Option Pricing in Markets with Informed Traders
Yuan Hu, Abootaleb Shirvani, Stoyan Stoyanov, Young Shin Kim, Frank J. Fabozzi, Svetlozar T. Rachev
aa r X i v : . [ q -f i n . M F ] A ug OPTION PRICING IN MARKETS WITH INFORMEDTRADERS
YUAN HU a , ABOOTALEB SHIRVANI b , STOYAN STOYANOV c , YOUNG SHIN KIM d ,FRANK J. FABOZZI e , and SVETLOZAR T. RACHEV fa Texas Tech University, [email protected] b Texas Tech University, [email protected] c Charles Schwab Corporation, [email protected] d Stony Brook University, [email protected] e EDHEC Business School, [email protected] f Texas Tech University, [email protected]
August 13, 2020
Abstract
The objective of this paper is to introduce the theory of option pricing for markets with in-formed traders within the framework of dynamic asset pricing theory. We introduce new models for optionpricing for informed traders in complete markets where we consider traders with information on the stockprice direction and stock return mean. The Black-Scholes-Merton option pricing theory is extended formarkets with informed traders, where price processes are following continuous-diffusions. By doing so, thediscontinuity puzzle in option pricing is resolved. Using market option data, we estimate the implied surfaceof the probability for a stock upturn, the implied mean stock return surface, and implied trader informationintensity surface.
Keywords
Theory of option pricing; markets with informed traders; European call option prices forinform traders.
The theory of option pricing (TOP), developed in the seminal works of Black & Scholes (1973) andMerton (1973), provides the theory of finance with the fundamentals to understand, model and apply theprocesses for pricing contingent claims. Several works provide a comprehensive exposition of TOP such asCochrane (2001), Duffie (2001), Skiadas (2009), Campbell (2000), C¸ elik (2012), and Munk (2013). Althoughit is impossible to overlook TOP’s enormous influence on the theory of finance and its applications, there aresome limitations of the original formulation of TOP due to several restrictive premises of the theory that areinconsistent with the findings of empirical studies on asset pricing processes. On the empirical side, it hasbeen found that there is long-range dependence in asset price time series, volatility clustering of asset returns,skewness of the distribution of asset returns, heavy tails of the distribution of asset returns, and multivariatetail dependencies in the vector of asset returns . Asset returns exhibiting these attributes are inconsistentwith the assumptions of TOP. On the theoretical side, the following assumptions are questionable: (1) marketparticipants have symmetric information , (2) prices are unpredictable , (3) asset prices are not driven byfractional processes exhibiting long-range dependence , and (4) markets do not exhibit chaotic, or irrational See Lo & MacKinley (1988), Rachev & Mittnik (2000), Schoutens (2003), Cont (2001), Cont & Tankov (2004), and Rachevet al. (2011). See Brunnermeier (2001) and Kelly & Ljungqvist (2012). See Campbell & Yogo (2005), Boucher (2006), Ang & Bekaert (2007), and Caporin et al. (2013). See Lo (1991), Campbell et al. (1997), Andersson (1998), Diebold & Inoue (2001), Nielsen (2010), Caporale & Gil-Alana(2014), Cheridito (2003), Cont (2005), Comte & Renault (1998), and Rostek (2009). . Studies have questioned these assumptions.There is a vast literature on asset pricing with asymmetric information, most notably the models proposedby both Kyle (1985) and Back (1992) . Both models assume a market with a continuous-time risky asset andasymmetric information. In the Kyle model there are three financial agents: the market maker, an insidertrader (who knows a payoff which will be revealed at a pre-specified future time), and an uninformed (noisy)trader. The market maker has to define a pricing rule in such a way that an equilibrium exists between thetraders. Back (1993) extended the model to continuous time. A second line of research focuses on the studyof markets with asymmetric information based on an enlargement of the filtration and the change of theprobability measure, the study by Aase et al. (2010) being one example.Our paper is close in spirit to Horst & Naujokat (2011). In our paper, traders operate in an imperfectmarket. The buyer and seller have different market information when making their option trades, but we dealwith perfectly liquid markets. In Horst & Naujokat (2011), the trades are executed in an illiquid market andoption traders “manipulate” the option portfolio value by impacting the slippage in trading (hedging) theunderlying. The common feature in both papers is that the option sellers use their hedged portfolio eitheras being more informed than the market (as in our paper), or “to increase their portfolio value by usingtheir impact on the dynamics of the underlying” in the paper by Horst & Naujokat (2011). Our approach isbased on dynamic asset pricing theory, while Horst and Naujpkat employ an equilibrium pricing approach.We derive option pricing formulas when some group of traders are in the possession of additional informa-tion about future asset prices. The information available to traders is multifaceted and any general definitionwill be restrictive in view of the traders’ particular trading activities. If traders have information on thestock price direction, we find that the fair option price follows the Black-Scholes-Merton formula with anadditional term which can be interpreted as a continuous dividend stream. If traders have better informationon the stock mean return, then the fair option price differs from the Black-Scholes-Merton formula only ifcontinuous-time trading is not allowed.The paper is organized as follows. In Section 2 we analyze the discontinuity puzzle in option pricing.Applying the option pricing model developed by Kim et al. (2019), we outline the resolution of the puzzle,and motivate our approach to the theory of option pricing for informed traders. We then estimate theimplied surfaces of the probability for upturn and implied mean return on market data. In Section 3, wederive option pricing when the hedger has information about the stock’s price direction. We estimate theimplied surface of a trader who possesses information on the probability for a stock’s upturn. In Section 4,we extend the results in Section 3, deriving an option pricing model when the hedger has information aboutthe stock mean return. In Section 5, the approach in Section 3 is generalized to cover markets with stockprices driven by continuous-diffusion processes. Section 6 concludes the paper. Our interest on the topic of option pricing in the presence of informed traders started with an attemptto remove an unnatural discontinuity of the derivative price valuation . Consider the Black-Scholes-Merton market ( S , B , C ) of risky asset (stock) S , riskless asset B , and Euro-pean Contingent Claim (ECC) C . The stock price dynamic follows a geometric Brownian motion (GBM) S t = S ( µ,σ ) t = S e ( µ − σ ) t + σB t , t ≥ , S > , µ > , σ > . (1) See Hsieh (1990), Jovanovic & Schinckus (2013), Rubinstein (2001), Shiller (2003), and Daniel & Titman (1999). See also Back & Baruch (2004), Back & Pedersen (1998), Caldentey & Stacchetti (2010), Cho (2003), and Collin-Dufresne& Fos (2015). See Kim et al. (2016, 2019), where the problem of discontinuity in option pricing was first discussed. We also refer to B as a riskless bank account, or equivalently as a riskless bond. P , determined by a stochastic basis (Ω , F = {F t } t ≥ , P ) with filtration F ,generated by the Brownian motion (BM) B t , t ≥
0. The bond price is given by β t = β e rt , t ≥ , β > , r ∈ (0 , µ ) . (2)The ECC with underlying asset S has terminal (expiration) time T >
0, and terminal payoff g ( S ( µ,σ ) T ).Consider, as an example, a call option with strike price K >
0. Then g ( S ( µ,σ ) T ) = max( S ( µ,σ ) T − K, C ( K ) t = C ( S ( µ,σ ) t , K, T − t, r, σ ) = C ( S ( r,σ ) t , K, T − t, r, σ ) is independent of the (instantaneous) stock mean return µ . If µ ↑ ∞ , the hedger(the trader taking the short position in the option contract), who can trade continuously in time with notransaction costs involved , will be indifferent to the large values of µ . Indeed, continuous-time trading withno transaction costs is a pure fiction in any real trading. Obviously, if µ = ∞ , C ( S ( ∞ ,σ ) t , K, T − t, r, σ ) = ∞ .However, by the Black-Scholes-Merton Theorem , for every fixed t ∈ [0 , T ), sup µ ∈ R C ( S ( µ,σ ) t , K, T − t, r, σ ) = C ( S ( r,σ ) t , K, T − t, r, σ ) < ∞ , and thus, we observe an unnatural discontinuity of the price of the option when µ ↑ ∞ . Similarly, if µ = −∞ , C ( S ( −∞ ,σ ) t , K, T − t, r, σ ) = 0. However according to the Black-Scholes-Mertonframework, for every fixed t ∈ [0 , T ) , inf µ ∈ R C ( S ( µ,σ ) t , K, T − t, r, σ ) = C ( S ( r,σ ) t , K, T − t, r, σ ) >
0, and thus,we again observe an unnatural discontinuity of the price of the derivative when µ ↓ −∞ .Therefore, we conclude that no real option trader will disregard the information about the mean stockreturn value µ in trading the option C . That information should be embedded in option price C ( K ) t = C ( S ( µ,σ ) t , K, T − t, r, σ ). Consider next the seminal Cox-Ross-Rubinstein (CRR) binomial option pricing model . The priceprocess given by (1) generates a triangular series of ∆ t log-returns, r k ∆ t = log S ( µ,σ ) k ∆ t − log S ( µ,σ )( k − t , k ∈ N n = { , ..., n } , n ∆ t = T . The returns are independent and identically distributed ( i.i.d. ) Gaussian randomvariables with mean µ k ∆ t = ( µ − σ )∆ t , and variance σ k ∆ t = σ ∆ t , and denoted as r k ∆ t d = N ( µ k ∆ t , σ k ∆ t ). Applying the CRR-model and the Donsker-Prokhorov Invariance Principle (DPIP) , we approximate r k ∆ t by r k ∆ t ; n = U ( CRR )∆ t ζ ( CRR ) k,n + D ( CRR )∆ t (1 − ζ ( CRR ) k,n ) , k ∈ N n , (3)where U ( CRR )∆ t = σ √ ∆ t, D ( CRR )∆ t = − σ √ ∆ t (4)for every fixed n ∈ N = { , , ... } , and { ζ ( CRR ) k,n , k ∈ N n } are i.i.d. Bernoulli random variables, ζ ( CRR ) k,n d = Ber ( p CRR ∆ t ) , P ( ζ ( CRR ) k,n = 1) = 1 − P ( ζ ( CRR ) k,n = 0) = p ( CRR )∆ t ∈ (0 ,
1) with success probability p ( CRR )∆ t = e µ ∆ t − e − σ √ ∆ t e σ √ ∆ t − e − σ √ ∆ t . (5)A widely used alternative to the CRR binomial pricing tree is the Jarrow-Rudd (JR) binomial model ,where in (3), (4) and (5), the triplet ( U ( CRR )∆ t , D ( CRR )∆ t , p ( CRR )∆ t ) is replaced by triplet ( U ( JR )∆ t , D ( JR )∆ t , p ( JR )∆ t ), U ( JR )∆ t = ( µ − σ )∆ t + σ √ ∆ t, D ( JR )∆ t = ( µ − σ )∆ t − σ √ ∆ t, p ( JR )∆ t = 12 (6) There is a vast literature on option price with transaction costs. Davis et al. (1993, 471) summarize the problems associatedwith continuous-time trading with no transaction costs. See Black & Scholes (1973) and Merton (1973). See Cox, Ross, & Rubinstein (1979) and Chapters 12 and 20 in Hull (2012) with µ = r . Here and in what follows, d = stands for “equal in distribution”, or “equal in probability law”. See Donsker (1951), Prokhorov (1956), Section 14 in Billingsley (1999), Chapter IX in Gikhman & Skorokhod (1969),Section 5.3.3 in Skorokhod (2005), and Davydov & Rotar (2008). Under (5), the risk -neutral probability in the CRR-binomial pricing model is q ( CRR )∆ t = e r ∆ t − e − σ √ ∆ t e σ √ ∆ t − e − σ √ ∆ t = + r − σ σ √ ∆ t with o (∆ t ) = 0, see Kim et al. (2016, 2019). See Jarrow & Rudd (1983, p. 179-190) and Section 20.4 in Hull (2012) with µ = r . . The CRR- and JR-binomial pricing tree constructions have manyadvantages, unfortunately, they have one common disadvantage – an option-price-discontinuity. To illustratethat, consider a one-period binomial model, where1. S > t = 0) one-share stock price,2. f is the unknown current option price,3. S T = ( S u w.p. p S d w.p. − p , where p ∈ (0 , u > d >
0, satisfying the no-arbitrage condition u > e rT > d ,4. the option payoff at maturity is f T = ( f ( u ) T = g ( S u ) w.p. p f ( d ) T = g ( S d ) w.p. − p (7)for some option payoff function g ( x ) , x > p ∈ (0 , t = 0 is given by f ( p ) = f ( ) = e − rT [ q f ( u ) T + (1 − q ) f ( d ) T ]with q = e rT − du − d , regardless of how close p is to 0 or 1. However, for p = 0 and p = 1, the option valuesare, respectively, f (0) = e − rT f ( d ) T and f (1) = e − rT f ( u ) T . The discontinuity gaps at p = 0 and p = 1, arerespectively, ( f (0) − lim p ↓ f ( p ) = e − rT q ( f ( d ) T − f ( u ) T ) = 0 ,f (1) − lim p ↑ f ( p ) = e − rT (1 − q )( f ( u ) T − f ( d ) T ) = 0 . (8)In contrast to the discontinuity gaps ( C ( S ( −∞ ,σ ) t , K, T − t, r, σ ) − inf µ ∈ R C ( S ( µ,σ ) t , K, T − t, r, σ ) < C ( S ( −∞ ,σ ) t , K, T − t, r, σ ) − sup µ ∈ R C ( S ( µ,σ ) t , K, T − t, r, σ ) = ∞ (9)reported in Section 2.1, where (9) could be explained by the (presumed) hedger’s ability to trade continuouslyin time with no transaction cost, in (8), the discontinuity gaps are present in one-period binomial pricing,and that makes the issue of option-price-discontinuity even more disturbing.The main reason for the discontinuity phenomenon ( the discontinuity puzzle in option pricing ) in (8)and (9) is that a trader ℵ , taking a short position in the option contract, is applying the CRR modeldisregarding any information about the mean return µ and probability for stock upturn p ∈ (0 , p = 1, the trader becomes fully aware that the stock price will be up; that is, this trader becomes a trader with complete information about the stock price direction. This jump from a noisy trader ℵ to afully informed trader ℵ ∞ seems unnatural.To resolve this issue, in this paper we will assume that the trader, called ℵ , knows, at time t = k ∆ t, k =0 , ..., n − , n ∆ t = T , with certain probability p ℵ ∆ t ∈ (0 , k ∆ t, ( k +1)∆ t ],or has information about the mean µ . We then have that ℵ is an informed trader if p ℵ ∆ t > , a misinformedtrader if p ℵ ∆ t < , and a noisy trader if p ℵ ∆ t = . To illustrate our approach in this paper, assume that ℵ isan informed trader who knows (at t = 0) with probability p ℵ ∈ ( ,
1) of the stock price direction at t = T .The stock price at T is given by S T = ( S u w.p. p S d w.p. − p , p ∈ (0 , . Under (6), the risk-neutral probability in the JR-binomial pricing model is q ( JR )∆ t = − θ √ ∆ t with o (∆ t ) = 0, where θ = µ − rσ is the market price of risk. If the risk-neutral probability q ( JR )∆ t = , is as in the original JR-model, then thecorresponding natural-world-probability p ( JR )∆ t can be either p ( JR )∆ t = − θ √ ∆ t , or p ( JR )∆ t = + θ √ ∆ t , see Kim et at (2016,2019). See, for example, Hull (2012, p.256). We will designate this trader as a noise trader ℵ . We will designate this trader, as a fully informed trader ℵ ∞ . chooses u and d , so that the stock return R T = S T S − E ( R T ) = µ T T , and variance V ar ( R T ) = σ T T, σ T >
0. The two moment conditions lead to u = 1 + µ T T + σ T q − p p T and d = 1 + µ T T − σ T q p − p T .Next, consider the option on the stock with price f at t = 0, and payoff at maturity t = T given by (7). Theself-financing portfolio comprised of a stock and bond , replicating the option value is P = a S + b = f .Then, f T = a S T + b (1 + r T T ). By the risk-neutrality, a S u + b (1 + r T T ) − f ( u ) T = a S d + b (1 + r T T ) − f ( d ) T = 0. This leads to a = f ( u ) T − f ( d ) T S ( u − d ) and b = r T T f ( d ) T u − f ( u ) T du − d . The option price at t = 0 is given by f = r T T ( q f ( u ) T + (1 − q ) f ( d ) T ), where q = p − θ T p p (1 − p ) T is the risk-neutral probability, and θ T = µ T − r T σ T is the market price of risk.Suppose ℵ takes a short position in the option contract with terminal payoff (7). If, at t = 0, ℵ believesthat the stock will move “upward”, he enters an N ℵ long forward contract. If, at t = 0, ℵ believes that thestock will move “downward”, he enters an N ℵ short forward contract. The probability of ℵ being correctin his guess on the stock price direction is p ℵ ∈ ( , ℵ replicates his short position in the option with the price process: S ℵ = S and S ℵ T = S u + N ℵ ( S u − S (1 + r T T )) w.p. p p ℵ ,S d + N ℵ ( S (1 + r T T ) − S d ) w.p. (1 − p ) p ℵ ,S u + N ℵ ( S (1 + r T T ) − S u ) w.p. p (1 − p ℵ ) ,S d + N ℵ ( S d − S (1 + r T T )) w.p. (1 − p )(1 − p ℵ ) . Then the mean and the variance of the stock return R ℵ T = S ℵ T S ℵ − E ( R ℵ T ) = µ T T + σ T √ T N ℵ (2 p ℵ − θ T √ T (2 p −
1) + 2 p p (1 − p )) ,V ar ( R ℵ T ) = σ T T [1 + N ℵ + N ℵ θ T T + 2 N ℵ (2 p ℵ − θ T p p (1 − p ) T + 1 − p )] − σ T T [ N ℵ (2 p ℵ − ( θ T √ T (2 p −
1) + 2 p p (1 − p )) ] . To simplify the exposition in this example, we set T = ∆ t , with o (∆ t ) = 0. Then, µ ∆ t = µ , σ ∆ t = σ , and r ∆ t = r . Assume that p ℵ = p ℵ ∆ t = (1 + ψ ℵ √ ∆ t ) for some ψ ℵ > Then E ( R ℵ ∆ t ) = ( µ +2 N ℵ σ p p (1 − p ) ψ ℵ )∆ t and V ar ( R ℵ ∆ t ) = σ (1 + N ℵ )∆ t . ℵ determines the optimal N ℵ = N ( ℵ ; opt ) as theone that maximizes the instantaneous market price of risk Θ( R ℵ ∆ t ) = E ( R ℵ ∆ t ) − r ∆ t √ V ar ( R ℵ ∆ t )∆ t = θ +2 N ℵ √ p (1 − p ) ψ ℵ √ N ℵ , θ = µ − rσ >
0. Choosing N ℵ = N ( ℵ ; opt ) = ψ ℵ √ p (1 − p ) θ ∆ t leads to Θ( R ℵ ∆ t ) = Θ ( opt ) ( R ℵ ∆ t ) = p θ + 4 ψ ℵ p (1 − p ).Furthermore, the optimal mean and variance of the return R ℵ ∆ t are E ( R ℵ ∆ t ) = µ ℵ ∆ t and V ar ( R ℵ ∆ t ) = σ ℵ ∆ t ,where µ ℵ = µ +4 σp (1 − p ) ψ ℵ θ and σ ℵ = σ q p (1 − p ) ψ ℵ θ . Next, ℵ hedges the stock price movements,upward and downward, using stock price process S ( ℵ ; opt )∆ t = ( S u ℵ ∆ t w.p. p S d ℵ ∆ t w.p. − p , where u ℵ ∆ t = 1 + µ ℵ ∆ t + σ ℵ q − p p ∆ t and d ℵ ∆ t = 1 + µ ℵ ∆ t − σ ℵ q p − p ∆ t . For ℵ , the option price is now f ℵ = r ∆ t ( q ℵ f ( u )∆ t + (1 − q ℵ ) f ( d )∆ t ), where q ℵ = p − θ ℵ p p (1 − p )∆ t , and θ ℵ = µ ℵ − rσ ℵ = p θ + 4 p (1 − p ) ψ ℵ .This results in an option price when the underlying stock is paying dividend D ℵ y > ℵ receives the dividendyield D ℵ y making use of his information about the stock’s price movement. The yield D ℵ y is determined by θ ℵ = µ ℵ − rσ ℵ = µ + D ℵ y − rσ , and is equal to D ℵ y = σ ( p θ + 4 p (1 − p ) ψ ℵ − θ ). If ℵ is a misinformed trader, We assume µ T > r T >
0, where r T > , T ]. The no-arbitrage condition requires u > r T T > d . Without loss of generality, we assume β = 1 in (2). ψ ℵ is ℵ ’s stock price direction information intensity . ℵ does not hedge the risk of his bet on the stock price direction being wrong. He hedges only the risk of stock’s upward ordownward movements.
5e does just the opposite of an informed trader, and what will be a profit for the informed trader will be aloss for the misinformed trader. Thus, in general, if p ℵ = p ℵ ∆ t = (1 + ψ ℵ √ ∆ t ) for some ψ ℵ ∈ R , the yield D ℵ y ∈ R , is given by D ℵ y = sign( ψ ℵ ) σ ( p θ + 4 p (1 − p ) ψ ℵ − θ ), wheresign( ψ ℵ ) = , if ψ ℵ > , if ψ ℵ = 0 − , if ψ ℵ < . We will elaborate on this approach to option pricing for informed traders in Sections 3, 4 and 5.
In this section we provide a summary of the Kim-Stoyanov-Rachev-Fabozzi (KSRF) binomial optionpricing (Kim et al., 2016 and 2019) which will be used in this paper as a basic model for discrete assetpricing.Consider again, a market of three assets: risky asset (stock) S , riskless asset (riskless bank account, risklessbond) B , and a derivative (option) C . In continuous time, the stock price dynamics S t = S ( µ,σ ) t , t ∈ [0 , T ]is given by (1). The bond price is given by (2), and the option contract C has continuous price process f t = f ( S t , t ) , t ∈ [0 , T ), and terminal payoff, f T = g ( S T ), where the real-valued function f ( x, t ) , x > , t ∈ [0 , T ) is sufficient smooth. The log-returns r k ∆ t = log S ( µ,σ ) k ∆ t − log S ( µ,σ )( k − t , k ∈ N n = { , ..., n } , n ∆ t = T are i.i.d. Gaussian random variables r k ∆ t d = N (( µ − σ )∆ t, σ ∆ t ). Following CRR binomial pricing model’sconstruction, KSRF introduce their binomial pricing tree. Consider the discrete filtration F ( n ) = {F k ; n = σ ( ζ ( p ∆ t )1 ,n , ..., ζ ( p ∆ t ) k,n ) , k ∈ N n , F n = { ∅ , Ω }} , where { ζ ( p ∆ t ) k,n , k ∈ N n } are i.i.d. Bernoulli random variableswith P ( ζ ( p ∆ t ) k,n = 1) = 1 − P ( ζ ( p ∆ t ) k,n = 0) = p ∆ t ∈ (0 , S ( p ∆ t )0 ,n = S , and for k = 1 , ..., n −
1, conditionally on F k ; n , S ( p ∆ t ) k +1 ,n = ( S ( p ∆ t ,u ) k +1 ,n = S ( p ∆ t ) k,n e U ∆ t , if ζ ( p ∆ t ) k +1 ,n = 1 S ( p ∆ t ,d ) k +1 ,n = S ( p ∆ t ) k,n e D ∆ t , if ζ ( p ∆ t ) k +1 ,n = 0 = S ( p ∆ t ) k,n ( e U ∆ t , w.p. p ∆ t e D ∆ t , w.p. − p ∆ t , (10)where U ∆ t = ( µ − σ − p ∆ t p ∆ t )∆ t + σ q − p ∆ t p ∆ t √ ∆ tD ∆ t = ( µ − σ p ∆ t − p ∆ t )∆ t − σ q p ∆ t − p ∆ t √ ∆ t . (11)With o (∆ t ) = 0, the binomial tree (10), has the equivalent form S ( p ∆ t ) k +1 ,n = S ( p ∆ t ,u ) k +1 ,n = S ( p ∆ t ) k,n (1 + µ ∆ t + σ q − p ∆ t p ∆ t √ ∆ t ) , if ζ ( p ∆ t ) k +1 ,n = 1 S ( p ∆ t ,d ) k +1 ,n = S ( p ∆ t ) k,n (1 + µ ∆ t − σ q p ∆ t − p ∆ t √ ∆ t ) , if ζ ( p ∆ t ) k +1 ,n = 0 . (12)If p ∆ t = p ( CRR )∆ t , and o (∆ t ) = 0, the KSRF pricing tree (10) and (11) becomes the CRR-pricing tree (3)and (4). If p ∆ t = , the KSRF pricing tree becomes the JR-pricing tree. By the DPIP, the D [0 , T ]-process, S ( n ) = { S ( n ) t = S ( n ; µ,σ ) t = S ( p ∆ t ) k,n , t ∈ [ k ∆ t, ( k + 1)∆ t ) , k = 0 , , ..., n − , S ( n ) T = S ( p ∆ t ) n,n } converges weakly in D [0 , T ] to S = { S t = S ( µ,σ ) t , t ∈ [0 , T ] } . The discrete dynamics of f t , t ∈ [0 , T ], on the lattice k ∆ t, k ∈ N n isdefined as follows: f k +1 ,n = ( f ( u ) k +1 ,n , if ζ ( p ∆ t ) k +1 ,n = 1 f ( d ) k +1 ,n , if ζ ( p ∆ t ) k +1 ,n = 0 , k = 0 , ..., n − , (13)and f n,n = g ( S T ) , f k,n = f ( S k ∆ t,k ∆ t ) , k = 0 , ..., n . At time instances k ∆ t, k = 0 , ..., n −
1, trader ℵ , taking ashort position in C , is forming a self-financing replicating risk-neutral portfolio P k ∆ t ; n = D k ∆ t S ( p ∆ t ) k,n − f k,n .6onditionally on F k,n , P ( k +1)∆ t ; n = D k ∆ t S ( p ∆ t )( k +1) ,n − f k +1 ,n . As demonstrated by KSRF, the risk-neutralitycondition implies that, conditionally on F k ; n , f k,n = e − r ∆ t ( q ( p ∆ t )∆ t f ( u ) k +1 ,n + (1 − q ( p ∆ t )∆ t ) f ( d ) k +1 ,n ) , k = 0 , ..., n − . (14)The risk-neutral probability q ∆ t in (14) is given by q ( p ∆ t )∆ t = exp {− ( r − µ )∆ t } − exp n − p ∆ t − p ∆ t σ ∆ t − σ q p ∆ t − p ∆ t ∆ t o exp n − − p ∆ t p ∆ t σ ∆ t − σ q − p ∆ t p ∆ t ∆ t o − exp n − p ∆ t − p ∆ t σ ∆ t − σ q p ∆ t − p ∆ t ∆ t o . (15)With o (∆ t ) = 0, q ∆ t has the form q ( p ∆ t )∆ t = p ∆ t − θ p p ∆ t (1 − p ∆ t ) √ ∆ t, (16)where θ = µ − rσ is the market price of risk. The risk-neutral pricing tree is given by S ( q ∆ t ) k +1 ,n = ( S ( q ∆ t ,u ) k +1 ,n = S ( q ∆ t ) k,n e U ∆ t , if ζ ( q ∆ t ) k +1 ,n = 1 S ( q ∆ t ,d ) k +1 ,n = S ( q ∆ t ) k,n e D ∆ t , if ζ ( q ∆ t ) k +1 ,n = 0 = S ( q ∆ t ) k,n ( e U ∆ t , w.p. q ∆ t ,e D ∆ t , w.p. − q ∆ t . (17)By the DPIP, the D [0 , T ]-process, S ( n ; Q ) = { S ( n ; Q ) t = S ( q ∆ t ) k,n , t ∈ [ k ∆ t, ( k + 1)∆ t ) , k = 0 , , ..., n − , S ( n ; Q ) T = S ( q ∆ t ) n,n } , converges weakly in D [0 , T ] to S ( n ; Q ) = { S ( Q ) t = S ( Q ; r,σ ) t , t ∈ [0 , T ] } , where S ( Q ) t = S ( Q ; r,σ ) t = S e ( r − σ ) t + σB ( Q ) t ,and B ( Q ) t , t ∈ [0 , T ], is a BM on (Ω , F = {F t } t ≥ , Q ). The probability measure Q ∼ P is the unique equivalentmartingale measure. The limiting continuous-time price process S ( Q ) t , t ∈ [0 , T ] is now independent of p ∆ t and µ . This is due to the assumption that ℵ can hedge his short position in the option contract continuouslyin time. However, if ℵ ’s hedging trading times are restricted to the time instances k ∆ t, k = 0 , , ..., n − p ∆ t and µ , due to (14) and (15). Furthermore, the discontinuity of theoption price at p ∆ t →
0, or p ∆ t →
1, does not exist anymore, because according to (16), lim p ∆ t ↑ q ∆ t = 1,and lim p ∆ t ↓ q ∆ t = 0. µ -surface and implied p ∆ t -surface In the previous section we showed the dependence of risk-neutral pricing tree (17) on stock mean return µ and the probability for stock upturn p ∆ t . Thus, similarly to the concept of implied volatility, we introducethe concept of implied µ -surface and implied p ∆ t -surface , illustrated in the following numerical example.The framework is based on KSRF binomial option pricing tree of (12), (14), and (16) in Section 2.3. Inthe simulation, we use daily trading frequencies of SPDR S&P 500 ETF(SPY) and corresponding Mini-SPX(XSP) call option prices as datasets. We use SPY to estimate the initial ˆ p , ˆ µ , and ˆ σ using one-yearback trading data with the time range from 5 / / / / of the starting date as the riskless rate r . To estimate ˆ p which is the probability of the sampleprice increasing for a fixed day, we use the proportion of the number of days with non-negative log-returnin one-year back trading period. We set ˆ σ as the sample standard deviation of sample return series, and∆ t = .The starting date for the option is 5 / / / / / / / / S = $286 .
28 with r = 0 . p = 0 .
56, ˆ µ = 1 . × − , and ˆ σ = 0 . See Chapter 6 in Duffie (2001). https://finance.yahoo.com/quote/SPY?p=SPY&.tsrc=fin-srch. The CBOE Mini-SPX (with ticker XSP) option contract is an index option product designed to track the underlying S&P500 Index with the size of 1/10 of the standard SPX options contract. See
7o get the implied µ -surface, we set µ as a free parameter, and then match the XSP option price C ( market ; C ) ( S , K, T, r, ˆ p, ˆ σ ) with the theoretical call option derived by (14). For the i th XSP contract in thesample, we have µ ( ℵ ; impled,i ) = arg min (cid:18) C ( ℵ ; C ,i ) ( S , K, T, r, ˆ p, ˆ σ ) − C ( market ; C ,i ) ( S , K, T, r, ˆ p, ˆ σ ) C ( market ; C ,i ) ( S , K, T, r, ˆ p, ˆ σ ) (cid:19) . Similarly, we switch the free parameter from µ to p to get the implied p -surface.As shown in Figure 1, the implied µ -surface is against “moneyness” ( M ) and time to maturity T in years. According to Figure 1, the µ ( ℵ ; implied ) ∈ ( − . , . M = 1 .
5, for example, the µ ( ℵ ; implied ) decreases from − .
02 to − .
04 for about three months and is stable for another eight months,then it recovers sharply to 0 .
04. At certain maturity time t ∈ [0 , T ] , T = 1 .
5, Figure1 indicates that the µ ( ℵ ; implied ) for the option trader slightly increases as M increases. And the increment is easier to captureafter one year.Figure 2 shows the result of p ( ℵ ; implied )∆ t which is the implied probability according to KSRF binomialoption pricing. Similarly, the implied p ∆ t -surface is plotted against M and T . In Section 2.2, we have that ℵ is an informed trader if p ℵ ∆ t > . More specifically, option traders are potentially more informed aboutfuture SPY returns better than spot traders. On the other hand, ℵ is a misinformed trader if p ℵ ∆ t < , whichdescribes the situation of spot traders more aware of the future movement of SPY than option traders. Ourresult indicates p ( ℵ ; implied )∆ t ∈ (0 . , . M , the implied probability of option traders decreasesas T increases. This fact indicates that option traders are informed in the near future rather than in thedistance future. For a fixed T , p ( ℵ ; implied )∆ t is roughly greater than 0 . M ∈ (1 , .
5) and less than 0 . M ∈ (0 . , M is high than when M issmall according to Figure 2.Figure 1: Implied µ –surface against time to maturity and moneyness. Here, we define moneyness M = KS , where K is the strike and S is the price. p ∆ t –surface against time to maturity and moneyness. To quantify the amount of information of an informed trader ℵ with p ℵ ∆ t > , we use Shannon’s entropy as an information measure. Shannon’s entropy of a Bernoulli random variable ζ ( p ) d = Ber ( p ) , P ( ζ ( p ) = 1) =1 − P ( ζ ( p ) = 0) = p ∈ (0 ,
1) is defined as H ( ζ ( p ) ) = − p ln p − (1 − p ) ln(1 − p ) , (18)and max
We assume that ℵ knows, with probability p ℵ ∆ t > , the stock price direction in period[ k ∆ t, ( k + 1)∆ t ]. We also assume that in the marketplace, there are a sufficient number of noisy traders, ℵ , whose trading activities are based on the assumption that p ∆ t = in (10). At any time instance, k ∆ t, k = 0 , ..., n, n ∆ t = T , ℵ makes independent bets, which are modeled as independent Bernoulli trials η ( ℵ ) k +1 ,n , k = 0 , ..., n − , P ( η ( ℵ ) k +1 ,n = 1) = 1 − P ( η ( ℵ ) k +1 ,n = 0) = p ℵ ∆ t ∈ ( , Sc ( up ) ) ζ ( p ∆ t ) k +1 ,n = 1; that is, S ( p ∆ t ) k +1 ,n = S ( p ∆ t ,u ) k +1 ,n = S ( p ∆ t ) k,n e U ∆ t , and ( Sc ( down ) ) ζ ( p ∆ t ) k +1 ,n = 0, thatis, S ( p ∆ t ) k +1 ,n = S ( p ∆ t ,d ) k +1 ,n = S ( p ∆ t ) k,n e D ∆ t . Now the filtration F ( n ) = {F k ; n = σ ( ζ ( p ∆ t )1 ,n , ..., ζ ( p ∆ t ) k,n ) , k ∈ N n , F n = { ∅ , Ω }} needs to be augmented with the sequence of ℵ ’s independent bets. We introduce the augmentedfiltration F ( n ; ℵ ) = {F ℵ k ; n = σ (( ζ ( p ∆ t )1 ,n , η ( ℵ )1 ,n ) , ..., ( ζ ( p ∆ t ) k,n , η ( ℵ ) k,n )) , k ∈ N n , F ( ℵ )0; n = { ∅ , Ω }} . At k ∆ t, k = 0 , ..., n − ℵ places his bets considering ( Sc ( up ) ) and ( Sc ( down ) ). If at k ∆ t , ℵ believes that( Sc ( up ) ) will happen, he takes a long position in ∆ ( ℵ ) k ∆ t = N ( ℵ ) S ( p ∆ t ) k,n -forward contracts for some N ( ℵ ) > According to the Efficient Market Hypothesis (EMH), asset price direction is unpredictable, see Fama (1970). However,some studies indicate that asset price direction is predictable (and, thus, questioning the EMH). See, among others, Shiller(2003 and 2013). We assume that ℵ is an informed trader, that is, p ℵ ∆ t > . We will develop a trading strategy for ℵ to utilize his informationon stock price direction. A misinformed trader will trade just the opposite of what an informed trader will do, and what willbe a profit for the informed trader will be a loss for the misinformed trader. Thus, it is sufficient to consider the case of ℵ beingan informed trader. We shall summarize the results for informed and misinformed traders at the end of Section 3.2. The short position in the forward contract could be taken by any trader who believes that S ( p ∆ t ) k +1 ,n = S ( p ∆ t ,d ) k +1 ,n = S ( p ∆ t ) k,n e D ∆ t is more likely to happen, or by a noisy trader ℵ . Parameter N ( ℵ ) will be optimized and will enter the formula for the positive yield ℵ will enjoy when trading options, seeSection 3.2. k + 1)∆ t . If at k ∆ t , ℵ believes that ( Sc ( down ) ) will happen, he takes ashort position in ∆ ( ℵ ) k ∆ t -forward contracts at maturity ( k + 1)∆ t . The overall payoff of ℵ ’s forward contractpositions is given by p ( ℵ ; forward ) k ∆ t → ( k +1)∆ t = ∆ ( ℵ ) k ∆ t ( S ( p ∆ t ,u ) k +1 ,n − S ( p ∆ t ) k,n e r ∆ t ) , if ζ ( p ∆ t ) k +1 ,n = 1 , η ( ℵ ) k +1 ,n = 1 , ( S ( p ∆ t ) k,n e r ∆ t − S ( p ∆ t ,d ) k +1 ,n ) , if ζ ( p ∆ t ) k +1 ,n = 0 , η ( ℵ ) k +1 ,n = 1 , ( S ( p ∆ t ) k,n e r ∆ t − S ( p ∆ t ,u ) k +1 ,n ) , if ζ ( p ∆ t ) k +1 ,n = 1 , η ( ℵ ) k +1 ,n = 0 , ( S ( p ∆ t ,d ) k +1 ,n − S ( p ∆ t ) k,n e r ∆ t ) , if ζ ( p ∆ t ) k +1 ,n = 0 , η ( ℵ ) k +1 ,n = 0 . (21)The conditional mean and variance of p ( ℵ ; forward ) k ∆ t → ( k +1)∆ t are given by E ( p ( ℵ ; forward ) k ∆ t → ( k +1)∆ t |F ( ℵ ) k ; n ) = N ( ℵ ) (2 p ℵ ∆ t − σ (cid:16) θ (2 p ∆ t − t + 2 p p ∆ t (1 − p ∆ t )∆ t (cid:17) ,V ar ( p ( ℵ ; forward ) k ∆ t → ( k +1)∆ t |F ( ℵ ) k ; n ) = N ( ℵ ) σ (cid:0) − p ℵ ∆ t − p ∆ t (1 − p ∆ t ) (cid:1) ∆ t. (22)where θ = µ − rσ . By the DPIP, we should have E ( p ( ℵ ; forward ) k ∆ t → ( k +1)∆ t |F ( ℵ ) k ; n ) = O (∆ t ) and V ar ( p ( ℵ ; forward ) k ∆ t → ( k +1)∆ t |F ( ℵ ) k ; n ) = O (∆ t ). To guarantee that, we set p ( ℵ )∆ t = (1 + λ ( ℵ ) √ p ∆ t (1 − p ∆ t ) √ ∆ t ), for some λ ( ℵ ) > The closer p ∆ t is to1, or 0, the more certain will be ℵ on stock price direction, and thus p ( ℵ )∆ t increases. Then, (22) simplifies to E ( p ( ℵ ; forward ) k ∆ t → ( k +1)∆ t |F ( ℵ ) k ; n ) = 2 N ( ℵ ) λ ( ℵ ) σ ∆ t,V ar ( p ( ℵ ; forward ) k ∆ t → ( k +1)∆ t |F ( ℵ ) k ; n ) = N ( ℵ ) σ ∆ t. (23)The instantaneous information ratio is given by IR ( p ( ℵ ; forward ) k ∆ t → ( k +1)∆ t |F ( ℵ ) k ; n ) = E ( p ( ℵ ; forward ) k ∆ t → ( k +1)∆ t |F ( ℵ ) k ; n ) √ ∆ t q V ar ( p ( ℵ ; forward ) k ∆ t → ( k +1)∆ t |F ( ℵ ) k ; n ) = 2 λ ( ℵ ) . (24) Suppose now that ℵ is taking a short position in the option contract within the BSM framework ( S , B , C ) .The stock price dynamics S t = S ( µ,σ ) t , t ≥
0, is given by (1), the bond price β t , t ≥ C has price f t = f ( S t , t ) , t ∈ [0 , T ] with terminal payoff f T = g ( S T ). When ℵ trades thestock S , hedging the short position in C , ℵ simultaneously runs his forward strategy. ℵ ’s trading strategy (acombination of the forward contact’s trading and trading the stock) leads to an enhanced price process, ofwhich dynamics can be expressed as follows: S ( ℵ ; C )0 ,n = S and S ( ℵ ; C ) k +1 ,n = S ( p ∆ t ,u ) k +1 ,n + N ( ℵ ) ( S ( p ∆ t ,u ) k +1 ,n − S ( p ∆ t ) k,n e r ∆ t ) , if ζ ( p ∆ t ) k +1 ,n = 1 , η ( ℵ ) k +1 ,n = 1 ,S ( p ∆ t ,d ) k +1 ,n + N ( ℵ ) ( S ( p ∆ t ) k,n e r ∆ t − S ( p ∆ t ,d ) k +1 ,n ) , if ζ ( p ∆ t ) k +1 ,n = 0 , η ( ℵ ) k +1 ,n = 1 ,S ( p ∆ t ,u ) k +1 ,n + N ( ℵ ) ( S ( p ∆ t ) k,n e r ∆ t − S ( p ∆ t ,u ) k +1 ,n ) , if ζ ( p ∆ t ) k +1 ,n = 1 , η ( ℵ ) k +1 ,n = 0 ,S ( p ∆ t ,d ) k +1 ,n + N ( ℵ ) ( S ( p ∆ t ,d ) k +1 ,n − S ( p ∆ t ) k,n e r ∆ t ) , if ζ ( p ∆ t ) k +1 ,n = 0 , η ( ℵ ) k +1 ,n = 0 , (25) The long position in the forward contract could be taken by any trader who believes that S ( p ∆ t ) k +1 ,n = S ( p ∆ t ,u ) k +1 ,n = S ( p ∆ t ) k,n e U ∆ t is more likely to happen, or by a noisy trader ℵ . The case of a misinformed trader can be considered in a similar manner. A misinformed trader with λ ( ℵ ) <
0, tradeslong-forward (resp. short-forward) when the informed trader with ( − λ ( ℵ ) ) >
0, trades short-forward (resp. long-forward). Anoisy trader will not trade any forward contracts, as he has no information about stock price direction. We have chosen the normalization p ∆ t (1 − p ∆ t ) for λ ( ℵ ) in p ( ℵ )∆ t = (1 + λ ( ℵ ) √ p ∆ t (1 − p ∆ t ) √ ∆ t ), so that IR ( p ( ℵ ; forward ) k ∆ t → ( k +1)∆ t |F ( ℵ ) k ; n ) = 2 λ ( ℵ ) is solely dependent on λ ( ℵ ) . The long position in the option contract is taken by a trader who trades the stock with stock dynamics given by (1). = 0 , , ..., n − , n ∆ t = T. It costs nothing to enter a forward contract at k ∆ t with terminal time ( k +1)∆ t .Then, E ( S ( ℵ ; C ) k +1 ,n S ( ℵ ; C ) k,n | S ( ℵ ; C ) k,n ) = 1 + (cid:16) µ + N ( ℵ ) ( µ − r )(2 p ∆ t − p ℵ ∆ t − (cid:17) ∆ t + 2 N ( ℵ ) σ p (1 − p ∆ t ) p ∆ t (2 p ℵ ∆ t − √ ∆ t. As already discussed in Section 3.1, we set 2 p ( ℵ )∆ t − λ ( ℵ ) √ p ∆ t (1 − p ∆ t ) √ ∆ t, λ ( ℵ ) >
0. Thus, the conditionalmean and variance of the log-return R ( ℵ ; C ) k,n = log( S ( ℵ ; C ) k +1 ,n S ( ℵ ; C ) k,n ) are E ( R ( ℵ ; C ) k,n | S ( ℵ ; C ) k,n ) = ( µ + 2 N ( ℵ ) σλ ( ℵ ) )∆ t and V ar ( R ( ℵ ; C ) k,n | S ( ℵ ; C ) k,n ) = σ (1 + N ( ℵ ) )∆ t . The instantaneous market price of risk is given byΘ( R ( ℵ ; C ) k,n | S ( ℵ ; C ) k,n ) = E ( R ( ℵ ; C ) k,n | S ( ℵ ; C ) k,n ) − r ∆ t √ ∆ t q V ar ( R ( ℵ ; C ) k,n | S ( ℵ ; C ) k,n ) = θ + 2 N ( ℵ ) λ ( ℵ ) √ N ( ℵ ) . (26)The optimal N ( ℵ ) , maximizing Θ( R ( ℵ ; C ) k,n | S ( ℵ ; C ) k,n ), is N ( ℵ ) = N ( ℵ ; opt ) = 2 λ ( ℵ ) θ > and the optimal instanta-neous market price of risk is Θ( R ( ℵ ; C ) k,n | S ( ℵ ; C ) k,n ) = Θ ( opt ) ( R ( ℵ ; C ) k,n | S ( ℵ ; C ) k,n ) = p θ + 4 λ ( ℵ ) . With N ( ℵ ) = N ( ℵ ; opt ) , E ( R ( ℵ ; C ) k,n | S ( ℵ ; C ) k,n ) = ( µ + 4 σ λ ( ℵ ) θ )∆ t,V ar ( R ( ℵ ; C ) k,n | S ( ℵ ; C ) k,n ) = σ (1 + 4 λ ( ℵ ) θ )∆ t. (27)Next, we consider the limiting behavior of S ( n, ℵ ; C ) = { S ( n, ℵ ; C ) t = S ( ℵ ; C ) k,n , t ∈ [ k ∆ t, ( k + 1)∆ t ) , k =0 , ..., n, S ( n, ℵ ; C ) T = S ( ℵ ; C ) n,n } as ∆ t ↓
0. We set p ∆ t = p ∈ (0 , S ( n, ℵ ; C ) converges weakly in D [0 , T ] to S ( ℵ ; C ) = { S ( ℵ ; C ) t , t ∈ [0 , T ] } as n ↑ ∞ , where S ( ℵ ; C ) t = S exp (cid:26) ( µ ( ℵ ; C ) − σ ( ℵ ; C ) ) t + σ ( ℵ ; C ) B t (cid:27) . (28)In (27), µ ( ℵ ; C ) = µ + 4 σ λ ( ℵ )2 θ , and σ ( ℵ ; C ) = σ q λ ( ℵ )2 θ . Now, in the limit ∆ t ↓ ℵ hedges the shortoption position using the price process S ( ℵ ; C ) . ℵ forms his instantaneous riskless replicating portfolio Π ( ℵ ; C ) t = a ( ℵ ; C ) t S ( ℵ ; C ) t + b ( ℵ ; C ) t β t = f t , t ∈ [0 , T ). As Π ( ℵ ; C ) t is self-financing portfolio, and thus, df t = d Π ( ℵ ; C ) t = a ( ℵ ; C ) t dS ( ℵ ; C ) t + b ( ℵ ; C ) t dβ t . By Itˆo’s formula, (cid:18) ∂f ( S t , t ) ∂t + µS t ∂f ( S t , t ) ∂x + 12 σ S t ∂ f ( S t , t ) ∂x (cid:19) dt + σS t ∂f ( S t , t ) ∂x dB t = a ( ℵ ; C ) t S ( ℵ ; C ) t ( µ ( ℵ ; C ) dt + σ ( ℵ ; F ) dB t ) + b ( ℵ ; C ) t rβ t dt. (29)Because the forward contract, which ℵ initiates at t , has zero value, then S ( ℵ ; C ) t = S t in (29). The no-arbitrage argument implies that a ( ℵ ; C ) t = ∂f ( S t ,t ) ∂x σσ ( ℵ ; C ) and b ( ℵ ; C ) t = β t (cid:16) f ( S t , t ) − a ( ℵ ; C ) t S ( ℵ ; C ) t (cid:17) . Thus, theBSM partial differential equation (PDE) for ℵ ’s option price f t = f ( x, t ) , x > , t ∈ [0 , T ), is given by ∂f ( x, t ) ∂t + ( r − D y ) x ∂f ( x, t ) ∂x + 12 σ x ∂ f ( x, t ) ∂x − rf ( x, t ) = 0 . (30) With every single share of the traded stock with price S ( p ∆ t ) k,n at k ∆ t , ℵ simultaneously enters N ( ℵ ) -forward contracts. Theforward contracts are long or short, depending on ℵ ’s views on stock price direction in time-period [ k ∆ t, ( k + 1)∆ t ]. By assumption, µ > r >
0, and thus, θ = µ − rσ > D y ∈ R has the form D y = D ( ℵ ; C ) y = σ ( √ θ + 4 λ ( ℵ ) − θ ) >
0. According to(30), the BSM formula for the European call option price for an informed trader ℵ , C t = C ( S t , t ) , t ∈ [0 , T ]with strike price K , is given by the standard option-price formula for the dividend-paying stock: C ( ℵ ; C ) ( S t , K, T − t, r, σ, D y ) = e − D y ( T − t ) N ( d ) S t − N ( d ) Ke − r ( T − t ) ,d = 1 σ √ T − t (cid:20) log( S t K ) + ( r + σ T − t ) (cid:21) ,d = d − σ √ T − t, t ∈ [0 , T ) . (31)In (31), D y = D ( ℵ ; C ) and N ( x ) , x ∈ R is the standard normal distribution function. For a misinformedtrader the yield is negative, and thus, in general, if p ( ℵ )∆ t = + λ ( ℵ ) √ p (1 − p ) √ ∆ t, λ ( ℵ ) ∈ R , the dividend yield D y in (31) is given by D y = D ( ℵ ; C ) y = sign( λ ( ℵ ) ) σ ( √ θ + 4 λ ( ℵ ) − θ ). λ ( ℵ ) Here we apply the Black-Sholes option pricing formula to construct the implied trader information in-tensity surface λ ( ℵ ; implied ) and implied probability p ( ℵ ; implied )∆ t = (cid:18) λ ( ℵ ; implied ) √ p (1 − p ) √ ∆ t (cid:19) . We calculate p ( ℵ ; implied )∆ t for options with different times to maturity and strike prices.To this end, we first estimate p , as the sample success probability ˆ p of the stock price being “up”for a fixed day. We use one-year of historical log-returns to calculate the sample mean ˆ µ as an estimatefor µ , and the historical sample standard deviation ˆ σ as an estimate for σ . Here, we compare the optionand spot trader’s information of stock returns by applying λ ( ℵ ; implied ) and p ( ℵ ; implied )∆ t . Rather than lookingat individual stocks, our analysis will focus on the aggregate stock market. In our analysis, we selected abroad-based market index, the S&P 500, as measured by the SPDR S&P 500, which is an exchange-tradedfund, as the proxy for the aggregate stock market. We use the 10-year Treasury yield as a proxy for therisk-free rate r . The database includes the period from November 2018 to November 2019. There were 252observations collected from Yahoo Finance.We use call option prices on 10 / / / / / / . / / / / r = 1 . C ( market ; C ,i ) ( S t , K, T − t ), i = 1 , . . . , T , we first construct the implied trader information intensity λ ( ℵ ) = λ ( ℵ ; implied ) -implied surface. λ ( ℵ ; implied ) -implied surface is graphed against both a standard measure of moneyness andtime to maturity (in years) in Figure 4.Figure 4 indicates that at each maturity, the implied trader information intensity of option tradersincreases as moneyness increases. Where the moneyness varies in (0 , . . , . λ ( N ; implied ) starts increasing from zero to 0 . p ( ℵ ; implied )∆ t -surface is graphed against both a standard measure of “moneyness”and time to maturity (in years) in Figure 5. Recall that values higher than 0 . p ( ℵ ; implied )∆ t , means thatoption traders have more information about the mean µ of SPY daily return or the option trader is theinformed trader. In other words, p ( ℵ ; implied )∆ t > . p ( ℵ ; implied )∆ t ≤ . . , . p ( ℵ ; implied )∆ t startsincreasing from 0 . . Option Pricing When The Trader Has Information on The In-stantaneous Mean Return
In this section we assume that trader ℵ has information about whether the instantaneous mean return ofstock S is above or below the market perceived value µ . First, we notice that this information, is equivalentto ℵ ’s information about the probability for stock price upturn. Consider the binomial stock price model: S ( p ∆ t )0 ,n = S , and conditionally on S ( p ∆ t ) k,n , S ( p ∆ t ) k +1 ,n = S ( p ∆ t ,u ) k +1 ,n = S ( p ∆ t ) k,n (1 + µ ∆ t + σ q − p p √ ∆ t ) , w.p. p + δ ( ℵ ) √ ∆ t,S ( p ∆ t ,d ) k +1 ,n = S ( p ∆ t ) k,n (1 + µ ∆ t − σ q p − p √ ∆ t ) , w.p. − p − δ ( ℵ ) √ ∆ t, (32) k = 1 , ..., n, n ∆ t = T , where δ ( ℵ ) ∈ ( , o (∆ t ) = 0. Then, E (cid:18) S ( p ∆ t ) k +1 ,n S ( p ∆ t ) k,n (cid:19) = 1 + µ ( ℵ ) ∆ t , where µ ( ℵ ) = µ + σ √ p (1 − p ) δ ( ℵ ) , and V ar (cid:18) S ( p ∆ t ) k +1 ,n S ( p ∆ t ) k,n (cid:19) = σ ∆ t . Thus, ℵ ’s belief that the true stock mean return is µ ( ℵ ) = µ + σ √ p (1 − p ) δ ( ℵ ) , δ ( ℵ ) = 0, rather than the market perceived value µ , is expressed by ℵ ’s belief thatthe true stock price dynamics is given by: S ( p ∆ t )0 ,n = S , and for k = 0 , , ..., n − , n ∆ t = T , S ( p ∆ t ) k +1 ,n = S ( p ∆ t ,u ) k +1 ,n = S ( p ∆ t ) k,n (1 + µ ( ℵ ) ∆ t + σ q − p p √ ∆ t ) , w.p. p S ( p ∆ t ,d ) k +1 ,n = S ( p ∆ t ) k,n (1 + µ ( ℵ ) ∆ t − σ q p − p √ ∆ t ) , w.p. − p (33)conditionally on S ( p ∆ t ) k,n . ℵ can use pricing model (33) or, equivalently , can use model (32). According to(32), ℵ believes that the true probability for an upturn is p ∆ t = p + δ ( ℵ ) √ ∆ t for some δ ( ℵ ) = 0, while themarket perceived stock price dynamics is given by (32) but with δ ( ℵ ) = 0. We introduce ℵ ’s strategy of trading forward contracts based on information on the instantaneous meanstock return. If at k ∆ t , ℵ believes that the stock mean return is µ ( ℵ ) = µ + σ √ p (1 − p ) δ ( ℵ ) , δ ( ℵ ) >
0, he enters∆ ( ℵ ,µ ) k ∆ t – long forwards, with ∆ ( ℵ ,µ ) k ∆ t = N ( ℵ ,µ ) S ( p ∆ t ) k,n , N ( ℵ ,µ ) >
0, and terminal time ( k + 1)∆ t . If ℵ believes that thestock mean return is µ ( ℵ ) = µ + σ √ p (1 − p ) δ ( ℵ ) , δ ( ℵ ) <
0, he enters ∆ ( ℵ ,µ ) k ∆ t – short forwards . In other words,if, at time instance k ∆ t , ℵ believes that the true probability for stock price upturn is p ∆ t = p + δ ( ℵ ) √ ∆ t with δ ( ℵ ) >
0, he bets that the stock price will be S ( p ∆ t ,u ) k +1 ,n = S ( p ∆ t ) k,n (1 + µ ∆ t + σ q − p p √ ∆ t ). In this case, ℵ enters ∆ ( ℵ ,µ ) k ∆ t – long forward contracts. If ℵ believes that the true probability for the stock price upturnis p ∆ t = p + δ ( ℵ ) √ ∆ t with δ ( ℵ ) <
0, he bets that the stock price will be S ( p ∆ t ) k,n (1 + µ ∆ t − σ q p − p √ ∆ t ).In this case, ℵ enters ∆ ( ℵ ,µ ) k ∆ t – short forward contracts. Following the same arguments as in Section 3.1, theconditional mean and variance of P ( ℵ ; forward,µ ) k ∆ t → ( k +1)∆ t are given by E ( P ( ℵ ; forward,µ ) k ∆ t → ( k +1)∆ t | S ( p ∆ t ) k,n ) = N ( ℵ ,µ ) (2 p ( ℵ ,µ )∆ t − σ ( θ (2 p k ∆ t − t + 2 p p ∆ t (1 − p ∆ t ) √ ∆ t ) ,V ar ( P ( ℵ ; forward,µ ) k ∆ t → ( k +1)∆ t | S ( p ∆ t ) k,n ) = N ( ℵ ,µ ) σ (cid:16) − p ( ℵ ,µ )∆ t − p ∆ t (1 − p ∆ t ) (cid:17) ∆ t. (34) Here “equivalently” means that the binomial option pricing trees in (32) and (33) generate the same limiting geometricBrownian motion, as ∆ t ↓ , n ∆ t = T . This argument follows from the DPIP. The probability of ℵ ’s guess on the stock price direction being correct is assumed to be p ( ℵ ,µ )∆ t ∈ (0 ,
15y the DPIP, for P ( ℵ ; forward,µ ) k ∆ t → ( k +1)∆ t , we must have E ( P ( ℵ ; forward,µ ) k ∆ t → ( k +1)∆ t ) = O (∆ t ). Assuming that ℵ is an informedtrader, we set p ( ℵ ,µ )∆ t = (1 + ρ ( ℵ ,µ ) √ p ∆ t (1 − p ∆ t ) √ ∆ t ), for some ρ ( ℵ ,µ ) >
0. Thus, with p ∆ t = p + δ ( ℵ ) √ ∆ t , (34)is simplified and has the form: E ( P ( ℵ ; forward,µ ) k ∆ t → ( k +1)∆ t ) = 2 N ( ℵ ,µ ) σρ ( ℵ ,µ ) ∆ t,V ar ( P ( ℵ ; forward,µ ) k ∆ t → ( k +1)∆ t ) = N ( ℵ ,µ ) σ ∆ t. (35)In (35), the mean E ( P ( ℵ ; forward,µ ) k ∆ t → ( k +1)∆ t ) and the variance V ar ( P ( ℵ ; forward,µ ) k ∆ t → ( k +1)∆ t ) do not depend on the actual valueof δ ( ℵ ) in µ ( ℵ ) = µ + σ √ p (1 − p ) δ ( ℵ ) . This is due to the fact that ℵ knows, with probability p ( ℵ ,µ )∆ t , whether µ ( ℵ ) is above or below µ ; that is, ℵ knows sign( δ ( ℵ ) ), but not the value | δ ( ℵ ) | . Comparing (22), (23) with (34) and (35), it becomes clear that ℵ ’s information (on whether the in-stantaneous mean stock return is above or below the market perceived value µ ) is equivalent to ℵ ’s infor-mation on whether the instantaneous upward probability is above or below the market perceived prob-ability p . Thus, when ℵ applies the forward strategy, the option pricing formula (31) is valid with D y = D ( ℵ ; C ,µ ) y = σ ( p θ + 4 ρ ( ℵ ,µ ) − θ ) >
0. As the yield D y = D ( ℵ ; C ,µ ) y does not depend on δ ( ℵ ) , theoption price (31) does not depend on Dev ( ℵ ,µ ) = µ ( ℵ ) − µ = σp (1 − p ) δ ( ℵ ) . However, consider the case wherethe binomial option pricing tree is given by: S ( ℵ ; C ,µ )0 ,n = S , and conditionally on S ( p ∆ t ) k,n , S ( ℵ ; C ,µ ) k +1 ,n = S ( p ∆ t ,u ) k +1 ,n + N ( ℵ ,µ ) ( S ( p ∆ t ,u ) k +1 ,n − S ( p ∆ t ) k,n e r ∆ t ) , w.p. p ∆ t p ( ℵ ,µ )∆ t ,S ( p ∆ t ,d ) k +1 ,n + N ( ℵ ,µ ) ( S ( p ∆ t ) k,n e r ∆ t − S ( p ∆ t ,d ) k +1 ,n ) , w.p. (1 − p ∆ t ) p ( ℵ ,µ )∆ t ,S ( p ∆ t ,u ) k +1 ,n + N ( ℵ ,µ ) ( S ( p ∆ t ) k,n e r ∆ t − S ( p ∆ t ,u ) k +1 ,n ) , w.p. p ∆ t (1 − p ( ℵ ,µ )∆ t ) ,S ( p ∆ t ,d ) k +1 ,n + N ( ℵ ,µ ) ( S ( p ∆ t ,d ) k +1 ,n − S ( p ∆ t ) k,n e r ∆ t ) , w.p. (1 − p ∆ t )(1 − p ( ℵ ,µ )∆ t ) , (36) k = 0 , , ..., n − , n ∆ t = T . According to (26), the optimal N ( ℵ ,µ ) is given by N ( ℵ ,µ ) = N ( ℵ ,µ ; opt ) = 2 ρ ( ℵ ,µ ) θ ,leading to E ( R ( ℵ ; C ,µ ) k,n | S ( ℵ ; C ,µ ) k,n ) = ( µ + 4 σ ρ ( ℵ ,µ ) µ − r )∆ t,V ar ( R ( ℵ ; C ,µ ) k,n | S ( ℵ ; C ,µ ) k,n ) = σ (1 + 4 σ ρ ( ℵ ,µ ) ( µ − r ) )∆ t, (37)where R ( ℵ ; C ,µ ) k,n = log( S ( ℵ ; C ,µ ) k +1 ,n S ( ℵ ; C ,µ ) k,n ). Consider the binomial option pricing tree: S ( p ∆ t ; δ,ρ )0 ,n = S , and conditionallyon S ( p ∆ t ; δ,ρ ) k,n , S ( p ∆ t ; δ,ρ ) k +1 ,n = S ( p ∆ t ,u ; δ,ρ ) k +1 ,n = S ( p ∆ t ; δ,ρ ) k,n (1 + v ( ℵ )∆ t ∆ t + S ( ℵ )∆ t q − p ∆ t p ∆ t √ ∆ t ) , w.p. p ∆ t ,S ( p ∆ t ,d ; δ,ρ ) k +1 ,n = S ( p ∆ t ; δ,ρ ) k,n (1 + v ( ℵ )∆ t ∆ − S ( ℵ )∆ t q p ∆ t − p ∆ t √ ∆ t ) , w.p. − p ∆ t , (38)for k = 0 , , ..., n −
1. In (38), v ( ℵ )∆ t = µ + 4 σ ρ ( ℵ ,µ )2 µ − r , S ( ℵ )∆ t = σ q σ ρ ( ℵ ,µ )2 ( µ − r ) . By the DPIP, the trees(36) and (38) have the same limiting pricing process as ∆ t →
0. The limiting process is independent of p ∆ t = p + δ ( ℵ ) √ ∆ t ∈ (0 , δ ( ℵ ) and µ ( ℵ ) = µ + σ √ p (1 − p ) δ ( ℵ ) will be lost.However, for a fixed trading frequency ∆ t , the risk–neutral probabilities q ( ℵ ; δ,ρ )∆ t and 1 − q ( ℵ ; δ,ρ )∆ t correspondingto the tree (38) are given by (16): q ( ℵ ; δ,ρ )∆ t = p ∆ t − θ ( ℵ ; δ,ρ )∆ t p p ∆ t (1 − p ∆ t ) √ ∆ t, (39)16here θ ( ℵ ; δ,ρ )∆ t = p θ + 4 ρ ( ℵ ,µ ) . Thus, the binomial option price process f k,n = e − r ∆ t ( q ( ℵ ; δ,ρ )∆ t f ( u ) k +1 ,n + (1 − q ( ℵ ; δ,ρ )∆ t ) f ( d ) k +1 ,n ) , k = 0 , ..., n (40)depends on δ ( ℵ ) , and thus on Dev ( ℵ ,µ ) = σ √ p (1 − p ) δ ( ℵ ) as well. Dev ( ℵ ,µ ) Similar to Section 2.4, we introduce the concept of implied
Dev ( ℵ ,µ ) -surface using the following numericalexample.Again, our dataset is collected from daily closing prices for the SPY and the call option contracts XSP,and we use the 10-year Treasury yield as an approximation of the riskless rate r . By setting 5 / / S = $286 .
28 and r = 0 . p = 0 . µ = 1 . × − , and ˆ σ = 0 . Dev ( ℵ ,µ ) = σ √ p (1 − p ) δ ( ℵ ) . After initiating parameters, the taskbecomes finding the implied δ ( ℵ ) -surface. According to (36), (38), (39), and (40), we build up a binomialoption pricing tree involving unknown parameters δ ( ℵ ) > ρ ( ℵ ,µ ) >
0. To construct the implied δ ( ℵ ) -surface, we want to fix ρ ( ℵ ,µ ) . With δ ( ℵ ) ∈ ( , δ ( ℵ ) j = 0 . , . , ...,
1, then we find the optimal ρ ( ℵ ,µ ; opt ) =arg min I X i =1 J X j =1 C ( ℵ ; C ,i ) ( S , K, T, r, ˆ p, ˆ σ, δ ( ℵ ) j , ρ ( ℵ ,µ ) ) − C ( market ; C ,i ) ( S , K, T, r, ˆ p, ˆ σ ) C ( market ; C ,i ) ( S , K, T, r, ˆ p, ˆ σ ) ! . In this numerical example, we set ρ ( ℵ ,µ ; opt ) = 0 .
49. Then, we calculate the implied δ ( ℵ ) -surface and implied Dev ( ℵ ) -surface.FigureFigure 6 shows the implied Dev ( ℵ ) -surface against “moneyness” and time to maturity T in years.Here, M ∈ [0 . , . T ∈ [0 . , . Dev ( ℵ ; implied ) ∈ [0 . , . Dev ( ℵ ; implied ) fluctuates between 0 .
02 and 0 .
03 on different call option contracts for SPY in this numericalexample. However, for fixed M , we can still capture the trend of increments of Dev ( ℵ ; implied ) as T increases.Recall the definition of implied Dev ( ℵ ) : Dev ( ℵ ,µ ) = µ ( ℵ ) − µ = σp (1 − p ) δ ( ℵ ) . This fact indicates that optiontraders believe the SPY will go “up” in the future. If ℵ has information whether the true volatility of stock S is above or below the market perceived value σ , the trader should trade the S -volatility . If ℵ has information whether the true interest rate is above orbelow the market perceived value r , the trader should invest in a money market ETF. As the volatility andinterest rate dynamics are generally mean reverting, we next extend our option pricing model for informedtraders in financial markets driven by a continuous-diffusion process. We start with the KSRF-binomial model for the continuous-diffusion price process. Consider the continuous-diffusion market ( S , B , C ) within the BSM framework with stock S , bond B , and ECC C . The stock price If S is SPDR (S&P ETF Trust, SPY, State Street Global Advisors), then VIXY (VIX Short-TermFutures ETF)-tracks S&P500 volatility, traded as S&P 500 VIX Short-Term Futures Index. Volatil-ity indices on stock indices, individual equities, currencies and interest rates are traded at the CBOE, Dev ( ℵ ,µ ) -surface against time to maturity and moneyness.dynamics follows a continuous-diffusion process S = { S t , t ∈ [0 , T ] } , where S t = S exp (cid:26)Z t ( µ u − σ u ) du + Z t σ u dB u (cid:27) , t ∈ [0 , T ] , S > , F = {F t } t ∈ [0 ,T ] , P ) with filtration F , generated by the BM B t , t ∈ [0 , T ]. The instantaneousmean function µ t > , t ∈ [0 , T ] and the volatility function σ t > , t ∈ [0 , T ] are deterministic and havecontinuous derivatives on [0 , T ]. The bond price is given by β t = β exp (cid:26)Z t r u du (cid:27) , t ∈ [0 , T ] , β > , (42)where the instantaneous riskless rate r t , t ∈ [0 , T ], has continuous derivative on [0 , T ], and 0 < r t < µ t , t ∈ [0 , T ]. The ECC with underlying asset S has terminal (expiration) time T >
0, and terminal payoff g ( S T ).Let n ∆ t = T, n ∈ N = { , , . . . } , and ǫ ( p ) k ∆ t , k = 1 , . . . , n be a sequence of independent Bernoulli randomvariables with P ( ǫ ( p )( k +1)∆ t = 1) = 1 − P ( ǫ ( p )( k +1)∆ t = 0) = p k ∆ t , k = 0 , , . . . , n −
1, where p t ∈ (0 , , t ∈ [0 , T ]has continuous first derivative. Consider the KSRF-binomial price dynamics : S ( p )( k +1)∆ t,n = S ( p,u )( k +1)∆ t,n = S ( p ) k ∆ t,n (1 + µ k ∆ t ∆ t + σ k ∆ t q − p k ∆ t p k ∆ t √ ∆ t ) , if ǫ ( p ) k ∆ t = 1 ,S ( p,d )( k +1)∆ t,n = S ( p ) k ∆ t,n (1 + µ k ∆ t ∆ − σ k ∆ t q p k ∆ t − p k ∆ t √ ∆ t ) , if ǫ ( p ) k ∆ t = 0 , (43)for k = 0 , , ..., n − S ( p )0 ,n = S . With o (∆ t ) = 0, (43) is a recombined tree, and E [( S ( p )( k +1)∆ t,n S ( p ) k ∆ t,n ) γ | S ( p ) k ∆ t,n ] = E [( S ( k +1)∆ t S k ∆ t ) γ | S k ∆ t ] = 1 + γ ( µ k ∆ t + γ − σ k ∆ t ) for all γ >
0. Set S ( n,p ) = { S ( n,p ) t , t ∈ [0 , T ] } , where S ( n,p ) t = S ( p ) k ∆ t,n for t ∈ [ k ∆ t, ( k + 1)∆ t ) , k = 0 , , ..., n − , S ( n,p ) T = S ( p ) n,n . Then, as n ↑ ∞ , S ( n,p ) weakly converges in D [0 , T ] to S . The risk-neutral probabilities q k ∆ t , k = 0 , , ..., n −
1, are q k ∆ t = p k ∆ t − θ k ∆ t p p k ∆ t (1 − p k ∆ t )∆ t , with o (∆ t ) = 0 and θ t = µ t − r t σ t , t ∈ [0 , T ]. See Proposition 3 in Davydov & Rotar (2008) and Kim et al. (2019). S ( q )( k +1)∆ t,n = S ( q,u )( k +1)∆ t,n = S ( q ) k ∆ t,n (1 + r k ∆ t ∆ t + σ k ∆ t q − q k ∆ t q k ∆ t √ ∆ t ) , if ǫ ( q )( k +1)∆ t = 1 ,S ( q,d )( k +1)∆ t,n = S ( q ) k ∆ t,n (1 + r k ∆ t ∆ t − σ k ∆ t q q k ∆ t − q k ∆ t √ ∆ t ) , if ǫ ( q )( k +1)∆ t = 0 , (44)for k = 0 , , ..., n − S ( q )0 ,n = S . In (44), ǫ ( q ) k ∆ t , k = 1 , ..., n is a sequence of independent Bernoulli randomvariables with P ( ǫ ( q )( k +1)∆ t = 1) = 1 − P ( ǫ ( q )( k +1)∆ t = 0) = q k ∆ t , k = 0 , , . . . , n −
1. Set S ( n,q ) = { S ( n,q ) t , t ∈ [0 , T ] } , where S ( n,q ) t = S ( q ) k,n for t ∈ [ k ∆ t, ( k + 1)∆ t ) , k = 0 , , ..., n − , S ( n,q ) T = S ( q ) n,n . Then, as n ↑ ∞ , S ( n,q ) weakly converges in D [0 , T ] to S ( Q ) = { S ( Q ) t , t ∈ [0 , T ] } , where S ( Q ) t = S exp { R t ( r u − σ u ) du + R t σ u dB Q u } , t ∈ [0 , T ], where B Q u is a BM on (Ω , F = {F t } t ≥ , Q ), and Q is the unique equivalent martingale measure. At any time k ∆ t, k = 0 , ..., n, n ∆ t = T , ℵ makes independent bets, which are modeled as independentBernoulli trials η ( ℵ )( k +1)∆ t,n , k = 0 , ..., n − , P ( η ( ℵ )( k +1)∆ t,n = 1) = 1 − P ( η ( ℵ )( k +1)∆ t,n = 0) = p ℵ k ∆ t ∈ ( , p ℵ t ∈ ( , , t ∈ [0 , T ] is assumed to have continuous first derivative on [0 , T ]. If at k ∆ t , ℵ believesthat ǫ ( p )( k +1)∆ t = 1 will happen, he takes a long position in ∆ ( ℵ ,p ) k ∆ t = N ( ℵ ,p ) k ∆ t S ( p ) k ∆ t,n – forward contracts for some N ( ℵ ,p ) k ∆ t >
0. The function N ( ℵ ,p ) t > , t ∈ [0 , T ] is assumed to have continuous first derivative on [0 , T ].The maturity of the forwards is ( k + 1)∆ t . If at k ∆ t , ℵ believes that ǫ ( p )( k +1)∆ t = 0 will happen, he takes ashort position in ∆ ( ℵ ,p ) k ∆ t – forward contracts at maturity ( k + 1)∆ t . The overall payoff of ℵ ’s forward contractpositions is given by P ( ℵ ,p ; forward ) k ∆ t → ( k +1)∆ t = ∆ ( ℵ ,p ) k ∆ t ( S ( p,u )( k +1)∆ t,n − S ( p ) k ∆ t,n e r ∆ t ∆ t ) , if ǫ ( p )( k +1)∆ t = 1 , η ( ℵ )( k +1)∆ t,n = 1 , ( S ( p ) k ∆ t,n e r ∆ t ∆ t − S ( p,d )( k +1)∆ t,n ) , if ǫ ( p )( k +1)∆ t = 0 , η ( ℵ )( k +1)∆ t,n = 1 , ( S ( p ) k ∆ t,n e r ∆ t ∆ t − S ( p,u )( k +1)∆ t,n ) , if ǫ ( p )( k +1)∆ t = 1 , η ( ℵ )( k +1)∆ t,n = 0 , ( S ( p,d )( k +1)∆ t,n − S ( p ) k ∆ t,n e r ∆ t ∆ t ) , if ǫ ( p )( k +1)∆ t = 0 , η ( ℵ )( k +1)∆ t,n = 0 . (45)The conditional mean and variance of P ( ℵ ,p ; forward ) k ∆ t → ( k +1)∆ t are given by E ( P ( ℵ ,p ; forward ) k ∆ t → ( k +1)∆ t | S ( p ) k ∆ t,n ) = N ( ℵ ,p ) k ∆ t (2 p ℵ k ∆ t − σ k ∆ t ( θ (2 p k ∆ t − t + 2 p p k ∆ t (1 − p k ∆ t ) √ ∆ t ) ,V ar ( P ( ℵ ,p ; forward ) k ∆ t → ( k +1)∆ t | S ( p ) k ∆ t,n ) = N ( ℵ ,p ) k ∆ t σ k ∆ t (cid:0) − p ℵ k ∆ t − p k ∆ t (1 − p k ∆ t ) (cid:1) ∆ t. By the DPIP , we should have E ( P ( ℵ ,p ; forward ) k ∆ t → ( k +1)∆ t | S ( p ) k ∆ t,n ) = O (∆ t ), and V ar ( P ( ℵ ,p ; forward ) k ∆ t → ( k +1)∆ t | S ( p ) k ∆ t,n ) = O (∆ t ).To guarantee that, we set p ℵ k ∆ t = (1 + ψ ( ℵ ) k ∆ t √ ∆ t ), for some ψ ( ℵ ) t > , t ∈ [0 , T ]. It is assumed that ψ ( ℵ ) t , t ∈ [0 , T ] has continuous first derivative on [0 , T ]. With p ℵ k ∆ t = (1 + ψ ( ℵ ) k ∆ t √ ∆ t ), and o (∆ t ) = 0, wehave E ( P ( ℵ ,p ; forward ) k ∆ t → ( k +1)∆ t | S ( p ) k ∆ t,n ) = 2 N ( ℵ ,p ) k ∆ t ψ ( ℵ ) k ∆ t σ k ∆ t p p k ∆ t (1 − p k ∆ t )∆ t,V ar ( P ( ℵ ,p ; forward ) k ∆ t → ( k +1)∆ t | S ( p ) k ∆ t,n ) = N ( ℵ ,p ) k ∆ t σ k ∆ t ∆ t. See Chapter 6 in Duffie (2001). See Davydov & Rotar (2008). The case of a misinformed trader can be considered in a similar manner. A misinformed trader with ψ ( ℵ ) <
0, tradeslong-forward (resp. short-forward) when the informed trader with ( − ψ ( ℵ ) ) >
0, trades short-forward (resp. long-forward). Anoisy trader will not trade any forward contracts, as he has no information about stock price direction. instantaneous information ratio is given by IR ( P ( ℵ ,p ; forward ) k ∆ t → ( k +1)∆ t | S ( p ) k ∆ t,n ) = E ( P ( ℵ ,p ; forward ) k ∆ t → ( k +1)∆ t | S ( p ) k ∆ t,n ) √ ∆ t q V ar ( P ( ℵ ,p ; forward ) k ∆ t → ( k +1)∆ t | S ( p ) k ∆ t,n )= 2 ψ ( ℵ ) k ∆ t p p k ∆ t (1 − p k ∆ t ) . Suppose now that ℵ is taking a short position in the option contract in the BSM market ( S , B , C ). Thestock price dynamics S t = S ( µ,σ ) t , t ≥
0, are given by (41), the bond price β t , t ≥ C has price f t = f ( S t , t ) , t ∈ [0 , T ] with terminal payoff f T = g ( S T ). When ℵ trades the stock S , hedging his short position in C , ℵ simultaneously executes his forward strategy. ℵ ’s trading strategy (acombination of forward trading with trading the stock) leads to an enhanced price process, the dynamics ofwhich can be expressed as follows: S ( ℵ ,p ; C ) k +1 ,n = S ( p,u )( k +1)∆ t,n + N ( ℵ ,p ) k ∆ t ( S ( p,u )( k +1)∆ t,n − S ( p ) k ∆ t,n e r k ∆ t ∆ t ) , if ζ ( p ∆ t ) k +1 ,n = 1 , η ( ℵ ) k +1 ,n = 1 ,S ( p,d )( k +1)∆ t,n + N ( ℵ ,p ) k ∆ t ( S ( p ) k ∆ t,n e r k ∆ t ∆ t − S ( p,d )( k +1)∆ t,n ) , if ζ ( p ∆ t ) k +1 ,n = 0 , η ( ℵ ) k +1 ,n = 1 ,S ( p,u )( k +1)∆ t,n + N ( ℵ ,p ) k ∆ t ( S ( p ) k ∆ t,n e r k ∆ t ∆ t − S ( p,u )( k +1)∆ t,n ) , if ζ ( p ∆ t ) k +1 ,n = 1 , η ( ℵ ) k +1 ,n = 0 ,S ( p,d )( k +1)∆ t,n + N ( ℵ ,p ) k ∆ t ( S ( p,d )( k +1)∆ t,n − S ( p ) k ∆ t,n e r k ∆ t ∆ t ) , if ζ ( p ∆ t ) k +1 ,n = 0 , η ( ℵ ) k +1 ,n = 0 , (46) k = 0 , , ..., n − , n ∆ t = T . At time k ∆ t , it costs nothing to enter a forward contract with terminal time( k + 1)∆ t . Thus, setting R ( ℵ ,p ; C ) k,n = log( S ( ℵ ,p ; C ) k +1 ,n S ( ℵ ,p ; C ) k,n ), it follows that E ( R ( ℵ ,p ; C ) k,n | S ( ℵ ,p ; C ) k,n ) = ( µ k ∆ t + N ( ℵ ,p ) k ∆ t ( µ k ∆ t − r k ∆ t )(2 p k ∆ t − p ℵ k ∆ t − t + 2 N ( ℵ ,p ) k ∆ t σ k ∆ t p p k ∆ t (1 − p k ∆ t )(2 p ℵ k ∆ t − √ ∆ t,V ar ( R ( ℵ ,p ; C ) k,n | S ( ℵ ,p ; C ) k,n ) = σ k ∆ t ((1 + N ( ℵ ,p ) k ∆ t ) − N ( ℵ ,p ) k ∆ t (2 p k ∆ t − p ℵ k ∆ t − t − σ k ∆ t N ( ℵ ,p ) k ∆ t p k ∆ t (1 − p k ∆ t )(2 p ℵ k ∆ t − ∆ t. As discussed in Section 5.2, we set p ℵ k ∆ t = (1 + ψ ( ℵ ) k ∆ t √ ∆ t ). Then with o (∆ t ) = 0, E ( R ( ℵ ,p ; C ) k,n | S ( ℵ ,p ; C ) k,n ) =( µ k ∆ t + 2 N ( ℵ ,p ) k ∆ t σ k ∆ t p p k ∆ t (1 − p k ∆ t ) ψ ( ℵ ) k ∆ t )∆ t , and V ar ( R ( ℵ ,p ; C ) k,n | S ( ℵ ,p ; C ) k,n ) = σ k ∆ t (1 + N ( ℵ ,p ) k ∆ t )∆ t . The in-stantaneous market price of risk is given byΘ( R ( ℵ ,p ; C ) k,n | S ( ℵ ,p ; C ) k,n ) = θ k ∆ t + 2 N ( ℵ ,p ) k ∆ t p p k ∆ t (1 − p k ∆ t ) ψ ( ℵ ) k ∆ t q N ( ℵ ,p ) k ∆ t . Then, the optimal N ( ℵ ,p ) k ∆ t maximizing Θ( R ( ℵ ,p ; C ) k,n | S ( ℵ ,p ; C ) k,n ), is N ( ℵ ,p ) k ∆ t = N ( ℵ ,p ; opt ) k ∆ t = 2 ψ ( ℵ ) k ∆ t θ k ∆ t p p k ∆ t (1 − p k ∆ t ), and the optimal instantaneous market price of risk isΘ( R ( ℵ ,p ; C ) k,n | S ( ℵ ,p ; C ) k,n ) = Θ ( opt ) ( R ( ℵ ,p ; C ) k,n | S ( ℵ ,p ; C ) k,n ) = q θ + 4 p k ∆ t (1 − p k ∆ t ) ψ ( ℵ ) k ∆ t . With N ( ℵ ,p ) k ∆ t = N ( ℵ ,p ; opt ) k ∆ t , E ( R ( ℵ ,p ; C ) k,n | S ( ℵ ; C ) k,n ) = ( µ k ∆ t +4 σ k ∆ t p k ∆ t (1 − p k ∆ t ) ψ ( ℵ )2 k ∆ t θ k ∆ t )∆ t , and V ar ( R ( ℵ ,p ; C ) k,n | S ( ℵ ; C ) k,n ) = σ k ∆ t (1 + 4 p k ∆ t (1 − p k ∆ t ) ψ ( ℵ )2 k ∆ t θ k ∆ t )∆ t . With every single share of traded stock with price S ( p ∆ t ) k,n at k ∆ t , ℵ simultaneously enters N ( ℵ ) – forward contracts. Theforward contracts are long or short, depending on ℵ ’s view on stock price direction in time period [ k ∆ t, ( k + 1)∆ t ]. By assumption, µ t > r t >
0, and thus, θ t = µ t − r t σ t > , t ∈ [0 , T ]. { S ( ℵ ,p ; C ) k,n , k = 0 , ..., n } as ∆ t ↓
0. By the DPIP, it follows that,in the limit ∆ t ↓ ℵ hedges his short derivative position using the price process S ( ℵ ,p ; C ) t , t ≥ S ( ℵ ,p ; C ) t = S exp { ( µ ( ℵ ,p ; C ) t − σ ( ℵ ,p ; C ) t ) t + σ ( ℵ ,p ; C ) t B t } , (47)where µ ( ℵ ,p ; C ) t = µ t + 4 σ t p t (1 − p t ) ψ ( ℵ )2 t θ t , σ ( ℵ ,p ; C ) t = σ t r p t (1 − p t ) ψ ( ℵ )2 t θ t . ℵ forms his instantaneousriskless replicating portfolio Π ( ℵ ,p ; C ) t = a ( ℵ ,p ; C ) t S ( ℵ ,p ; C ) t + b ( ℵ ,p ; C ) t β t = f t = f ( S ( ℵ ,p ; C ) t , t ) , t ∈ [0 , T ) with df t = d Π ( ℵ ,p ; C ) t = a ( ℵ ,p ; C ) t dS ( ℵ ,p ; C ) t + b ( ℵ ,p ; C ) t dβ t . As in Section 3.2, f ( x, t ) , x > , t ∈ [0 , T ), satisfies the PDE ∂f ( x, t ) ∂t + ( r t − D ( ℵ ,p ; C ) y,t ) x ∂f ( x, t ) ∂x + 12 σ t x ∂ f ( x, t ) ∂x − r t f ( x, t ) = 0 , (48)where x > , t ∈ [0 , T ). And f ( x, T ) = g ( x ) is the boundary condition . The dividend yield D ( ℵ ,p ; C ) y,t in (48)is given by D ( ℵ ,p ; C ) y,t = σ t ( θ ( ℵ ,p ; C ) t − θ t ), where θ ( ℵ ,p ; C ) t = µ ( ℵ ,p ; C ) t − r t σ ( ℵ ,p ; C ) t = θ t + − θ t q θ t + 4 p t (1 − p t ) ψ ( ℵ ) t > θ t The PDE (48) has Feynman-Kac probabilistic solution : f ( x, t ) = E Q [ e − R Tt r s ds X T | X t = x ] , (49)where the process X t , t ∈ [0 , T ] is defined on a stochastic basis (Ω , F = {F t } t ≥ , Q ) with filtration F , generatedby the BM B Q t , t ≥
0, and satisfies the stochastic differential equation: dX t = ( r t − D ( ℵ ,p ; C ) y,t ) X t dt + σ t X t dB Q t , t ∈ [0 , T ] . (50)The yield D ( ℵ ,p ; C ) y,t for a misinformed trader is negative, and thus, in general, if we set p ℵ k ∆ t = (1+ ψ ( ℵ ) k ∆ t √ ∆ t ),for some function ψ ( ℵ ) t ∈ R, t ∈ [0 , T ], with continuous first derivative on [0 , T ], the dividend yield D ( ℵ ,p ; C ) y,t in (50) is given by D ( ℵ ,p ; C ) y,t = sign( ψ ( ℵ ) t ) − θ t q θ t + 4 p t (1 − p t ) ψ ( ℵ ) t . In the literature on binomial option pricing, valuation is performed in four steps. In the first step, thebinomial model in the natural world is used, where the probability for the underlying stock upturn and stockmean return are model parameters. Then, the risk-neutral probabilities are found, which should depend onthose two parameters, as shown in Kim et al. (2016, 2019). The second step involve obtaining continuous-time model using the Donsker-Prokhorov invariance principle to derive the continuous-time dynamics of theunderlying stock in the natural world. It is in this step that the probability for a stock upturn is naturallylost. Deriving the continuous-time risk-neutral valuation is the third step, where due to the presumed abilityof the hedger to trade continuously with no transaction costs, the second parameter, the stock mean return,also disappears. In the fourth step, returning to the risk-neutral option price dynamics in the binomialdiscrete-time model, the risk-neutral probability now depends neither on the probability for a stock upturnnor on the mean return. In trinomial and multinomial option pricing models, the first three steps areabandoned, and only the last step is considered, leaving silent the issue of which discrete-pricing model inthe natural world led to the discrete model in the risk-neutral world. Appendix E, formula (E.8) in Duffie (2001).
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