On financial models with price impact Dmitry Kramkov (with Peter Bank) 2 preprints on the webpage: http://www.math.cmu.edu/kramkov/publications.html Carnegie Mellon University and University of Oxford Analysis, Stochastics, and Applications, In Honor of Walter Schacherymayer Vienna, July 12-17, 2010 1 / 24
Outline Model for a “small” trader Price impact in Mathematical Finance Price impact in Financial Economics Trading at market indifference prices Conclusion 2 / 24
Model for an economic agent Hereafter, a “financial model” for a single agent is understood as a map: Q �→ X ( Q ) , where Q = ( Q t ): predictable process of the number of stocks X ( Q ) = ( X t ( Q )): predictable process of cash balance complementing Q is a self-financing way. Remark “Equilibrium” model for an economy (a collection of agents) is a map Initial Endowments − → Equilibrium Allocations. 3 / 24
Model for a “small” trader Input: price process S = ( S t ) for traded stock. Key assumption: trader’s actions do not affect S . For a simple strategy with a process of stock quantities : N � Q t = θ n 1 ( t n − 1 , t n ] , n =1 where θ n ∈ L 0 ( F t n − 1 ), the cash balance process � � X t ( Q ) = ( θ n − θ n +1 ) S t n = θ n ( S t n − S t n − 1 ) − θ n max ( t )+1 S t nmax ( t ) . t n ≤ t t n ≤ t Mathematical challenge: define X ( Q ) for general Q . Hereafter we shall focus on Q with LCRL trajectories. 4 / 24
Passage to continuous time trading Closability: the convergence of simple ( Q n ) to LCRL Q in ucp ( Q n − Q ) ∗ | Q n T = sup t − Q t | → 0 t ∈ [0 , T ] implies the existence of X ( Q ) such that ( X ( Q n ) − X ( Q )) ∗ T → 0 . By the Bichteler-Dellacherie theorem: Closability holds ⇔ S is a semimartingale. For a semimartingale S we can extend the map Q �→ X ( Q ) from simple to general Q arriving to stochastic integrals : � t X t ( Q ) = Q u dS u − Q t S t . 0 5 / 24
Fundamental Theorems of Asset Pricing R : the family of all equivalent local martingale measures R for S ; 1st FTAP: (Delbaen and Schachermayer, 1994, 1998) Absence of Arbitrage (NFLVR) ⇔ R � = ∅ . 2nd FTAP: (Harrison and Pliska, 1983), (Jacod, 1979) Completeness ⇔ |R| = 1. Remark Need to impose admissibility requirement on strategies; subtle point for investment with real-line utilities. 6 / 24
Model with price impact Model: Q �→ X ( Q ). Price impact: Q → S ( Q ). Logical requirements: 1. Allow for general continuous-time trading strategies; no need for admissibility. 2. Closability: ( Q n − Q ) ∗ ⇒ ( X ( Q n ) − X ( Q )) ∗ T → 0 = T → 0. 3. Obtain the “small” trader model in the limit: � t Q u dS u (0) − Q t S t (0)) + o ( ǫ ) , ǫ → 0 . X t ( ǫ Q ) = ǫ ( 0 Practical need: e.g. optimal liquidation problem; (Almgren and Chriss, 2001), (Schied and Sch¨ oneborn, 2009). 7 / 24
Approach in Mathematical Finance The form of the maps: Q �→ X ( Q ) and Q �→ S ( Q ) is postulated (exogenous); survey (Gokay, Roch, and Soner, 2010). Reaction functions: (Frey and Stremme, 1997), (Platen and Schweizer, 1998), (Papanicolaou and Sircar, 1998), (Bank and Baum, 2004), . . . , Supply curves: (C ¸etin, Jarrow, and Protter, 2004), . . . Dependence on the “past”: X t ( Q ) = X t ( Q t ), S t ( Q ) = S t ( Q t ), where Q t � ( Q min( s , t ) ) 0 ≤ s ≤ T . Logical problem: absence of closability property | Q n | ≤ 1 ∃ ( Q n ) : X T ( Q n ) �→ 0 . but n 8 / 24
Approach in Financial Economics The form of the maps: Q �→ X ( Q ) and Q �→ S ( Q ) is derived as an output of an equilibrium (endogenous); book (O’Hara, 1995), survey (Amihud, Mendelson, and Pedersen, 2005). S T ( Q ) = ψ : be the (exogenous) terminal price of the stock. ⇒ Recall that for ”small” agent case absence of arbitrage = S t = S t (0) = E R [ ψ |F t ] , for some martingale measure R . Economic nature of R : equilibrium ; e.g. Pareto equilibrium. 9 / 24
Pareto allocation Consider an economy with M market makers with utility functions u m , m = 1 , . . . , M . Definition Random variables α = ( α m ) 1 ≤ m ≤ M form a Pareto allocation if there is no other allocation β = ( β m ) 1 ≤ m ≤ M of the same total endowment: M M β m = � � α m , m =1 m =1 which leaves all market makers better off: E [ u m ( β m )] > E [ u m ( α m )] for all 1 ≤ m ≤ M . 10 / 24
Pricing measure of Pareto allocation Pricing measure is defined by the marginal rate of substitution: u ′ m ( α m ) d R d P = m ( α m )] , m = 1 , . . . , M . E [ u ′ (Marginal) price process of the traded contingent claims ψ : S t = E R [ ψ |F t ] Remark A trading of very small quantities at S does not change the expected utilities of market makers in the first order. 11 / 24
Two sources of price impact The expression S t ( Q ) = E R t ( Q ) [ ψ |F t ( Q )] , suggests that the price impact is due to two common aspects of market’s microstructure: 1. Information : Q �→ F t ( Q ); insider: (Glosten and Milgrom, 1985), (Kyle, 1985), and (Back and Baruch, 2004). 2. Inventory : Q �→ R t ( Q ). In Pareto’s framework u ′ m ( α m d R t ( Q ) t ( Q )) = t ( Q ))] , m = 1 , . . . , M , m ( α m d P E [ u ′ this reflects how α t ( Q ), the Pareto optimal allocation of the total wealth or “inventory”, induced by the strategy Q affects the valuation of marginal trades. 12 / 24
Information in inventory models The information (for the market makers and the large trader) has Symmetric (common) part: the same (exogenous) filtration which is not affected by strategy Q : F t ( Q ) = F t , 0 ≤ t ≤ T . Asymmetric part: the knowledge at time t about the subsequent evolution ( Q s ) t ≤ s ≤ T of the investor’s strategy, conditionally to the forthcoming random outcome on [ t , T ]. 13 / 24
Price impact inspired by Arrow-Debreu equilibrium References: (Grossman and Miller, 1988), (Garleanu, Pedersen, and Poteshman, 2009), (PhD thesis, German 2009). Information: the market makers have full knowledge of the investor’s future strategy Q (contingent on ω ) Pricing measures and allocations do not depend on time: R t ( Q ) = R ( Q ) , α t ( Q ) = α ( Q ) , 0 ≤ t ≤ T , and are determined by the budget equations: E R ( Q ) [ α m (0)] = E R ( Q ) [ α m ( Q )] , m = 1 , . . . , M , and the clearing condition: � T M M � α m ( Q ) = � α m (0) + Q t dS t ( Q ) . 0 m =1 m =1 14 / 24
Price impact inspired by Arrow-Debreu equilibrium For the case of exponential utilities, when u m ( x ) = − exp( − a m x ) , a m > 0 , m = 1 , . . . , M . the stock price depends only on the “future” of the strategy: S t ( Q ) = S t ( Q − Q t ) , 0 ≤ t ≤ T . Remark Optimal strategy for the large trader � Arrow-Debreu or even Pareto allocation for the total economy: market makers and trader. ◮ Benefit of the “first move” for the large investor; ◮ Information asymmetry regarding the knowledge of preferences of the large trader. 15 / 24
Trading at market indifference prices Bertrand competition: the market makers trade infinitesimal quantities at Pareto prices (most aggressively without losing in utility); (Stoll, 1978). Information: The market makers do not anticipate (or can not predict the direction of) future trades of the large economic agent. � Two strategies coinciding on [0 , t ] and different on [ t , T ] will produce the same effect on the market up to time t : R t ( Q ) = R t ( Q t ) , α t ( Q ) = α t ( Q t ) , X t ( Q ) = X t ( Q t ) , S t ( Q ) = S t ( Q t ) . 16 / 24
Financial model 1. Uncertainty and the flow of information are modeled, as usual, by a filtered probability space (Ω , F , ( F t ) 0 ≤ t ≤ T , P ). 2. Traded securities are European contingent claims with maturity T and payments ψ = ( ψ i ). 3. Prices are quoted by a finite number of market makers. 3.1 Utility functions ( u m ( x )) x ∈ R , m =1 ,..., M (defined on real line ) are continuously differentiable, strictly increasing, strictly concave, and bounded above: u m ( ∞ ) = 0 , m = 1 , . . . , M . 3.2 Initial (random) endowments α 0 = ( α m 0 ) 1 ≤ m ≤ M ( F -measurable random variables) form a Pareto optimal allocation . 17 / 24
Simple strategy Consider a simple strategy with the process of quantities: N � Q t = θ n 1 ( t n − 1 , t n ] , n =1 where θ n is F t n − 1 -measurable. We shall define the corresponding cash balance process : N � X t ( Q ) = ξ n 1 ( t n − 1 , t n ] , n =1 where ξ n is F t n − 1 -measurable. 18 / 24
Trading at initial time 1. The market makers start with the initial Pareto allocation α 0 = ( α m 0 ) 1 ≤ m ≤ M of the total (random) endowment: M � α m Σ 0 := 0 . m =1 2. After the trade in θ 1 shares at the cost ξ 1 , the total endowment becomes Σ 1 = Σ 0 − ξ 1 − θ 1 ψ. 3. Σ 1 is redistributed as a Pareto allocation α 1 = ( α m 1 ) 1 ≤ m ≤ M . 4. Key condition: the expected utilities of market makers do not change , that is, E [ u m ( α m 1 )] = E [ u m ( α m 0 )] , 1 ≤ m ≤ M . 19 / 24
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