Rational Finance Approach to Behavioral Option Pricing
RRational Finance Approach to Behavioral Option Pricing
Jiexin Dai a , Abootaleb Shirvani a , and Frank J. Fabozzi ba The Department of Mathematics and Statistics, Texas Tech University b EDHEC Business School
Abstract:
When pricing options, there may be different views on the instantaneous mean return ofthe underlying price process. According to Black (1972), where there exist heterogeneous views on theinstantaneous mean return, this will result in arbitrage opportunities. Behavioral finance proponents arguethat such heterogenous views are likely to occur and this will not impact option pricing models proposedby rational dynamic asset pricing theory and will not give rise to volatility smiles. To rectify this, a leadingadvocate of behavioral finance has proposed a behavioral option pricing model. As there may be unexploredlinks between the behavioral and rational approaches to option pricing, in this paper we revisit Shefrin(2008) option pricing model as an example and suggest one approach to modify this behavioral financeoption pricing formula to be consistent with rational dynamic asset pricing theory by introducing arbitragetransaction costs which offset the gains from arbitrage trades.
Keywords:
Rational dynamic asset pricing theory; behavioral option pricing; arbitrage costs
Proponents of behavioral finance have identified several market anomalies (i.e., empirical findings thatare inconsistent with theories formulated by traditional finance). Since the late 1970s (see Heukelom (2014))there have been numerous papers that seek to explain these anomalies by building upon the foundationalideas of behavioral economics proposed by Daniel Kahneman and Amos Tversky. The debate regardingwhether financial market agents are rational or irrational in making financial decisions is ongoing . MarkRubinstein, an advocate of rational markets, criticized the behavioral finance approach to asset pricing.Rubinstein (2001) argued that as a trained financial economist, he was taught that the Prime Directivein pricing is to explain asset prices by using rational models. This is not to say that pricing cannot relyon irrational investor behavior if rational models fail to price assets properly (i.e., fail to correctly pricemarket prices). In his opinion, the behavioralist literature has lost all the constraints of this directive. MeirStatman, one of the leading academics who has contributed to the behavioral finance camp, has a differentperspective. Statman (1995) argues that standard finance is indeed so weighted down with anomalies thatit makes much sense to continue the reconstruction of financial theory on behavioral lines.Proponents of the behavioral finance did not merely critique rational finance models on asset pricing,option pricing, and portfolio selection. They have proposed behavioral finance based on irrational behavior.Meir Statman and Hersh Shefrin have taken the lead on formulating such models. For example, Shefrinand Statman (1993) first proposed option pricing model based on prospect theory. ? argued that we livein a world of heterogeneous beliefs and option markets are particular vulnerable in this respect. Later,Shefrin (2008) proposed an equilibrium approach to option pricing in which the representative agents viewthe return from the underlying asset as a mixture of two different normal distributed returns representingthe heterogeneous views on the asset return of the buyer and the seller of the option. More specifically, See, for example, Zeckhauser (1986), Hirshleifer (2001), Shiller (2003), Barberis and Thaler (2003), Brav et al. (2004),Curtis (2004), Parisi and Smith (2005) in Chapter 21, and Thaler (2005). a r X i v : . [ q -f i n . C P ] M a y hefrin defines a market model with two investors sharing two price processes with common Brownianmotion as market driver, the same volatility parameters and different instantaneous mean returns. Sincethe mixture of two different log-normal distributions is not infinitely divisible , the price process of therepresentative agent is not a semi-martingale, which according to Black (1972), Shefrins proposed modelallows for arbitrage opportunities. Rockenbach (2004) reports the arbitrage-free option pricing is invalidatedbased on mental accounts, which result is consistent with behavioral portfolio theory of Shefrin and Statman(see Shefrin and Statman (2000)). Pena et al. (2010) derived a behavioral Black-Scholes option pricing modelbut the underlying price process is not a semi-martingale, thus, according to the fundamental theorem ofasset pricing, this model leads to arbitrage (see Delbaen and Schachermayer (1994)). Barberis et al. (2019)incorporate prospect theory into asset pricing models trying to explain market anomalies, they modifiedexpected utility from the cumulative prospective individual through value function and probability weightingfunction. However, after transformations under the value function and probability weighting function, theunderlying price process is no longer infinitely divisible and for that reason the prospect theory based assetpricing models lead to arbitrages.Within Rational Dynamic Asset Pricing Theory (RDAPT) the most important problem is the characterizationof economically rational consistent models for financial markets (see Duffie (2010)). In RDATP, the centralassumption is that there is no-arbitrage: a market participant should not engage in a contract in whichthe market participant can lose an infinite amount of money in a frictionless market. Regardless of howirrational a representative agent might be, one should not be so irrational as to be subject to an infinite loss,and the assumption of infinitely divisibility is crucial to no-arbitrage option pricing (see Bayraktar et al.(2016)). Brav et al. (2004) mentioned perhaps every equilibrium prediction that assumes the survival of theirrationality may require a (shadow) prediction from some rational model. Indeed, one should incorporatehuman beings’ behavior to the modeling the dynamic of asset prices. However, as Miller (1986) pointedout that for individual investors unlike those institutional investors, there may be numerous nontriviallife-related concerns associated with each trading activity. But the purpose of proposing financial modelsis neither paying too much attention on human behaviors to get lost in millions of details nor completelyignore human behaviors. Rather, focusing on pervasive market forces is the direction of proposing abstractasset pricing model. We are proposing abstract asset pricing models to incorporating human behaviors butnot forgetting our principal concern. No-arbitrage, as our principal concern, will drive any mispricing causedby irrational traders’ bad investment strategy to zero. There is no reason for behavioralists to object tothe no-arbitrage assumption which is as a fundamental notion for both the rational or behavioral financecamps. If this assumption is not satisfied, agents using financial asset pricing formulas allowing for arbitragecould suffer tremendous losses. If a trader applies a behavioral option pricing model such as Shefrin (2008)by being long in the contract, there will be a rational trader who will take the short position and applyarbitrage strategy. Indeed, in Shefrin’s option pricing approach there is no proposed hedging strategy theoption seller (i.e., the short) could use and the reason is that there is no such strategy possible.In this paper, we use Shefrin’s option pricing model as an example, and suggest one approach to adjustthis option pricing formulas for option traders having heterogeneous views on the underlying pricing processso that those formulas are consistent with the RDAPT. Specifically, we impose trading costs (so-calledarb-costs) which will offset the gains from possible arbitrage opportunities in the market. Equilibriumoption pricing models when traders have heterogeneous beliefs have been studied in the RDART literature.Our approach to option pricing in the presence of heterogeneous beliefs can be roughly explained as follows:for the hedger to realize arbitrage profits, the hedger must be able to trade at a high speed, imposingarb-costs on the velocity of trades can offset the gains the hedger accumulates when applying the arbitragetrade. In standard hedging when no arbitrage occurs, the arb-costs are not significant.We have organized the paper as follows. In Section 2 we identify the flaw in Shefrin’s behavioral approachto option pricing. In Section 3 we introduce an arb-cost on the binomial tree of Shefrins behavioral marketmodel and derive the risk-neutral price dynamics as well as the resulting risk-free rate. Our numerical See Steutel and Van Harn (2003) See Brav et al. (2004): individual investors who hold modest amounts of stock directly and who, unlike institutional andother large investors, do not rely heavily on professional portfolio advisers.
Shefrin constructed asset pricing model by defining a sole agent (the representative investor) as therepresentative of society as a whole, capable of aggregating the behavior of large numbers of economicagents. He then extended the asset pricing model that was constructed based on the representative investorwho was not rational in order to include heterogeneous beliefs. Shefrin constructed an option pricing modelby introducing two investors. Both investors agree on the risk-free asset process and volatility of the riskyasset, but disagree on the drift term for the risky asset. Investor 1 believe that the stock price S obeys theprocess dSS = µ dt + σdZ (1)where Z is a Wiener process. Investor 2 believe that the stock price S obeys the process dSS = µ dt + σdZ (2)Shefrin claims the option will be priced according to BlackScholes in this setting. Furthermore, he concludesthat heterogeneity will neither impact option prices nor give rise to volatility smiles.In general, the above claim from Shefrin (2008) is not true. To see that, suppose investor 1 takes thelong position in the European option contract C with:(1) price process C ( t ) = f ( S ( t ) , t ) , t ≥ f ( S ( T ) , T ) = g ( T ), where f ( x, t ) , x > , t ≥ C is given by the It formula: dC ( t ) = df ( S ( t ) , t )= (cid:18) ∂f ( S ( t ) , t ) ∂t + ∂f ( S ( t ) , t ) ∂x µ S ( t ) + 12 ∂ f ( S ( t ) , t ) ∂x σ S ( t ) (cid:19) dt + ∂f ( S ( t ) , t ) ∂x σS ( t ) dZ ( t ) (3)Suppose investor 2 takes the short position in C , and forms a self-financing strategy C ( t ) = f ( S ( t ) , t ) = a ( t ) S ( t ) + b ( t ) β ( t ) , t ≥ β ( t ) = β (0) e rt , t ≥ r . The the dynamic of the replicatingportfolio is given by dC ( t ) = df ( S ( t ) , t )= a ( t ) dS ( t ) + b ( t ) dβ ( t )= ( a ( t ) µ S ( t ) + b ( t ) rβ ( t )) dt + a ( t ) σS ( t ) dZ ( t ) (5)Equating the expression for dC ( t ) leads to a ( t ) = ∂f ( S ( t ) ,t ) ∂x , and ∂f ( S ( t ) , t ) ∂t + ∂f ( S ( t ) , t ) ∂x µ S ( t ) + 12 ∂ f ( S ( t ) , t ) ∂x σ S ( t ) = a ( t ) µ S ( t ) + r ( f ( S ( t ) , t ) − a ( t ) S ( t )) (6)Setting S ( t ) = x , leads to the following partial differential equation (PDE): ∂f ( x, t ) ∂t + ∂f ( x, t ) ∂x ( µ − µ + r ) x − rf ( x, t ) + 12 ∂ f ( x, t ) ∂x σ x = 0 (7)which is the Black-Scholes formula in the homogeneous case µ = µ . For µ (cid:54) = µ , the above PDE will leadto option pricing with arbitrage opportunities. 3 Rational Option Pricing with Transaction Cost
We now start with the description of Shefrins model, see Shefrin (2008), Chapter 8, but from the viewpointof RDAPT. Consider a financial market with two investors sharing an aggregate consumption (AC) ω (0) > t (0) = 0. At any subsequent period t ( k +1) = ( k + 1)∆ t, k = 0 , , . . . , n − , t ( n ) = T < ∞ , n ↑ ∞ ,the aggregate amount available will unfold through a binomial process, growing by u (∆ t ) > d (∆ t ) < (cid:105) ( j ) , j = 1 , p ( j ) (∆ t ) for an upwardmovement of the AC-process and 1 − p ( j ) (∆ t ) for a downturn movement; that is, the discrete AC-dynamicsis given by ω ( j ) (cid:16) t ( k +1) (cid:17) = (cid:26) ω ( j ) (cid:0) t ( k ) (cid:1) u (∆ t ) , w · p · p ( j ) (∆ t ) ω ( j ) (cid:0) t ( k ) (cid:1) d (∆ t ) , w · p · − p ( j ) (∆ t ) (8)Following Shefrin (2005), Chapter 8, suppose trader (cid:105) ( S ) (resp. (cid:105) ( V ) trades a risky asset (stock) M on abinomial lattice with price dynamic S k ∆ t , k ∈ N (0) , with S > V k ∆ t , k ∈ N (0) , with V > . (cid:20) S ( k +1)∆ t V ( k +1)∆ t (cid:21) = (cid:20) S ( k +1)∆ t : up = S k ∆ t (1 + µ ∆ t + σ √ ∆ t ) V ( k +1)∆ t ; up = V k ∆ t (1 + m ∆ t + v √ ∆ t ) (cid:21) w · p · (cid:20) S ( k +1)∆ t : down = S k ∆ t (1 + µ ∆ t − σ √ ∆ t ) V ( k +1)∆ t : down = V k ∆ t (1 + m ∆ t − v √ ∆ t ) (cid:21) w · p · (9) k ∈ N (0) , ∆ t > , µ ∈ R , m ∈ R , σ > , v > . For every fixed T >
0, the the bivariate binomialtree ( S k ∆ t , V k ∆ t ) k ∈ ,...,N ∆ t generates a bivariate polygon process with trajectories in the Prokhorov space C (cid:0) [0 , T ] (cid:1) which converges weakly to the following bivariate geometric Brownian motion ( S t , V t ) t ∈ [0 ,T ]6 : S t = S e (cid:16) µ − σ (cid:17) t + σB ( t ) , V t = V e (cid:16) m − v (cid:17) t + vB ( t ) , t ∈ [0 , T ] (10)where B ( t ) , t ≥
0, is a Brownian motion generating a stochastic basis (Ω , F , F = ( F t , t ≥ , P ).As we pointed out in Section 2, the above model is not free of arbitrage. To rectify this model, we willintroduce a new model with transaction costs to eliminate the possible gains from arbitrage trading in theabove model.To this end, let us consider a perpetual European derivative contract G . Following Shefrin’s model, G has price process G k ∆ t = G ( S k ∆ t , V k ∆ t ) , k ∈ N (0) . Next, we shall derive the trading dynamics of arepresentative investor N who is observing historical trading activities of (cid:105) ( S ) and (cid:105) ( V ) . We assume N (as arepresentative agent) has taken simultaneously both the long and the short position in G . N trades S t (resp. V ( t )) as (cid:105) ( S ) (resp. (cid:105) ( V ) ) would do. N forms a self-financing strategy ( a k ∆ t , b k ∆ t ) , k ∈ N that generating aself-financing portfolio P ( t ) = a ( t ) S ( t ) + b ( t ) V ( t ). Thus, G (cid:16) t ( k ) (cid:17) = g (cid:16) s (cid:16) t ( k ) (cid:17) , V (cid:16) t ( k ) (cid:17)(cid:17) = P (cid:16) t ( k ) (cid:17) = a (cid:16) t ( k ) (cid:17) S (cid:16) t ( k ) (cid:17) + b (cid:16) t ( k ) (cid:17) V (cid:16) t ( k ) (cid:17) (11)when N trade as (cid:105) ( S ) (resp. (cid:105) ( V ) ), at any time interval (cid:2) t ( k ) , t ( k +1) (cid:1) the trade is subject to transaction cost (cid:32) S (cid:0) t ( k +1) (cid:1) S (cid:0) t ( k ) (cid:1) (cid:33) ρ ( S ) resp · (cid:32) V (cid:0) t ( k +1) (cid:1) v (cid:0) t ( k ) (cid:1) (cid:33) ρ ( V ) (12) This binomial tree (11) was introduced in Kim at al (2016) (see also Jarrow and Rudd (2008)) as an extension of theclassical CRR-model (Cox et al. (1979)). We use this more general binomial pricing tree, because we require the bivariatepricing tree to be driven by one risk factor, and with that requirement, CRR-model is not appropriate. Here, and in what follows, all terms of order o (∆ t ) are assumed to be 0. The proof is similar to that in Davydov and Rotar (2008)Kim et al. (2016), Theorem 2, and thus is omitted. Transaction costs include commissions, execution costs and opportunity costs. The investment costs have a fixed componentand a variable componentCollins and Fabozzi (1991). ρ ( S ) = C µσ (resp. ρ ( V ) = C µσ ) and C is an absolute constant.Next, N choose (cid:0) a (cid:0) t ( k ) (cid:1) , b (cid:0) t ( k ) (cid:1)(cid:1) , so that − g (cid:16) S (cid:16) t ( k +1) (cid:17) , V (cid:16) t ( k +1) (cid:17)(cid:17) + a (cid:16) t ( k ) (cid:17) S (cid:16) t ( k +1) (cid:17) (cid:32) S (cid:0) t ( k +1) (cid:1) s (cid:0) t ( k ) (cid:1) (cid:33) ρ ( S ) + b (cid:16) t ( k ) (cid:17) V (cid:16) t ( k +1) (cid:17) (cid:32) V (cid:0) t ( k +1) (cid:1) v (cid:0) t ( k + k ) (cid:1) (cid:33) ρ ( v ) = 0(13)That is, at t ( k +1) the hedged portfolio plus the short position in G has value zero for all states of the world,and thus its value at t ( k ) should also be zero. Thus, g (cid:16) S (cid:16) t ( k ) (cid:17) , V (cid:16) t ( k ) (cid:17)(cid:17) = P (cid:16) t ( k ) (cid:17) = a (cid:16) t ( k ) (cid:17) S (cid:16) t ( k ) (cid:17) + b (cid:16) t ( k ) (cid:17) V (cid:16) t ( k ) (cid:17) (14)With ρ ( S ) = C µσ (resp. ρ ( V ) = C µσ ), we obtain the binomial option price dynamics: g (cid:16) s (cid:16) t ( k ) (cid:17) , V (cid:16) t ( k ) (cid:17)(cid:17) == Q (∆ t ) g (cid:16) S (cid:16) t ( k +1 ,up ) (cid:17) , V (cid:16) t ( k +1 ,up ) (cid:17)(cid:17) + (cid:16) − Q (∆ t ) (cid:17) g (cid:16) S (cid:16) t ( k +1 ,down ) (cid:17) , V (cid:16) t ( k +1 ,down ) (cid:17)(cid:17) (15)and the risk-neutral probabilities ( N ’s state-price probabilities) are Q (∆ t ) and 1 − Q (∆ t ) , where Q (∆ t ) = 12 − µ (cid:0) C µσ (cid:1) (cid:0) C σ (cid:1) − m (cid:0) C mv (cid:1) (cid:0) C v (cid:1) σ − v + C ( µ − m )) √ ∆ t (16)Note that even if σ = v (which is an arbitrage pricing model if (cid:105) ( S ) and (cid:105) ( V ) trades without arb-cost), as soonas µ (cid:54) = m , risk-neutral probabilities exist, and thus the introduction of transaction costs 1 + ρ ( S ) ln S ( t ( k +1) ) S ( t ( k ) )(resp. 1 + ρ ( V ) ln V ( t ( k +1) ) V ( t ( k ) ) ) has offset the arbitrage gains. From Kim et al. (2016) Section 3.2, and Black(1972), Q (∆ t ) should have the representation: Q (∆ t ) = 12 − µ ( ∗ ) − r ( ∗ ) σ ( ∗ ) √ ∆ t = 12 − m ( ∗ ) − r ( ∗ ) v ( ∗ ) √ ∆ t (17)where, r ( ∗ ) := µ ( ∗ ) v ( ∗ ) − m ( ∗ ) σ ( ∗ ) v ( ∗ ) − σ ( ∗ ) (18) µ ( ∗ ) := µ (cid:16) C µσ (cid:17) (cid:16) C σ (cid:17) (19) m ( ∗ ) := m (cid:16) C mv (cid:17) (1 + C v ) (20) σ ( ∗ ) := σ + C µ (21) v ( ∗ ) := v + C m (22) r ( ∗ ) is N ’s risk-neutral rate, and µ ( ∗ ) , m ( ∗ ) , σ ( ∗ ) , and v ( ∗ ) are the adjusted (for arb-cost) drift and volatilityparameters. Now the price process ( S t , V t ) t ∈ [0 ,T ] as seen by N in the risk-neutral world (Ω , F , F = ( F t , t ≥ , Q ), Q ∼ P has a dynamic given by S t = S e (cid:16) r ( ∗ ) − σ ( ∗ )2 (cid:17) t + σ ( ∗ ) B ( ∗ ) ( t ) , V t = V e (cid:16) r ( ∗ ) − v ( ∗ )2 (cid:17) t + v ( ∗ ) B ( ∗ ) ( t ) , t ∈ [0 , T ] (23) B ( ∗ ) ( t ) , t ≥ Q , and an arithmetic Brownian motion on P with B ( ∗ ) ( t ) = B ( t )+ θ ( ∗ ) t .The parameter θ ( ∗ ) = µ ( ∗ ) − r ( ∗ ) σ ( ∗ ) = m ( ∗ ) − r ( ∗ ) v ( ∗ ) is the market price of risk in N ’s market model with arb-costs.Note that the risk-neutral probability without arb-costs is Q (∆ t ; no arb − cost ) := 12 − µ − m σ − v ) √ ∆ t (24)5hus, Q (∆ t ; no arb − cost ) is the risk-neutral probability in Black’s (1972) model, and Q (∆ t ; no trans cost ) := 12 − µ − rσ √ ∆ t (25)where µ − rσ = m − rv = µ − mσ − v and r = µv − mσv − σ . N ’s model can be viewed as an extension of the Black (1972)model when our special type of transaction costs is introduced. In this section, we apply the method introduced in Section 3 to a cross-sectional data analysis. Followingthe set-up in Shefrin’s option pricing model: two representative spot traders have different views on onestock price process. We use the SPDR S&P 500 ETF (SPY) and Vanguard S&P 500 ETF (IVV) optionprices as a data analysis example since both SPY and IVV track the Standard & Poor 500 index which hasbeen considered as a benchmark for the U.S. equity. We use the historical call option price for the SPYand IVV to calibrate the implied risk-free interest rate ( r ∗ ) and the implied volatility ( σ ∗ ) to calculate theoptimized absolute constant C as in equation 14. Then, the value for r ∗ across time was calculated by usinghistorical return data for SPY and IVV. Next, having the value of r ∗ over time, we plot the option pricingimplied volatility surfaces for SPY and IVV. Lastly, arbitrage cost ( C ) surfaces and the spread arbitrage costcoefficient surfaces for SPY and IVV were plotted in order to compare the liquidity spread surfaces. The data used in this study are the daily stock price of SPDY S&P 500 (SPY) and iShare Core S&P 500(IVV) obtained from Yahoo Finance from 09/09/2010 to 10/22/2019. The Treasury yield 10-years (TNX)is used as the riskless asset and the price data is obtained from Yahoo Finance. The European call optionprices of SPY on 10/22/2019 with different time-to-maturity and strike prices were obtained from ChicagoBoard Options Exchange (CBOE) . C As mentioned in Section 3, we use equation 14 to calibrate C . The price process of SPY in risk-neutralworld is S t = S e (cid:18) r ( · ) − σ ( · )22 (cid:19) t + σ ( · ) B ( · ) ( t ) (26)where B ( ∗ ) ( t ) , t ≥ Q , and an arithmetic Brownian motion on P with B ( ∗ ) ( t ) = B ( t ) + θ ( ∗ ) t .We use call option prices for the SPY and the price data is obtained from the CBOE on 10/22/2019 withdifferent expiration dates and strike prices. The SPY stock price is used to calibrate the implied risk-freeinterest rate ( r ∗ ) and implied volatility ( σ ∗ ). The expiration date for the call option on SPY varies from10/25/2019 to 01/21/2022, the prices per contract varies from $0 .
005 to $274 .
105 and the strike price variesfrom $25 to $430 among 3,330 different call option contracts.The SPY stock price as the underlying for the call option was $299 .
03 quoted on CBOE on 10/22/2019.The sample mean and sample standard deviation for the SPY prices from 09/09/2010 to 10/22/2019 asthe underlying on the call option were used as the estimated values for µ and σ in equation 14. By usingBlack-Scholes-Merton model, the optimized implied risk-free interest rate ( r ∗ ) and implied volatility ( σ ∗ ) As mentioned in Section 3, the parameter θ ( ∗ ) = µ ( ∗ ) − r ( ∗ ) σ ( ∗ ) = m ( ∗ ) − r ( ∗ ) v ( ∗ ) is the market price of risk in N ’s market modelwith arb-costs. r ( ∗ ) with window size per yearfor the SPY call option price were calibrated and we obtained r ( ∗ ) = 0 . σ ( ∗ ) = 0 . σ = 0 . µ = 0 . and the following equation (3.14): σ ( ∗ ) := σ + C µ (27)We obtained the optimized value of C as a constant is 299.4773 and this value was used to calibrate therisk-free rate over time in the next subsection. r ( ∗ ) The risk-free rate r ( ∗ ) is calibrated by using the historical stock return data of SPY and IVV as theunderlying from 01/03/2011 to 10/22/2019. We calibrated the values of µ ( ∗ ) , m ( ∗ ) , σ ( ∗ ) , and v ( ∗ ) by applyingequations (12), (13), and (14), respectively , We use the rolling method to find daily r ( ∗ ) with a fixed windowsize of 252 days. The smoothing plot of the risk-free rate is shown in Figure 1. As mentioned in Section 3,this is representative investor’s ( N ) risk-neutral rate. By having the value of the implied risk-free rate r ( ∗ ) across time, we use option price data to plot theBlack-Scholes implied volatility surface for various strike prices and maturities at a point in time for SPYand IVV, respectively, as shown in Figure 2a and Figure 2b. The implied volatility surfaces flatten out asthe time to maturity increases for both SPY and IVV.In the plots of the arbitrage coefficient C (ACS) surfaces as shown in Figure 3a and Figure 3b, we usedthe mid-point of the bid and ask prices for the SPY and IVV.We next use the spread prices of options in the construction of the two arbitrage cost surfaces as shown sample standard deviation and sample mean of daily returns of SPY from 09/09/2010 to 10/22/2019. C as a constant is 299.4773 was applied in each equation. a) The plot of implied volatility for SPY(b) The plot of implied volatility for IVV Figure 2: The term-structure of implied volatility of SPY and IVVin Figure 4a and Figure 4b in order to compare the two arbitrage cost surfaces of SPY and IVV. Becauseof the competitive nature of the market, equilibrium bid-ask spreads should reflect the expected costs ofproviding liquidity services to the market, and the differences in bid-ask spreads can be directly related todifferences in the costs faced by investors (representative investors) across options.In order to visualize any differences of the spread arbitrage cost surfaces, we plot the two surfaces asshown in Figure 5.As we can observe, the size of the spread arbitrage cost coefficient of the SPY and IVV options in Figure 5are highly overlapped and flatten out as the time to maturity increases, which indicate the two market playersagreed on the transaction costs when an option contract is put in place. The differs in size of the bid-askspread arbitrage cost surfaces from SPY to IVV have been observed in higher moneyness with a relativeshort time-to-maturity of the option contracts, this causes is mainly been considered as the difference inliquidity of each option rather than the difference of the volatility surfaces.Next, the combined arbitrage cost surface for spread prices was plotted by finding the optimized arbitragecost coefficient value C which minimized the price differences of options bid and ask prices and the calibratedBlack-Scholes bid and ask prices as shown in Figure 6. This combined ACS is determined by the marketrepresentative investor ( N , as mentioned in Section 3), whose views are heavily influenced by the marketliquidity. Spread price is equal to ask price minus bid prices. a) The plot of arbitrage cost surface for SPY(b) The plot of arbitrage cost surface for IVV Figure 3: The arbitrage cost surfaces of the call options of SPY and IVV9 a) The plot of spread arbitrage cost surface for SPY(b) The plot of spread arbitrage cost surface for IVV
Figure 4: The spread arbitrage cost surfaces of the call options of SPY and IVVFigure 5: Spread arbitrage cost surface of call options of SPY and IVV10igure 6: The plot of combined arbitrage cost surface of call options of SPY and IVV
In this paper, we correct the statements made by Shefrin (2005) as they pertain to behavioral optionpricing. We pointed out that option pricing formulas in Shefrin (2005) has a flaw from the perspectiveof rational dynamic asset pricing theory. In order to correct Shefrins (2005) behavioral approach to optionpricing, we introduce arbitrage-costs so that the generated arbitrage gains would be eliminated by arbitrage-costs.We derive the risk-free rate in this setting that generalizes the Black (1972) approach to rational dynamicmarket with two risky assets driven by the same Brownian motion. In our numerical example, we applied theproposed model to SPY call option prices to calibrate the optimized arbitrage cost coefficient and plottedimplied the risk-free rate over time. By comparing the spread bid-ask arbitrage cost surfaces of SPY and IVVcall options, we conclude the arbitrage cost coefficient C is one and the same for both market participants(SPY and IVV). Lastly, we plot the combined ACS of the market representative investor N . We believe theeffect from belief bias of option traders should be canceled out when the average belief is unbiased, and anypotential arbitrage gains shall be eliminated by transaction cost.11 eferences Barberis, N., Jin, L. J., and Wang, B. (2019). Prospect theory and stock market anomalies.
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