A model for a large investor trading at market indifference prices Dmitry Kramkov (joint work with Peter Bank) Carnegie Mellon University and University of Oxford 5th Oxford-Princeton Workshop on Financial Mathematics & Stochastic Analysis March 27-28, 2009 1 / 26
Outline Model for a “small” trader Features of a “large” trader model Literature (very incomplete!) Trading at market indifference prices Asymptotic analysis: summary of results Conclusion 2 / 26
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 0 = t 0 < · · · < t N = T and θ n ∈ L 0 ( F t n − 1 ), the terminal value N � V T = V T ( Q ) = θ n ( S t n − S t n − 1 ) n =1 Mathematical challenge: define terminal wealth V T for general predictable process Q = ( Q t ). 3 / 26
Passage to continuous time trading Two steps: 1. Establish that S is a semimartingale 1.1 ⇔ ∃ limit of discrete sums, when sequence ( Q n ) of simple integrand converges uniformly (Bechteler-Dellacherie) 1.2 ⇐ Absence of arbitrage for simple strategies (NFLBR) (Delbaen & Schachermayer (1994)). 2. If S is a semimartingale, then we can extend the map Q → V T ( Q ) from simple to general (predictable) strategies Q arriving to stochastic integrals : � T V T ( Q ) = Q t dS t . 0 4 / 26
Basic results for the “small” trader model Fundamental Theorems of Asset Pricing: 1. Absence of arbitrage for general admissible strategies (NFLVR) ⇔ S is a local martingale under an equivalent probability measure (Delbaen & Schachermayer (1994)). 2. Completeness ⇔ Uniqueness of a martingale measure for S (Harrison & Pliska (1983), Jacod (1979)). Arbitrage-free pricing formula: in complete financial model the arbitrage-free price for a European option with maturity T and payoff ψ is given by p = E ∗ [ ψ ] , where P ∗ is the unique martingale measure. 5 / 26
“Desirable” features of a “large” trader model Logical requirements: 1. Allow for general continuous-time trading strategies. 2. Obtain the “small” trader model in the limit: � T Q t dS (0) V T ( ǫ Q ) = ǫ + o ( ǫ ) , ǫ → 0 . t 0 Practical goal: computation of liquidity or price impact corrections to prices of derivatives: 1 p ( ǫ ) = ǫ E ∗ [ ψ ] + 2 ǫ 2 C ( ψ ) + o ( ǫ 2 ) . � �� � liquidity correction Here p ( ǫ ) is a “price” for ǫ contingent claims ψ . Of course, we expect to have C ( ψ ) ≤ 0 for all ψ and < 0 for some ψ. 6 / 26
Literature (very incomplete!) Model is an input: Jarrow (1992), (1994); Frey and Stremme (1997); Platen and Schweizer (1998); Papanicolaou and Sircar (1998); Cuoco and Cvitanic (1998); Cvitanic and Ma (1996); Schonbucher and Wilmott (2000); Cetin, Jarrow and Protter (2002); Bank and Baum (2003); Cetin, Jarrow, Protter and Warachka (2006), Cetin, Touzi, and Soner (2008) , . . . Model is an output (a result of equilibrium): Kyle (1985), Back (1990), Gˆ arleanu, Pedersen, Poteshman (1997). . . 7 / 26
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 . 8 / 26
Pareto allocation Definition A vector of random variables α = ( α m ) 1 ≤ m ≤ M is called 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 would leave all market makers not worse and at least one of them better off in the sense that E [ u m ( β m )] ≥ E [ u m ( α m )] for all 1 ≤ m ≤ M , and E [ u m ( β m )] > E [ u m ( α m )] for some 1 ≤ m ≤ M . 9 / 26
Pricing measure of Pareto allocation First-order condition: We have an equivalence between 1. α = ( α m ) 1 ≤ m ≤ M is a Pareto allocation. 2. The ratios of the marginal utilities are non-random: u ′ m ( α m ) n ( α n ) = const( m , n ) . u ′ Pricing measure is defined by the marginal rate of substitution: u ′ m ( α m ) d Q d P = 1 ≤ m ≤ M . m ( α m )] , E [ u ′ (Marginal) price process of the traded contingent claims ψ : S t = E Q [ ψ |F t ] 10 / 26
Simple strategy Strategy: a process of quantities Q = ( Q t ) of ψ . Goal: specify the terminal value V T = V T ( Q ). 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 = ξ n 1 ( t n − 1 , t n ] , n =1 where ξ n is F t n − 1 -measurable. 11 / 26
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 . 12 / 26
Trading at time t n 1. The market makers arrive to time t n with F t n − 1 -Pareto allocation α n of the total endowment: Σ n = Σ 0 − ξ n − θ n ψ. 2. After the trade in θ n +1 − θ n shares at the cost ξ n +1 − ξ n , the total endowment becomes Σ n +1 =Σ n − ( ξ n +1 − ξ n ) − ( θ n +1 − θ n ) ψ =Σ 0 − ξ n +1 − θ n +1 ψ. 3. Σ n +1 is redistributed as F t n - Pareto allocation α n +1 . 4. Key condition: the conditional expected utilities of market makers do not change , that is, E [ u m ( α m n +1 ) |F t n ] = E [ u m ( α m n ) |F t n ] , 1 ≤ m ≤ M . 13 / 26
Final step The large trader arrives at maturity t N = T with 1. quantity Q T = θ N of the traded contingent claims ψ . 2. cash amount X T = ξ N . Hence, finally, his terminal wealth is given by V T := X T + Q T ψ. Proposition For any simple strategy Q the cash balance process X = X ( Q ) and the terminal wealth V T = V T ( Q ) are well-defined. Mathematical challenge: define terminal wealth V T for general strategy Q . 14 / 26
More on economic assumptions The model is essentially based on two economic assumptions: Market efficiency After each trade the market makers form a complete Pareto optimal allocation . ⇔ They can trade anything with each other (not only ψ )! 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 . ⇔ The agent can split any order in a sequence of very small trades at marginal prices. ⇔ The expected utilities of market makers do not change. Remark From the investor’s point of view this is the most “friendly” type of interaction with market makers. 15 / 26
Comparison with Arrow-Debreu equilibrium Economic assumptions behind a large trader model based on Arrow-Debreu equilibrium: Market efficiency (Same as above) After re-balance the market makers form a Pareto optimal allocation . ⇔ They can trade anything between each other (not only ψ )! Information The market makers have perfect knowledge of strategy Q . ⇔ Changes in Pareto allocations occur only at initial time. ⇒ Expected utilities of market makers increase as the result of trade. 16 / 26
Model based on Arrow-Debreu equilibrium Given a strategy Q the market makers immediately change the initial Pareto allocation α 0 to another Pareto allocation � α = � α ( Q ) with pricing measure � P , the price process � S t := E � P [ ψ |F t ] and total endowment M � � α m Σ := � m =1 such that � T Σ 0 − � Q t d � Σ = S t , 0 and the following “clearing” conditions hold true: P [ α m α m ] , E � 0 ] = E � P [ � 1 ≤ m ≤ M . 17 / 26
Back to our model Mathematical challenge: define terminal wealth for general Q . Question Let Q and ( Q n ) n ≥ 1 be simple strategies such that ( Q n − Q ) ∗ | Q n T � sup t − Q t | → 0 , in probability . (1) 0 ≤ t ≤ T Do their terminal gains converge: V n T → V T , in probability? (2) Question Let Q be a predictable process (with LCRL trajectories) and ( Q n ) n ≥ 1 be simple strategies such that (1) holds. Is there a random variable V T such that (2) holds as well? 18 / 26
Process of Pareto allocations Consider a simple strategy N � Q t = θ n 1 ( t n − 1 , t n ] , n =1 where θ n is F t n − 1 -measurable and denote by N � A t = α n 1 ( t n − 1 , t n ] n =1 the corresponding (non-adapted!) process of Pareto allocations. Remark The Pareto allocation A t contains all information at time t but is not F t -measurable ( infinite-dimensional sufficient statistic). 19 / 26
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