*3.3. Pricing Performance*

In this subsection, we present the empirical results for the calibrated models. In order to investigate the impacts of the systematic and idiosyncratic volatility and jump risks on equity option pricing, we took the 2-FSV, 2-SV, and 2-SVJ models as benchmark models to evaluate the pricing performance of our proposed model.

Figures 1–10 exhibit the predicted prices of the four model specifications and market prices listed on 9 May 2019, with 11, 16, 21, 26, 31, 51, 71, 96, 116, and 181 trading days to expiry, respectively. Here, the predicted prices (out-of-sample pricing) were calculated by the in-sample calibration parameters reported in Table 1. One can clearly observe from the left panels of Figures 1–10 that the option prices obtained by theoretical models were generally closer to the market prices for different strike prices. To further investigate the pricing performance of the four models, the right panels of Figures 1–10 show the relative price differences (relative errors) between the theoretical model prices and market prices.<sup>5</sup> For simplicity, we refer to a call option as deep out-of-the-money (DOTM) if *S*/*K* ≤ 0.90; out-of-the-money (OTM) if 0.90 < *S*/*K* ≤ 0.97; at-the-money (ATM) if 0.97 < *S*/*K* ≤ 1.03; in-the-money (ITM) if 0.97 < *S*/*K* ≤ 1.10; and deep in-the-money (ITM) if 1.10 < *S*/*K*. Moreover, we considered options less than 60 days to expiration as short term; options with 60–120 days to expiration as medium term; and options larger than 120 days to expiration as long term. For the options with 11, 16, 21, 26, 31, and 51 trading days to expiry, the relative pricing errors produced by our proposed model were all significantly lower than those of 2-FSV, 2-SV, and 2-SVJ models in the case of DOTM options, while the relative errors of all models were slightly higher.

It is also worth noting that the pricing performance of the stochastic model with jump behavior was much better than that of the model without jump in the case of deep out-of-money. For the options with 71, 96, 116, and 181 trading days to expiry, we did not find the same conclusions as the above short term options. In conclusion, the pricing performance of equity option valuation model considering market and idiosyncratic volatility and jump risks was significantly improved for short term and DOTM options.

**Figure 1.** The comparison of predicted prices of four model specifications and market prices on 9 May 2019, with maturity *T* = 24 May 2019.

<sup>5</sup> The relative error is defined by |*Cmodel*−*Cmarket*| *Cmarket* × 100%, where *Cmodel* and *Cmarket* denote the theoretical model option prices and the real market prices, respectively.

**Figure 2.** The comparison of predicted prices of four model specifications and market prices on 9 May 2019, with maturity *T* = 31 May 2019.

**Figure 3.** The comparison of predicted prices of four model specifications and market prices on 9 May 2019, with maturity *T* = 7 June 2019.

**Figure 4.** The comparison of predicted prices of four model specifications and market prices on 9 May 2019, with maturity *T* = 14 June 2019.

**Figure 5.** The comparison of predicted prices of four model specifications and market prices on 9 May 2019, with maturity *T* = 21 June 2019.

**Figure 6.** The comparison of predicted prices of four model specifications and market prices on 9 May 2019, with maturity *T* = 19 July 2019.

**Figure 7.** The comparison of predicted prices of four model specifications and market prices on 9 May 2019, with maturity *T* = 16 August 2019.

**Figure 8.** The comparison of predicted prices of four model specifications and market prices on 9 May 2019, with maturity *T* = 20 September 2019.

**Figure 9.** The comparison of predicted prices of four model specifications and market prices on 9 May 2019, with maturity *T* = 18 October 2019.

**Figure 10.** The comparison of predicted prices of four model specifications and market prices on 9 May 2019, with maturity *T* = 17 January 2019.

To summarize the model calibration results, we also adopted the RMSE as a measure of the goodness of fit. Table 2 reports the out-of-sample pricing errors for the four models across different maturities. Note from Table 2 that our proposed model generally outperformed the other three models in terms of out-of-sample pricing errors. In fact, the same was true for in-sample, whose pricing errors were generally lower than those of the out-of-sample. We will not repeat them here. To measure the extent to which a model was better or worse than another, we defined the improvement rate as the relative differences between the pricing errors from the benchmark model and our proposed model, i.e.,

$$\text{Improvement rate} = \frac{\text{RMSE}\_{benchmark} - \text{RMSE}\_{our}}{\text{RMSE}\_{benchmark}} \times 100\%$$

where RMSE*our* and RMSE*benchmark* denote the RMSE implied by our model and benchmark model, respectively. A positive (or negative) value of improvement rate meant that our model yielded lower (or higher) pricing errors than benchmark model, implying that the pricing performance of the former was better (or worse) than that of the latter by a percentage of that value.

From the last column of Table 2, we can see that our model was superior to the 2-SVJ model across different maturities, which meant that it was necessary to consider the market factor structure in equity option pricing. From the third last column of Table 2, the improvement rate indicated that our model slightly outperformed the 2-FSV model in terms of short term options, but was worse than that of both medium and long term. In spite of this, our empirical study presented here could at least illustrate that the equity option pricing model considering systematic and idiosyncratic volatility and jump risks may offer a good competitor of the models of Bates (2000), Christoffersen et al. (2009), or Christoffersen et al. (2018) for some other equity option markets.


**Table 2.** Out-of-sample pricing errors. Note: This table shows the out-of-sample pricing errors across different maturities. Pricing errors are reported as the root mean squared errors (RMSE) of option prices for four models.
