.
For a given model, we compute the price of each option using the same day’s implied
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parameters and implied stock volatility. Note that we use the European call options for Standard & Poor’s 500 index (S&P500). The data are obtained from DataStream with the sample period from each Wednesday during January 1, 2007 to August 31, 2017. The call options with 6 to 180 days-to-expiration, 0.92 to 1.08 moneyness (S/K), call prices more than $0.375 and nonarbitrary opportunities are reserved. After filtering sample, there are a total number of 22828 available observations for call options.
After filtering sample, there are a total number of 22828 available observations for call options. The sample is divided into 36 categories. That is, we have 6 types of days-to-expiration and 6 types of moneyness (S/K). First type of days-to-expiration A is extremely short-term ( ≤ 30 days), second type of days-to-expiration B is short-term (30-60 days), third type of days-to-expiration C is near-term (60-90 days), fourth type of days-to-expiration D is middle-maturity (90-120 days), fifth of days-to-expiration E is long-term (120-150 days), and sixth of days-to-expiration F is extremely long-term (≥ 150 days). Next, first type of moneyness G is deep-out-of-the-money (S/K ≤ 0.94), second type of moneyness H is out-of-the-money (0.94 ≤ S/K < 0.97), third and fourth type of moneyness I and J are at-the-money (0.97 ≤ S/K < 1.03), fifth type of moneyness J is in-the-money (1.03 ≤ S/K < 1.06), sixth type of moneyness K is deep-in-the-money (1.06 < S/K). The Table 10 lists the absolute pricing error which is the sample average of the absolute difference between the market price and the model price for each call in a given moneyness-maturity categories. The percentage pricing error is the sample average of the market price minus the model price, divided by the market price for each call in a given moneyness-maturity categories. moneyness categories.
We follow Ornthanalai (2014) to compute the in-sample option pricing performance with the time-series means and standard deviations of weekly root-mean-square errors of the implied volatility (RIVMSE) in equation (6.23) for each model in Panel A of Table 8. In this paper, we divided L´evy jump type into two categories finite activity (Mer-ton jumps and Double exponential jumps) and infinite activity jumps (Variance gamma jumps and inverse Gaussian jumps). We divided option performance into three parts out of the market (OTM), at the moneyness (ATM) and in the-market (ITM). We first examine the option performance in pricing OTM options, i.e., the moneyness categories
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G and H. According to both errors measures in Table 8 and Table 10, we firstly compare the models with the same type jumps and stochastic volatility model (SV).
First, we force on the type of Merton jumps, the ranking of the four models is con-sistent with our prior, the SV-MJ-VCJ model outperforms all others, followed by the SV-MJ-VIJ, the SV-MJ and finally the SV model, because the complex models will have smaller pricing errors than the simple models in sample testing. For example, in the Panel A of Table 8, SV-MJ-VCJ model produces an average root-mean-square errors of the implied volatility of 12.66% versus 12.82% by the SV-MJ-VIJ model, 13.27% by the SV-MJ model, and 13.64% by the SV model.
For the panel A of Table 10, in the category the deep-out-of-the-money G (S/K ≤ 0.94) and the near-term C (60-90 days) options, the average absolute pricing error by the SV-MJ-VCJ model is $2.41 versus $2.77 by the SV-MJ-VIJ model, $2.89 by the SV-MJ model, and $3.44 by the SV model. Clearly, the performance improvement is significant for each moneyness and maturity category in the Panel A of Table 8 and Table 10 from the SV to the SV-MJ, to the SV-MJ-VIJ and to the SV-MJ-VCJ model. This pricing performance ranking of the four models can also be seen using the average percentage pricing errors, as in given in the Panel A of Table 10.
Second, in the type of double-exponential jumps, the ranking of the four models is also consistent with our prior, the SV-DEJ-VCJ model outperforms all others, followd by the SV-DEJ-VIJ, the SV-DEJ and finally the SV model. For example, in the Panel A of Table 8, SV-DEJ-VCJ model produces an average root-mean-square errors of the implied volatility of 12.32% versus 12.45% by the SV-DEJ-VIJ model, 12.88% by the SV-DEJ model, and 13.64% by the SV model. For the panel A of Table 10, in the category the out-of-the-money H (0.94 ≤ S/K < 0.97) and the middle-maturity D (90-120 days) op-tions, the average absolute pricing error by the SV-DEJ-VCJ model is$3.65 versus $4.10 by the SV-DEJ-VIJ model, $4.59 by the SV-DEJ model, and $4.71 by the SV model.
Clearly, the performance improvement is significant for some moneyness and maturity category in the Panel A of Table 8 and Table 10 from the SV to the SV-DEJ, to the SV-DEJ-VIJ and to the SV-DEJ-VCJ model.
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Third, in the infinite activity jumps type, the ranking of the four models is not entirely consistent with our prior, the complex model SV-VG does not outperforms all others. For example, in the Panel A of Table 8, SV-VG model produces an average root-mean-square errors of the implied volatility of 13.09% versus 12.69% by the SV-NIG model, and 13.64%
by the SV model. However, for the panel A of Table 10, in the category the out-of-the-money H (0.94 ≤ S/K < 0.97) and the middle-maturity D (90-120 days) options, the average absolute pricing error by the SV-VG model is $3.15 versus $3.73 by the SV-NIG model, and $4.71 by the SV model. Clearly, the performance improvement is significant for some moneyness and maturity category in the Panel A of Table 10 from the SV to the to the SV-VG, to the SV-NIG, but not in Table 8.
For ATM options , i.e., the moneyness categories I and J, we find that we cannot rank the performance in the same jump type. However, we can compare the in sample average absolute pricing error with the different jump type. For example, in the category the at the money I (0.97 ≤ S/K < 1.00) and the near-term C (60-90 days) options, the average absolute pricing error by the SV-DEJ-VCJ model is $1.71 versus $1.82 by the SV-MJ-VCJ model, SV-DEJ-VIJ model is $1.70 versus $1.77 by the SV-MJ-VIJ model, and the SV-NIG $1.84, $1.94 by the SV-DEJ model, $1.96 by the SV-VG model, and
$2.05 by the SV-MJ model. Therefore, the model with the double exponential jumps type will better than the Merton jumps type when the models have the volatility jump.
However, when the models does not have the volatility jump, the infinite activity jumps type models are the best model, especially for the SV-NIG model. For ITM options , i.e., the moneyness categories K and L, we also compare the in sample average absolute pricing error with the different jump type. For example, in the category the in the money K (1.03 ≤ S/K < 1.06) and the short-term C (60-90 days) options, the average absolute pricing error of the double expontential jumps type models are between 4.32 and 4.51, the Merton jumps type models are between 5.20 and 5.65, the infinite activity jumps type models are between 4.76 and 5.52. Clearly, the double exponential jumps type models have the best performance for the ITM moneyness and each maturity category in the Panel A of Table 8.
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However, some patterns of mispricing can be find across all moneyness-maturity cate-gories. First, all nine models produce negative percentage pricing errors for options with moneyness (S/K ≤ 0.97), and positive percentage errors for options with moneyness (S/K > 1.00). This means that the models systematically overprice OTM call options and under price ITM call options. Second, the model absolute pricing error and percent-age pricing error have a slight U-shape relationship , i.e., smile, with moneyness as the call goes from deep out of the money to deep in the money, regardless of time to expira-tion. For example, the absolute pricing error of the SV model is$1.34 in the category G ( (S/K ≤ 0.94), deep out of the-money), $1.66 in the category H ( (0.94 < S/K ≤ 0.97), out of the money), $2.01 in the categories I, J ( (0.97 < S/K ≤ 1.03), at the money),
$2.61 in the category K ( (1.03 < S/K ≤ 0.94), in the money) and $2.52 in the category L ( (1.06 < S/K), deep in the money). Clearly, the absolute pricing error of the SV model is slight smile from deep out of the money to deep in the money in the Panel A of Table 10.
Panel B of Table 8 reports Diebold-Mariano (DM) pairwise statistics discussed in section 6.6 for weekly RIVRMSE. The DM statistics measure the difference between the squared pricing error of the Benchmark model X and the Test model Y. Note that a positive and significant value for DM statistic means that X has a larger RIVRMSE than Y. Looking at the first row of Panel B of Table 8 reveals that all test statistics are positive and significant, regardless the SV-MJ model and the SV-VG model, suggesting that all jump models significantly outperform the SV model. Looking at the pairwise tests under column “SV-DEJ-VCJ”, the SV-DEJ-VCJ model outperforms other models. That is, the test statistics of the SV-DEJ-VCJ model are all positive and significant. Moreover, the pairwise test statistic under column “SV-NIG” and row “SV-DEJ-VCJ” shows that the SV-NIG model underperforms the SV-DEJ-VCJ model, i.e., the test statistic is -3.4427.
Overall, we find strong evidence that among the jump models, the SV-DEJ-VCJ model has the smallest squared option pricing error.