Note that this log likelihood function under the measure in (6.21) is only used to examine the performance of the model. When we optimize the joint likelihood function, we still use the log likelihood function in (6.18).
Moreover, in order to computer the log-likelihood for fitting call options LQoption,t, we assume that the option pricing errors related to the implied volatility follow a normal distribution,
εt,j = IVt Ot,jBS − IVt Ot,jM odel , j = 1, ..., Nt,
Assume that the implied volatility error εt,j ∼ N 0, σ2ε,t, where σε,t2 is the variance of the Black-Scholes implied volatility of the market option price for each day. Thus, the complete log-likelihood of option price:
LQoption,t = ln
where IVt Ot,jBS is the Black-Scholes implied volatility of the j-th market-observed call op-tion price Ot,jBS = C (St,j, Kt,j, τt,j, rt,j), IVt OM odelt,j is the Black-Scholes implied volatility of the j-th call option price Ot,jM odel = C (Ψt|St,j, Kt,j, τt,j, rt,j) which is computed using the model, where the parameters St,j, Kt,j, τt,j, rt,j are the underlying S&P500 index price, strike, days-to-expiration, riskless rate and Ψt is the parameter vector containing the model’s risk-neutral parameter and risk premiums ht,1 and ht,2.
6.7 Model Diagnostics and Comparisons
Aftering obtaining the estimated model parameters and tracking the dynamic of the latent variables, we can examine the performance of all nine models in capturing the
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joint dynamics of spot price and call option prices.
First, in order to check the goodness of fit of the models under the P measure, we follow Li, Wells, and Yu (2008, 2011) and Kou (2017) to use the Kolmogorov-Smirnov (KS) test of the hypothesis to examine whether the standardized residuals of a fitted returns dynamic model follow an standard normal distribution. That is,
yt+1= yt+1− yt− µ∆ − Jt+1y
√vt∆ ∼ N (0, 1)
Second, in order to capture the performance of different models under the Q measure, we follow Li et al. (2011) and Ornthanalai (2014) to use the Diebold and Mariano (DM, 1995) test to compare weekly root-mean-square errors of the implied volatility (RIVMSE) and in and out of option pricing errors betweens models. Consider a sample path d (t) = e2X(t) − e2Y(t) for t = 1, ..., T of a loss-differential series between two models X and Y , where eX(t) represent weekly RIVRMSE and option pricing errors for models X and Y , respectively. Under the null hypothesis that the two models have the same squared pricing errors are E (e2X(t)) = E (e2Y(t)), or E (d (t)) = 0. DM(1995) show that if {d (t)}Tt=1 is covariance stationary and short memory, then we have the asymptotic distribution of the sample mean loss differential
√
T d − µ¯ d d
→ N (0, 2πfd(0)) where ¯d = 1/TPT
t=1[e2X(t) − e2Y(t)], fd(0) = 1/2π P∞
q=−∞Υd(q)
is the spectral den-sity of the the loss differential as frequency 0, Υd(q) = E [(dt− µd) (dt−q− µd)] is the autocovariance of the loss differential at displacement q, and µd is the population mean loss differential. In the large samples, ¯d is approximately normally distributed with mean µd and 2πfd(0) /T . Thus, under null hypothesis of equal squared errors, the following DM statistic
DM =
d¯ q
2π ˆfd(0) /T
→ N (0, 1)d
is distributed asymptotically as N (0, 1), where ˆfd(0) is a consistent estimate of fd(0) is defined by
fˆd(0) = 1 2π
T −1
X
q=−(T −1)
I
q h − 1
Υˆd(q)
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is the lag window and h − 1 is the truncation lag which implied that only h − 1 sample autocovariance need to be used in the estimation of fd(0). Thus,fˆd(0) = 1
However, there are two main issues of DM test, when we use the DM test to capture the performance of different models under the Q measure. First, the simulation experiments in Diebold and Mariano (DM, 1995) show that the normal distribution can be a very poor approximate of DM test’s finite sample null distribution. Their result show that the DM test statistic can have wrong size, rejecting the null too often. Thus, we follow Harvey, Leybourne, and Newbold (1997) which suggest that improved small-sample properties can be obtained by making a bias correction to the DM test statistic and comparing the modified statistic with a Student-t distribution with T − 1 degrees of freedom, replacing the standard normal distribution. Thus, the modified test statistic is obtained as
HLN − DM =
Second, in empirical analysis, we do not observe the true option pricing errors {eX(t)}Tt=1 and {eY(t)}Tt=1. Instead, we only know the estimated pricing errors {ˆeX(t)}Tt=1 and {ˆeY(t)}Tt=1. Due to parameter estimation uncertainty, E [ˆei(t)] 6= E [ei(t)], for i = XandY . To address this issue, we follow Li et al. (2011) to use modified pricing er-rors q
T
T −niˆei(t) in our implementation the DM test, where ni represents the the number of parameter for each model. This approach is based on the fact that in both linear and nonlinear regression, T −n1
i
PT
t=1eˆ2i (t) is an unbiased estimator of E (ˆe2(t)) as T → ∞, for i = X and Y .
To compare the performance of different models under the Q measure, we use the DM test statistic to measure wether one model has significantly smaller weekly root-mean-square errors of the implied volatility (RIVMSE) than another model. We follow Orn-thanalai (2014) to compute the in-sample option pricing performance with the time-series
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means and standard deviations of weekly root-mean-square errors of the implied volatility (RIVMSE) for each model. That is,
RIV M SEt= v u u t
1 mt
mt
X
j=1
IVt Ot,jBS − IVt OM odelt,j IVt OBSt,j
!2
, t = 1, ..., T, (6.23)
where mtis the number of option prices in the market, IVt OBSt,j is the Black-Scholes im-plied volatility of the j-th market-observed call option price Ot,jBS = C (St,j, Kt,j, τt,j, rt,j), IVt Ot,jM odel is the Black-Scholes implied volatility of the j-th call option price Ot,jM odel= C (Ψt|St,j, Kt,j, τt,j, rt,j), IVt Ot,jM odel
which is computed using the model, where the parameters St,j, Kt,j, τt,j, rt,j are the underlying S&P500 index price, strike, days-to-expiration, riskless rate and Ψt is the parameter vector containing the model risk-neutral parameter and risk premiums h2,t and h3,t. t,j ∼ N 0, σ,t2 , where σ2,t is the variance of the Black-Scholes implied volatility of the market option price for each day. Let eX(t) and eY(t) represent weekly RIVRMSE for models X and Y where t = 558 is the number of weeks from January 1, 2007 through August 31, 2017.