It takes two to Tango: Estimation of the zero-risk premium strike of a call option via joint physical and pricing density modeling

Höcht, Stephan; Madan, Dilip B.; Schoutens, Wim; Verschueren, Eva Article It takes two to Tango: Estimation of the zero-risk premium strike of a call option via joint physical and pricing density modeling Risks Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Höcht, Stephan; Madan, Dilip B.; Schoutens, Wim; Verschueren, Eva (2021) : It takes two to Tango: Estimation of the zero-risk premium strike of a call option via joint physical and pricing density modeling, Risks, ISSN 2227-9091, MDPI, Basel, Vol. 9, Iss. 11, pp. 1-19, https://doi.org/10.3390/risks9110196 This Version is available at: https://hdl.handle.net/10419/258279 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ risks Article It Takes Two to Tango: Estimation of the Zero-Risk Premium Strike of a Call Option via Joint Physical and Pricing Density Modeling Stephan Höcht 1, Dilip B. Madan 2, Wim Schoutens 3,* and Eva Verschueren 4   Citation: Höcht, Stephan, Dilip B. Madan, Wim Schoutens, and Eva Verschueren. 2021. It Takes Two to Tango: Estimation of the Zero-Risk Premium Strike of a Call Option via Joint Physical and Pricing Density Modeling. Risks 9: 196. https:// doi.org/10.3390/risks9110196 Academic Editor: Hailiang Yang Received: 16 September 2021 Accepted: 29 October 2021 Published: 4 November 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1Assenagon GmbH, Prannerstraße 8, 80333 München, Germany; [email protected] 2Robert H. Smith School of Business, University of Maryland, College Park, MD 20742, USA; [email protected] 3Department of Mathematics, University of Leuven, Celestijnenlaan 200B, 3001 Leuven, Belgium 4Department of Accounting, Finance and Insurance, University of Leuven, Naamsestraat 69, 3000 Leuven, Belgium; [email protected] *Correspondence: [email protected] Abstract: It is generally said that out-of-the-money call options are expensive and one can ask the question from which moneyness level this is the case. Expensive actually means that the price one pays for the option is more than the discounted average payoff one receives. If so, the option bears a negative risk premium. The objective of this paper is to investigate the zero-risk premium moneyness level of a European call option, i.e., the strike where expectations on the option’s payoff in both the P - and Q -world are equal. To fully exploit the insights of the option market we deploy the Tilted Bilateral Gamma pricing model to jointly estimate the physical and pricing measure from option prices. We illustrate the proposed pricing strategy on the option surface of stock indices, assessing the stability and position of the zero-risk premium strike of a European call option. With small fluctuations around a slightly in-the-money level, on average, the zero-risk premium strike appears to follow a rather stable pattern over time. Keywords: pricing density; physical density; bilateral gamma; tilted bilateral gamma; call option; risk premium 1. Introduction Each event in the financial market is characterized by both its likelihood and its price, which is why financial engineers make a distinction between the so-called P -world and Q -world. The P -world is the physical world in which payoffs are realized. A probability measure in this world estimates the real probability on the occurrence of a particular event. However, the Q -world is an artificial setting under which one determines the price. Probabilities under the pricing measure Q do not describe real-world probabilities but they reflect prices, the price a representative market player is willing to pay for getting a dollar in a particular state of the market. For a contingent claim, the (discounted) expected realized payoff is the (discounted) expectation of the payoff in the P -world, whereas the arbitrage-free price is the discounted expectation of the payoff in the Q -world (Harrison and Pliska 1981). A contingent claim is considered expensive when expectations in the Q -world exceed those in the P -world. To capture the difference in expectation under the P - and Q -probability measures, the concept of a risk premium is introduced and here modeled as risk premium =expectedPpayoff −expectedQpayoff expectedQpayoff , (1) Risks 2021,9, 196. https://doi.org/10.3390/risks9110196 https://www.mdpi.com/journal/risks Risks 2021,9, 196 2 of 19 i.e., the ratio of the difference between the expected payoff in the P -world and the expected payoff in the Q -world to the expected payoff in the Q -world. This coincides with the definition of an expected net return as used in Coval and Shumway (2001). An expensive claim then bears a negative risk premium whereas an inexpensive claim bears a positive risk premium. The pricing kernel relates the price of a claim to its expected payoff under measure P , i.e., it informs on how to transform subjective probabilities into pricing ones (Cochrane 2005) .Coval and Shumway (2001) show that, under the assumption of a monotonically declining pricing kernel, risk premia of European call options are always positive, above the risk-free rate, while these of European put options lie below the risk-free rate. Moreover, risk premia for both type of options are increasing in the strike price. In contrast, a growing body of literature reports on negative average realized returns, decreasing with moneyness, for out-of-the-money call options, questioning the accuracy of using a declining pricing kernel. Cuesdeanu and Jackwerth (2018b) review the literature on and confirm the existence of this so-called pricing kernel puzzle, the disagreement between theoretical predictions of standard option pricing theory and empirics. Among others, Cuesdeanu and Jackwerth (2018a), Sichert (2020) and Volkmann (2021) have recently reported evidence on a locally increasing, U-shaped pricing kernel. In this article, we revisit risk premia in European (call) options under the assumption of a U-shaped pricing kernel. To the best of our knowledge, we are the first to focus on the option with a zero-risk premium, which is, for a fixed maturity, completely determined by the so-called zero-risk premium strike. In a general setting, that includes a U-shaped kernel, we prove the existence of this zero-risk premium strike for European call options and the nonexistence for European put options, which immediately justifies our focus on call options. We additionally show that the zero-risk premium strike is unique, i.e., it indicates the transition point from inexpensive to expensive call options. The theoretical results in this paper are accompanied by an empirical study based on the S&P500 and DAX stock index. In order to calculate the risk premium for call options on these indices, we need information on the physical and pricing probability measure. Today, the rich variety of traded vanilla options provide us with valuable information on the Q -measure and so the pricing distribution of an asset’s return. The estimation of a pricing density from option data is often preceded by the allocation of an option pricing model. In 1973, Black, Scholes and Merton made a significant breakthrough in asset modeling when publishing what has come to be known as the Black-Scholes market model (Black and Scholes 1973;Merton 1973) . Later on, alternative pricing models such as the Variance Gamma model (Madan et al. 1998;Madan and Seneta 1990) are successfully introduced to improve on the ideas of Black, Scholes and Merton. More recently, Küchler and Tappe (2008) suggested the four-parameter class of Bilateral Gamma processes as an improvement on the Variance Gamma framework to model the fluctuations of the financial market. Instead of using option data, a physical return density is often inferred from historical time series data on the return of an asset. However, historical data is backward looking and only extended with one new observation each day. Leveraging the distributional wealth of the option market, we therefore elaborate on the methodology of Madan et al. (2020) to extract physical distributional information from option data. To this purpose, we deploy the Tilted Bilateral Gamma option pricing model, which proceeds from the Bilateral Gamma model by imposing a U-shaped pricing kernel on the physical probability measure. This allows for the simultaneous extraction of model parameters according to both the physical and the pricing probability measure. A calibration of the Tilted Bilateral Gamma model ultimately results in an estimate of the zero-risk premium strike over time. Within the covered sample period, the zero-risk premium strike of a one-month held-to-maturity European call option seems to be located slightly in-the-money, on average, though close to the at-the-money level. Besides, we see that risk premia for European call options on the S&P500 stock index are slightly increasing Risks 2021,9, 196 3 of 19 with moneyness, for far in-the-money options, but decreasing with moneyness for close to at-the-money and out-of-the-money contracts. The decreasing part is theoretically grounded by Bakshi et al. (2010), who show that, within the framework of a U-shaped pricing kernel, risk premia on call options are decreasing in the strike price, for strikes beyond a certain threshold. Combining these results with the observed level of the zero-risk premium strike, we thus find evidence that not only out-of-the-money call options are expensive, but often also the options with an in-the-money strike, close to the at-the-money level. The pattern of the risk premium roughly matches the realized average option returns over the available sample period. With that result, we first support the above literature that advocates the U-shaped pricing kernel as a possible answer to the empirically observed returns on the market. Second, we add to the literature that focuses on matching risk premia in European call options across strike prices with realized average returns 1 . Recently, McKeon (2019) used a theoretical, simulation-based derivation of risk premia in call options, where prices of options are calculated according to the original Black-Scholes model. The general pattern obtained for the risk premia roughly matches the one obtained in the Tilted Bilateral Gamma framework. Furthermore, Hu and Liu (2021) compare realized average returns with risk premia in call and put options across strike prices, implied by various option pricing models. The authors show that standard models involving an equity risk premium only have difficulties in describing realized returns, whereas a stochastic volatility model in which volatility risk is priced fits the average option returns reasonably well. From that, we conclude that not only pricing models based on the assumption of a U-shaped kernel are able to match the average realized returns on the market. The outline of the rest of the paper is as follows. Section 2formalizes the definition of a risk premium and in particular a zero-risk premium strike and confirms its existence under certain modeling assumptions. The theory behind the Tilted Bilateral Gamma model is presented in Section 3, as well as the calibration methodology, which results in a joint estimation of physical and pricing distributional information from option prices. Section 4 elaborates on a numerical example, based on option surfaces of the S&P500 and DAX index. It reports on the empirical evolution and position of the zero-risk premium strike over time. Finally, Section 5concludes. 2. The Zero-Risk Premium Strike of a European Call Option Consider an asset S , with level St at time t . Let RT=ln(St+T)−ln(St) be the T -period rate of return on this same asset. The payoff from buying a European call ( EC ) option on asset S, at time t, with strike Kand maturity T, is generally given by payoff EC(K,T) = (St+T−K)+= (SteRT−K)+, =(SteRT−Kif SteRT≥K 0 if SteRT≤K.(2) 2.1. Definition of a Zero-Risk Premium Strike In the physical world, or P -world, the market performance of an asset is modeled according to a physical probability density function. Using Equation (2) , the expected payoff of the European call option under the physical return density fRT:R→R+ of asset Sis determined as expectedPpayoff EC(K,T) = EP[(SteRT−K)+], =Z+∞ −∞(Stex−K)+fRT(x)dx. (3) The discounted value of the expectation in Equation (3) results in the expected realized payoff at the time of buying the option. Risks 2021,9, 196 4 of 19 In the pricing world, or Q -world, the performance of asset S is modeled using the corresponding pricing probability density function. As such, the expected payoff under the pricing measure Qis determined as expectedQpayoff EC(K,T) = EQ[(SteRT−K)+], =Z+∞ −∞(Stex−K)+gRT(x)dx, (4) where gRT:R→R+ is the pricing return density of asset S . Note that the arbitrage-free price of the option is given by the discounted value of the expectation in Equation (4). Connecting the option’s expected payoff under the physical measure P to the corresponding expected payoff under the pricing measure Q naturally leads to the concept of a risk premium, defined as risk premium EC(K,T) = EP[(SteRT−K)+]−EQ[(SteRT−K)+] EQ[(SteRT−K)+], =EP[(SteRT−K)+] EQ[(SteRT−K)+]−1, (5) consistent with the definition of an expected net return, as used in Coval and Shumway (2001) . From Equation (5) we see that the risk premium is determined by the gap between the P - and Q -probability measures and so decided upon the shape and location of the pricing density with respect to the physical density. The risk premium is the return one can expect from buying an held-to-maturity European call option at time t and it can be seen as a compensation directly related to the uncertainty on the future asset level. For a fixed maturity T, we are interested in identifying the strike Kt,Tsuch that EP[(SteRT−Kt,T)+] = EQ[(SteRT−Kt,T)+], (6) i.e., the strike where expectations on the payoff of a European call option are equal under both the P - and Q -probability measures. This strike thus determines the European call option with a zero-risk premium. We also accept this as the definition of the zero-risk premium strike and refer to it as Kt,T , recognizing the dependency upon the fixed maturity T on the one hand and the moment of buying, time t , on the other hand. The zerorisk premium option with maturity T is equivalently defined by the moneyness level kt,T=Kt,T/St. 2.2. Conditions on the Existence of a Zero-Risk Premium Strike In what follows, we discuss the conditions that guarantee a solution to Equation (6) and so the existence of a call option’s zero-risk premium strike. We also briefly touch upon the European put option case to further substantiate our focus on call options. The zero-risk premium strike of a European call option is defined by Equation (6) , which we can rewrite in terms of the asset Sinstead of the return RTas Z+∞ Kt,T (x−Kt,T)fS(x)dx =Z+∞ Kt,T (x−Kt,T)gS(x)dx, (7) using the T -period physical and pricing density, respectively, fS:R+→R+ and gS: R+→R+, of asset Sand the expressions in Equations (3) and (4). As opposed to a European call option, the payoff from buying a European put ( EP ) option on asset S, at time t, is generally given by payoff EP(K,T) = (K−St+T)+= (K−SteRT)+, =(K−SteRTif SteRT≤K 0 if SteRT≥K.(8) Risks 2021,9, 196 5 of 19 The equivalent of Equation (7) for a European put option with the same features then becomes ZKt,T 0(Kt,T−x)fS(x)dx =ZKt,T 0(Kt,T−x)gS(x)dx. (9) Representing the corresponding T -period cumulative distribution functions of fS and gS as FS:R+→( 0, 1 ) and GS:R+→( 0, 1 ) , integration by parts of Equations (7) and (9) , respectively, leads to Z+∞ Kt,T (1−FS(x))dx =Z+∞ Kt,T (1−GS(x))dx, (10) for the European call option and ZKt,T 0FS(x)dx =ZKt,T 0GS(x)dx, (11) for the European put option. Based on Equations (10) and (11), we now define c(K) = Z+∞ K(1−FS(x))dx −Z+∞ K(1−GS(x))dx, (12) p(K) = ZK 0FS(x)dx −ZK 0GS(x)dx. (13) Typically for equity such as stocks and indices, the fundamental drift of the asset will exceed the risk-free rate of return to reflect risk compensation. Under the conditions of arbitrage-free pricing we then have c(0)>0. (14) Using the above assumption, we present Proposition 1. Proposition 1. If for all x ∈(0, +∞)it holds that FS(x)≤GS(x), there will not exist a zero-risk premium strike for the European call option on asset S , neither for the European put option on this same asset. Proof. The strike K is a zero-risk premium strike for the European call option on asset S if c(K) = 0. Likewise, K is a zero-risk premium strike for the European put option on asset S if p(K) = 0. It is therefore sufficient to show that there exists no such strike for both functions cand p. The expressions in Equations (12) and (13) result in, respectively, c(∞) = 0 and p(0) = 0. Besides, it is easy to see that for each K∈[0, +∞): c0(K) = FS(K)−GS(K) = p0(K), (15) and so c0(0) = p0(0) = c0(∞) = p0(∞) = 0. (16) Risks 2021,9, 196 6 of 19 Since cand phave the same derivative, it holds that c(K) = −Z+∞ Kc0(u)du =−Z+∞ Kp0(u)du =p(K)−p(∞), (17) p(K) = ZK 0p0(u)du =ZK 0c0(u)du =c(K)−c(0). (18) Now, using Equation (14), Equation (18) results in p(∞) = −c(0)<0. (19) The condition that ∀x∈( 0, +∞):FS(x)≤GS(x) now easily translates in both c0(K)≤ 0 and p0(K)≤ 0 for each value of K∈( 0, +∞) , using the equality in Equation (15) . c0≤ 0 together with c( 0 )> 0 and c(∞) = 0 leads to the conclusion that c can never be zero, meaning that there exists no solution to Equation (7) and no zero-risk premium strike for the call option. p0≤ 0 together with p( 0 ) = 0 and p(∞)< 0 leads to the conclusion that p is always negative and so no zero-risk premium strike for the put option exists either, which ends the proof. A graphical clarification can be found in Figure 1a,b. First, note that the condition in Proposition 1can be translated into FS first-order stochastically dominating GS (Denuit et al. 2005). Second, in Proposition A1, in Appendix A , we show that the positioning of the density functions as in Figure 1c, i.e., exactly one point of intersection, results in first order stochastic dominance of the respective cumulative density functions. No zero-risk premium for both the European call and European put option will exist in that situation. Next, we derive a sufficient condition on the existence of a zero-risk premium strike for call options. In realistic circumstances, the premium to be paid for an insurance against hitting an asset level close to zero is higher than the probability of occurrence. In other words, price dominates probability in the left tail. Under this assumption, we present the following Proposition 2. Proposition 2. If the cumulative distribution functions FS and GS of asset S cross exactly once, meaning that there is a unique x ∈(0, +∞)such that 0<FS(x) = GS(x)<1, there exists a zero-risk premium strike for the European call option on this asset. Moreover, the zero-risk premium strike is unique. Under the same condition, there will not exist a zero-risk premium strike for the European put option on asset S. Proof. Since it is assumed that price dominates probability in the left tail, it is expected for all xclose to zero that fS(x)−gS(x)<0. (20) Suppose that FS and GS cross exactly once at strike Kc , i.e., FS(Kc) = GS(Kc) . Combining the results in Equations (15) and (20), it then holds that ∀0<K<Kc:c0(K) = p0(K) = FS(K)−GS(K)≤0, (21) and c and p are decreasing functions for all K smaller than Kc . Besides, Equation (15) results in ∀K>Kc:c0(K) = p0(K) = FS(K)−GS(K)≥0, (22) and both cand pare increasing functions for all Klarger than Kc. Risks 2021,9, 196 7 of 19 From Equation (12) we see that c(Kc) = Z+∞ Kc (1−FS(x))dx −Z+∞ Kc (1−GS(x))dx, =Z+∞ Kc (GS(x)−FS(x))dx <0, and c is also negative for all K≥Kc . As c(∞) = 0 and c only increases for K>Kc , there will not exist a K∈[Kc , +∞) such that c(K) = 0. However, as c decreases over all 0 <K<Kc , c( 0 )> 0 and c(Kc)< 0, there exists a unique Kt,T∈( 0, Kc) such that c(Kt,T) = 0. This Kt,T is called the zero-risk premium strike for the European call option on asset S. As p( 0 ) = 0 and p only decreases for all K∈( 0, Kc) , there will not exist a K in this region such that p(K) = 0. Furthermore, as p only increases over all K>Kc and p(∞)< 0, there will not exist a K∈[Kc , +∞) such that p(K) = 0 and thus no zero-risk premium strike for the European put option on asset S . A graphical clarification can again be found in Figure 1d,e. The density functions, resulting from the cumulative distribution functions in Figure 1d , are added in Figure 1f. In Proposition A2, in Appendix A, we show that the situation as presented in Figure 1f , i.e., two density functions crossing exactly twice, results in cumulative distribution functions that meet the conditions in Proposition 2. In that case, a unique zero-risk premium strike for the European call option exists. (a) (b) (c) (d) (e) (f) Figure 1. ( a ) Cumulative distribution functions FS and GS of asset S under, respectively, the physical measure P and the pricing measure Q . For all x∈( 0, +∞) it holds that FS(x)≤GS(x) . ( b ) Functions c and p resulting from the cumulative distribution functions in Figure 1a. ( c ) Density functions fS and gS resulting from the cumulative distribution functions in Figure 1a. ( d ) Cumulative distribution functions FS and GS under the condition that there exists exactly one x∈( 0, +∞) such that FS(x) = GS(x) . ( e ) Functions c and p resulting from the cumulative distribution functions in Figure 1d. ( f ) Density functions fS and gS resulting from the cumulative distribution functions in Figure 1d. ( a ) FS and GS ; ( b ); c and p ( c ) fS and gS; (d)FSand GS; (e)cand p; (f)fSand gS. Risks 2021,9, 196 8 of 19 A necessary condition for the existence of a zero-risk premium strike for European put options is that the cumulative distribution functions under the P - and Q -probability measures intersect at least twice. One can show that, in the specific situation of two intersection points, there exists at most one zero-risk premium strike for both the European call and European put option. The proof is similar to that of Proposition 2. 3. Joint Density Estimation Methodology An accurate estimation of both the physical density and the pricing density of the underlying asset are crucial in determining the risk premium of a European call option. In what follows, we impose a U-shape on the measure change between the P - and Q - probability measures, which gives rise to the pricing strategy of Madan et al. (2020) . Next, we detail how this strategy, combined with the option pricing formula of Carr and Madan (1999), results in a probability density estimate under both the Pand Q-measures. 3.1. The Pricing Density as U-Shaped Perturbation of the Physical Density The pricing density of an asset’s return arises naturally from the corresponding physical density, acknowledging a U-shaped pricing kernel. Following Cochrane (2005), we accept the existence of a pricing kernel m(R) such that the price pt at time t of a security paying out a cash-flow c f (R)after a period of length Tequals pt=exp (−rT)EP[m(R)c f (R)], (23) with r the T -period risk-free rate of return. That way, the price of a European call option at time tis represented as price EC(K,T) = exp (−rT)Z+∞ −∞(Stex−K)+m(x)fRT(x)dx. (24) The pricing kernel thus relates the price of a security to its expected payoff under measure P , i.e., it reflects a representative market player’s assessment on different states of the market: it is more valuable to earn a dollar in a state of the market where the own wealth is low (Cuesdeanu and Jackwerth 2018b). Following Madan et al. (2020), we construct the U-shaped pricing kernel m , connecting the pricing density g to the physical density f , as the weighted sum of two exponential functions. We define g(x) = C·h(1−p)·e−ηx+p·eζxi·f(x). (25) The constant Cis needed to ensure that gis a proper density function and so C−1=Z∞ −∞h(1−p)·e−ηx+p·eζxi·f(x)dx. (26) Investors’ preferences are thus characterized introducing the parameters η∈( 0, ∞) , ζ ∈( 0, ∞) and p∈( 0, 1 ) . The first parameter η represents the risk aversion coefficient for being in a long position and likewise, ζ represents the risk aversion coefficient for being in a short position. The last parameter p weighs the importance of the declining part of the U-shape against the importance of the inclining part. Note that it will be meaningful to calculate the risk premium and especially the zerorisk premium strike of a call option under the assumption of a U-shaped pricing kernel. Indeed, this kernel lifts both tails of the physical density, which results in exactly two points of intersection of the physical and corresponding pricing density. An example is given later on, in Section 4.2. Combining the results in Propositions A2 and 2, the existence and uniqueness of the zero-risk premium strike is confirmed. Risks 2021,9, 196 15 of 19 5. Conclusions The risk premium of a European call option is defined as the relative difference in expected payoff under the P - and Q -probability measures. Historically, Coval and Shumway (2001) showed that this risk premium is increasing with moneyness, and always above the riskfree rate, under the assumption of a monotonically decreasing pricing kernel. However, this does not match with the empirically examined pattern of average realized returns on European call options. One often encounters negative returns, particularly for out-of-themoney options. Within the framework of a U-shaped pricing kernel, we revisit risk premia in European call options and we especially focus on the option with a zero-risk premium, defined by the so-called zero-risk premium strike. We prove the uniqueness of this strike, i.e., it indicates the transition point from which on call options are considered expensive. In order to calculate this zero-risk premium strike, pricing and physical distributional information on the return of the underlying asset is needed. While historical time series are classically used to estimate a physical distribution, we use evidence from the option market to extract information on both the physical and corresponding pricing distribution. To that purpose, we deploy the Tilted Bilateral Gamma pricing model, first introduced by Madan et al. (2020) . Leveraging the distributional wealth of the option market, a calibration of this model on an option surface allows us to simultaneously extract information on both physical and pricing densities. Based on an empirical study on the S&P500 and DAX stock index, we conclude that the zero-risk premium strike, over the covered sample period, is typically located slightly in-the-money. With small fluctuations around a mean level, the zero-risk premium strike appears to follow a rather stable pattern over time. The research is important from a practical point of view since the joint calibration of the P - and Q -measures may lead to specific option positionings. One can trade events that are cheap in respect to the ratio of P to Q ; P reflects the likelihood of occurrence of an event in the real world, whereas Q reflects the price to be paid to bet on the realization of the corresponding event. Further, the monitoring of the zero-risk premium strike over time may be of use from a risk-management point of view as it could be informative on the market’s perception of certain risks. Author Contributions: Writing—original draft preparation, E.V.; writing—review and editing, S.H., D.B.M. and W.S. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Option price data were obtained from OptionMetrics and are available at https://optionmetrics.com/ (S&P500: accessed on 31 August 2018. DAX: accessed on 17 April 2020). Conflicts of Interest: The authors declare no conflict of interest. Abbreviations The following abbreviations are used in this manuscript: BS Black-Scholes BG Bilateral Gamma φCharacteristic function e exp EExpected value EC European call option Risks 2021,9, 196 16 of 19 EP European put option FFT Fast Fourier Transform fPhysical density FPhysical cumulative probability function gPricing density GPricing cumulative probability function KStrike PPhysical measure QPricing measure rRisk-free rate RReturn on asset S RMSE Root Mean Squared Error SAsset TMaturity TBG Tilted Bilateral Gamma VG Variance Gamma Appendix A Proposition A1. f and g are two probability density functions, with respective support (af , bf) and (ag,bg). Given is that −∞<ag≤af<bg≤bf<∞, and there exists a unique c ∈(ag,bf)such that      f(x)<g(x)x∈(ag,c) f(x) = g(x)6=0x=c f(x)>g(x)x∈(c,bf). If F and G are the cumulative distribution functions of, respectively, f and g , then F will first order stochastically dominate G, i.e., F(x)≤G(x), for all x ∈(−∞,+∞). Proof. First of all, it is clear that for all x∈(−∞ , ag] , it holds that F(x) = G(x) = 0, and for all x∈[bf,+∞), we have F(x) = G(x) = 1. Second, for all x∈(ag,c]we have F(x) = Zx −∞f(y)dy <Zx −∞g(y)dy =G(x). (A1) Now, suppose F and G intersect at least once, i.e., there exists a point xc∈(ag , bf) such that F(xc) = G(xc) . According to the result in Equation (A1) , xc must be strictly larger than c. Since fand gboth integrate to 1, we have Zxc −∞f(x)dx +Z+∞ xc f(x)dx =Zxc −∞g(x)dx +Z+∞ xc g(x)dx, F(xc) + Z+∞ xc f(x)dx =G(xc) + Z+∞ xc g(x)dx, and so Z+∞ xc f(x)dx =Z+∞ xc g(x)dx. This results in Zbg xc [g(x)−f(x)]dx =Zbf bg f(x)dx. (A2) Risks 2021,9, 196 17 of 19 However, since f(x)>g(x) for all x∈(c , bg) , we have a strictly negative result in the left hand side of Equation (A2) and a positive result in the right hand side. We conclude that Equation (A2) cannot be valid and F(x)<G(x) must hold for all x∈(ag , bf) . This means that Fstochastically dominates G. Proposition A2. f and g are two density functions, with respective support (af , bf) and (ag , bg) . Given is that −∞<ag≤af<bf≤bg<∞, and there exist exactly two points, c1and c2, such that c1<c2and                f(x)<g(x)x∈(ag,c1) f(x) = g(x)6=0x=c1 f(x)>g(x)x∈(c1,c2) f(x) = g(x)6=0x=c2 f(x)<g(x)x∈(c2,bg). If F and G are the cumulative distribution functions of, respectively, f and g , there exists a unique point xcsuch that 0<F(xc) = G(xc)<1. Proof. First, for all x∈(ag,c1]we have F(x) = Zx −∞f(y)dy <Zx −∞g(y)dy =G(x). (A3) Second, for x smaller than, but close to bg , we have that f(x)<g(x) . Since g has a fatter right tail, G will reach the value of 1 slower than F and so G(x)<F(x) , for these values of x . Since G is above F when reaching values close to 0, but below F when reaching values close to 1, it is clear that Fand Gmust intersect at least once in a point xc. Since f(x)<g(x) for all x∈(ag , c1) , we have that F is flatter than G until c1 , and so xc must be larger than c1 . Furthermore, F is steeper than G for all x∈(c1 , c2) , so F possibly intersects G in this region, but only once, at a unique point. For x larger than c2 , F is again flatter than G , and will not cross it in this region. Since xc must exist, this means that xc∈(c1,c2)holds and this point is unique. Appendix B In Figure A1, we report on the realized return of held-to-maturity call options on the S&P500 index, with varying moneyness level and maturity. The realized return is calculated as Return =Payofft+T−Pricet Pricet, with t varying over all business days between 2 January 2018 and 29 August 2018. The given return is the average return over all values of t. In general, we observe an overall declining behavior, for increasing moneyness, with some minor deviations for longer maturities. For the options with a one and two months maturity, the zero-risk premium strike lies around the at-the-money-level. This coincides with our findings of Section 4.3. Note that for the longer maturity of 6 months, the realized average returns are all negative for the considered moneyness range. Risks 2021,9, 196 18 of 19 Figure A1. The average realized return of held-to-maturity call options on the S&P500 index, over the sample period from 2 January 2018 to 29 August 2018. 4 different maturity levels are encountered, and moneyness varies from 0.8 to 1.2. Note 1 A more extensive literature focuses on the other side of the pricing kernel puzzle, i.e., on abnormal put option returns that cannot be explained by standard option models. See, e.g., Broadie et al. (2009), Bondarenko (2014) and Bernales et al. (2020). Recently, we also see some interest in the relationship between risk premia in options and volatility in the underlying asset, see Chaudhury (2017) and Hu and Jakobs (2020). References Bakshi, Gurdip S., Dilip B. Madan, and George Panayotov. 2010. Returns of claims on the upside and the viability of U-shaped pricing kernels. Journal of Financial Economics 97: 130–54. [CrossRef] Bernales, Alejandro, Gonzalo Cortazar, Luka Salamunic, and George Skiadopoulos. 2020. Learning and Index Option Returns. Journal of Business & Economic Statistics 38: 327–39. Black, Fisher, and Myron Scholes. 1973. The pricing of options and corporate liabilities. Journal of Political Economy 18: 637–54. [CrossRef] Bondarenko, Oleg. 2014. Why are put options so expensive? Quarterly Journal of Finance 4: 145–95. [CrossRef] Broadie, Mark, Mikhail Chernov, and Michael S. Johannes. 2009. Understanding index option returns. The Review of Financial Studies 22: 4493–529. [CrossRef] Carr, Peter, and Dilip Madan. 1999. Option valuation using the fast Fourier transform. The Journal of Computational Finance 2: 61–73. [CrossRef] Chaudhury, Mo. 2017. Volatility and Expected Option Returns: A note. Economics Letters 152: 1–4. [CrossRef] Cochrane, John H. 2005. Asset Pricing, Revised ed. Princeton: Princeton University Press. Coval, Joshua D., and Tyler Shumway. 2001. Expected option returns. Journal of Finance 56: 983–1009. [CrossRef] Cuesdeanu, Horatio, and Jens C. Jackwerth. 2018a. The pricing kernel puzzle in forward looking data. Review of Derivatives Research 21: 394–419. [CrossRef] Cuesdeanu, Horatio, and Jens C. Jackwerth. 2018b. The pricing kernel puzzle: Survey and outlook. Annals of Finance 14: 289–329. [CrossRef] Denuit, Michel, Jan Dhaene, Marc Goovaerts, and Rob Kaas. 2005. Actuarial Theory for Dependent Risks: Measures, Orders and Models. West Sussex: Wiley. Harrison, Michael, and Stanley Pliska. 1981. Martingales and Stochastic Integrals in the Theory of Continuous Trading. Stochastic Processes and Their Applications 11: 215–60. [CrossRef] Hu, Guanglian, and Kris Jakobs. 2020. Volatility and Expected Option Returns. Journal of Financial and Quantitative Analysis 55: 1025–60. [CrossRef] Hu, Guanglian, and Yuguo Liu. 2021. The Pricing of Volatility and Jump Risks in the Cross-Section of Index Option Returns. Journal of Financial and Quantitative Analysis. forthcomming. [CrossRef] Küchler, Uwe, and Stefan Tappe. 2008. Bilateral Gamma distributions and processes in financial mathematics. Stochastic Processes and Their Applications 118: 261–83. [CrossRef] Risks 2021,9, 196 19 of 19 Madan, Dilip B., Peter P. Carr, and Eric C. Chang. 1998. The Variance Gamma process and option pricing. Review of Finance 2: 79–105. [CrossRef] Madan, Dilip B., and Wim Schoutens. 2016. Applied Conic Finance, 1st ed. Cambridge: Cambridge University Press. Madan, Dilip B., Wim Schoutens, and King Wang. 2020. Bilateral multiple Gamma returns: Their risks and rewards. International Journal of Financial Engineering 7: 2050008. [CrossRef] Madan, Dilip B., and Eugene Seneta. 1990. The Variance Gamma (V.G.) model for share market returns. The Journal of Business 63: 511–24. [CrossRef] McKeon, Ryan. 2019. Time Variation in Options Expected Returns. Presented at the 2019 Derivative Markets Conference. Available online: https://acfr.aut.ac.nz/__data/assets/pdf_file/0010/265339/Time-Variation-in-Option-Expected-Returns-Spring-2019 .pdf (accessed on 21 October 2021). Merton, Robert C. 1973. Theory of rational option pricing. The Bell Journal of Economics and Management Science 4: 141–83. [CrossRef] Sichert, Tobias. 2020. The Pricing Kernel Is U-Shaped. Available online: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3095 551 (accessed on 21 October 2021). Stoll, Hans R. 1969. The relationship between put and call option prices. The Journal of Finance 14: 801–24. [CrossRef] Volkmann, David. 2021. Explaining S&P500 option returns: An implied risk-adjusted approach. Central European Journal of Operations Research 29: 665–85.