Replication under Price Impact and Martingale Representation Property

Transcription

1 Replication under Price Impact and Martingale Representation Property Dmitry Kramkov joint work with Sergio Pulido (Évry, Paris) Carnegie Mellon University Workshop on Equilibrium Theory, Carnegie Mellon, June 15, / 21

2 Classical model for a small agent Input: price process S = (S t ) for traded stock. Usually, S is a solution of SDE: ds t = µ(t, S t )dt + σ(t, S t )db t, S = x, where B is a Brownian motion. Key assumption: trader s actions do not affect S. Strategy: predictable S-integrable process γ = (γ t ) of the number of stocks. Gain from strategy: G t = t γds = (γ S) t. 2 / 21

3 Model with price impact Price impact: γ S(γ). Practice & Mathematical Finance: the price impact is postulated (exogenous). Financial Economics: the price impact is an output of equilibrium (endogenous). Main idea: Let ψ be stock s dividend paid at maturity T, so that, S T (γ) = ψ. Recall that for the small agent model, (NA) Q() P : S t () = E Q() [ψ F t ]. For γ we similarly expect to have S t (γ) = E Q(γ) [ψ F t ], where the measure Q(γ) P is obtained from an equilibrium. 3 / 21

4 Optimal investment Input: financial market and investor s preferences 1. S = (S t ): stocks prices; 2. U(x) = 1 a e ax, x R: utility function; a > is investor s risk-aversion. Output: optimal process γ = (γ t ) of the number of stocks: [ T ] γ = arg max E U( ζds) ζ [ T ] = arg min E exp( a ζds). ζ Martingale characterization: γ = (γ t ) is optimal S is a local martingale and γ S is a UI martingale under Q given by dq T T dp = const U ( γds) = const exp( a γds). 4 / 21

5 Price impact model Input: dividends, market makers preferences, demand 1. ψ: stocks dividends paid at maturity T ; 2. U(x) = 1 a e ax, x R: representative utility; a > is aggregate risk-aversion (harmonic mean), a market s liquidity. 3. γ = (γ t ): demand process (number of stocks owned by the market). Output: stocks prices S = (S t ) such that S T = ψ and [ T ] γ = arg max E U( ζds) ζ [ T ] = arg min E exp( a ζds) ζ S t = E Q [ψ F t ] with dq dp = const exp( a T γds). 5 / 21

6 References Single period: Sanford J. Grossman and Merton H. Miller. Liquidity and market structure. The Journal of Finance, Discrete time: Nicolae Garleanu, Lasse Heje Pedersen, and Allen M. Poteshman. Demand-based option pricing. Rev. Financ. Stud., 29. Simple strategies in continuous time: David German. Pricing in an equilibrium based model for a large investor. Math. Financ. Econ., 211. General strategies in continuous time: K. and Sergio Pulido. 1. A system of quadratic BSDEs arising in a price impact model. Ann. Appl. Probab., Stability and analytic expansions of local solutions of systems of quadratic BSDEs with applications to a price impact model. SIAM J. Financial Math., / 21

7 Brownian framework Assumption The filtration is generated by a Brownian motion B = (B t ): F t = F B t, t [, T ]. Stocks prices evolve as ds = σλdt + σdb, S T = ψ, where λ = (λ t ): the market price of risk; σ = (σ t ): the volatility. Denote also R t = 1 T ] [exp( a a log E γds) F t, t [, T ], the certainty equivalence value (CEV) of remaining gain. t 7 / 21

8 BSDE for S = S(ψ, a, γ) Theorem (K.,Pulido AAP-16) For dividends ψ, risk-aversion a >, and demand γ the following items are equivalent: 1. S = S(ψ, a, γ) is a price process, σ is the volatility, λ is the market price of risk, and R is the CEV process. 2. (R, S, η, θ) with η = λ aσγ and θ = aσ solves the BSDE: ar t = 1 2 T t as t = aψ T ( θγ 2 η 2 )ds T t θ(η + θγ)ds t T t ηdb, θdb, and the products Z(γ S) and ZS are UI martingales, where Z E( λ B) = E( (η + θγ) B). 8 / 21

9 BMO norms For a continuous martingale M with M =, M BMO ess sup{e [ M T M τ 2 ] F τ } 1/2, τ where the supremum is taken with respect to all stopping times τ. For an integrable random variable ξ with E[ξ] = set ξ BMO (E [ξ F t ]) t [,T ] BMO For a predictable process ζ = (ζ t ) set ( [ T ]) ζ BMO ess sup E ζ s 2 1/2 ds F τ, τ where the supremum is taken with respect to all stopping times τ. By Ito s isometry, ζ BMO = ζdb BMO. τ 9 / 21

10 Existence and uniqueness Theorem (K.,Pulido AAP-16) There is a constant c > such that if a γ ψ E [ψ] BMO c, then the prices S = S(ψ, a, γ) exist and unique. Proposition (K.,Pulido AAP-16) There are bounded γ and ψ such that a γ ψ E [ψ] = 1 and such that the prices S = S(ψ, a, γ) either do not exist or are not unique. 1 / 21

11 Asymptotic expansion Theorem (K.,Pulido JFM-16) Assume that γ ψ E [ψ] BMO <. Then there is a constant K = K(ψ, γ) such that where S(a, γ) (S() + as (1) (γ)) BMO Ka 2, a, t S t () = E [ψ F t ] = E [ψ] + σ()db ( small agent s model) [ T ] S (1) t (γ) = E σ 2 ()γds F t (first-order correction) t 11 / 21

12 Replication problem Denote [ Exp {ξ : E e t ξ ] <, t > }. Replication problem: for a contingent claim ξ Exp find the initial wealth p R and a demand γ = (γ t ) such that p Lemma (uniqueness, easy) If p and γ satisfy (1), then T γds(γ) = ξ. (1) p = E Q [ξ], S t (γ) = E Q [ψ F t ], where dq dp = const eaξ. 12 / 21

13 Completeness assumption for a = Assumption (S()-model is complete) The small agent model with price process S t () = E [ψ F t ] is complete, that is, for for every contingent claim ξ with E [ ξ ] < the martingale P t () = E [ξ F t ], admits integral representation: t P t () = E [ξ] + γds(), for some predictable S()-integrable process γ. Example (No existence) Even if S()-model is complete, for any a > one can find not replicable contingent claims ξ such that ξ / 21

14 Approximate replication Theorem (Approximate replication for fixed a > ) Suppose that S()-model is complete, that ψ L p for some p > 1 and ξ Exp. Then for every ɛ (, 1 2 ] there are p(ɛ) R and a demand γ(ɛ) such that Moreover, in this case, T a ξ (p(ɛ) γ(ɛ)ds γ(ɛ) ) ɛ. p(ɛ) p 2ɛp, S γ(ɛ) S 2ɛ S, where p = E Q [ξ], S t = E Q [ψ F t ], dq dp = const eaξ. 14 / 21

15 Generic replication Theorem (Generic replication for variable a > ) Suppose that S()-model is complete, that ψ L p for some p > 1 and ξ Exp. Then there is at most countable set I (, ) such that for risk-aversions a I the contingent claim ξ is replicable: Moreover, in this case, T ξ = p(a) γ(a)ds γ(a). p(a) = E Q(a) [ξ], S γ(a) t = S t (a) = E Q(a) [ψ F t ], where dq(a) dp = const eaξ. 15 / 21

16 The Martingale Representation Property We work on a filtered probability space (Ω, F, (F t ) t, P) satisfying the usual conditions. Definition Let Q P and S be a d-dimensional local martingale under Q. We say that S has the Martingale Representation Property (MRP) if every local martingale M under Q is of the form M = M + γ S = M + γ ds, where γ is a predictable S-integrable process with values in R d. 16 / 21

17 Classical (forward) results Let Q P and S be a d-dimensional local martingale under Q. Jacod s theorem (2FTAP): S has MRP if and only if Q is the only equivalent local martingale measure for S. Forward dynamics. Suppose that B Q is a n-dimensional Brownian motion under Q, F t = Ft BQ and S t = S + t σ s db Q s, where σ = (σ ij t ) is a predictable process with values in R n d. Then S has the MRP if and only if rank σ t (ω) = n, dt dp(ω) a.s.. 17 / 21

18 Density of probability measures with MRP Let ψ = (ψ i ) i=1,...,d be a d-dimensional random variable. We denote by Q(ψ) the family of probability measures Q P such that 1. E Q [ ψ ] <, 2. S Q t = E Q [ψ F t ], t, has the MRP. Theorem Suppose that Q(ψ). Then for every R P such that E R [ ψ ] < and every ɛ > there is Q Q(ψ) such that dq dr 1 ɛ. 18 / 21

19 Analytic maps with values in a Banach space Let X be a Banach space and U be an open connected set in R l. We recall that a map x X (x) of U to X is analytic if for every y U there exists ɛ = ɛ(y) > and (Y n = Y n (y)) n in X such that X (x) = Y n (x y) n, y x < ɛ n= where the series converges in X. 19 / 21

20 Generic property Theorem Let U be an open connected set in R l, x U, and x ζ(x) and x ξ(x) be continuous maps of U {x } to L 1 (R) and L 1 (R d ), respectively, whose restrictions to U are analytic. For every x U {x } we assume that ζ(x) > and define a probability measure Q(x) and a Q(x)-martingale S(x) by dq(x) dp = ζ(x) E[ζ(x)], S t(x) = E Q(x) t If S(x ) has the MRP, then the exception set [ ] ξ(x). ζ(x) I = {x U : S(x) does not have the MRP} has the Lebesgue measure zero. If, in addition, U is an interval in R, then the set I is at most countable 2 / 21

21 Summary We study the continuous-time version of a price impact model, which goes back to Grossman and Miller (1986); inverse to optimal investment. Stock price S(γ) depend on demand γ through a solution to a to multi-dimensional quadratic BSDE. While exact replication may not be possible, the model has approximate and generic completeness properties. Prices for contingent claims are quite explicit (= utility-based prices). The model is supported by general results on the existence of MRP in backward setup. Other applications of these results are forthcoming. 21 / 21