Refinamiento de la teoría de evaluación cognitiva.

The violation of MR may affect the estimation of RNMs via two channels. The first channel is via the theoretical formulae used for their computation, and the second channel is via the IVs used in the process of obtaining a continuum of option prices by interpolation given that in practice option prices trade for discrete rather than a continuum of strikes. To quantify how these two channels may affect the estimation of RNMs, we conduct a simulation exercise.

We assume that the stock price follows the log-normal distribution, equation (2.32), and the stock may violate MR (i.e.,rS 6=rf).11 The current stock price is set toSt= 100, the risk-free rate is rf = 3%, the time-to-maturity is τ = 1/12. For sim- plicity, we assume no dividend payment. The volatility parameter is set to σ= 30%. We consider three cases of the expected stock return rS: rS =rf, rS =rf + 3%, and

rS =rf −3%. Since the log stock return, log(ST/St), follows a normal distribution, the true MFIV, RNS, RNK are σ = 30%, zero, and three, respectively, regardless of the value of rS.

We generate a realistic set of option prices as follows. We assume that options trade at K = 94,95, . . . ,108. These strikes approximately cover the delta range from 11_{This setup is similar in spirit toDennis and Mayhew(2009), who analyze how microstructural}

0.2 to 0.8, corresponding to the delta range in the OptionMetrics Volatility Surface file, which we use in the subsequent empirical analysis. We calculate these option prices based on equations (2.34) and (2.35). Then, we calculate the standard BS-IV of these options. Unless rS = rf, the standard BS-IV curve is not flat (Proposition 2.4.3). On the other hand, the robust IV curve is flat at the level of σ = 30%.

In line with the standard estimation procedures in the literature (e.g., Stilger et al.,2017), we interpolate the discrete option IVs by the cubic Hermite polynomial in the moneyness-IV metric and extrapolate IVs horizontally beyond the highest and lowest strikes. We follow Stilger et al. (2017) and interpolate and extrapolate call IVs and put IVs, separately. Then, we convert the interpolated IV curves to 1001 option prices over the equally-spaced strikes in the moneyness range [1/3,3]. Finally, we numerically calculate the integrals in the BKM formulae and calculate the RNMs. To perform these procedures, we have two choices regarding the type of IV to be interpolated and two choices regarding the type of the RNM formulae. These yield four estimation specifications, OS-, OR-, GS-, and GR-RNMs. The first letter of the prefix stands for the type of the formulae (the “Original” BKM formulae or the “Generalized” BKM formulae) and the second letter stands for the type of the IVs used for the IV interpolation (the “Standard” IVs or the “Robust” IVs).

Table 2.2 reports the result of the estimated RNMs by the four specifications. We can see that MFIV and RNK are not affected by the violation of MR in our simulated situation. On the other hand, RNS is affected both by the choice of the BKM formula and the employed IVs. Unless both the generalized formulae and the robust IV are employed, the estimated RNS differs from the true value of zero. This result shows that both the explicit consideration of CFER in the BKM formulae and the use of the robust IV for the inter- and extrapolation matter for the estimation of RNS.

[Table 2.2 about here.]

Why does the input IV to the interpolation and extrapolation process matter? Roughly speaking, this is because the extrapolation of the standard IV results in biases in the extrapolated option prices, and these biases transmit to the power-nlog return contract.

To fix ideas, we consider a positive CFER case. In this case, the call and put standard IV curves are given by the two dotted lines in Figure2.3. In the extrapolation

step, the IV curves are horizontally extended beyond the observed moneyness range (i.e., the two solid lines in Figure 2.3). Since the extrapolated call (put) line is above (below) the true standard IV line, this means that the extrapolation of the standard IV curve results in the overestimation (underestimation) of the extrapolated call (put) option prices in the case CFER is positive.

[Figure 2.3 about here.]

Then, how the overestimation of OTM call options and underestimation of OTM put options transmit to the power-n log return contract prices? We answer this question based on the following Lemma about the sign of the weighting function

η(K;St, n) in equation (2.26).

Lemma 2.4.1. For odd-degree power-n log return contracts, η(K;St, n) satisfies the following inequalities. η(K;St, n)              <0 K/St<1, ≥0 1≤K/St≤en−1, <0 K/St> en−1. (2.43)

For even-degree power-n log return contracts, η(K;St, n), satisfies the following in- equalities. η(K;St, n)      ≥0 K/St ≤en−1, <0 K/St > en−1. (2.44)

Proof. See Appendix 2.A.7. 2

To simplify our discussion, we ignore the respective last cases in equations (2.43) and (2.44) about the far deep call OTM region K/St > en−1. This simplification is innocuous for empirical applications because call option prices in this deep OTM region are typically negligible and do not much affect the calculation of the call integral in equation (2.26).

Equation (2.43) shows that the η function of the odd-degree power-n contract is negative in the put OTM region and positive in the call OTM region. Therefore, the undervaluation of OTM put options and the overvaluation of OTM call options both result in an upward bias in the power-3 contract price which enters in the calculation

of RNS. Therefore, a positive CFER results in an upward bias in the estimated RNS via the bias in OTM option prices arisen during the extrapolation process.

On the other hand, equation (2.44) shows that the η function of the even-degree power-n contract is always non-negative. Consequently, the overvaluation in OTM call prices and undervaluation in OTM put prices have the opposite effect on the power-2 and power-4 contract prices, leaving these prices largely unaffected. This explains why MFIV and RNK are largely unaffected by the bias arisen during the extrapolation process.