Hedging, Arbitrage, and Optimality with Superlinear Frictions

Superlinear Frictions

Hedging, Arbitrage, and Optimalitywith Superlinear Frictions

Paolo Guasoni1,2 Miklós Rásonyi3,4

Boston University1

Dublin City University2

MTA Alfréd Rényi Institute of Mathematics, Budapest3

University of Edinburgh4

Analyse stochastique pour la modélisation des risquesCIRM, September 10th, 2014


Market Depth

• Depth:the size of an order flow innovation required to change prices a givenamount (Kyle, 1985)

• Documented empirically:Amihud (2002), Admati and Pfleiderer (1988), Cho (2007)

• In Illiquid Portfolio Choice:Rogers and Singh (2010), Garleanu and Pedersen (2013)

• In Optimal Liquidation:Almgren and Chriss, (2001), Bertsimas and Lo (1998), Schied andSchöneborn (2009)

• Unlike frictionless markets, trading affects prices.• Unlike transaction costs, prices depend on direction and speed.


Theory

• Usual questions: arbitrage, hedging, optimality.• Discrete time:

Astic and Touzi (2007), Pennanen and Penner (2010),Dolinsky and Soner (2013)

• Continuous Time:Cetin, Soner, and Touzi (2010), Cetin and Rogers (2007).

• Which trading strategies?• Attainable payoffs?


Results in a Nutshell

• Risky asset: cadlag process St .

• Number of shares Φt =∫ t

0 φsds. Effect of friction G on wealth Xt :

dXt = ΦtdSt −G(φt )dt

• G superlinear. Trading twice as fast for half the time more expensive.• Typical example G(x) = λx2 from Kyle’s equilibrium model.• Absolutely continuous strategies are closed.

(Limits of their payoffs are also their payoffs.)• Market bound: all payoffs dominated by one random variable!• Martingale (not local) property of (shadow) wealth processes.• No doubling strategies.• Superhedging: price is sup of expected shadow values, minus a penalty.• Reduces to frictionless and transaction costs for G = 0 and G(x) = ε|x |.


Feasible Strategies

Zero safe rate. S0 ≡ 1. Risky assets Sit cadlag adapted processes.

DefinitionA feasible strategy is a process φ in the class

A :=

φ : φ is a Rd -valued, optional process,

∫ T

0|φu|du <∞ a.s.

.

Unlike usual admissible strategies...• ...the definition of feasible strategy does not depend on the asset• ...and wealth can be unbounded from below• ...but the number of shares must change at a finite rate.


FrictionAssumption (Friction)

Let G : Ω× [0,T ]× Rd → R+ be a O ⊗ B(Rd )-measurable function, such thatG(ω, t , ·) is convex with G(ω, t , x) ≥ G(ω, t ,0) for all ω, t , x.Gt (x) short for G(ω, t , x)

• Positions in risky assets and cash

V it (z, φ) :=z i +

∫ t

0φi

udu 1 ≤ i ≤ d ,

V 0t (z, φ) :=z0 −

∫ t

0φuSudu −

∫ t

0Gu(φu)du.

• Doing something costs more than doing nothing. Participation cost.• Convexity: speed expensive. Patience pays off.• For one risky asset and Gt (0) = 0, equivalent to execution price equal to:

St = St + Gt (φt )/φt


Superlinear Friction

Assumption

There is α > 1 and an optional process H such that

inft∈[0,T ]

Ht > 0 a.s.,

Gt (x) ≥ Ht |x |α, for all ω, t , x ,∫ T

0

(sup|x|≤N

Gt (x)

)dt <∞ a.s. for all N > 0,

supt∈[0,T ]

Gt (0) ≤ K a.s. for some constant K .

• Superlinearity: trading twice as fast (uniformly) more than twice expensive.• Frictions never disappear...• ...but remain finite in finite time.


Market BoundLemmaUnder the superlinearity assumption, any feasible φ ∈ A satisfies

V 0T (z, φ) ≤ z0 +

∫ T

0G∗t (−St )dt <∞ a.s.

G∗t (y) := supx∈Rd (xy −Gt (x)) is the dual friction (Dolinsky and Soner, 2013).

Proof.

z0 −∫ t

0φuSudu −

∫ t

0Gu(φu)du ≤ z0

t +

∫ t

0G∗u(−Su)du

and G∗t (y) ≤ supr∈R (ry − Ht |r |α) = α−1α α

11−α H

11−α

t |y |α

α−1 .

• Without frictions or with transaction costs, no-arbitrage conditions makethe payoff space bounded in L0. Superlinear frictions do even better.

• All payoffs below the market bound. Almost surely.


Shadow ProbabilitiesDefinition

• P denotes the set of probabilities Q ∼ P such that

EQ

∫ T

0Hβ/(β−α)

t (1 + |St |)βα/(α−β)dt <∞.

• P denotes the set of probability measures Q ∈ P such that

EQ

∫ T

0|St |dt <∞ and EQ

∫ T

0sup|x|≤N

Gt (x)dt <∞ for all N ≥ 1.

• For a (possibly multivariate) random variable W , define

P(W ) := Q ∈ P : EQ |W | <∞, P(W ) := Q ∈ P : EQ |W | <∞.

• Think of these sets as martingale probabilities for some execution price St– and integrable enough.


Trading Volume Bound

LemmaLet Q ∈ P and φ ∈ A be such that EQξ− <∞, where

ξ := −∫ T

0Stφtdt −

∫ T

0Gt (φt )dt .

Then

EQ

∫ T

0|φt |β(1 + |St |)βdt <∞.

• Excessive trading may be hazardous for your wealth.• Bounded Losses imply bounded trading volume...• Follows from careful use of Hölder’s inequality.


Closed Payoff Space

Space of payoffs superhedged by feasible strategies

C := [VT (0, φ) : φ ∈ A − L0(Rd+1+ )] ∩ L0(Rd+1)

Proposition

The set C ∩ L1(Q) is closed in L1(Q) for all Q ∈ P such that∫ T

0 |St |dt isQ-integrable.

Corollary

T the set C is closed in probability.

• Absolutely continuous strategies are the only strategies.

• Similar to proof that h :∫ 1

0 (ht )2dt < 1 is relatively compact in L1.


Superreplication

For x ∈ Rd+1, define x ∈ Rd by x i = (x i/x0)1x0 6=0.

TheoremLet W ∈ L0(Rd+1), z ∈ Rd+1. There exists φ ∈ A such that VT (z, φ) ≥W a.s. ifand only if

Z0z ≥ EQ(ZT W )− EQ

∫ T

0Z 0

t G∗t (Zt − St )dt , (1)

for all Q ∈ P and for all Rd+1+ -valued bounded Q-martingales Z with Z 0

0 = 1satsifying Z i

t = 0, i = 1, . . . ,d on Z 0t = 0.

• Multivariate claims: assets not convertible instantaneously.• Take Q = P for simplicity. Then any positive martingale Z gives a shadow

price, penalized for how far Z is from S.• If Gt (0) = 0 and Z = S (i.e. SZ 0 is a martingale), then penalty is zero.


ExamplesSuperreplication is expensive.

Example

Let µ ∈ R, σ,S0 > 0, St := S0e(µ−σ2/2)t+σWt , Gt (x) = λ2 Stx2, where Wt is a

Brownian motion. Then a cash payoff equal to ST cannot be superreplicatedfrom any initial capital.

• Cannot hedge what may exceed the market bound!• What is superreplicable, then?

Example

Let St > 0 a.s. for all t and Gt (x) := λ2 Stx2. Then, for all k > 0, the contract

that at time T pays 1λ

∫ T0 (√

1 + 2kλ/St − 1)dt units of the risky asset issuperreplicable from initial cash position kT .

• Buy asset at rate k and get that payoff.


Arbitrage

• Superreplication theorem does not require absence of arbitrage...• ...so it helps characterize it.

DefinitionAn arbitrage of the second kind is a strategy φ ∈ A, such that VT (c, φ) ≥ 0 forsome c < 0.

• in other words, a negative superreplication price for a positive payoff.

TheoremAbsence of arbitrage of the second kind holds if and only if, for all ε > 0, thereexists Q ∈ P and an Rd+1

+ -valued Q-martingale Z with ZT ∈ Lγ(Q) such thatEQ∫ T

0 Z 0t G∗t (Zt − St )dt < ε, where 0 < β < α and 1/β + 1/γ = 1.

• Enough to find shadow prices that are very close to martingale measures.• Processes with conditional full support satisfy this criterion for H constant.


Utility Maximization

TheoremLet U : R→ R be concave and nondecreasing, W a random variable, and letE |U(c + B + W )| <∞ hold for the market bound B =

∫ T0 G∗t (−St )dt. There is

φ∗ ∈ A′(U, c) such that

EU(V 0T (c, φ∗) + W ) = sup

φ∈A′(u,c)EU(V 0

T (c, φ) + W ),

whereA′(U, c) = φ ∈ A : V i

T (c, φ) = 0, i = 1, . . . ,d , EU−(V 0T (c, φ) + W ) <∞.

• Optimal strategies exist under general conditions.• Friction generates compactness. Cannot buy too much of anything.


First-order ConditionTheorem

a) Let U be concave with U ′ strictly decreasing, and for some C > 0 and δ > 1

U(x) ≤ −C|x |δ, x ≤ 0,

b) assume that U ′ is strictly increasing, where U is the conjugate of U,c) let W be a bounded random variable;d) let Q ∈ P be such that dQ/dP ∈ Lη,where (1/η) + (1/δ) = 1;e) let G′t (·) exist continuous P × Leb-a.s. and be strictly increasing;

f) let Z be a càdlàg process with ZT ∈ Lγ′

for some γ′ > γ and let φ∗ be afeasible strategy such that, for some y∗ > 0, the following conditions hold:

i) Z is a Q-martingale;ii) U ′(V 0

T (x , φ∗) + W ) = y∗(dQ/dP) a.s.;

iii) Zt = St + G′t (φ∗t ) a.s. in P × Leb;

iv) EQ

(V 0

T (x , φ∗)−

∫ T0 G∗t (Zt − St)dt

)= x.

Then φ∗ is optimal for the problem maxφ∈A′(U,c) E[U(V 0

T (x , φ) + W )].


Cash is a Martingale• Milestone for optimality: pass from a.s. upper bound

V 0T (x , φ) ≤ x −

∫ T

0Ztφtdt +

∫ T

0G∗t (Zt − St )dt

• to risk-neutral upper bound

0 ≤ EQ

[(x − V 0

T (x , φ) +

∫ T

0G∗t (Zt − St )dt

)].

LemmaUnder the assumptions of the previous Theorem, any φ ∈ A′(U, c) satisfies

EQ

∫ T

0φtZtdt = 0.

• Shadow wealth is a martingale.


Conclusion

• Superlinear frictions:execution prices increase arbitrarily with trading speed.

• Market bound: one payoff dominates them all.• Finite losses imply finite volume.• Absolutely continuous strategies generate closed payoff space.• Utility maximization: first order conditions for payoff and for shadow price.


Thank You!Questions?

Hedging, Arbitrage, and Optimality with Superlinear Frictions

Economy & Finance

Transcript of Hedging, Arbitrage, and Optimality with Superlinear Frictions