7.4 Model Performance
7.4.2 Out-of-Sample Pricing Performance
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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.
7.4.2 Out-of-Sample Pricing Performance
To examine out-sample cross-sectional pricing performance for each model, we compute out-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
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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, (7.11)
After the model parameters are calibrated by minimizing the the sum of squared errors of implied volatility, the out-sample average absolute pricing errors can be represented as follows:
εAt+1,out−of −sample= 1 mt
mt
X
j=1
|Ot+1,jBS − OM odelt+1,j |, t = 1, ..., T − 1, (7.12)
And the out-of-sample average relative pricing errors can be represented as follows:
εRt+1,out−of −sample = 1 mt+1
mt+1
X
j=1
OBSt+1,j− OM odelt+1,j OM odelt+1,j
!
, t = 1, ..., T − 1, (7.13)
where mt+1 is the number of option prices at time t + 1 in the market, IVt+1 OBSt+1,j is the Black-Scholes implied volatility of the j-th market-observed call option price Ot+1,jBS = C (St+1,j, Kt+1,j, rt+1,j, τt+1,j) at time t + 1, IVt+1 OM odelt+1,j is the Black-Scholes implied volatility of the j-th call option price Ot+1,jM odel = C (Θt|St+1,j, Kt+1,j, rt+1,j, τt+1,j) which is computed using the specific model at time t + 1, where the parameters St+1,j, Kt+1,j, rt+1,j, τt+1,j are the underlying S&P500 index, strike, days-to-expiration, riskless rate at the time t + 1 and Θtis the parameter vector containing the model risk-neutral parameter and risk premiums at the time t.
Table 11 reports the out-of-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 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. 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
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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 11 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.
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 G and H. According to both errors measures in Table 11, we firstly compare the models with the same type jumps and stochastic volatility model (SV).
First, we force on the type of Merton jump, 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. For example, in the panel A of Table 11, 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$2.87 versus
$3.22 by the SV-MJ-VIJ model, $3.31 by the SV-MJ model, and $3.78 by the SV model.
Clearly, the performance improvement is significant for each moneyness and maturity category in the Panel A of Table 11 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 11.
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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, followed by the SV-DEJ-VIJ, the SV-DEJ and finally the SV model. For example, in the panel A of Table 11, 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-DEJ-VCJ model is $4.72 versus $5.03 by the SV-DEJ-VIJ model, $5.43 by the SV-DEJ model, and $5.75 by the SV model. Clearly, the performance improvement is significant for some moneyness and maturity category in the Panel A of Table 11 from the SV to the SV-DEJ, to the SV-DEJ-VIJ and to the SV-DEJ-VCJ model.
Third, in the type of infinite activity jumps, the ranking of the four models is also consistent with our prior, the SV-VG model outperforms all others. For example, in the Panel A of Table 11, in the category out-of-the-money H (0.94 ≤ S/K < 0.97) and middle-maturity D (90-120 days) options, the average absolute pricing error by the SV-VG model is $4.41, $4.84 by the SV-NIG model, and $5.75 by the SV model. Clearly, the performance improvement is significant for some moneyness and maturity category in the Panel A of Table 11 from the SV to the to the SV-VG, to the SV-NIG.
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 out-of-the-money H (0.94 ≤ S/K < 0.97) 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 expontential 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
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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 exponential jumps type models are between 5.16 and 5.30, the Merton jumps type models are between 5.86 and 6.18, the infinite activity jumps type models are between 5.53 and 6.02. Clearly, the double expontential jumps type models have the best performance for the ITM moneyness and each maturity category in the Panel A of Table 11.
However, some patterns of mispricing can be find across all moneyness-maturity cat-egories. 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 underprice ITM call options. Second, the model’s absolute pricing error and percentage pricing error have a 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 expiration. For example, in the extremely short-term A (< 30 days) options, the absolute pricing error of the SV model is $1.92 in the category G ( S/K ≤ 0.94, deep out of the-money), $2.29 in the category H ( 0, 94 < S/K ≤ 0.97, out of the money), $3.15 in the categories I ( 0.97 < S/K ≤ 1.00, at the money), $3.11 in the categories J ( 1.00 < S/K ≤ 1.03, at the money),$2.99 in the category K (1.03 < S/K ≤ 1.06 in the money) and $2.67 in the category L ( 1.06 < S/K, deep in the money). Clearly, the absolute pricing error of the SV model is a smile from deep out of the money to deep in the money in the Panel A of Table 11.
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Chapter 8 Conclusion
We attempt to answer three questions about implication of information in both the spot and option markets: (i) On average, what the proportion of the stochastic volatility and return jumps account for the total return variations in S&P 500 index, respectively? In particular, which one has more influence than the other does on the total return varia-tions? (ii) Is the fitting performance of infinite-activity jump models better than that of finite-activity jump models both in the spot and option markets? (iii) When will investors require significantly higher risk premiums? Specifically, were there significant changes in volatility risk premiums and in jump risk premiums before, during or after the financial crisis?
To answer the first question is that most of the return variations are explained by the stochastic volatility. In fact, the return jump accounts for the higher percentage than the stochastic volatility at the beginning of financial crisis. To answer the second question, we adopt the particle filtering with expectation-maximization algorithm and dynamic joint estimation to obtain the stochastic volatility model with double-exponential jumps and correlated jumps in volatility (SV-DEJ-VCJ) and the stochastic volatility model with normal inverse Gaussian jumps (SV-NIG) fit S&P 500 index returns and options well in different criterions, respectively. Finally, for the third question, we observe that both the volatility and jump risk premiums significantly increase after the financial crisis periods, that is, the panic in the post-crisis period causes more expected returns.
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Appendix A
Change Measure: Stochastic Volatility Model
The dynamic process of the stochastic volatility model, under the physical measure (P) is
dyt=
µ − 1
2vt
dt +√
vtdWy,tP, dvt= κ (θ − vt) dt + σv√
vtdWv,tP,
(A.1)
where µ is the drift term which is a constant. The stochastic volatility vtfollows a mean-reverting square root process with long term level θ > 0, the speed of adjustment is κ > 0, and the variation coefficient of stochastic volatility is σv > 0. Moreover, dWy,tP and dWv,tP are a pair of correlated Wiener processes with correlation coefficient ρdt, i.e., dWy,tP ∼ N (0, dt), dWv,tP ∼ N (0, dt), and Corr(dWy,tP, dWv,tP) = ρdt. Therefore, by the Cholesky decomposition, the correlated Wiener processes can be decomposed into two independent Wiener processes, i.e., dWy,tP = dB1,tP and dWv,tP = ρdB1,tP +p1 − ρ2dB2,tP. Thus, the equation (A.1) can be rewritten as following
dyt=
µ − 1
2vt
dt +√
vtdB1,tP , dvt= κ (θ − vt) dt + σv√
vt
dBP1,t+p
1 − ρ2dB2,tP ,
(A.2)
Let dQdPt
t be the Radon-Nikod´ym derivative with Esscher transform parameters θ1 and θ2 for the Brownian motions.
dQt
dPt = eR0tθ1
√vudB1,uP +Rt 0θ2
√vudB2,uP
E e
Rt 0θ1
√vudBP1,u+Rt 0θ2
√vudBP2,u
‧
Since the new Brownian motions dB1,tP and dB2,tP are independent, we can alter the prob-ability measure separately. That is,
dQt
1≤i≤n∆ti Obviously, the distributions of two Brownian motions under the risk-neutral measure (Q) are
dB1,tQ = dB1,tP − θ1√
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with
θ1 = − (µ − r)
√vt = h1
√vt, θ2 = −ρθ1− h2 p1 − ρ2
Thus, under the risk-neutral measure Q, the equation (A.1) can be rewritten as following dyt =
r − 1
2vt
dt +√
vtdW1,tQ, dvt = ˜κ ˜θ − vt
dt + σv
√vtdW2,tQ,
(A.3)
where µ is the drift term which is a constant. The stochastic volatility vtfollows a mean-reverting square root process with long term level ˜θ = κ+hκθ
2, the speed of adjustment is
˜
κ = κ + h2, the variation coefficient of the diffusion volatility is σv, h1 is the price risk premium and h2 is the volatility risk premium. Moreover, dWy,tQ and dWv,tQ, are a pair
κ = κ + h2, the variation coefficient of the diffusion volatility is σv, h1 is the price risk premium and h2 is the volatility risk premium. Moreover, dWy,tQ and dWv,tQ, are a pair