On financial models with price impact Dmitry Kramkov (with Peter - - PowerPoint PPT Presentation

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

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Outline

Model for a “small” trader Price impact in Mathematical Finance Price impact in Financial Economics Trading at market indifference prices Conclusion

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Model for an economic agent

Hereafter, a “financial model” for a single agent is understood as a map: Q → X(Q), where Q = (Qt): predictable process of the number of stocks X(Q) = (Xt(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.

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Model for a “small” trader

Input: price process S = (St) for traded stock. Key assumption: trader’s actions do not affect S. For a simple strategy with a process of stock quantities: Qt =

N

• n=1

θn1(tn−1,tn], where θn ∈ L0(Ftn−1), the cash balance process Xt(Q) =

• tn≤t

(θn−θn+1)Stn =

• tn≤t

θn(Stn−Stn−1)−θnmax(t)+1Stnmax (t). Mathematical challenge: define X(Q) for general Q. Hereafter we shall focus on Q with LCRL trajectories.

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Passage to continuous time trading

Closability: the convergence of simple (Qn) to LCRL Q in ucp (Qn − Q)∗

T =

sup

t∈[0,T]

|Qn

t − Qt| → 0

implies the existence of X(Q) such that (X(Qn) − 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: Xt(Q) = t QudSu − QtSt.

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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.

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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: (Qn − Q)∗

T → 0 =

⇒ (X(Qn) − X(Q))∗

T → 0.

• 3. Obtain the “small” trader model in the limit:

Xt(ǫQ) = ǫ( t QudSu(0) − QtSt(0)) + o(ǫ), ǫ → 0. Practical need: e.g. optimal liquidation problem; (Almgren and Chriss, 2001), (Schied and Sch¨

• neborn, 2009).

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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”: Xt(Q) = Xt(Qt), St(Q) = St(Qt), where Qt (Qmin(s,t))0≤s≤T. Logical problem: absence of closability property ∃(Qn) : |Qn| ≤ 1 n but XT(Qn) → 0.

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Approach in Financial Economics

The form of the maps: Q → X(Q) and Q → S(Q) is derived as an

• utput of an equilibrium (endogenous); book (O’Hara, 1995),

survey (Amihud, Mendelson, and Pedersen, 2005). ST(Q) = ψ: be the (exogenous) terminal price of the stock. Recall that for ”small” agent case absence of arbitrage = ⇒ St = St(0) = ER[ψ|Ft], for some martingale measure R. Economic nature of R: equilibrium; e.g. Pareto equilibrium.

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Pareto allocation

Consider an economy with M market makers with utility functions um, m = 1, . . . , M.

Definition

Random variables α = (αm)1≤m≤M form a Pareto allocation if there is no other allocation β = (βm)1≤m≤M of the same total endowment:

M

• m=1

βm =

M

• m=1

αm, which leaves all market makers better off: E[um(βm)] > E[um(αm)] for all 1 ≤ m ≤ M.

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Pricing measure of Pareto allocation

Pricing measure is defined by the marginal rate of substitution: dR dP = u′

m(αm)

E[u′

m(αm)],

m = 1, . . . , M. (Marginal) price process of the traded contingent claims ψ: St = ER[ψ|Ft]

Remark

A trading of very small quantities at S does not change the expected utilities of market makers in the first order.

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Two sources of price impact

The expression St(Q) = ERt(Q)[ψ|Ft(Q)], suggests that the price impact is due to two common aspects of market’s microstructure:

• 1. Information: Q → Ft(Q); insider: (Glosten and Milgrom,

1985), (Kyle, 1985), and (Back and Baruch, 2004).

• 2. Inventory: Q → Rt(Q). In Pareto’s framework

dRt(Q) dP = u′

m(αm t (Q))

E[u′

m(αm t (Q))],

m = 1, . . . , M, this reflects how αt(Q), the Pareto optimal allocation of the total wealth or “inventory”, induced by the strategy Q affects the valuation of marginal trades.

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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: Ft(Q) = Ft, 0 ≤ t ≤ T. Asymmetric part: the knowledge at time t about the subsequent evolution (Qs)t≤s≤T of the investor’s strategy, conditionally to the forthcoming random outcome on [t, T].

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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: Rt(Q) = R(Q), αt(Q) = α(Q), 0 ≤ t ≤ T, and are determined by the budget equations: ER(Q)[αm(0)] = ER(Q)[αm(Q)], m = 1, . . . , M, and the clearing condition:

M

• m=1

αm(Q) =

M

• m=1

αm(0) + T Qt dSt(Q).

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Price impact inspired by Arrow-Debreu equilibrium

For the case of exponential utilities, when um(x) = − exp(−amx), am > 0, m = 1, . . . , M. the stock price depends only on the “future” of the strategy: St(Q) = St(Q − Qt), 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.

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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: Rt(Q) = Rt(Qt), αt(Q) = αt(Qt), Xt(Q) = Xt(Qt), St(Q) = St(Qt).

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Financial model

• 1. Uncertainty and the flow of information are modeled, as usual,

by a filtered probability space (Ω, F, (Ft)0≤t≤T, P).

• 2. Traded securities are European contingent claims with

maturity T and payments ψ = (ψi).

• 3. Prices are quoted by a finite number of market makers.

3.1 Utility functions (um(x))x∈R,m=1,...,M (defined on real line) are continuously differentiable, strictly increasing, strictly concave, and bounded above: um(∞) = 0, m = 1, . . . , M. 3.2 Initial (random) endowments α0 = (αm

0 )1≤m≤M (F-measurable

random variables) form a Pareto optimal allocation.

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Simple strategy

Consider a simple strategy with the process of quantities: Qt =

N

• n=1

θn1(tn−1,tn], where θn is Ftn−1-measurable. We shall define the corresponding cash balance process: Xt(Q) =

N

• n=1

ξn1(tn−1,tn], where ξn is Ftn−1-measurable.

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Trading at initial time

• 1. The market makers start with the initial Pareto allocation

α0 = (αm

0 )1≤m≤M of the total (random) endowment:

Σ0 :=

M

• m=1

αm

0 .

• 2. After the trade in θ1 shares at the cost ξ1, the total

endowment becomes Σ1 = Σ0 − ξ1 − θ1ψ.

• 3. Σ1 is redistributed as a Pareto allocation α1 = (αm

1 )1≤m≤M.

• 4. Key condition: the expected utilities of market makers do

not change, that is, E[um(αm

1 )] = E[um(αm 0 )],

1 ≤ m ≤ M.

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Trading at time tn

• 1. The market makers arrive to time tn with Ftn−1-Pareto

allocation αn of the total endowment: Σn = Σ0 − ξn − θnψ.

• 2. After the trade in θn+1 − θn shares at the cost ξn+1 − ξn, the

total endowment becomes Σn+1 =Σn − (ξn+1 − ξn) − (θn+1 − θn)ψ =Σ0 − ξn+1 − θn+1ψ.

• 3. Σn+1 is redistributed as Ftn-Pareto allocation αn+1.
• 4. Key condition: the conditional expected utilities of market

makers do not change, that is, E[um(αm

n+1)|Ftn] = E[um(αm n )|Ftn],

1 ≤ m ≤ M.

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Simple strategy and questions

Theorem (“Easy”)

For any simple strategy Q the cash balance process X = X(Q) is well-defined. Closability A: Let Q and (Qn)n≥1 be simple strategies such that (Qn − Q)∗

T

sup

0≤t≤T

|Qn

t − Qt| → 0.

(1) Do their cash balance processes converge: (X(Qn) − X(Q))∗

T → 0?

(2) Closability B: Let Q be a predictable process (with LCRL trajectories) and (Qn)n≥1 be simple strategies such that (1) holds. Is there an LCRL process X(Q) such that (2) holds as well?

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Standing Assumptions

Assumption

The utility functions of market makers are in C2 and their risk-aversion coefficient: 1 c < am(x) −u′′

m(x)

u′

m(x) < c < ∞.

Assumption

The filtration is generated by a Brownian motion W = (W i).

Assumption

The total initial endowment Σ0 and payoffs of stocks ψ have all exponential moments.

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Final results

M: number of market makers; J: number of stocks.

Theorem

If |a(k)

m (x)| ≤ c < ∞, k ≤ M+J 2

+ 1, then (A) holds.

Theorem

If all utilities um(x) are exponential, then (B) holds.

Theorem

If |a′

m(x)| ≤ c < ∞ and Σ0 and ψ are Malliavin differentiable, then

(B) holds. For example, Σ0 = f (XT), ψ = g(XT) where X is a strong solution of an SDE.

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Concluding remarks

◮ The price impact models coming from Financial Economics

start with economic primitives.

◮ They have “good” qualitative structure:

◮ No need for admissibility conditions to preclude an arbitrage. ◮ Natural closability properties holds.

◮ Many mathematical challenges!

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