A Rational Finance Explanation of the Stock Predictability Puzzle
aa r X i v : . [ q -f i n . M F ] N ov A Rational Finance Explanation of the StockPredictability Puzzle
Abootaleb Shirvani a , Svetlozar T. Rachev b , and Frank J. Fabozzi ca Department of Mathematics and Statistics, Texas Tech University, [email protected] b Department of Mathematics and Statistics, Texas Tech University, [email protected] c EDHEC Business School, [email protected]
Abstract
In this paper, we address one of the main puzzles in finance observed in thestock market by proponents of behavioral finance: the stock predictability puzzle. We offer astatistical model within the context of rational finance which can be used without relying onbehavioral finance assumptions to model the predictability of stock returns. We incorporatethe predictability of stock returns into the well-known Black-Scholes option pricing formula.Empirically, we analyze the option and spot trader’s market predictability of stock prices bydefining a forward-looking measure which we call “implied excess predictability”. The em-pirical results indicate the effect of option trader’s predictability of stock returns on the priceof stock options is an increasing function of moneyness, while this effect is decreasing for spottraders. These empirical results indicate potential asymmetric predictability of stock pricesby spot and option traders. We show in pricing options with the strike price significantlyhigher or lower than the stock price, the predictability of the underlying stock’s return shouldbe incorporated into the option pricing formula. In pricing options that have moneyness closeto one, stock return predictability is not incorporated into the option pricing model becausestock return predictability is the same for both types of traders. In other words, spot tradersand option traders are equally informed about the future value of the stock market in thiscase. Comparing different volatility measures, we find that the difference between implied andrealized variances or variance risk premium can potentially be used as a stock return predictor.
Keywords
Predictability of stock returns; behavioral finance; rational dynamic stockpricing theory; option pricing; Stratonovich integral.1
Introduction
In an efficient market, price discovery should be instantaneous and contemporaneous. Empirical evidence suggests that the excess aggregate stock market returns are predictable.Using monthly, real, equal-weighted New York Stock Exchange returns from 1941–1986,Fama and French (1988) found that the dividend–price ratio can explain 27% of the variationof cumulative stock returns over the subsequent four years. Campbell and Shiller (1988)specify econometric models of dividend discounting that imply that price dividend ratiospredict stock returns. These two studies were among the first to identify this as the “stockpredictability puzzle.”There are a good number of more recent empirical studies that have investigated thepredictability of stock returns. Some believe stock return predictability is attribute to changesin business conditions, while others attribute it to market inefficiency.The majority of work on the predictability of stock returns is based on statistical, macro,and fundamental factor analyses models. Recently, a good numbers of studies in behavioralfinance have examined behavioral factors that could lead to the predictability of stock returns.The behavioral factors that proponents of behavioral finance have suggested that can lead tostock return predictability are (1) sentiment, (2) overconfidence, (3) optimism and wishfulthinking, (4) conservatism, euphoria and gloom, (5) self-deception, (6) cursedness, (7) beliefperseverance, and (8) anchoring. Motivated by the empirical findings that stock returns are predictable, some researchershave investigated the impact of stock return predictability on the prices of related assets.Lo and Wang (1995), for example, discussed the effect of stock return predictability on theprice of stock options. They showed that even a small amount of predictability could havea significant impact on option pricing. Liao and Chen (2006) demonstrated that the effectof autocorrelated returns on European option prices is significant. Huang et al. (2009) andPaschke and Prokopczusk (2010) offer even more recent examples of studies about the impactof stock returns predictability on the valuation of options. The upshot of these studies is that toobtain more realistic stock prices, it is essential to model and analyze stock return predictabilityand incorporate its impact into stock log-returns and option pricing models.Modeling and analyzing the stock return predictability is crucial for stock and risk See Kumar and Chaturvedul (2013). See for example, Kandel and Stambaugh (1996), Neely and Weller (2000), Malkiel (2003),Barberis and Thaler (2003), Shiller (2003), Avramov (2003), Wachter and Warusawitharana (2009), Pesaran(2010), Zhou (2010), and Bekiros (2013). The models that have been used are (1) Conditional Capital AssetPricing Model, (2) vector autoregressive models, (3) Bayesian statistical factor analysis, (4) posterior moments ofthe predictable regression coefficients, (5) posterior odds, (6) the information in stock prices, (7) business cycleseffects, (8) stock predictability of future returns from initial dividend yields, (9) firm characteristics as stock returnpredictors, (10) anomalies, (11) predictive power of scaled-price ratios such as book-to-market and earnings-to-price, forward spread, and short rate, (12) variance risk premia and variance spillovers, (13) momentum, marketmemory and reversals, and (14) early announcements and others. See Lewellen (2000), Barberis and Thaler (2003), Ferson (2006), Peleg (2000, Chapter 1), andDaniel and Hirshleifer (2015). Lo and Wang (1995) introduced a model to price options when stock returnsare predictable. Their model is based on a specially designed multivariate trending Orn-stein–Uhlenbeck (O-U) process includes many parameters. The trending O-U processes withsmall dimensions such as univariate and bivariate processes are not realistic as noted byLo and Wang (1995). Moreover, in their model, predictability is induced by the drift parame-ter, which is not a parameter in the classical Black-Scholes model.In this paper, we propose a method to model the prediction of stock prices by adjustingthe stock predictability as a parameter with the Black and Scholes (1973) and Merton (1973)model framework by using the Stratonovich integral. In our model, predictability is viewed asthe dividend yield, which we refer to dividend yield due to predictability , and is incorporatedinto the option pricing formula. We derive an option pricing model by incorporating thepredictable stock returns within the classical Black-Scholes-Merton (BSM) framework.Next, we define implied excess predictability to compare an option trader’s predictabilityof stock returns with that of a trader in the cash market (i.e., spot trader). Using the observedprice of European call options based on the SPDR S&P 500 ETF (SPY), we plot the impliedexcess predictability against “moneyness” and time to maturity. The pattern of the impliedexcess predictability surface shows that at each maturity, an option trader’s predictability ofthe SPY is an increasing function of moneyness. The turning point of the surface is where themoneyness is close to . . The effect of option trader’s predictability of stock returns on theprice of stock options increases when the moneyness increases from . to . . Conversely„when the moneyness decreases from . to . , the effect of spot trader’s predictability ofstock returns on the price of stock options decreases. These empirical results indicate potentialasymmetric predictability of stock prices by spot and option traders.We demonstrate that in pricing an option with significant intrinsic value, stock returnpredictability should be incorporated into the BSM model. In pricing options that havemoneyness close to one stock predictability is not incorporated into the BSM model becausestock predictability is the same for both types of traders. In other words, spot traders and optiontraders are equally informed about stock market in this case. We show a popular stock marketvolatility index — the CBOE volatility index (VIX ) – is potentially more informative than theother volatility measures (historical, realized, and time series estimation method volatility) forpredicting stock returns. The variance risk premium – the difference between implied varianceand realized variance – can potentially predict stock market returns.This paper is organized as follows. The next section describes our methodology for model-ing the prediction of stock prices. Then we derive an option pricing formula by incorporatingthe predictability of stock returns into the model. Section 3 describes the results of our modelusing the S&P 500 index options. We then analyze and compare the prediction of stock marketreturns by option and spot traders. Section 5 summarizes our findings. See Shirvani et al. (2019). See Kloeden et al. (2000, Chapter 2), Øksendal (2003, Chapter 5), and Syga (2015). The Predictability of Stock pricing
A major issue raised by the proponents of behavioral finance is that prices are often pre-dictable. . More precisely, given a stochastic basis ( Ω , F , F = (F t , t ≥ ) , P ) a price process S ( t ) t ≥ , defined on ( Ω , F , P ) is not necessarily F -adapted, it is adapted to an augmentedfiltration F (∗) ⊃ Ð t ≥ o F t , with F (∗) ⊂F .Admitting the fact that stock returns are predictable, we propose a method to model theprediction of stock returns by adjusting the predictability of stock returns. Our option pricingmodel is close to the idea put forth by Shiller (2003) of “smart money versus ordinary investors.”To model the predictability of stock prices, we use the Stratonovich integral : ∫ T θ ( t ) ◦( ) dB ( t ) = lim = t ( ) < t ( ) < ··· < t ( k ) = T , t ( j ) = j ∆ t , ∆ t ↓ Í k − j = θ (cid:16) t ( j + ) + t ( j ) (cid:17) (cid:16) B ( t ( j + ) ) − B ( t ( j ) ) (cid:17) . (1)In (1), B ( t ) , t ≥ , is a Brownian motion generating a stochastic basis ( Ω , F , F = (F t , t ≥ ) , P ) , θ ( t ) t ≥ is F -adapted left-continuous and locally bounded process. An important propertyof the Stratonovich integral is that it “looks into the future,” and therefore, price processesbased on the Stratonovich integral possess predictability properties. In sharp contrast, the Itôintegral: ∫ T θ ( t ) dB ( t ) = lim = t ( ) < t ( ) < ··· < t ( k ) = T , t ( j ) = j ∆ t , ∆ t ↓ k − Õ j = θ (cid:16) t ( j ) (cid:17) (cid:16) B ( t ( j + ) ) − B ( t ( j ) ) (cid:17) (2)“does not look in the future,” and thus Itô prices are not predictable. Combining both integrals(1) and (2) within a Stratonovich α -integral with α ∈ [ , ] we obtain: ∫ T θ ( t ) ◦ ( α ) dB ( t ) = lim = t ( ) < t ( ) < ··· < t ( k ) = T , t ( j ) = j ∆ t , ∆ t ↓ Í k − j = θ (cid:16) t ( j ) ( − α ) + α t ( j + ) (cid:17) (cid:16) B ( t ( j + ) ) − B ( t ( j ) ) (cid:17) = α ∫ T θ ( t ) ◦( ) dB ( t ) + ( − α ) ∫ T θ ( t ) dB ( t ) . (3)Consider a market with two assets: ( i ) a risky asset (stock) S with potentially predictiveprice process S ( t ) , t ≥ , following Itô stochastic differential equation (SDE): dS ( t ) = µ ( t , S ( t )) dt + σ ( t , S ( t )) dB ( t ) , t ≥ , S ( ) > , (4)where µ ( t , S ( t )) = µ t S ( t ) , and σ ( t , S ( t )) = σ t S ( t ) , For the regularity conditions implyingexistence and uniqueness of the strong solution of (3), see Duffie (2001, Chapter 6). By theItô formula, stock price dynamics is given by See, for example, Daniel and Hirshleifer (2015). See Kloeden et al. (2000, Chapter 2), Øksendal (2003, Chapter 5), and Syga (2015). ( t ) = S ( ) exp (cid:26)∫ t (cid:18) µ s − σ s (cid:19) ds + ∫ t σ s dB ( s ) (cid:27) , S ( ) > , t ≥ . ( ii ) riskless asset (bond) B with price process β ( t ) , t ≥ , defined by d β ( t ) = r t β ( t ) , r t = r ( t , S ( t )) , β ( ) > , (5)that is, β ( t ) = β ( t ) exp (cid:16)∫ t r s ds (cid:17) t ≥ . Consider a European Contingent Claim (ECC) C with price process C ( t ) = C ( t , S ( t )) ,where C ( t , x ) , t ≥ , x > , has continuous derivatives ∂ C ( t , x ) ∂ t and ∂ C ( t , x ) ∂ x . C ’s terminal timeis T > , and C ‘s terminal payoff is C ( T ) = C ( T , S ( T )) = g ( S ( T )) , for some continuous g : ( , ∞) → R .Assume that a trader i ( l ) takes a long position in C . Furthermore, when i ( l ) trades stock S with possibly superior or inferior to (4), the following Stratonovich α SDE: dS ( t ) = µ ( t , S ( t ))) dt + σ ( t , S ( t )) ◦ ( α ) dB ( t ) , t ≥ , S ( ) > , α ∈ [ , ] . (6)Thus, the Stratonovich SDE dS ( t ) = µ ( t , S ( t )) dt + σ ( t , S ( t )) ◦ ( α ) dB ( t ) , is equivalent to the Itô SDE dS ( t ) = (cid:16) µ ( t , S ( t )) + ασ ( t , S ( t )) ∂σ ( t , S ( t )) ∂ x (cid:17) dt + σ ( t , S ( t )) dB ( t ) = µ ( α ) t S ( t ) dt + σ t S ( t ) dB ( t ) , µ ( α ) t = µ t + ασ t , t ≥ , t ≥ . (7)Assume that a trader i ( s ) takes a short position in C trading in the contract where i ( l ) hadtaken the long position. i ( l ) and i ( s ) have entered the contract C as the only participants at theclosing bid-ask traded C -contract. i ( s ) observes only the dynamics of S traded by i ( l ) andgiven by (3). Furthermore, when i ( s ) trades stock S , with dynamics following Stratonovich γ SDE: dS ( t ) = µ ( t , S ( t )) dt + σ ( t , S ( t )) ◦ ( γ ) dB ( t ) , t ≥ , S ( ) > , (8)for some γ ∈ [ , ] ; that is, dS ( t ) = µ ( γ ) t S ( t ) dt + σ t S ( t ) dB ( t ) , µ ( α ) t = µ t + γσ t t ≥ , t ≥ . (9)The C -dynamics as traded by i ( l ) is determined by the Itô formula: We assume that i ( l ) and i ( s ) are the two trading parties in a bid-ask trade of C providing the smallest bid-askspread, which ultimately ends up with the trade transaction of C . C ( t , S ( t )) = n ∂ C ( t , S ( t )) ∂ t + ∂ C ( t , S ( t )) ∂ x µ ( γ ) t S ( t ) + ∂ C ( t , S ( t )) ∂ x σ t S ( t ) o dt + ∂ C ( t , S ( t )) ∂ x σ t S ( t ) dB ( t ) . (10)To hedge the risky position, i ( s ) forms a replicating self-financing strategy given by thepair a ( t ) , b ( t ) , t ≥ , where C ( t , S ( t )) = a ( t ) S ( t ) + b ( t ) β ( t ) with dC ( t , S ( t )) = a ( t ) dS ( t ) + b ( t ) d β ( t ) . Thus, dC ( t , S ( t )) = (cid:16) a ( t ) µ ( γ ) t + b ( t ) r t β ( t ) (cid:17) S ( t ) dt + a ( t ) σ ( t , S ( t )) dB ( t ) (11)From (10) and (11), i ( s ) obtains a ( t ) = ∂ C ( t , S ( t )) ∂ x , and b ( t ) β ( t ) = C ( t , S ( t )) − ∂ C ( t , S ( t )) ∂ x S ( t ) . Equating the terms with dt and setting S ( t ) = x , results in the following partial differentialequation (PDE): = ∂ C ( t , x ) ∂ t + ∂ C ( t , x ) ∂ x (cid:16) r t − p σ t (cid:17) x − r t C ( t , x ) + ∂ C ( t , x ) ∂ x σ t x , p = γ − α. (12)We call p ∈ [− , ] the excess predictability of S traded by i ( s ) over the S -dynamic, when S is traded by i ( l ) . In the classical Black-Scholes model, dividends were not accounted for inthe model. If we assume that the stock S provides a continuous dividend yield of p σ t (i.e.,the dividend paid over interval ( t , t + dt ] equals p σ t S t ) we obtain the Black-Scholes partialdifferential equation given by (12). Borrowing this idea, stock with continuously compoundeddividend yield p σ t , we denote D y ( t ) = p σ t as the dividend yield due to predictability . Asthe payment of dividends impacts the option price of the underlying stock, the stock returnpredictability impacts the price of options. Depending on the sign of p , D y ( t ) could be positiveor negative. When p = , we obtain the classical Black-Scholes equation.In particular, C -price dynamics is given by C ( t ) = E Q t n e − ∫ Tt r u du g ( S ( T ))} o , t ∈ [ , T ) , (13)where Q is the equivalent martingale measure for the dividend-stock-price. That is, Q ∼ P , andthe discounted gain process G ( Y ) ( t ) = X ( Y ) ( t ) + D ( Y ) ( t ) is a Q -martingale, S ( t ) = S ( ) exp (cid:26)∫ t (cid:18) µ s − σ s (cid:19) ds + ∫ t σ s dB ( s ) (cid:27) , S ( ) > , t ≥ . Y ( t ) = β ( t ) , t ≥ , X ( Y ) ( t ) = X ( t ) Y ( t ) , and dD ( Y ) ( t ) = Y ( t ) dD ( t ) . The dynamics of S on Q is given by dS ( t ) = (cid:0) r t − D y ( t , x ) (cid:1) S ( t ) dt + dB ( t ) , where r t is the risk-free rate at time t .In conclusion, with (13) we are able to incorporate the predictability of stock returns into See Duffie (2001, Section 6). C is a European call option with maturity T and strike K , and g ( S ( T )) = max ( S ( T ) − K , ) . Then for time to maturity, τ = T − t , the value of a call option for adividend-paying underlying stock in terms of the Black–Scholes parameters is C ( t ) = c ( S ( t ) , τ, K , r t , σ t , p ) = S ( t ) e − D y ( t ) τ Φ ( d + ) − K e − r t τ Φ ( d − ) , (14)where D y ( t ) = p σ t , Φ denotes the standard normal cumulative distribution function, and d ± = ln (cid:18) S ( t ) e − Dy ( t ) τ Ke − rt τ (cid:19) ± σ t τσ t √ τ . Given put–call parity, the price of a put option, P ( t ) is P ( t ) = C ( t ) + D y ( t ) − S t + K e − r t τ . In this section, we compare the option and spot trader’s predictability of stock returnsby defining the implied excess predictability . Implied excess predictability is a metric thatcaptures the view of the option and spot trader of the likelihood moves in the stock price. Itcan be used to predict the of stock price from two perspectives. An important characteristicof implied excess predictability is that it is forward looking. It compares the predictability ofmarkets for the given underlying stock market index from two perspectives. Recall that impliedexcess predictability is calibrated from the BSM option price formula.We denote by p the excess predictability of S traded by i ( s ) over the S -dynamic, when S is traded by i ( l ) . To study i ( s ) ’s stock return predictability (option trader) compared to i ( l ) (spot trader), we define implied excess predictability p = p (cid:16) S ( t ) K , τ (cid:17) as a function of moneyness S ( t ) K and time to maturity τ as the solution of c ( S ( t ) , τ, K , r t , σ t , p ) = C ( market ) ( t , S ( t ) , τ, K ) , (15)where C ( market ) ( t , S ( t ) , τ, K ) is the call option prices of SPY .We assume that SPY-daily closing prices follow S ( t ) = S ( ) exp (cid:26)∫ t ν s ds + ∫ t σ s dB ( s ) (cid:27) , S ( ) > , t = k ∆ t , k ∈ N + = { , , . . . } , (16)where ν s = µ s − σ s . Thus, the SPY-daily return series is given by R ( t + ∆ t ) = ln (cid:18) S ( t + ∆ t ) S ( t ) (cid:19) = ∫ t + ∆ tt ν s ds + ∫ t + ∆ tt σ s dB ( s ) , t = k ∆ t , k ∈ N + . (17) See, for example, Hull (2009), Chapter 13. https://nance.yahoo.com/quote/SPY/options?p=SPY . Q , the SPY daily return is given by dS ( t ) = (cid:0) r t − D y ( t , x ) (cid:1) S ( t ) dt + dB ( t ) . The valueof a call option for the time to maturity, τ = T − t , is given by (14). We calculate the impliedexcess predictability by taking the option’s market price, entering it into the (15) formula, andback-solving for the value of p .Here, we compare the option and spot trader’s predictability of stock returns by using theimplied excess predictability. Rather than looking at individual stocks, our analysis will focuson the aggregate stock market. In our case, the SPY is the proxy we use for the aggregate stockmarket. We compare the predictability of markets for the given underlying stock market indexfrom two perspectives, by doing so it provides important insight about the view of option andspot traders regarding the future price of the stock market.We use call option prices from / / to / / with different expiration datesand strike prices. The expiration date varies from / / to / / , and the strike pricevaries from to among different call option contracts. The midpoint of the bid and askis used in the computation. As the underlying of the call option, the SPY index price was . on / / . We use the 10-year Treasury yield curve rate on / / as therisk-free rate r t , here r t = . .As an estimates for σ t , we use the following four metrics: (1) daily closing values of V I X t /√ ; (2) historical volatility based on one-year historical data; (3) realized volatilityover one-year historical data; and (4) estimated volatility over one-year by modeling timeseries with classical methods ARIMA ( , ) -GARCH ( , ) with the Student’s t distribution asan innovation distribution. The minimum estimated value for σ t is derived where the realizedvolatility is applied and the maximum estimated value is derived where the daily closing valuesof VIX is used .Since implied excess predictability surfaces of all models are very similar, we plot theexcess predictability surface when σ t is estimated from realized volatility. The implied excesspredictability surface is graphed against both a standard measure of “moneyness” and timeto maturity (in year) in Figure 1. Recall that a high value for p (close to one) means excesspredictability of SPY daily return traded by i ( s ) over the predictability of SPY traded by i ( l ) .In other words, p = means that option traders potentially predict the future of the SPY returnsbetter than the spot trader. The opposite is true when p = − . Recall that the implied excesspredictability surface is an increasing function of σ t .Figure 1 indicates that at each maturity, implied excess predictability of option tradersincrease as moneyness increases. Where the moneyness varies in ( , . ) , the surface is flat atpoint − , indicating higher predictability of spot traders comparing to option traders. Thus,to price significant out-the-money options, the value of p in the model should be − . Wherethe moneyness varies in ( . , . ) , in-the-money options, the value of p starts increasingfrom . , and flats out at point . This finding indicates that option traders can potentiallypredict market changes better than spot traders when the option is in-the-money. In this case,for pricing in-the-money option, the value of p in the log-return model should be .The turning point of the surface is where the moneyness is close to . . When the Figure 1: Implied dividend yield against time to maturity and moneyness.
Figure 2: Relative difference of excess predictability VIX-Realized model against time tomaturity and Moneyness.moneyness varies in ( . , . ) , p varies in (− . , . ) . This is the range that spot and optiontraders are equally informed about the market, and the predictability of the market is equalfor both traders. Thus, to price options with no significant intrinsic value, the classical BSMequation can be used.As we mentioned, the other four surfaces are very similar. Here, instead of plotting the9our similar surfaces, we plot the relative difference of the excess predictability of each surfaceto the surface derived from realized variance, denoted by p i − p , where i = , , . Here (1) p refers to the excess predictability surface when σ t is imputed from the VIX index, (2) p is where σ t imputed by historical volatility, and (3) p is where σ t is estimated by time seriesmodels. Figures 2-4 show the relative difference of excess predictability for each surface. Inall surfaces, where the moneyness varies in ( . , . ) , the relative difference is significant.At each value for moneyness in ( . , . ) , the relative difference of excess predictabilityincreases as time to maturity increases.Zhou (2010) defined variance risk premium at time t as the difference between the ex-anterisk-neutral expectation and the objective or statistical expectation of the return variance overthe [ t , t + ] time interval, V AR t = E Qt ( V ar ( r t + )) − E Pt ( V ar ( r t + )) , (18)which is not directly observable in practice. In practice, the risk-neutral expectation of the returnvariance, E Qt ( V ar ( r t + )) , is typically replaced by the VIX index and statistical expectation ofthe return variance, E Pt ( V ar ( r t + )) , is estimated by realized variance.Zhou (2010) showed that the difference between implied variance and realized variance((i.e., variance risk premium) can be used for the short-term predictability of equity returns,bond returns, forward premiums, and credit spreads. Comparing Figures 2-4, the most signif-icant relative difference of excess predictability is observed in Figure 2. It indicates that theVIX index contains more information about the stock market compared to the other metrics.Figure 3, the historical-realized surface, has the minimum relative difference.Recall that the maximum and minimum values for σ t are derived from the VIX and realizedvolatility. As we observed, the most significant relative difference of excess predictability,Figure 2, is derived where the VIX and realized volatility are used in the model. Thus, bycomparing equation (18) for different volatility measures and the fact that excess predictabilityis an increasing function of σ t , suggests that the variance risk premium measure potentiallycontains more information compared to the other variance measures for predicting stock marketreturns. The historical volatility measure is the poorest metric. In this paper, we studied the predictability of stock returns within the framework of rationalfinance rather than relying on behavioral finance explanations. We proposed a method tomodel stock returns by incorporating the predictability of stock returns in the model andthen deriving an option pricing formula. To compare the predictability of stock returns byoption traders and spot traders, we constructed a forward-looking measure that comparesthe option and spot trader’s stock returns predictability which we called the “implied excesspredictability measure.” The empirical results indicate that to price a significant in-the-moneyand out-the-money option, the option’s and spot trader’s predictability of stock returns should10 .10.05 10.1 0.9 0.8 0.250.15 0.7 0.20.2 0.6 0.150.5 0.10.4 0.050.3
Figure 3: Relative difference of excess predictability Historical-Realized model against timeto maturity and Moneyness.
Figure 4: Relative difference of excess predictability time series-Realized model against timeto maturity and Moneyness.be incorporated into the BSM model. For options with a small intrinsic values, spot traders andoption traders are equally informed, and the predictability of the market is equal for both traders.In this case, the classical BSM model can be used for option pricing without incorporatingstock return predictability. Finally, we showed that the difference between implied variance11nd realized variance, which we called variance risk premium, is an informative measure forpredicting the market in contrast to other volatility measures.
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