Equilibrium price and optimal insider trading strategy under stochastic liquidity with long memory
aa r X i v : . [ q -f i n . M F ] J a n Equilibrium price and optimal insider trading strategy understochastic liquidity with long memory ∗ Ben-zhang Yang a , Xinjiang He b , Nan-jing Huang a, † a. Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, P.R. Chinab. School of Mathematics and Applied Statistics, University of Wollongong NSW 2522, Australia Abstract:
In this paper, the Kyle model of insider trading is extended by characterizing thetrading volume with long memory and allowing the noise trading volatility to follow a generalstochastic process. Under this newly revised model, the equilibrium conditions are determined,with which the optimal insider trading strategy, price impact and price volatility are obtainedexplicitly. The volatility of the price volatility appears excessive, which is a result of the factthat a more aggressive trading strategy is chosen by the insider when uninformed volume ishigher. The optimal trading strategy turns out to possess the property of long memory, and theprice impact is also affected by the fractional noise.
Key Words:
Asset pricing; equilibrium; insider trading; optimal investment; liquidity;fractional Brownian motion.
Equilibrium asset pricing is always an important and ongoing topic for any financial market, and it has beenextensively studied by a number of authors [9, 11, 18, 28, 31]. It also can be used to deal with insider tradingproblem and the history can be dated back to 1985, when Kyle [22] developed a model, in which a largetrader is assumed to possess long-lived private information about the value of a stock that will be revealedat some known date and optimally trades continuously into the stock to maximize his/her expected profits,while risk-neutral market makers try to infer the information possessed by the insider from the aggregateorder flow. The resulted equilibrium price dynamic actually responds linearly to the order flow instead ofbeing fully revealed, since the order flow is also driven by uninformed noise traders aiming solely at liquiditypurposes. Albeit appealing, the Kyle model also suffers from two major drawbacks; Kyle’s lambda, which isa measurement of the equilibrium price impact of the order flow is constant, and the price volatility turnsout to be constant and independent of the noise trading volatility.To further investigate the impact of asymmetric information on asset prices, volatility, volume, andmarket liquidity, Admati and Pfleiderer [1], Foster and Viswanathan [13, 14] were the first to modify theKyle model, extending it to dynamic economies with myopic agents so that the price volatility is no longer ∗ This work was supported by the National Natural Science Foundation of China (11471230, 11671282). † Corresponding author. E-mail address: [email protected]
In this section, some useful concepts and results are presented. We assume that (Ω , F T , P ) is a probabilityspace, with F t , σ ( X s ; 0 ≤ s ≤ t ) being the smallest σ -filed with respect to which the random variable X s ismeasurable for all s ∈ [0 , t ]. Let L ∞ (Ω , F T , P ) represent the space of all bounded and F -measurable randomvariables and M c,loc denote the space of all continuous local martingales. Assume that h X i t is defined as thequadratic variation process of X t and C stands for the space of the continuous functions x = ( x t ; 0 ≤ t ≤ , P and P N B ( R d +1 ) be the predictable σ -field and the product σ -filed formed from the σ -fields P and B ( R d +1 ), respectively. Lemma 2.1. (Theorem 4.6 in Chapter 3 of [21]) Let M = { M t , F t ; 0 ≤ t ≤ ∞} ∈ M c,loc satisfy lim t →∞ h M i t = ∞ , a.s. Define, for each ≤ s < ∞ , the stopping time as T ( s ) = inf { t ≥ h M i t > s } . Then the following time-changed process B s , M T ( s ) , G s , F T ( s ) ; 0 ≤ s < ∞ is a standard Brownian motion. In particular, the filtration {G s } satisifies the usual conditions and M t = B h M i t ; 0 ≤ t < ∞ , a.s. Lemma 2.2. (Problem 9.3 in Chapter 2 of [21]) Let W = { W t , F t ; 0 < t < ∞} be a standard Brownianmotion. Then lim t →∞ W t t = 0 , a.s. Lemma 2.3. (Theorem 4.6 in [25]) Let the non-anticipative functionals a ( t, x ) , b ( t, x ) , t ∈ [0 , , x ∈ C ,satisfy the Lipschitz condition | a ( t, x ) − a ( t, y ) | + | b ( t, x ) − b ( t, y ) | ≤ L Z t | x s − y s | dK ( s ) + L | x t − y t | and a ( t, x ) + b ( t, x ) ≤ L Z t (1 + x s ) dK ( s ) + L (1 + x t ) , where L and L are constants, K ( s ) is a non-decreasing right continuous function with < K ( s ) < , and x, y ∈ C . Let η = η ( ω ) be an F -measurable random variable with P ( | η ( ω ) < ∞| ) = 1 . Then the equation dx t = a ( t, x ) dt + b ( t, x ) dW t , x = η, has a unique strong solution ξ = ( ξ t , F t ) . ℓ : R → R + be a strictly positive and continuous function. We say that ℓ belongs to the class L if itsatisfies: for all −∞ < a ≤ ≤ b < + ∞ , the following ODEs L t = a − Z Tt ℓ ( L s ) ds, U t = b + Z Tt ℓ ( U s ) ds (1)have global bounded solutions on [0 , T ]. Lemma 2.4. ([24]) ℓ ∈ L if and only if Z −∞ dxℓ ( x ) = Z ∞ dxℓ ( x ) = ∞ . Moreover, when ℓ ∈ L , the equations presented in (1) have unique solutions. Lemma 2.5. ([24]) Assume that ξ ∈ L ∞ (Ω , F T , P ) and f is a P N B ( R d +1 ) measurable function such that f ( t, ω, · , · ) is continuous for all t and ω , and there exists some finite constant C such that | f ( t, ω, y, z ) | ≤ ℓ ( y ) + C | z | for all t , ω , y and z . If ℓ ∈ L , then the BSDE (backward stochastic differential equation) Y t = ξ + Z Tt f ( s, ω, Y s , Λ s ) ds − Z Tt Λ s dW s (2) has a maximal bounded solution ( Y, Λ) . Moreover, Y is a continuous process and L ≤ L t ≤ Y t ( ω ) ≤ U t ≤ U holds for all t and ω , where ( L, U ) are the unique solutions of (1) with b = k ξ k ∞ and a = −k ξ k ∞ . Lemma 2.6. ([24]) Assume that f and h are two P N B ( R d +1 ) measurable functions such that(i) for all t , ω ; f ( t, ω, · , · ) is continuous;(ii) for all t , ω , y , z ; f ( t, ω, y, z ) ≥ h ( t, ω, y, z ) ;(iii) there exists a constant C > such that, for all t , ω , y , z ; | f ( t, ω, y, z ) | < ℓ ( y ) + C | z | where ℓ ∈ L ;(iv) ξ , η ∈ L ∞ (Ω , F T , P ) and ξ ≤ η P − a.s. If ( X, Z ) is a maximal bounded solution of (2) with terminal value ξ and coefficient f , and ( Y, Γ) is a boundedsolution of (2) with terminal value η and coefficient h , then Y ≤ X . Definition 2.1. ([26]) A fractional Brownian motion B Ht with Hurst index H is a centered Gaussian processsuch that its covariance function R ( t, s ) = E [ B Ht B Hs ] is given by R ( t, s ) = 12 ( | t | H + | s | H − | t − s | H ) , where < H < . Remark 2.1. If H = , R ( t, s ) = min( t, s ) and B Ht is the usual standard Brownian motion. If H = , B Ht is neither a semimartingale nor a Markov process. It is a process of long memory in the following sense[29]: If ρ n = E [ B H ( B Hn +1 − B Hn )] , then the series P ∞ n =0 ρ n is either divergent or convergent with very laterate. Equilibrium price and optimal strategy
We start by presenting several fundamental assumptions made in our model. For a certain stock price process P t , there is an insider who is risk-neutral and knows its terminal stock value v . The insider tries to maximizethe expectation of his/her terminal profit as follows: J t = max { θ s } s ≥ t ∈A E " Z Tt ( v − P s ) θ s ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F Yt , v . (3)Here, following [4], we assume that the insider chooses an absolutely continuous trading rule θ which belongsto an admissible set A = ( θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E "Z T θ s ds < ∞ ) . (4)Moreover, the aggregate order flow arrival Y t (caused by the insider’s demand and the noise-trader’s demand)is assumed to follow dY t = θ t dt + σ t dB Ht , (5)where B Ht is a fractional Brownian motion independent of v , which implies that the volume has remarkablymemory characteristics, H is restricted within ( ,
1] as it is of interest in finance to investigate the effect of thelong-range dependence exhibited by the volatility, F Yt is the filtration generated by the historic informationof the aggregate order flow Y t = { Y u } u ≤ t . Another assumption we made here is that that σ t follows dσ t = m ( t, σ t ) σ t dt + ν ( t, σ t ) σ t dM t , (6)where M t is a martingale, the long term rate m ( t, σ t ) and volatility ν ( t, σ t ) are dependent on the past historyof the volatility σ t , instead of depending on the history of Y .On the other hand, the market maker is also risk-neutral, and has a prior information that v follows thenormal distribution N ( P , Σ ), instead of having the knowledge of the terminal value v . The market makerabsorbs the total order flow by trading against it at a price which is determined to achieve break even onaverage. As the market maker is risk-neural, equilibrium break-even requires that the market clearing priceis P t = E (cid:2) v |F Yt (cid:3) . (7)Finally, both the market maker and the insider are assumed to perfectly observe the history of σ , whichindicates that the filtration F Yt here contains both history of the order flow Y t and volatility σ t . This isdifferent from what is assumed in the Kyle model, where only equilibrium prices are observable, since theinsider there is able to recover the total order flow, while only observing prices are not sufficient for therecovery of the noise trading volatility, as the volatility of the uninformed order flow in our model is nolonger constant, but a stochastic variable.It should be remarked here that if the uncertain term were general Brownian motion without memories,then the model would degenerate to the one studied by Collin-Duersne and Fos [11]. Furthermore, if thenoise trading σ t were constant, then the model would become exactly the same as the classical Kyle-Backmodel [22]. 5 .1 Equilibrium price process Definition 3.1.
An equilibrium occurs when ( P T , θ t ) satisfy the condition (7) while solving the insider’soptimal problem (3) . To solve this equilibrium, we proceed in a three-step process: a) the stock price dynamic is presented,being consistent with the market maker’s risk-neutral filtration, which is conditional on a conjectured strategyrule followed by the insider; b) the insider’s optimal problem (3) is solved under the assumed dynamic (7);c) solving the conjecture rule (7) is shown to be consistent with the optimal solution of (3), which showsthat we have reached the equilibrium defined in Definition 3.1.We start by presenting a few lemmas, which will be used when deriving the optimal trading strategy ofthe insider. Lemma 3.1 below shows that the change of the stock price process P t is linear in the order flow,if the insider takes a trading strategy that is linear in his/her profit with memory. Lemma 3.1.
If the insider adopts the following memorable trading strategy θ t = β t ( v − P t ) − ( H −
12 ) ψ t (8) for some F Yt adapted process β t with ψ ( t ) = Z t ( t − u + ε ) H − dW u for any ε > , then the stock price defined in (7) follows dP t = λ t dY t , (9) where λ t is the price impact defined by λ t = β t Σ t ε H − σ ( t ) , (10) and Σ t is the following conditional variance Σ t = E (cid:2) ( v − P t ) (cid:12)(cid:12) F Yt (cid:3) . (11) Moreover, the dynamic of Σ t can be formulated as follows: d Σ t = − λ t ε H − σ t dt. (12) Proof.
If the stochastic process ψ t is defined by ψ t = Z t ( t − u + ε ) H − dW u , then the stochastic theorem of Fubini yields Z t ψ s ds = Z t Z s ( s − u + ε ) H − dW u ds = Z t (cid:20)Z us ( s − u + ε ) H − ds (cid:21) dW u = 1 H − (cid:20)Z t ( t − u + ε ) H − dW u − ε H − W t (cid:21) . B ε,Ht = Z t ( t − s + ε ) H − dW s . Then, based on the expression of the fractional Brownian motion B Ht = Z t ( t − s ) H − dW s , we can obtain Z t ψ s ds = 1 H − h B ε,Ht − ε H − W t i . (13)This leads to B ε,Ht = ( H −
12 ) Z t ψ s ds + ε H − W t , and thus B ε,Ht is a semimartingale. Moreover, B ε,Ht uniformly converges to B Ht for t ∈ [0 , T ] in L (Ω) when ε tends to 0. Therefore, we hereafter use B ε,Ht to approximate the original fractional Brownian motion B Ht ,which is a common practice in mathematical finance now [27, 30].Now, with the standard Gaussian projection theorem (Theorems 12.6 and 12.7 in [25]), we can derive P t + dt = E[ v | Y t , Y t + dt , σ t , σ t + da ]= E[ v | Y t , σ t ] + Cov( v, Y t + dt − Y t | Y t , σ t )V( Y t + dt − Y t | Y t , σ t ) × (cid:0) Y t + dt − Y t − E[ Y t + dt − Y t | Y t , σ t ] (cid:1) = P t + E[( v − P t )( θ t + ( H − ) ψ t ) | Y t , σ t ] dtε H − σ t dt ( Y t + dt − Y t )= P t + E[ β t ( v − P t ) | Y t , σ t ] dtε H − σ t dt ( Y t + dt − Y t )= P t + β t Σ t ε H − σ t dY t . (14)Here, the second equality uses the fact that σ t is independent of the asset value in the order flow, the thirdequality is obtained from the fact that the expected change in the order flow is zero for the conjecturedpolicy θ t , and the last equality follows from (11).Finally, applying the projection theorem yieldsVar[ v | Y t , Y t + dt , σ t , σ t + da ] = Var[ v | Y t , σ t ] − (cid:18) β t Σ t ε H − σ t (cid:19) Var[ Y t + dt − Y t | Y t , σ t ] , (15)which leads to Σ t + dt = Σ t − ε H − λ t σ t , (16)where λ t is defined by (10). This completes the proof.As our model now exhibits long memory, we also need to introduce the new market depth process andderive the corresponding equilibrium price process, before we are able to find the optimal solution to theproblem (3). Lemma 3.2.
Define G t by setting λ t = q Σ t G t and assume that the market depth (i.e. Kyle’s lambda) process λ t is martingale. If Σ T = 0 , σ t is uniformly bounded above by σ and below by σ > , then there exists aunique bounded solution G t satisfying σ ε H − ( T − t ) ≤ G t ≤ σ ε H − ( T − t ) . (17)7 roof. By setting E[ d λ t ] = 0 and using (12), one hasE[ dG t ] = − ε H − σ t √ G t dt (18)with the terminal condition G T = 0, which implies that G t satisfies p G t = E " Z Tt ε H − σ s √ G s ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F σt . (19)Thus, √ G t solves the BSDE dy t = − f ( t, y t ) dt − Λ t dM t , y T = 0 , where f ( t, y ) = ε H − σ t y . If we define ℓ ( y ) = ε H − σ | y | , then it is not difficult to find that f ( t, y t ) ≤ ℓ ( y t ) for all ( t, ω ) and Z ∞ ℓ ( x ) dx = Z −∞ ℓ ( x ) dx. As a result, according to Lemmas 2.4 and 2.5, there exist two solutions L ( t ) and U ( t ) solving L t = − R Tt ℓ ( L s ) ds and U t = − R Tt ℓ ( U s ) ds , respectively, such that L t ≤ G t ≤ U t . Moreover, it is easy to de-rive that U t = − L t = ε H − σ ( T − t ) , yielding the upper bound of G t . On the other hand, if we consider the solution to the following backwardequation dx t = − ε H − σ x t dt − e Λ t dM t with terminal condition x T = 0, then we can actually obtain x t = ε H − σ √ T − t by setting e Λ t = 0. Considering f ( t, y t ) ≥ ε H − σ y , ∀ ( t, ω ) , the comparison result of Lemma 2.6 leads to y t ≥ x t which further yields the lower bound on the maximalsolution for G t .If we define g t = √ G t and assume that there are two uniformly bounded solutions g ( t ) and g ( t ), thenthe difference between the two solutions, ∆ t = g t − g t , satisfies∆ t = E " Z Tt − ∆ s ε H − σ s g s g s ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F σt . If we denote a t = ε H − σ s p g s g s , then a t ≥ (cid:16) − R t a s ds (cid:17) ∆ t is a bounded continuous martingale. Therefore, we can finally reach theconclusion that ∆ t = 0 since ∆ T = g T − g T = 0. This completes the proof. Lemma 3.3.
The price process P t driven by (9) and (10) converges almost surely to v at time T . roof. It is straightforward that dP t = λ t dY t = r Σ t G t θ t dt + σ t dB ε,Ht = r Σ t G t (cid:20)(cid:18) θ t + ( H − ψ t ) (cid:19) dt + ε H − σ t dW t (cid:21) = r Σ t G t h β t ( v − P t ) dt + ε H − σ t dW t i = v − P t G t ε H − σ t dt + r Σ t G t ε H − σ t dW t (20)and d Σ t = − Σ t G t ε H − σ t dt, Σ T = 0 . (21)If we consider the process X t = P t − v , then X t = e − R t ε H − σ uGu du X + Z t e − R ts ε H − σ uGu du r Σ s G s ε H − σ s dW s , I + I . From (17), it is not difficult to figure out ε H − σ σ log (cid:18) TT − t (cid:19) ≤ Z t ε H − σ u G u du. ≤ ε H − σ σ log (cid:18) TT − t (cid:19) , which directly leads to lim t → T I ( t ) = 0. Moreover, I ( t ) can be alternatively expressed as I ( t ) = e − R t ε H − σ uGu du M t , where M t = Z t e R s ε H − σ uGu du r Σ s G s ε H − σ s dZ s is a Brownian martingale, whose quadratic variation is equal to h M i t = Z t e R s ε H − σ uGu du Σ s G s ε H − σ s ds = Σ Z t e R s ε H − σ uGu du ε H − σ s G s ds = Σ (cid:18) e R t ε H − σ uGu du − (cid:19) . According to Lemma 2.1, there exists a standard Brownian motion B t such that the continuous martingalecan be viewed as a time-changed Brownian motion, i.e., M t = B h M i t . Applying the strong law of largenumbers for Brownian motion specified in Lemma 2.2, we finally arrive at the desired resultlim t → T I ( t ) = lim t → T e − R t ε H − σ uGu du M t = lim t → T B h M i t h M i t Σ = lim τ →∞ B τ τ + τ = 0 . (22)This completes the proof. Remark 3.1.
If we define the mean-reversion rate of P t as κ t = ε H − σ t G t , hen we can obtain the mean-reverting form of P t as dP t = κ t ( v − P t ) dt + p Σ e − R t κs ds √ κ t dW t (23) and the insider would adopt the following memorable trading strategy θ t = κ t λ t ( v − P t ) − ( H −
12 ) ψ t . (24) Remark 3.2.
There is also a useful result about the limiting distribution of the standard price process h t = P t − v √ Σ t . Itˆo’s lemma can yield dh t = − ε H − σ t G t h t dt + ε H − σ t √ G t dZ t , (25) which implies that h t is a time-change Ornstein-Uhlenbeck (O-U) process with the following stochastic timechange process: τ t = Z t ε H − σ s G s ds, being independent of the filtration generated by Z t . Furthermore, since E [ h T ] = 0 and E [ h T ] = 1 , the limitingdistribution of h T is a standard normal distribution. With the results presented above, we are now able to show that the market depth process is a martingaleand a new bound can be established for G t , the details of which are illustrated below. Lemma 3.4.
Market depth process λ t is a martingale which is orthogonal to the order flow. Moreover, theprice impact process λ t is a submartingale.Proof. From the definition of G t specified in Lemma 3.2, it is straightforward to deduce d p G t + ε H − σ t √ G t dt = d M t , (26)where M t = E " Z T ε H − σ t √ G t dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ t . Since M t ≤ ε H − σ σ √ T , which is a direct result of (17), M t is actually a martingale adapted to the filtration generated by the noisetrading volatility process. Using the definition of market depth process, one can easily obtain d λ t = d r Σ t G t = 1Σ t d p G t − √ G t t d Σ t = 1Σ t d M t , (27)which clearly shows that λ t is a martingale. Furthermore, since Z t and M t are independent with each other,we have d M t dZ t = 0 and so d λ t dY t = 0. Finally, a further computation using Jensen’s inequality yields1E[ λ s |F t ] ≤ E (cid:20) λ s (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) = 1 λ t , indicating that λ t ≤ E [ λ s |F t ]. This completes the proof.10 emark 3.3. It should be remarked that this contradicts to the results from most of the previous literature.While the price impact is constant in the original the Kyle model and either a martingale or a supermartingalein various well-known extensions of the Kyle model [3, 4, 5, 6, 8], our result is consistent with what ispresented in [11], where the insider trader has an opportunity to wait for a better liquidity to trade. Indeed,the price impact must increase on average to encourage the insider to trade early and give up his opportunityto wait for better liquidity states under the framework of stochastic liquidity.
Lemma 3.5.
If a bounded solution G t exists, then G t ≤ ε H − E "Z Tt σ s ds . Proof.
Applying Itˆo’s formula to G t directly yields d p G t = 2 p G t d p G t + d [ p G t ] t = − ε H − σ t dt + 2 p G t d M t + d [ p G t ] t = − ε H − σ t dt + 2 p G t d M t + Σ t d (cid:20) λ (cid:21) t . (28)Integrating (28) from t to T and taking the expectation on both sides of the resulted equation, we can getthe desired result. With all necessary results presented in the previous subsection, we are now ready to show that the strategy(24) is indeed the optimal solution to the target problem (3). The main results are summarized in thefollowing proposition.
Proposition 3.1.
If price dynamic is given by (7) , (10) and the volatility is uniformly bounded above by σ and below by σ > , then the optimal value process defined in (3) can be derived as J t = ( v − P t ) + Σ t λ t + (cid:18) H − (cid:19) Z t ψ s ( v − P s ) ds − (cid:18) H − (cid:19) A, (29) and the optimal trading strategy has the following expression θ ∗ t = ε H − σ t λ t G t ( v − P t ) − ( H −
12 ) ψ t . (30) Here, the constant A can be computed from A = E " Z T ψ t ( v − P t ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F Y , v , (31) where the dynamic of P t is specified in (23) .Proof. We conjecture that the target value function can be expressed as (29). Applying Itˆo’s lemma to J t yields dJ t = ( v − P t ) + Σ t dλ t + 1 λ t (cid:18) − ( v − P t ) dP t + 12 ( dP t ) (cid:19) + 12 λ t d Σ t + (cid:18) H − (cid:19) ψ t ( v − P t )= ( v − P t ) + Σ t √ Σ t d M t + 12 λ t d Σ t + 1 λ t (cid:20) − ( v − P t ) λ t (cid:16) θ t dt + ε H − σ t dW t (cid:17) + 12 ε H − σ t dt (cid:21) = ( v − P t ) + Σ t √ Σ t d M t − θ t ( v − P t ) dt − ε H − σ t ( v − P t ) dW t , (32)11here the second equality is obtained using (20) in Lemma 3.3 and the third equality is a consequence of(12) in Lemma 3.1. Integrating (32) from 0 to T , we further obtain J T − J + Z T θ t ( v − P t ) dt = Z T ( v − P t ) + Σ t √ Σ t d M t + Z T ε H − σ t ( v − P t ) dW t . (33)If we define I ( t ) = Z t ε H − σ s ( v − P s ) dW s , and I ( t ) = Z t ( v − P s ) + Σ s √ Σ s d M s , then they are actually martingales for any admissible strategy. To prove this, we start from (23) and (24)by expressing P t as P t = P + Z t λ s b θ s ds + Z t ε H − σ s λ s dW s with b θ s = θ s + (cid:0) H − (cid:1) ψ s . Clearly, one has Z t ε H − σ s λ s dW s = Σ − Σ t < + ∞ and E (cid:20)Z t λ s b θ s ds (cid:21) ≤ E (cid:20)Z t λ s ds Z t b θ s ds (cid:21) ≤ E (cid:20)Z t σ Σ s G s σ s ds Z t b θ s ds (cid:21) = E (cid:20) Σ − Σ t ε H − σ Z t b θ s ds (cid:21) ≤ E (cid:20) Σ ε H − σ Z t b θ s ds (cid:21) ≤ ε H − σ E (cid:20)Z t b θ s ds (cid:21) ≤ ε H − σ E (cid:20)Z t θ s ds + ( H −
12 ) Z t ψ s ds (cid:21) < ∞ , where the third equality follows from (21) and the last inequality is a result of (4) and (13). This impliesthat P t has finite variance. Considering that σ t is uniformly bounded, we can certainly obtainE "Z T ε H − σ t ( v − P t ) dt < ∞ , and thus I ( t ) is a martingale. On the other hand, it follows from Lemma 3.4 and Remark 3.2 thatE (cid:20) ( v − P t ) Σ t (cid:21) < Σ E[ h t ] < ∞ , which further leads to E "Z T ( v − P s ) Σ s d hM s i < ∞ . In this case, Z T ( v − P t ) √ Σ t d M t is also a martingale, since M t is a uniformly bounded martingale. As a result, considering Σ t is a decreasingprocess, I ( t ) is a martingale.Now, if we take expectation on both sides of (33), then J = E "Z T θ t ( v − P t ) dt + J T (34)12or any θ ∈ A . Since ( v − P t ) +Σ t λ t ≥
0, the expectation of J T , E[ J T ], can be calculated from (29) by setting t = T and it is straightforward that E[ J T ] ≥
0, yieldingE "Z T θ t ( v − P t ) dt ≤ J . This implies that J will be the optimal value function if there exists a trading strategy θ ∗ t , being consistentwith (10), such that E[ J T ] = 0. Indeed, from Lemma 3.2 and (31), we getE[ J T ] = E (cid:20) ( v − P T ) λ T + √ Σ T G T (cid:21) = E (cid:20) ( v − P T ) λ T (cid:21) ≤ vuut E [( v − P T ) G T ] E "(cid:18) v − P T √ Σ T (cid:19) = 0 . (35)Therefore, we have proved the optimality of the value function and that of the trading strategy.From Proposition 3.1 and Remark 3.2, it is clear that the equilibrium price admits a bridge process thatconverges to the value v which is only known to the insider at maturity T . This guarantees that all privateinformation will have been incorporated into equilibrium prices at maturity, and the result presented in [3]that the equilibrium price in the continuous-time Kyle model follows a standard Brownian Bridge with theconstant volatility is generalized. It can also be observed from our model that only if the mean-reversionrate κ t is stochastic will the equilibrium price volatility be stochastic.Proposition 3.1 further indicates that while the optimal trading strategy for the insider is to tradeproportionally to the undervaluation of the asset v − P t at a rate that is inversely related to his/her priceimpact λ t , it is a monotonic increasing function of the current “state of liquidity”, which is measured bythe relative difference between the current noise trading variance σ t and the expected noise trading variance G t , i.e., σ t G t . It should also be remarked that due to the presence of the fractional Brownian motion, ouroptimal strategy θ ∗ ( t ), embracing the stochastic term ( H − ) ψ t , also shows long memory, a property that isnot demonstrated in [11], implying that the introduction of the fractional Brownian motion has a significantimpact on the choice of the optimal strategy.Some further remarks are made below for aggregate execution or slippage costs incurred by uninformedliquidity traders. Remark 3.4.
The total losses between and T by noise traders can be derived through Z T ( P t + dt − v ) σ t dB ε,Ht = Z T ( P t + dP t − v ) σ t dB ε,Ht = Z T ε H − λ t σ t dt + Z T ( P t − v ) σ t dB ε,Ht . (36) The first component is the pure execution or slippage cost caused by the situation that market orders submittedat time t in the Kyle model will get executed at date t + dt at a price set by competitive market makers. Thesecond component is a fundamental loss resulted from noise traders purchasing a security with long memory,whose fundamental value v is unknown to them. From this, using a similar definition in [11], aggregateexecution or slippage costs incurred by uninformed liquidity traders here can be obtained as Z T σ t dB ε,Ht dP t = Z T ε H − λ t σ t dt, (37) which is stochastic, being path-dependent, and is affected by the fractional noise. emark 3.5. The unconditional expected profits of the insider can be determined through E "Z T ( v − P t ) θ t dt = E "Z T ε H − σ t √ Σ t G t ( v − P t ) dt − ( H −
12 )E "Z T ( v − P t ) ψ t dt = E "Z T ε H − σ t √ Σ t G t Σ t dt − ( H −
12 )E "Z T ( v − P t ) ψ t dt = E "Z T ε H − λ t σ t dt − ( H −
12 )E "Z T ( v − P t ) ψ t dt , where the first equality is obtained with the substitution of θ ∗ t and the second one is derived using the lawof iterated expectations. Being different from the results in [11], the unconditional expected profits of theinsider is no longer equal to the unconditional expected execution costs paid by noise traders, and instead,they have an additional component due to the introduction of the fractional Brownian motion, implyingthat the property of long memory possessed by the aggregate order flow has an influence on the insider’sunconditional expected profits. In this section, two special cases are considered to further investigate the effect of the noise trading volatilityprocesses on the equilibrium, and they are distinguished by whether the growth rate of noise trading isstochastic
This subsection will discuss the case where the growth rate of the noise trading volatility process in (6) isdeterministic (the volatility of that takes a general form). Under this particular assumption, a closed-formsolution for G t can be derived, based on which the equilibrium price process, the equilibrium trading strategy,the equilibrium volatility, and the equilibrium price impact can be obtained. The corresponding results arepresented in the following proposition. Proposition 4.1.
Suppose that the growth rate of the noise trading volatility is deterministic such that D t = Z Tt e R ut m s ds du (38) is bounded for all t ∈ [0 , T ] . Then the solution to (19) can be expressed as G t = ε H − σ t D t (39) and the stock price dynamic has the following form dP t = 1 D t ( v − P t ) dt + e R t m s ds σ v dW t , (40) where σ v = Σ D . In equilibrium, the price impact can be represented by λ t = e R t m s ds ε H − σ v σ t (41)14 nd the optimal trading strategy of the insider can be formulated as θ ∗ t = 1 λ t D t ( v − P t ) − ( H −
12 ) ψ t . (42) Furthermore, the expected trading rate of the insider is E[ θ | v, F ] = e R t m s ds σ ( v − P ) σ v D . (43) Proof.
With the utilization the martingale property, it is straightforward from (6) thatE [ σ u |F t ] = σ t e R ut m s ds . (44)In this case, if we assume that the solution to G t takes the form of (39), then it follows from (19) that p D t = Z Tt e R ut m s ds √ D u du. Clearly, our guess is correct if D t specified in the proposition satisfies this integral equation, which is exactlythe case here. Considering the uniqueness we established above, (39) is indeed the expression of the targetsolution.The expected trading rate of the insider can be computed fromE[ θ t | v, F ] = E (cid:20) v − P t √ Σ t ε H − σ t √ G t − ( H −
12 ) ψ t (cid:21) = E (cid:20) v − P t √ Σ t ε H − σ t √ G t (cid:21) = v − P √ Σ e − R t Ds ds σ ε H − e R t m s ds √ D t , (45)which is a direct result of (19), the dynamic of h t specified in Lemma 3.4, and ψ t being a martingale. Thus,using the identity e − R t Ds ds = r D t D e − R t m s ds , directly yields the desired result.The other results in the proposition actually follow from Proposition 3.1, and this has completed theproof.It should be remarked that our model has successfully taken into consideration the effect of long memory,after the introduction of the fractional Brownian motion. Of course, this model takes [11] as a special casewhen H = is set in(20), and it will further degenerate to the continuous-time Kyle model [3] when σ t = σ with m t = v t = 0 and D t = T − t . In this case, with σ v = Σ T being the annualized variance of themarket maker’s prior, both of the price volatility and price impact are constant, being equal to σ v and σ v σ ,respectively.It should also be noted that the price volatility and the posterior variance of the fundamental value Σ t are deterministic, as a result of the growth rate of the noise trading volatility being deterministic, while theprice impact is stochastic and negatively correlated with the noise trading volatility. On the other hand, theoptimal trading trading strategy of the insider is not only negatively and positively dependent on the priceimpact λ t and the liquidity state κ t , respectively, it also exhibits the property of long memory, due to theintroduction of the fractional Brownian motion. It is also interesting to notice that the fractional noise hasno influence on the expected trading rate of the insider, which is also reasonable as this is in the sense ofaverage. 15 λ t H=0.6H=0.8H=1
Figure 1: The impact of H on λ t . Parameter values are: Σ = 0 . , σ = 1, m s = 1, T=1, ε = 0 . T λ t ǫ =0.01 ǫ =0.1 ǫ =0,5 Figure 2: The impact of ε on λ t . Parameter values are: Σ = 0 . , σ = 1, m s = 1, T=1, H = 0 . λ t (Kyle’s lambda) changes withrespect to the long-memory parameter H and the approximation factor ε is numeircally shown. In specific,Fig. 1 displays that the long-memory parameter H has a positive influence on the price impact, as a higher H value contributes to a higher price impact. This can be understood by the fact that more information isobserved from the order flow when the volatility shows greater long range dependence. Moreover, the priceimpact is more sensitive to the time when H increases, while it is almost a constant when H is low, whichis expected as the approximate fractional Brownian motion degenerates to the standard Brownian motionwhen H approaches and the long memory property no longer exists. On the other hand, an apposite trendcan be observed in Fig. 2 that the smaller the approximation factor ε is, the larger the price impact willbe. This is also reasonable since there will be less fractional noise when ε decreases, implying that moreinformation will be revealed, leading to a larger price impact.However, it needs to be stressed that although the price impact is stochastic, the price volatility isstill a deterministic function, and neither contemporaneous relation between volume changes and the pricevolatility nor that between the price impact and price volatility can be generated. Such relations can only16e generated when the growth rate of the noise trading volatility is stochastic. In the next subsection, aframework generating both stochastic price volatility and a meaningful correlation between the price volatilityand volume will be introduced. In this subsection, a special case where the noise trading volatility follows a two-state continuous Markovchain is presented, and this introduces state-dependent predictability, resulting in successfully capturing thestochastic expected growth rate in the noise trading volatility.We start by specifying the dynamic of σ t as dσ t = (cid:0) σ H − σ t (cid:1) dN L ( t ) − ( σ t − σ L ) dN H ( t ) , (46)where σ L and σ H are two fixed values satisfying σ L < σ H , the initial level of the volatility σ ∈ { σ H , σ L } ,and N i ( t ) is a standard Poisson counting process with jump intensity λ i for i = H , L . In this case, thesolution to (19) is presented in the following proposition. Proposition 4.2.
The unique bounded solution to (19) is given by G t = { σ t = σ L } ε H − G L ( T − t ) + { σ t = σ H } ε H − G H ( T − t ) , where G L and G H are the solutions to the following ODEs (ordinary differential equations) dG L ( τ ) = ( σ L ) + 2 λ L (cid:18)q G H ( τ ) G L ( τ ) − G L ( τ ) (cid:19) ,dG H ( τ ) = ( σ H ) + 2 λ H (cid:18)q G H ( τ ) G L ( τ ) − G H ( τ ) (cid:19) (47) with the boundary conditions G L (0) = G H (0) = 0 .Proof. If we define G ( t, σ t ) = { σ t = σ L } G L ( T − t ) + { σ t = σ H } G H ( T − t )and M ( t ) = p G ( t, σ t ) + Z t σ u p G ( u, σ u ) du, then it is not difficult to find that M ( t ) is a pure jump martingale, which implies that M ( t ) = E[ M ( T ) |F t ],leading to p G ( t, σ t ) + Z t σ u p G ( u, σ u ) du = E " p G ( T, σ T ) + Z T σ u p G ( u, σ u ) du (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t . Considering the boundary conditions G L (0) = G H (0) = 0, it is straightforward that p G ( t, σ t ) = E " Z Tt σ u p G ( u, σ u ) du (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t , and thus q ε H − G ( t, σ t ) = ε H − p G ( t, σ t ) = E " Z Tt ε H − σ u p ε H − G ( u, σ u ) du (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F t . This has completed the proof. 17ccording to Remark 3.1, the price dynamic can be written as dP t = κ ( t, σ t )( v − P t ) dt + σ P ( t ) dW t , (48)where σ P ( t ) = p Σ e − R t κ ( s,σ s ) ds p κ ( t, σ t )and κ ( t, σ t ) = { σ t = σ L } κ L ( T − t ) + { σ t = σ H } κ H ( T − t )with κ i ( T − t ) = ( σ i ) G i ( T − t ) , i = L, H.
Clearly, it can be easily observed that the price now follows a mean-reverting process with stochastic volatility,with both of the mean-reversion speed and the volatility being controlled by the Markov chain. Moreover, ahigh value of the noise trading volatility always contributes a higher mean-reversion speed, which is becausethe insider is expected to trade more aggressively. It should also be noted that both the mean-reversion speedand the volatility would approach infinity when the time becomes closer to expiry, which can be explained bythe fact that agents trade more and more aggressively when it is approaching expiry since they do not wantto leave any money on the table. A positive relationship between volume changes and the price volatility canalso be witnessed, as the price volatility always jumps in the same direction as the noise trading volatilitydoes.
While all the discussions above are based on the assumption that the aggregate order flow and noise tradingvolatility are conditionally uncorrelated for the simplicity of the illustration, this assumption can in fact berelaxed. To illustrate this, let us consider a more general model with the total order flow Y ′ t being definedas follows: dY ∗ t = θ t dt + σ t dB ε,Ht + η ( t, σ t , Y ∗ t ) dM t . (49)If we further define dY t = dY ∗ t − η ( t, σ t , Y ∗ t ) v ( t, σ t ) ( dσ t − m ( t, σ t ) dt ) = η t dt + σ t dB ε,Ht , (50)which is exactly the same as what is presented in (50), then it can be easily shown that all our results aboveare unchanged under this generalized framework, since observing ( Y ∗ t , σ t ) is equivalent to observing Y t , σ t for all market participants. However, one should also notice that our equilibrium proof does depend on theassumption that the history of Y t has no influence on the future dynamic of σ t . On the other hand, sincethe price change is now linear in Y t , it is no longer linear in the total order flow, and instead it is only linearin the component of the order flow that is informative about the insider’s actions.Inspired by the ideas of [2, 10, 11, 16], we also model the time series of price changes as subordinatedto the normal distribution. To capture the heteroscedasticity in returns, microeconomic foundations forsuch a subordinate process modeling the stock return are provided under our model, with the directingprocess being endogenous and related to the trading volume, and the results are presented in the followingproposition. 18 roposition 4.3. If we define the positive increasing stochastic directing process as τ t = T (cid:18) − e − R t ε H − σ uGu du (cid:19) and assume σ v = Σ T , then Σ t = σ v ( τ T − τ t ) and dP t = v − P t τ T − τ t dτ t + σ v dW ′ τ t for some Browinian motion W ′ independent of M , with its definition as σ v dW ′ τ = ε H − λ t σ t dW τ t .Proof. The definition of the time-change yields dτ t = − σ v d Σ t = ε H − Σ t σ t σ v G t = ε H − ( τ T − τ t ) σ t G t , the substitution of which into the equilibrium price process along with Lemma 3.3 and Remark 3.2 leads tothe desired result.From Proposition 4.3, it is clear that the equilibrium price is a time-changed Brownian Bridge, similar towhat is presented in the Kyle-Back model, where the price process is a standard Brownian Bridge. However,our equilibrium can not be derived simply using a time-change of that model, which is a result of the factthat the price impact is constant in the Kyle-Back model, while it is a stochastic process in our case. Onemay also find that price is a time-changed Brownian motion, belonging to the class of subordinate processesproposed in [10], in the market maker’s filtration. Moreover, our model gives an endogenous expression forthe directing process τ t , which depends on the (uninformed) volume dynamic and the fractional noise, whilehaving no requirement on the specification of a latent information process to generate stochastic volatility[2]. An obvious advantage of this model is its generality, as the directing process can be determined forany dynamic of volume, typical examples of which include Normal, Poisson, and Log-normal, implying thatmajor stylized facts can be jointly taken into consideration, which is an important property according to[16]. In this paper, we propose a modified the Kyle model for dynamic insider trading, with the noise tradingvolatility and trading volume being respectively governed by a general stochastic process and a fractionalstochastic process. Under equilibrium conditions, the resulted equilibrium price process exhibits excessivevolatility because of the insider trading more aggressively when uninformed volume is higher. The optimalinsider trading strategy displays long memory, and the price impact is negatively correlated with the noisetrading volatility, which is also affected by the fractional noise.The model makes many simplifying assumptions that could be relaxed to further our standing of howinformation flows into prices and how price volatility, price impact and trading volume change. For instance,we can consider the investment under varying time horizon instead of fixed horizon and assume that thepresence of the insider is common knowledge for the market. We leave these extensions for future research.19 eferences [1] A. Admati, P. Pfleiderer, A theory of intraday patterns: volume and price variability,
Rev. Financ.Stud. , (1988), 3-40.[2] T.G. Andersen, Return volatility and trading volume: an information flow interpretation of stochasticvolatility, J. Finance ,
51 (1) (1996), 169-204.[3] K. Back, Insider trading in continuous time,
Rev. Financ. Stud. , (1992), 387-409.[4] K. Back, H. Pedersen, Long-lived information and intraday patterns, J. Financ. Mark. , (1998), 385-402.[5] K. Back, S. Baruch, Information in securities markets: Kyle meets Glosten and Milgrom, Econometrica ,
72 (2) (2004), 433-465.[6] S. Baruch, Insider trading and risk aversion,
J. Financ. Mark. , (2002), 451-464.[7] T. Bollerslev, D. Jubinski, Equity trading volume and volatility: latent information arrivals and commonlong-run dependencies, J. Bus. Econ. Stat. , (1999), 9-21.[8] R. Caldentey, E. Stacchetti, Insider trading with a random deadline, Econometrica,
78 (1) (2010),245-283.[9] G. Chang, S. Suresh, Asset prices and default-free term structure in an equilibrium model of default,
J.Bus. , (2005), 1215-1266.[10] P.K. Clark, A subordinated stochastic process model with finite variance for speculative prices, Econo-metrica ,
41 (1) (1973), 135-155.[11] P. Collin-Dufresne, V. Fos, Insider trading, stochastic liquidity, and equilibrium prices,
Econometrica ,
84 (4) (2016), 1441-1475.[12] F. Diebold, A. Inoue, Long memory and regime switching,
J. Econometrics , (2001), 131-159.[13] F.D. Foster, S. Viswanathan, A theory of the interday variations in volume, variance, and trading costsin securities markets, Rev. Financ. Stud. , (1990), 593-624.[14] F.D. Foster, S. Viswanathan, Variations in trading volume, return volatility, and trading costs: evidenceon recent price formation models, J. Finance ,
48 (1) (1993), 187-211.[15] F.D. Foster, S. Viswanathan, Can speculative trading explain the volume-volatility relation?
J. Bus.Econ. Stat. ,
13 (4) (1995), 379-396.[16] A.R. Gallant, P.E. Rossi, G. Tauchen, Stock prices and volume,
Rev. Financ. Stud. , (1992),199-242.[17] C. Granger, N. Hyung, Occasional structural breaks and long memory with an application to the S&P500 absolute stock returns, J. Empir. Financ. , (2004), 399-421.[18] P. Guasoni, M.H. Weber, Rebalancing multiple assets with mutual price impact, J. Optim. TheoryAppl. ,
179 (2) (2018), 618-653. 2019] X.J. He, S.P. Zhu, An analytical approximation formula for European option pricing under a newstochastic volatility model with regime-switching,
J. Econom. Dynam. Control. , (2016), 77-85.[20] N. Hyung, S. Poon, C. Granger, A Source of Long Memory in Volatility. Working Paper, University ofCalifornia, San Diego, 2006.[21] I. Karatzas, S.E. Sherve, Brownian Motion and Stochastic Calculs, Berlin: Springer-Verlag, 1991.[22] A. Kyle, Continuous auctions and insider trading, Econometrica ,
53 (6) (1985), 1315-1335.[23] V. Naik, M. Lee, General equilibrium pricing of options on the market portfolio with discontinuousreturns.
Rev. Financ. Stud. , (1990), 493-521.[24] J.P. Lepeltier, J.S. Martin, Existence for BSDE with superlinear-quadratic coefficient, Stoch. Stoch.Rep. ,
63 (3-4) (1992), 227-240.[25] R. Liptser, A. Shiryaev, Statistics of Random Processes II: Applications (Second ed.), Berlin: Springer-Verlag, 2001.[26] B. Mandelbrot, J. van Ness, Fractional Brownian motions, fractional noises and applications,
SIAMRev. ,
10 (4) (1968), 422-437.[27] M. Mr´azek, J. Posp´ıˇsil, T. Sobotka, On calibration of stochastic and fractional stochastic volatilitymodels,
Eur. J. Oper. Res. (2016), 1036-1046.[28] J. Liu, J. Pan, T. Wang, An equilibrium model of rare-event premiums and its implication for optionsmirks.
Rev. Financ. Stud. , (2005), 131-164.[29] A.N. Shiryaev, On arbitrage and replication for fractal models, Preprint, Moscow University and SteklovInstitute, 1999.[30] T.H. Thao, An approximate approach to fractional analysis for finance. Nonlinear Anal. RW , (2006),124-132.[31] B.Z. Yang, J. Yue, N.J. Huang, Variance swaps under L´evy process with stochastic volatility andstochastic interest rate in incomplete market. arXiv:1712.10105[q-fin.PR].[32] J. Yue, N.J. Huang, Neutral and indifference pricing with stochastic correlation and volatility, J. Ind.Manag. Optim ,
14 (1) (2018), 199-229.[33] J. Yue, N.J. Huang, Fractional Wishart processes and ε -fractional Wishart processes with applications, Comput. Math. Appl. ,
75 (8) (2018), 2955-2977.[34] S.P. Zhu, X.J. He, A new closed-form formula for pricing European options under a skew Brownianmotion,
Eur. J. Financ. ,
24 (12)24 (12)