UTILITY INDIFFERENCE PRICING WITH HIGH RISK AVERSIONAND SMALL LINEAR PRICE IMPACT

YAN DOLINSKY∗AND SHIR MOSHE†

Abstract. We consider the Bachelier model with linear price impact. Exponential utilityindifference prices are studied for vanilla European options and we compute their non-trivial scalinglimit for a vanishing price impact which is inversely proportional to the risk aversion. Moreover, wefind explicitly a family of portfolios which are asymptotically optimal.

Key words. Utility Indifference Pricing, Linear Price Impact, Asymptotic Analysis

1. Introduction. In financial markets, trading moves prices against the trader:buying faster increases execution prices, and selling faster decreases them. This aspectof liquidity, known as market depth [3] or price-impact has recently received increasingattention (see, for instance, [2, 4, 6, 8, 9, 11, 12, 13, 14, 15, 16, 17] and the referencestherein).

In [13] the authors showed that for a reasonable market model, in the presence ofprice impact, super–replication is prohibitively costly. Namely, in the presence of priceimpact, even in market models such as the Bachelier model or the Black–Scholes model(which are complete in the frictionless setup) there is no practical way to construct ahedging strategy which eliminates all risk from a financial position. This brings us toutility indifference pricing.

In this paper we study utility indifference pricing for vanilla European options inthe multi–dimensional Bachelier model with linear price impact. Our main result iscomputing the asymptotic behavior of the exponential utility indifference prices wherethe risk aversion goes to infinity at a rate which is inversely proportional to the linearprice impact which goes to zero. In addition we provide a family of asymptoticallyoptimal hedging strategies.

This type of scaling limits goes back to the seminal work of Barles and Soner[7] which determines the scaling limit of utility indifference prices of vanilla optionsfor small proportional transaction costs and high risk aversion. The present noteprovides an analogous analysis for the case of linear price impact which results inquadratic transaction costs, albeit using probabilistic techniques rather than takinga PDE approach as pursued in [7].

We divide the proof of our main result, namely Theorem 2.1 into two main steps:the proof of the lower bound and the proof of the upper bound.

The proof of the lower bound goes through a dual representation of the certaintyequivalent. In the dual problem, there is only one player: a maximizing (adverse)player that controls the probability measure. A key ingredient in the proof of thelower bound is a construction of a family of probability measures which attain (in theasymptotic sense) the desired limit. This is done in Proposition 3.2.

The proof of the upper bound does not use duality and is based on a directargument (Proposition 4.1). More precisely, we construct a family of trading strategiesfor which the expected utility converges to the scaling limit. Roughly speaking, thesestrategies are given by a reversion towards the ∆–hedging strategy which correspondsto a modified European option and a modified stock price.

∗Department of Statistics, Hebrew University of Jerusalem. yan.dolinsky@mail.huji.ac.il.†Department of Statistics, Hebrew University of Jerusalem. shir.tapiro@mail.huji.ac.il.‡Both authors supported by the GIF Grant 1489-304.6/2019 and the ISF grant 230/21.

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1

The rest of the paper is organized as follows. In the next section we introduce thesetup and formulate the main results. In Section 3 we discuss the dual representationand prove the lower bound. In Section 4 we prove the upper bound.

2. Preliminaries and Main Results. Let T <∞ be the time horizon and letW =

(W 1t , ...,W

dt

)t∈[0,T ]

be a standard d-dimensional Brownian motion defined on the

filtered probability space (Ω,F , (Ft)0≤t≤T ,P) where the filtration (Ft)0≤t≤T satisfiesthe usual assumptions (right continuity and completeness). We consider a simplefinancial market with a riskless savings account bearing zero interest (for simplicity)and with d-risky asset S =

(S1t , ..., S

dt

)t∈[0,T ]

with Bachelier price dynamics

(2.1) Sit = si0 + µit+

d∑j=1

σijW jt

where s0 = (s10, ..., s

d0) ∈ Rd is the initial position of the risky assets, µ = (µ1, ...µd) ∈

Rd is a constant vector (drift) and σ = σij1≤i,j≤d ∈ Md(R) is a constant non-singular matrix (volatility). Without loss of generality we assume that the constantnonsingular matrix σ is a positive definite matrix.

Following [1], we model the investor’s market impact, in a temporary linear formand, thus, when at time t the investor turns over her position at the i–asset Φit at therate φit := Φit the execution price is St + Λ

2 φit for some constant Λ > 0. As a result,

the profits and losses from a trading strategy φ =(φ1t , ..., φ

dt

)t∈[0,T ]

with the initial

position Φ0 =(Φ1

0, ...,Φd0

)are given by

(2.2) V Φ0,φt :=

∫ t

0

〈Φu, dSu〉 −Λ

2

d∑i=1

∫ t

0

||φv||2dv, t ∈ [0, T ]

where, for convenience, we assume that the investor marks to market her positionΦt = Φ0 +

∫ t0φvdv in the risky asset that she has acquired by time t. As usual, 〈·, ·〉

and || · ||, denotes the standard scalar product and the Euclidean norm, respectively.In our setup, the natural class of admissible strategies is

A :=

φ =

(φ1t , ..., φ

dt

)t∈[0,T ]

: φ is FW -adapted with

∫ T

0

||φt||2dt <∞ a.s.

.

Remark 2.1. Let us notice that by scaling the risky assets, there is no loss ofgenerality in assuming that the constant Λ > 0 which represents the linear priceimpact is the same for all risky assets.

Next, consider a vanilla European option with the payoff f (ST ) where f : Rd →R+ is a Lipschitz continuous function. The investor will assess the quality of a hedgeby the resulting expected utility. Assuming exponential utility with constant absoluterisk aversion α > 0, the utility indifference price and the certainty equivalent price ofthe claim f (ST ) (see, e.g., [10] for details on indifference prices) do not depend onthe investor’s initial wealth and, respectively, take the well-known forms

(2.3) π(Λ, α,Φ0, f) :=1

αlog

infφ∈A EP

[exp

(α(f(ST )− V Φ0,φ

T

))]infφ∈A EP

[exp

(−αV Φ0,φ

T

)]

2

and

c(Λ, α,Φ0, f) :=1

αlog

(infφ∈A

EP

[exp

(α(f(ST )− V Φ0,φ

T

))]).

If the risk aversion α > 0 is fixed, then by applying standard density argumentswe obtain that for Λ ↓ 0, the above indifference price converges to the unique price ofthe continuous time complete (frictionless) market given by (2.1). A more interestinglimit emerges, however, if we re-scale the investor’s risk-aversion in the form α := A/Λ.Before we formulate the limit theorem we need some preparations.

For a given A > 0 introduce the functions

(2.4) gA(x) := supy∈Rd

[f(x+ y)− 〈yσ

−1, y〉2√A

], x = (x1, ..., xd) ∈ Rd

and

(2.5) uA(t, x) := EP[gA(x+WT−tσ)

], (t, x) ∈ [0, T ]× Rd

where the vectors x, y,W are considered as row vectors and for any row vector z ∈ Rd,zσ−1, zσ ∈ Rd are the standard matrix products. The term uA(t, St) represents theprice at time t of a European option with the payoff gA(ST ) in the complete marketgiven by (2.1). It is well known that u ∈ C1,2([0, T )× Rd) solves the PDE

(2.6)∂uA

∂t+tr(σ2D2

xuA)

2= 0 in [0, T )× R

where tr(·) is the trace of the square matrix · and D2xu

A is the Hessian matrix with

respect to x = (x1, ..., xd) which is given by [D2xu

A]ij := ∂2uA

∂xi∂xj , 1 ≤ i, j ≤ d.For a given A,Λ > 0 consider the d–dimensional (random) ODE (Φ is a row

vector)

(2.7) φt := Φt =

√A

Λ

(Dxu

A(t, St −

√AΦtσ

)− Φt

)σ, t ∈ [0, T )

where DxuA :=

(∂uA

∂x1 , ...,∂uA

∂xd

)∈ Rd is the gradient with respect to x. From the

linear growth of f it follows that for any ε > 0 the function DxuA, D2

xuA are uniformly

bounded in the domain [0, T−ε]×Rd. In particular DxuA is Lipschitz continuous with

respect to x in the domain [0, T−ε]×Rd. Hence, from the standard theory of ODE (seeChapter II, Section 6 in [18]) we obtain that for a given initial value Φ0 there exists

a unique solution to (2.7) which we denote by (ΦA,Λt )0≤t<T . Next, the Lipschitzcontinuity of f implies that gA is a Lipschitz continuous function (with the sameconstant as f), and so Dxu

A is uniformly bounded in [0, T )×Rd. This together with

the mean reverting structure of the ODE (2.7) yields that limt→T−ΦA,Λt exists and

finite a.s. Thus, we can extend ΦA,Λ to the interval [0, T ] by ΦA,Λt := limt→T−ΦA,Λt

and we define φA,Λ ∈ A by φA,Λt := ΦA,Λt for t < T and φA,ΛT = 0. Obviously,

ΦA,Λt := Φ0 +

∫ t

0

φA,λv dv, t ∈ [0, T ].

Remark 2.2. In words, the ODE (2.7) says that the solution ΦA,Λ is tracking the∆–hedging strategy which corresponds to the modified payoff gA and the shifted stockprice St −

√AΦA,Λt σ. We notice that the shift depends on the solution ΦA,Λ.

3

We arrive at the main result of the paper which provides an explicit formula forthe asymptotic behavior of the certainty equivalent and an optimal family (shouldnot be unique) of hedging strategies in the asymptotic sense.

Theorem 2.1. For vanishing linear price impact Λ ↓ 0 and re-scaled high risk-aversion A/Λ with A > 0 fixed, the certainty equivalent of f(ST ) has the scalinglimit

(2.8) limΛ↓0

c(Λ, A/Λ,Φ0, f) = uA(

0, s0 −√AΦ0σ

)+

√A〈Φ0σ,Φ0〉

2.

Moreover, we have,

limΛ↓0

Λ

Alog

(EP

[exp

(A

Λ

(f(ST )− V Φ0,φ

A,Λ

T

))])= uA

(0, s0 −

√AΦ0σ

)+

√A〈Φ0σ,Φ0〉

2.

From Theorem 2.1 we obtain immediately the following corollary which says thatthe asymptotic value of the utility indifference prices is equal to the price of thevanilla European option with the payoff gA(ST ) and the shifted initial stock prices0 −

√AΦ0σ.

Corollary 2.2. For vanishing linear price impact Λ ↓ 0 and re-scaled high risk-aversion A/Λ with A > 0 fixed, the utility indifference price of f(ST ) has the scalinglimit

limΛ↓0

π(Λ, A/Λ,Φ0, f) = uA(

0, s0 −√AΦ0σ

).

Proof. Apply (2.8) and take f ≡ 0 for the denominator of (2.3).

We end this section with the following example.

Example 2.3. Consider a European option with the payoff

f(x) = (〈a, x〉+ b)+, x ∈ Rd

for some constant a ∈ Rd and b ∈ R. Then we have

gA(x) := supy∈Rd

[(〈a, x+ y〉+ b)

+ −⟨yσ−1, y

⟩2√A

], x ∈ Rd.

Clearly, the quadratic pattern y → 〈a, y〉 − 〈yσ−1,y〉

2√A

attains its maximum at y∗ :=√Aaσ. This together with the obvious inequality gA ≥ f ≥ 0 yields that

gA(x) =

(〈a, x+ y∗〉+ b−

⟨y∗σ−1, y∗

⟩2√A

)+

=

(〈a, x〉+ b+

√A 〈aσ, a〉

2

)+

.

4

Fig. 2.1. We consider a one dimensional Bachelier model with σ = 1 and µ = 5. The maturitydate is T = 1 and the initial stock price is s0 = 8. Let f(ST ) = (ST − 8)+, i.e. we take at themoney call option. For A = 1 and Λ = 0.001 we simulate a solution to the ODE (2.7) with the

initial value Φ0 = 0. The blue line is a simulation of the map t → DxuA(t, St −

√AΦΛ,A

t σ)

and

the yellow line is a simulation (for the same Brownian path) of the map t→ ΦΛ,At .

3. The Dual Problem and the Lower Bound. In this section we establishthe inequality ≥ in (2.8).

We start with the following lemma.

Lemma 3.1. Denoting by Q the set of all probability measures Q ∼ P with finiteentropy EQ

[log(dQdP)]<∞ relative to P, we have

c(Λ, α,Φ0, f)

≥ supQ∈Q

EQ

[f (ST )− 〈Φ0, ST − s0〉 −

1

αlog

(dQdP

)(3.1)

− 1

2Λ

∫ T

0

||St − EQ (ST |Ft)||2 dt].

Proof. The proof rests on the classical Legendre-Fenchel duality inequality

(3.2) pq ≤ ep + q(log q − 1), p ∈ R, q > 0.

Let φ ∈ A and Q ∈ Q. From the Girsanov theorem it follows that there exists aprocess θ = (θ1

t , ..., θdt )t∈[0,T ] such that WQ

t := Wt−∫ t

0θvdv, t ∈ [0, T ] is a Q–Brownian

motion. Moreover, from the equality

Z :=dQdP

= exp

(∫ T

0

〈θt, dWt〉 −1

2

∫ T

0

||θt||2dt

)5

and the fact that Q ∈ Q we obtain that

EQ [logZ] =1

2EQ

[∫ T

0

||θt||2dt

]<∞

and so from (2.1)

(3.3) EQ

[sup

0≤t≤T||St||2

]<∞.

Without loss of generality we assume that EP

[eα(f(ST )−V Φ0,φ

T

)]< ∞. Thus, from

(3.2) we obtain

(3.4) αEQ

[(f(ST )− V Φ0,φ

T

)+]≤ EP

[eα(f(ST )−V Φ0,φ

T

)]+ EQ [logZ] <∞.

Next, from (2.2) and the integration by parts formula it follows that

(3.5) V Φ0,φT = 〈Φ0, ST − s0〉+

∫ T

0

(〈φt, ST − St〉 −

Λ

2||φt||2

)dt.

Hence, from the simple inequality

〈φt, ST − St〉 −Λ

4||φt||2 ≤

1

Λ||ST − St||2

we obtain

f(ST )− V Φ0,φT ≥ −V Φ0,φ

T

≥ −〈Φ0, ST − s0〉+

∫ T

0

(Λ

4||φt||2 −

1

Λ||ST − St||2

)dt.

This together with (3.3)–(3.4) gives that

(3.6) EQ

[∫ T

0

||φt||2dt

]<∞.

From (3.2)–(3.3) and (3.5)–(3.6) we obtain that for any γ > 0

EP

[eα(f(ST )−V Φ0,φ

T

)]≥ EP

[αγZ

(f(ST )− 〈Φ0, ST − s0〉 −

∫ T

0

(〈φt, ST − St〉 −

Λ

2||φt||2

)dt

)]− EP [γZ (log(γZ)− 1)]

= αγEQ [f(ST )− 〈Φ0, ST − s0〉]

− αγEQ

[∫ T

0

(〈φt,EQ[ST |Ft]− St〉 −

Λ

2||φt||2

)dt

]− γ(log γ − 1)− γEQ [logZ]

≥ αγEQ

[f(ST )− 〈Φ0, ST − s0〉 −

1

2Λ

∫ T

0

||EQ[ST |Ft]− St||2 dt

]− γ(log γ − 1)− γEQ [logZ](3.7)

6

where the last inequality follows from the maximization of the quadratic pattern

φt → 〈φt,EQ[ST |Ft]− St〉 −Λ

2||φt||2, t ∈ [0, T ].

Optimizing (3.7) in γ > 0 we arrive at

1

αlog

(EP

[eα(f(ST )−V Φ0,φ

T

)])≥ EQ

[f(ST )− 〈Φ0, ST − s0〉 −

1

2Λ

∫ T

0

||EQ[ST |Ft]− St||2 dt−1

αlogZ

].

Since φ ∈ A and Q ∈ Q were arbitrary we complete the proof.

Remark 3.1. By mimicking the arguments of Proposition A.2 in [5] to the mul-tidimensional case one can show that the inequality in (3.1) is in fact an equality.Namely, there is no duality gap. However, since we need only the lower bound of theduality then we just provide a self contained proof for (3.1).

Next, we fix A > 0 and prove the following key result.

Proposition 3.2. Let h : Rd → Rd be a bounded and measurable function andlet Y = h(WT ). Then,

lim infΛ↓0

c (Λ, A/Λ,Φ0, f) ≥ EP

[f (s0 +WTσ − Y ) + 〈Φ0, Y 〉 −

1

2√A

⟨Y, Y σ−1

⟩].

Proof. For a matrix valued process Ξ = Ξijt 1≤i,j≤d,0≤t≤T denote∫ t

0Ξs · dWs

the row vector (Z1, ..., Zd) which is given by Zi =∫ t

0

∑dj=1 Ξijs dW

js . By applying the

martingale representation theorem and standard density arguments it follows that

without loss of generality we can assume that Y = y +∫ T

0H(s,Ws) · dWs where

y ∈ Rd is a constant vector and H : [0, T ]×Rd →Md(R) is a continuous and boundedfunction which satisfies H[T−δ,T ]×Rd ≡ 0 for some δ > 0.

The proof of Proposition 3.2 is done in three steps. In the first step we fix Λ > 0and construct a sequence of probability measures Qn ∼ P, n ∈ N with finite entropy.In the second step we estimate (for the constructed probability measures) the asymp-totic value of the components in the right hand side of (3.1) for α := A/Λ. In the laststep we summarize the computations and apply Lemma 3.1.

Step I: Recall the drift vector µ = (µ1, ..., µd) which appears in (2.1). For anyΛ > 0 and 0 ≤ s ≤ t ≤ T define

KΛt,s := cosh

(√A(T − t)σ/Λ

)(sinh

(√A(T − s)σ/Λ

))−1

and aΛt := µσ−1 +

√A

ΛyKΛ

t,0

where for any square matrix ξ

cosh(ξ) :=exp(ξ) + exp(−ξ)

2, sinh(ξ) :=

exp(ξ)− exp(−ξ)2

and exp(ξ) is the matrix exponential of ξ.

7

Fix Λ > 0. For any n ∈ N define the processes (which are also depend on Λ)

Wn = (Wn,1t , ...,Wn,d

t )t∈[0,T ] and θn = (θn,1t , ..., θn,dt )t∈[0,T ] by the following recursiverelations. For t ∈ [0, T/n]

θnt := aΛt , Wn

t := Wt +

∫ t

0

θnv dv

and for k = 1, ..., n− 1, t ∈ (kT/n, (k + 1)T/n]

θnt := aΛt + I||κnt ||<nκ

nt , Wn

t := Wt +

∫ t

0

θnv dv

where I denotes the indicator function and

κnt :=

√A

Λ

∫ kTn

0

KΛt,sH(s,Wn

s ) · dWns , t ∈ (kT/n, (k + 1)T/n].

Clearly, for any n the process θn is a bounded process, and so from the Girsanovtheorem there exists a probability measure Qn such that Wn is a Qn–Brownian mo-tion. For these probability measures we have the weak convergence (on the space ofcontinuous function Cd[0, T ])

(3.8) Qn θn ⇒ P (aΛ + κΛ

)where κΛ = (κΛ

t )t∈[0,T ] is given by κΛt :=

√A

Λ

∫ t0KΛt,sH(s,Ws) · dWs.

Step II: From the Fubini theorem, (2.1), (3.8) and and the simple equality√A

Λ

∫ TsKΛt,sdt = σ−1 ∀s ∈ [0, T ], we get the weak convergence

(3.9) Qn ST ⇒ P (s0 +WTσ − Y ) .

Since H is bounded and H[T−δ,T ]×Rd ≡ 0 then the term Kt,sH(s,Ws) is uniformlybounded and so

(3.10) supn∈N

EQn

[sup

0≤t≤T||κnt ||p

]<∞, ∀p > 0.

In particular, Qn ST , n ∈ N are uniformly integrable. Thus, (3.9) implies (f has alinear growth)

(3.11) limn→∞

EQn [f (ST )− 〈Φ0, ST − s0〉] = EP [f (s0 +WTσ − Y ) + 〈Φ0, Y 〉] .

Next, let H ′ be the transpose of H. From the Fubini theorem, the Ito isometry,the growth bound (3.10) and the equality EQn [κnt ] = 0 it follows that

Λ

Alimn→∞

EQn

[log

(dQn

dP

)]=

Λ

2Alimn→∞

EQn

[∫ T

0

||θnt ||2dt

]

=Λ

2AEP

[∫ T

0

||aΛt + κΛ

t ||2dt

]≤ c1Λ +

1

2Λ

∫ T

0

||yKΛt,0||2dt

+1

2ΛEP

[tr

(∫ T

0

H ′(s,Ws)

(∫ T

s

(KΛt,s)

2dt

)H(s,Ws)ds

)](3.12)

8

for some constant c1 > 0 which does not depend on Λ.Finally, we estimate the last term in the right hand side of (3.1). From (2.1)

St − EQn [ST | Ft] = yGΛt + EQn

[∫ T

t

I||κnv ||<nκnvσdv | Ft

], t ∈ [0, T ]

for

GΛt :=

√A

Λσ

∫ T

t

KΛv,0dv = sinh

(√A(T − t)σ/Λ

)(sinh

(√ATσ/Λ

))−1

.

This together with (3.10) and the equality EQn [κnt ] = 0 yields

1

2Λlimn→∞

EQn

[∫ T

0

||St − EQn [ST | Ft]||2 dt

]

=1

2Λ

∫ T

0

∣∣∣∣yGΛt

∣∣∣∣2 dt+1

2Λlimn→∞

EQn

∫ T

0

∣∣∣∣∣∣∣∣∣∣EQn

[∫ T

t

κnvσdv | Ft

]∣∣∣∣∣∣∣∣∣∣2

dt

=

1

2Λ

∫ T

0

∣∣∣∣yGΛt

∣∣∣∣2 dt+1

2ΛEP

∫ T

0

∣∣∣∣∣∣∣∣∣∣EP

[∫ T

t

κΛv σdv | Ft

]∣∣∣∣∣∣∣∣∣∣2

dt

.(3.13)

From the Fubini theorem

EP

[∫ T

t

κΛv dv |Ft

]

=

√A

ΛEP

[∫ T

0

(∫ T

t∨sKΛv,sdv

)H(s,Ws) · dWs | Ft

]

=

√A

Λ

∫ t

0

(∫ T

t

KΛv,sdv

)H(s,Ws) · dWs.

Hence, the Ito isometry yields

1

2ΛEP

∫ T

0

∣∣∣∣∣∣∣∣∣∣EP

[∫ T

t

κΛv σdv |Ft

]∣∣∣∣∣∣∣∣∣∣2

dt

=

1

2ΛEP

[tr

(∫ T

0

H ′(s,Ws)

(∫ T

s

(Lλt,s

)2dt

)H(s,Ws)ds

)](3.14)

where

LΛt,s :=

√A

Λσ

∫ T

t

KΛv,sdv = sinh

(√A(T − t)σ/Λ

)(sinh

(√A(T − s)σ/Λ

))−1

.

Step III: In this step we take Λ ↓ 0. First, from the Ito isometry

(3.15) EP[⟨Y, Y σ−1

⟩]= 〈y, yσ−1〉+ EP

[tr

(∫ T

0

H ′(s,Ws)σ−1H(s,Ws)ds

)].

9

Next, since σ is positive definite, then for any ε > 0 we have the uniform convergence

limΛ↓0

(sup

0≤t≤T−ε

∣∣∣∣∣∣∣∣∣∣ 1

2Λ

∫ T

s

(Kλt,s

)2dt− σ−1

4√A

∣∣∣∣∣∣∣∣∣∣)

= limΛ↓0

(sup

0≤t≤T−ε

∣∣∣∣∣∣∣∣∣∣ 1

2Λ

∫ T

s

(Lλt,s

)2dt− σ−1

4√A

∣∣∣∣∣∣∣∣∣∣)

= 0.

Hence, by combining (3.1) and (3.11)–(3.15) (recall that H[T−δ,T ]×Rd ≡ 0) we obtain

lim infΛ↓0

c (Λ, A/Λ,Φ0, f)

≥ lim infΛ↓0

limn→∞

EQn

[f (ST )− 〈Φ0, ST − s0〉 −

1

αlog

(dQn

dP

)− 1

2Λ

∫ T

0

||St − EQn (ST |Ft)||2 dt]

≥ EP

[f (s0 +WTσ − Y ) + 〈Φ0, Y 〉 −

1

2√A

⟨Y, Y σ−1

⟩]as required.

We now have all the pieces in place that we need for the completion of the proofof the inequality ” ≥ ” in (2.8).

Proof. Recall the definition of gA given by (2.4). Choose ε > 0. From theLipschitz continuity of f it follows that there exists a finitely valued (and hencebounded and measurable) function ζ : Rd → Rd such that

(3.16) gA(x) < ε+ f (x− ζ(x))− 〈ζ(x)σ−1, ζ(x)〉2√A

, ∀x ∈ Rd.

By applying Proposition 3.2 for the random variable

Y := ζ(s0 −√AΦ0 +WTσ) +

√AΦ0σ

and (3.16) for x := s0 −√AΦ0σ +WTσ we obtain

lim infΛ↓0

c(Λ, A/Λ,Φ0, f)

≥ EP

[gA(s0 −

√AΦ0σ +WTσ

)+

⟨Y σ−1 −

√AΦ0, Y −

√AΦ0σ

⟩2√A

+ 〈Φ0, Y 〉 −1

2√A

⟨Y, Y σ−1

⟩ ]− ε

= uA(

0, s0 −√AΦ0σ

)+

√A〈Φ0σ,Φ0〉

2− ε

where the equality follows from the definition of uA given by (2.5). By taking ε ↓ 0we complete the proof.

4. Proof of the Upper Bound. In order to complete the proof of Theorem2.1 it remains to establish the following result.

10

Proposition 4.1. Fix A > 0 and recall the trading strategies ΦA,Λ, Λ > 0 whichare given by (2.7). Then,

limΛ↓0

Λ

Alog

(EP

[exp

(A

Λ

(f(ST )− V Φ0,φ

A,Λ

T

))])≤ uA

(0, s0 −

√AΦ0σ

)+

√A〈Φ0σ,Φ0〉

2.

Proof. Introduce the d–dimensional process

ΘAt := Dxu

A(t, St −

√AΦtσ

), t ∈ [0, T ].

From the ODE (2.7) it follows that for any Λ > 0

ΦA,Λt = Φ0 · exp(−√Atσ/Λ

)+

√A

Λ

∫ t

0

ΘΛv σ exp

(√A(v − t)σ/Λ

)dv, t ∈ [0, T ].

Since σ is positive definite and DxuA is uniformly bonded (f is Lipschitz), then there

exists a constant C > 0 such that

(4.1) sup0≤t≤T

||ΦA,Λt || ≤ C, ∀Λ > 0.

Next, for any Λ > 0 define the process Mλ = (MΛt )t∈[0,T ] by

MΛt := exp

(A

Λ

(uA(t, St −

√AΦA,Λt σ

)+

√A〈ΦA,Λt σ,ΦA,Λt 〉

2− V Φ0,φ

A,Λ

t

)).

From the Ito formula, (2.2) and (2.6)

dMΛt

MΛt

=A

Λ

⟨Dxu

A(t, St −

√AΦA,Λt σ

)− ΦA,Λt , dSt

⟩+

A2

2Λ2

∣∣∣∣∣∣(DxuA(t, St −

√AΦA,Λt σ

)− ΦA,Λt

)σ∣∣∣∣∣∣2 dt

− A3/2

Λ

⟨φA,Λt ,

(Dxu

A(t, St −

√AΦA,Λt σ

)− ΦA,Λt

)σ − Λ

2√AφA,Λt

⟩dt

=√A⟨φA,Λt σ−1, dSt

⟩where the last equality follows from (2.7). Hence, from (2.1)

exp(−√A⟨

ΦA,Λt − ΦA,Λ0 , µσ−1⟩)

MΛt , t ∈ [0, T ]

is a local–martingale, and so from the obvious inequality MΛ > 0 we conclude thatthis process is a super–martingale.

Finally, (2.4) yields that f(x) ≤ gA (x− yσ) + 〈yσ,y〉2√A

for all x, y ∈ Rd. This

together with the growth bound (4.1) and the above super–martingale property gives(observe that uA(T, ·) = gA(·)) that for any Λ > 0

Λ

Alog

(EP

[exp

(A

Λ

(f(ST )− V Φ0,φ

Λ

T

))])≤ Λ

Alog(EP[MΛ

T ])≤ Λ

Alog(MΛ

0

)+

Λ√A

2CT ||µσ−1||

= uA(

0, s0 −√AΦ0σ

)+

√A〈Φ0σ,Φ0〉

2+

Λ√A

2CT ||µσ−1||

11

and the result follows by taking Λ ↓ 0.

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