a discontinuous mispricing model under asymmetric information

European Journal of Operational Research 243 (2015) 944–955

Contents lists available at ScienceDirect

European Journal of Operational Research

journal homepage: www.elsevier.com/locate/ejor

Decision Support

A discontinuous mispricing model under asymmetric information

Winston S. Buckley a,∗, Hongwei Long b

a Department of Mathematical Sciences, Bentley University, Waltham, MA 02452, USAb Department of Mathematical Sciences, Florida Atlantic University, Boca Raton, FL 33431-0991, USA

a r t i c l e i n f o

Article history:

Received 23 October 2013

Accepted 25 December 2014

Available online 7 January 2015

Keywords:

Mispricing

Levy jumps

Asymmetric information

Optimal portfolio

Expected utility

a b s t r a c t

We study a discontinuous mispricing model of a risky asset under asymmetric information where jumps in

the asset price and mispricing are modelled by Lévy processes. By contracting the filtration of the informed

investor, we obtain optimal portfolios and maximum expected utilities for the informed and uninformed

investors. We also discuss their asymptotic properties, which can be estimated using the instantaneous

centralized moments of return. We find that optimal and asymptotic utilities are increased due to jumps in

mispricing for the uninformed investor but the informed investor still has excess utility, provided there is not

too little or too much mispricing.

© 2015 Elsevier B.V. All rights reserved.

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1 The literature related to mean-reversion in asset prices is very large and includes

stocks: Poterba and Summers (1988), Depenya and Gil-Alana (2002), Gropp (2004),

Mukherji (2011, 2012); currencies: Engel and Hamilton (1990), Cheung and Lai (1994),

1. Introduction

Asymmetric information models assume that there are two types

of investors in the market: informed and uninformed. The informed

investors (e.g., institutional investors with internal research capabili-

ties) observe both fundamental and market prices, while uninformed

investors (e.g., retail investors who rely on public information in or-

der to make investment choices) observe market prices only. The

uninformed investors are viewed as liquidity traders or hedgers. The

prevalence of informed traders affects liquidity, transaction costs, and

trading volumes. The informed investors partially reveal information

through trades, which can cause higher permanent price changes.

Asymmetric information asset pricing models rely on a noisy rational

expectation equilibrium in which prices only partially reveal the in-

formed investor’s information. Many empirical studies confirm that

information asymmetry is priced and imply that liquidity is a primary

channel that links information asymmetry to prices (see, e.g., Admati,

1985; Easley & O’Hara, 2004; Grossman & Stiglitz, 1980; Hellwig,

1980; Kelly & Ljungqvist, 2012; Wang, 1993).

Mispricing is the difference between the asset’s market price and

fundamental value. The fundamental value can be defined as the

market price that would prevail if all the market participants were

perfectly informed investors. Because of the mean-reverting nature

of the mispricing process, it is typically modelled by a continuous

Ornstein–Uhlenbeck (O–U) process, while the price of the risky asset

is usually modelled by a continuous geometric Brownian motion (see,

e.g., Buckley, Brown, & Marshall, 2012; Guasoni, 2006; Wang, 1993).

∗ Corresponding author.

E-mail addresses: [email protected] (W. S. Buckley), [email protected]

(H. Long).

S

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v

(

http://dx.doi.org/10.1016/j.ejor.2014.12.045

0377-2217/© 2015 Elsevier B.V. All rights reserved.

he mean-reverting speed or equivalently, the mean reverting-time,

s a proxy for mispricing. Mean-reversion1 is well-documented in the

mpirical financial literature and applies to asset returns, stock prices,

urrencies/exchange rates, interest rates, commodities, indexes, stock

ndex futures, and options.

This paper addresses how asymmetric information, mispricing,

nd jumps in both the price of a risky asset and its mispricing affect

he optimal portfolio strategies and maximum expected utilities of

wo distinct classes of rational long-horizon investors in an economy

here preference is logarithmic. For the purpose of this exposition,

e take the risky asset to be stock, but the model can be applied to any

sset (e.g. currencies) where prices are mean-reverting. The investors

ssume that the risky asset has a fundamental or true value (for ex-

mple, expected discounted future dividends) as well as a market

rice. When prices move away from its fundamental value, they al-

ays revert to it (see, e.g., LeRoy & Porter, 1981; Poterba & Summers,

988; Shiller, 1981; Summers, 1986). What do we mean by funda-

ental value? Allen, Morris, and Postlewaite (1993) posit that the

undamental value of an asset is the present value of the stream of

he market value of dividends or services generated by this asset. The

undamental value can also be defined as the market price that would

revail if the cost of gathering and processing information is zero for

weeney (2006), Serban (2010); indices: Miller, Muthuswamy, and Whaley (1994),

alvers, Wu, and Gilliland (2000), Balvers and Wu (2006), Caporale and Gil-Alana

2008), Kim, Stern, and Stern (2009), Chen and Kim (2011), Spierdijk, Bikker, and

an den Hoek (2012), Malin and Bornholt (2013); index futures: Monoyios and Sarno

2002), Miao, Lin, and Chao (2014).


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W. S. Buckley, H. Long / European Journal of Operational Research 243 (2015) 944–955 945

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ll investors. Any of the definitions above will suffice. However, we

dopt the caveat in Allen et al. (1993) and take fundamental value to

ean the value of an asset in normal use, as opposed to some value

t may have as a speculative instrument.

Because mispricing is the difference between the asset’s market

rice and its fundamental value, if the fundamental value or asset

rice jumps, then it follows that the mispricing will jump as well,

rovided the jump components are not identical and cancel each

ther. This is particularly evident in the case of independently driven

ump processes (see, e.g., Applebaum, 2004). All recent studies (e.g.,

uckley et al., 2012; Buckley, Long, & Perera, 2014; Guasoni, 2006)

ssume that mispricing is a continuous O–U process. However, in this

aper, mispricing is no longer purely continuous. Instead, it jumps

nd is driven by a mean-reverting O–U process which has a continu-

us component as well as a discontinuous component generated by

pure-jump Lévy process. As in Buckley et al. (2014), the price of

he risky asset is still subject to Levy jumps. We solve this model for

oth investors and present explicit formulas for their optimal port-

olios and maximum expected logarithmic utilities under reasonable

ssumptions.

Portfolio allocation problems have been extensively studied since

he seminal work of Markowitz (1952). Merton’s (1971) model is

he benchmark of optimal asset allocation in the continuous-time

ramework. There is a rich plethora of portfolio optimization papers2

n various settings. We refer to Buckley et al. (2012, 2014) for related

iterature review and discussion on recent progress on this topic.

In this paper, we find that the optimal portfolio of each investor

ontains excess risky asset that depends on the Lévy measures of both

ump processes (asset price and mispricing), the diffusive volatility,

nd the level of mispricing, as represented by the mean-reversion

peed or time. Under quadratic approximation of the portfolios, we

how that excess asset holdings are dependent on the first two instan-

aneous centralized moments of return (see, e.g., Cvitanic et al., 2008).

n addition, the maximum expected utility from terminal wealth for

ach investor is increased by the presence of jumps in the mispric-

ng. This suggests that investors are better off when mispricing jumps

han when it changes continuously.

Further, we show that the uninformed investor obtains excess

tility from jumps in mispricing. However, as proven in prior studies

y Guasoni (2006) and Buckley et al. (2012, 2014), the informed in-

estor still has positive excess utility over the uninformed investor.

his implies that the informed investor gets more utility from the dif-

usive component driving the asset price process. We also show that

he asymptotic excess optimal expected utility from terminal wealth

f the informed investor has a similar structure to that presented in

uasoni (2006) and Buckley et al. (2012, 2014), but is increased by a

actor directly attributed to the jumps in the mispricing. As in the case

f jumps in stock price only, the mean-reversion rate λ is replaced

y a smaller adjusted mean-reversion rate λ, which depends on the

olatility of the asset price only.

There is also an equivalent representation of excess utility by way

f the mean-reversion speed λ of an adjusted continuous O–U mis-

ricing process which also depends on the volatility of the long-run

sset price. In this framework, the excess optimal expected utility of

he informed investor is greater (less) than its continuous Merton

1971) geometric Brownian motion counterpart if the product of

he diffusive variance and the quadratic variation of the mispricing

rocess is greater (less) than the second instantaneous centralized

oment of return of the jump component of the asset price process.

otwithstanding the presence of asymmetric information, mispricing

2 See Mansini and Speranza (1999), Kellerer, Mansini, and Speranza (2000), Soyer and

anyeri (2006), Celikyurt and Ozekici (2007), Corazza and Favaretto (2007), Cvitanic,

olimenis, and Zapatero (2008), Lin and Liu (2008), Canakoglu and Ozekici (2010),

u, Lari-Lavassani, and Li (2010), Huang, Zhu, Fabozzi, and Fukushima (2010), Yu,

akahashi, Inoue, and Wang (2010), and Buckley et al. (2012, 2014).

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nd jumps, our results show that it still pays to be more informed in

he long-run, unless there is too little or too much mispricing. More-

ver, our results nest those contained in Guasoni (2006) and Buckley

t al. (2014).

The practical, financial, economic and operational implication of

his paper is that when asymmetric information and jumps exist

n both asset price and mispricing, informed and uninformed long-

orizon investors will maximize their expected logarithmic utilities

rom terminal wealth by holding portfolios that contain excess asset

oldings which depend not only on the level of information asymme-

ry and preference, but also on the nature and frequency of the jumps

n both asset price and mispricing as dictated by the governing Lévy

easures. To the best of our knowledge, this paper is the first to study

he utility and portfolio implications for the risky asset by investors

hen asymmetric information, logarithmic preferences, jumps in as-

et price and mispricing are intertwined, and therefore contributes to

he finance and operational research literature.

Our model is related to heterogeneous agents models (HAM),

here investors observe the same information but have different be-

iefs. However, there are two important distinctions. First, switching

s not allowed between investor classes, i.e., an uninformed investor is

ot allowed to become informed, and vice versa. Second, one investor

nows more than the other (see, e.g., Chiarella, Dieci, & He, 2009 and

e, 2012 for a review of the HAM literature).

As in Buckley et al. (2014), our model is different from insider

rading models3 which use enlargement of filtrations to obtain the

ptimal portfolio and utility of the insider trader/informed investor.

n contrast to insider trading models, we specify the price dynamics

f the informed investor in the larger filtration, and then obtain the

ynamics for the uninformed investor by contracting (or restricting)

he larger filtration. We then compute and compare optimal portfolios

nd expected logarithmic utilities for each investor relative to their

espective filtrations.

The remainder of the paper is organized as follows. The model

s introduced in Section 2, including information flows/filtration and

sset price dynamics of investors. In Section 3, expected utilities are

aximized, and optimal portfolios are computed and estimated us-

ng instantaneous centralized moments of returns. Asymptotic and

xcess utilities are presented in Section 4. We conclude in Section 5

y giving directions of possible future research. All proofs are given

n Appendix A.

. The discontinuous mispricing model

The economy consists of two assets—a risk-free asset B, called

ank account or money market (or U.S. Treasury bill), and a risky

sset S, called stock. The risk-free asset earns a continuously com-

ounded risk-free interest rate rt , and has price Bt = exp(∫ t

0 rs ds)

.

he continuous component of the stock’s percentage appreciation

ate is μt , at time t ∈ [0, T], where T > 0 is the investment horizon.

he stock is subject to volatility σt > 0. The market parameters are

t, rt, σt, t ∈ [0, T], and are assumed to be deterministic functions.

he stock’s Sharpe ratio or market price of risk θt = μt−rtσt

is square

ntegrable. The risky asset has price S, and lives on a probability space

Ω, F , P) on which is defined two independent standard Brownian

otions W = (Wt)t≥0 and B = (Bt)t≥0. The stock is viewed by investors

n disjoint classes populated by uninformed and informed investors,

ndexed by i = 0 and i = 1, respectively. Investors have filtrations Kit

3 See, e.g., Pikovsky and Karatzas (1996), Amendinger, Imkeller, and Schweizer

1998), Amendinger, Becherer, and Schweizer (2003), Ankirchner, Dereich, and

mkeller (2006), Ankirchner and Imkeller (2007), who solve this problem for loga-

ithmic and power utilities. Grorud (2000) studies insider trading in a discontinuous

arket, while Goll and Kallsen (2003) give a complete explicit characterization of

og-optimal portfolios without constraints.

946 W. S. Buckley, H. Long / European Journal of Operational Research 243 (2015) 944–955

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with K0t ⊂ K1

t ⊂ F , t ∈ [0, T]. All random objects are defined on a

filtered probability space (Ω,F ,Ki, P).Based on the classical Black–Scholes geometric Brownian motion

model (cf. Black & Scholes, 1973) and Merton jump-diffusion model

(cf. Merton, 1971, 1976) as well as some recent models discussed in

Kou (2002), Guasoni (2006) and Buckley et al. (2014), we propose

a new and more general model for the risky asset by incorporat-

ing mean-reversion, mispricing, asymmetric information, and Lévy

jumps. This new model will be more flexible and help capture the

asset price movements including sudden jumps in both the funda-

mental value (through the mispricing process) and market price. It

will also greatly enhance our understanding of the link between as-

set mispricing and asymmetric information in financial markets. We

therefore extend the theory on mispricing models under asymmetric

information in Lévy markets to include the situation where both stock

price and mispricing jump.

We model the jumps in the mispricing by a pure jump Lévy process

Z = (Zt)t≥0 with E(Zt) = 0 and E(Z2t ) < ∞, for all t ∈ [0, T]. Jumps in

the stock price are driven by an independent pure-jump Lévy process

X = (Xt)t≥0. Both jump processes are governed by Poisson random

measures NS and NU defined on R+ × B(R − {0}) that are linked to the

stock (S) and mispricing (U), respectively. These random measures

count the jumps of the stock price S (equivalently, X) and mispricing U

(equivalently, Z), respectively, in the time interval (0, t). The respec-

tive Lévy measures are vS(dx) = ENS(1, dx) and vU(dz) = ENU(1, dz).

We use NU(dt, dz)�= NU(dt, dz)− vU(dz)dt to denote the compensated

Poisson measure.

2.1. The model

The risky asset or stock has price S, with log-return dynamics

d(log St) =(μt − 1

2σ 2

t

)dt + σt dYt + dXt, t ∈ [0, T] (1)

Yt = p Wt + q Ut, p2 + q2 = 1, p ≥ 0, q ≥ 0, (2)

dUt = −λ Ut dt + dLt, λ > 0, U0 = 0, (3)

Lt = Bt + Zt, (4)

Xt =∫ t

0

∫R

x NS(dt, dx), (5)

Zt =∫ t

0

∫R

z NU(dt, dz), E(Zt) = 0, E(Z2

t

)< ∞. (6)

W and B are independent standard Brownian motions. U = (Ut) is a

mean-reverting Ornstein–Uhlenbeck process with mean-reversion

rate λ. Because of the exponential decay associated with mean-

reverting processes, its half-life is H = ln 2λ

. Half-life is a measure of

the slowness of a mean-reverting process, and is measured in trad-

ing days. It may be defined as the average time it takes the mean-

reverting process to get pulled half-way back to its long-term mean.

In this context, half-life is a very important number—it gives an es-

timate of how long we should expect the mispricing to remain far

from zero. Thus, a half-life of 10 days means that it takes 20 trad-

ing days on average for the mispricing to revert to zero. Noting that

half-life is inversely proportion to the mean-reversion rate, we sim-

ply use δ = 1λ

as the measure of mispricing. Thus, half-life H, λ and

δ are equivalent measures of mispricing. Large values of the mean-

reversion time δ (equivalently, small values of λ) correspond to high

levels of mispricing—that is, the mispricing process U takes a long

time to revert to zero, and as such, indicates that mispricing is persis-

tent. The jumps in the mispricing Ut are controlled by the pure-jump

Lévy process Zt , although it also has a continuous component Bt .

Standing assumption. We assume that NS and NU are independent

random measures if NS �= NU . That is, X and Z are independent pure-

jump Lévy processes with E(Zt) = 0, for all t ∈ [0, T] and∫

zvU(dz) <
R
. From (6),

t =∫ t

0

∫R

zNU(ds, dz) =∫ t

0

∫R

zNU(ds, dz)− t

∫R

zvU(dz).

learly Z = (Zt) is a martingale provided∫

R zvU(dz) < ∞. From Eqs. (3)

nd (4), we get the dynamics

Ut = −λ Utdt + dBt + dZt, (7)

hich admits the unique solution

t =�= UBt + UZ

t , (8)

here

Bt =

∫ t

0

e−λ(t−s)dBs and UZt =

∫ t

0

e−λ(t−s)dZs. (9)

ote, in this case, that the mispricing process U is a linear combi-

ation of two independent O–U processes: a continuous componentB driven by the Brownian motion B, which is identical to the mis-

ricing process used by Buckley et al. (2014) and Guasoni (2006),

nd a jump component UZ driven by the pure-jump martingale Z.

t is easy to show that for all λ > 0 and t ∈ [0, T], E(UBt ) = 0 and

(UBt )

2 = 1−e−2λ t

2λ. We have a similar result for the process UZ .

roposition 1. Let λ > 0. For each t ∈ [0, T], E(UZt ) = 0,

ar(UZt ) = 1 − e−2λ t

2λ

∫R

z2 vU(dz),

nd limt→∞

Var(UZt ) = 1

2λ

∫R

z2 vU(dz).

We have a similar result for the mean-reverting mispricing

rocess U.

roposition 2. Let λ > 0. For each t ∈ [0, T], E(Ut) = 0,

ar(Ut) =(

1 − e−2λ t

2λ

) (1 +

∫R

z2 vU(dz)

),

nd limt→∞

Var(Ut) = 1

2λ

(1 +

∫R

z2 vU(dz)

).

t now follows from Eqs. (2) and (8) that

t = p Wt + q Ut = p Wt + q

∫ t

0

e−λ(t−s)dBs + q

∫ t

0

e−λ(t−s)dZs,

s a stochastic process consisting of three independent components:

wo are continuous, while one is a pure-jump Lévy process. Taking

ifferentials and importing Eq. (7) yield

Yt = p dWt + q dUt = p dWt + q(−λUtdt + dBt + dZt)

= p dWt + q dBt − λqUtdt + q dZt

= p dWt + q dBt − λqUBt dt − λUZ

t dt + q dZt

= dB1t + υ1,B

t dt + υZt dt + q dZt = dB1

t + υ1t dt + q dZt, (10)

here

B1t

�= p Wt + q Bt, (11)

1t

�= υ1,Bt + υZ

t , υ1,Bt

�= −λ q UBt , υZ

t

�= −λ q UZt . (12)

.2. Filtration of the investors

We expand the filtration developed in Buckley et al. (2012, 2014)

nd Guasoni (2006) to include the information generated by X = (Xt)nd the martingale Z = (Zt). We assume that all filtrations obey the

sual hypothesis. Namely, they are right-continuous and complete.

et F it (i = 1, 0) be the information flow/filtration for the informed

nd uninformed investors in the models where there are no jumps in


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ispricing or asset price (see Buckley et al., 2012 or Guasoni, 2006).

hese filtrations are defined as

1t

�= σ (Ws, Bs : s ≤ t) = FS, Ut and F0

t

�= σ (Ys : s ≤ t) = FYt .

he filtration of the informed investor in our new framework is K1 =K1

t )t≥0 defined by

1t = F1

t ∨ σ(Zu, Xu : u ≤ t), t ∈ [0, T].

The uninformed investor observes the stock price only and does

ot know what the mispricing is, if it exists. Uninformed investors

ave filtration K0, where K0t ⊂ F is defined by the prescription:

0t = F0

t ∨ σ(Zu, Xu : u ≤ t), t ∈ [0, T],

hich is contained in K1t , the filtration of the informed investor.

We shall develop the dynamics for these investors, starting with

he log-return dynamics for the stock given by Eqs. (1)–(6). Recall that

t = UBt + UZ

t .

efinition 1. Define the continuous part of the process Y by

ct

�= Yt − q UZt = p Wt + q UB

t . (13)

e have the following important result.

emma 1. There exist an K0-Brownian motion B0, and a process υ0, Bt ≡

φ0, Bt , adapted to K0

t , such that for each t ∈ [0, T] and q ∈ [0, 1],

0t = Yc

t +∫ t

0

φ0, Bs ds ≡ Yt − q UZ

t −∫ t

0

υ0, Bs ds, (14)

here λ > 0 is the mean-reversion speed of U, p2 + q2 = 1 and

0, Bt = −λ

∫ t

0

e−λ(t−s)(1 + γ (s))dB0s , (15)

(s) = 1 − p2

1 + p tanh (λps)− 1, p ∈ [0, 1].

rom Eq. (14), we have Yt = B0t + ∫ t

0 υ0, Bs ds + q UZ

t . Therefore,

Yt = dB0t + υ0, B

t dt + q dUZt = dB0

t + υ0, Bt dt − λ q UZ

t dt + qdZt

= dB0t + υ0, B

t dt + υZt dt + qdZt = dB0

t + υ0t dt + qdZt, (16)

here

0t

�= υ0,Bt + υZ

t , υ0,Bt

�= −λ

∫ t

0

e−λ(t−s)(1 + γ (s))dB0s ,

Zt

�= −λ q UZt . (17)

Note that (16) has the same form as (10). We now extract Proposi-

ion 2 in Buckley et al. (2012) concerning υ i, Bt , the number of standard

eviations from the mean return μt , since it will be required in the

equel.

roposition 3. Let λ > 0, i ∈ {0, 1}, p ∈ [0, 1], p2 + q2 = 1 and t ∈0, T].

Let υ0, Bt = −λ

∫ t0 e−λ(t−s)(1 + γ (s))dB0

s and υ1, Bt = −λ q Ut. Then

(0) E[υ i, Bt ] = 0.

(1) Varυ0, Bt = E[υ0, B

t ]2 = λ2∫ t

0 e−2λ(t−s)(1 + γ (s))2ds ≤ 12 λ q2.

(2) Varυ1, Bt = E[υ1, B

t ]2 = λ2 (1 − p2)(1 − e−2λt) ≤ 1

2 λ q2.

(3) Varυ i, Bt = E[υ i, B

t ]2 → λ2 (1 − p)(1 + (−1)i+1p) : as t → ∞.

(4)∫ T

0 E[υ i, Bt ]2dt λ

2 (1 − p)(1 + (−1)i+1p)T as T −→ ∞.

(5) As T −→ ∞, the asymptotic excess cumulative variance of the

vis is∫ T

0

E[υ1, Bt ]2dt −

∫ T

0

E[υ0, Bt ]2dt λ p (1 − p)T ≤ λ

4T.

We have a useful result for the process υ i, which is a generalization

f the process used in Buckley et al. (2012, 2014) and Guasoni (2006).

roposition 4. Let λ > 0, t ∈ [0, T], i ∈ {0, 1} and υ it = υ i,B

t + υZt . Then

or p, q ∈ [0, 1] with p2 + q2 = 1, we have

(1) E[υ it] = 0.

(2) Var[υ1t ] = λ

2 (1 − e−2λ t)(1 + p)(1 + (−1)1p)(1 + ∫R z2 vU(dz)).

(3) Var[υ0t ] = λ2

∫ t0 e−2λ(t−s)(1 + γ (s))2ds +

λ2 q2(

1−e−2λ t

2λ

) ∫R z2 vU(dz).

Moreover,

limt→∞

E[υ it]

2 = λ

2(1 + p)(1 + (−1)ip)

(1 +

∫R

z2 vU(dz)

).

.3. Asset price dynamics for the investors

Let i ∈ {0, 1}. Define μ∗t

�= μt − 12 σ 2

t . The dynamics for the stock

s given by (1) as d(log St) = μ∗t dt + σt dYt + dXt, t ∈ [0, T]. Imposing

2), (3) and (10) yields

(log St) = μ∗t dt + σt

(dBi

t + υ itdt + q dZt

) + dXt

= μ∗t dt + υ i

tσtdt + σtdBit + q σtdZt + dXt

=(μ∗

t − qσt

∫R

xυU(dx)+ υ itσt

)dt + σtdBi

t

+∫

R

q σtzNU(dt, dz)+∫

R

xNS(dt, dx)

= μ∗,it dt + σtdBi

t +∫

R

q σtzNU(dt, dz)+∫

R

xNS(dt, dx),

(18)

here

∗,it = μ∗

t − qσt

∫R

zυU(dz)+ υ itσt

= μt + υ itσt − qσt

∫R

zυU(dz)− 1

2σ 2

t . (19)

emark 1. If NS = NU = N then v = vU = vS, and (18) becomes

(log St) = μ∗,it dt + σt dBi

t +∫

R

x (qσt + 1)N(dt, dx). (20)

e now quote a very useful result on page 47 in Applebaum (2004).

emma 2 (Proposition 1.3.12, Applebaum (2004)). If (N1(t), t ≥ 0)nd (N2(t), t ≥ 0) are two independent Poisson processes defined on the

ame probability space (,F , P) with arrival times (T(j)n , n ∈ N), for

= 1, 2, respectively, then P(T(1)m = T(2)

n for some m, n ∈ N) = 0.

his result tells us that independent Poisson processes do not jump

t the same time. Thus, for random measures NU and NS and Borel

et A ∈ B(R − {0}), NU(t, A) and NS(t, A) are independent Poisson pro-

esses and do not jump together by Lemma 2. We have the following

mportant result.

heorem 1. Let i ∈ {0, 1}. The log-return dynamics (1), for the ith in-

estor, is

(log St) = μ∗, it dt + σtdBi

t +∫

R

qσt z NU(dt, dz)+∫

R

x NS(dt, dx),

(21)

nd its percentage return dynamics is

dSt

St−= μi

tdt + σtdBit +

∫R

(eqσt z − 1)NU(dt, dz)

+∫

R

(ex − 1)NS(dt, dx), (22)

here

∗, it

�= μit − 1

2σ 2

t , μit

�= μt + υ itσt − qσt

∫zυU(dz).


3

s

A

πs

c

p

t

E

T

o

i

t

m

a

3

s

a

t

G

w

i

d

S

t∫D

G

L

T

v

n

T

t

t

π

I

π

w

KR

The asset’s percentage return dynamics (22) consists of four com-

ponents: the first is continuous; the second is random and driven

by a standard Brownian motion; the third is driven by jumps in the

mispricing process and the fourth component is driven by jumps in

the asset price. However the total mean return is driven only by three

components: the first is continuous, the second is due to the jumps

in mispricing, and the third is due to jumps in the asset price.

3. Optimal portfolios and expected utilities

Random objects for the ith investor live on a filtered space

(Ω,F ,Ki, P) for i ∈ {0, 1}.

Definition 2 (Portfolio process). The process π ≡ π i : [0, T] −→ R is

called the portfolio process of the ith investor, if π it(ω) is Ki

t-adapted

for each ω ∈ and E∫ T

0 (π itσt)2dt < ∞.

Note that π it is really π i

t(ω), where ω ∈ , and hence, is a random

process. π it is the proportion of the wealth of the ith investor that is

invested in the risky asset at time t ∈ [0, T]. The remainder, 1 − π it , is

invested in the risk-free asset or money market. Where it is clear, we

drop the superscript “i” and simply use π for the portfolio process.

Definition 3 (The wealth processes). The wealth process for the ith

investor is Vi, π , x : [0, T] −→ R, where Vi, π, xt is the value of the port-

folio consisting of the risky asset and money market at time t, when

π it is invested in the risky asset and the initial capital is x > 0.

For brevity, we denote this process by Vi, π = (Vi, πt ) or simply

Vi = (Vit), t ∈ [0, T], when the context is clear. The initial capital is

Vi0 = x > 0.

Definition 4 (Admissible portfolio). π is called an admissible portfo-

lio if Vπt > 0 almost surely, for all t ∈ [0, T].

3.1. Maximization of logarithmic utility from terminal wealth

Let π ≡ π i ≡ (π it), t ∈ [0, T], be the portfolio process of the ith

investor. Let θ i = μi−rσ be the Sharpe ratio. The corresponding wealth

process with initial capital x > 0 is: Vit ≡ Vi,π

t ≡ Vi,π,xt . We now give

the dynamics for Vi.

Theorem 2. The percentage return of the wealth Vi of the ith investor

with stock price dynamics (1) is:

dVit

Vit−

= (rt + πt σt θ

it

)dt + πt σt dBi

t +∫

R

πt(eqσt z − 1)NU(dt, dz)

+∫

R

πt(ex − 1)NS(dt, dx). (23)

The discounted wealth process is

Vit = Vi

t exp

(−

∫ t

0

rsds

)= Vi

0 exp

(∫ t

0

(πsσsθ

is − 1

2π2

s σ 2s

)ds +

∫ t

0

πsσsdBis

)×�0≤u≤t(1 + πu(e

qσu Zu − 1))× �0≤s≤t(1 + πs(e Xs − 1)).

(24)

Remark 2. We use the subscripts “u” and “s” in Eq. (24) to distin-

guish the jumps due to mispricing U and asset price S, respectively.

The budget constraint (23) consists of four components: the first is

continuous; the second is random and drive by a standard Brownian

motion; the third is driven by jumps in the mispricing process and

the fourth component is driven by jumps in the asset price. Note that

volatility plays a key role in the component driven by the mispricing

process, while the component driven by the jumps in asset price is

independent of volatility. The discounted wealth process has identi-

cal dependencies. This will be the argument used to maximize the

expected logarithmic utility function.

.2. Admissible set for each investor

Let Vi0 = x. We seek a portfolio process π i = (πt)i

t≥0 in an admis-

ible set Ai(x), defined by

i(x)={π i: π i isKi − predictable such that V i, π i

t > 0 a.s.∀t ∈ [0, T]}.i is predictable if it is measurable with respect to the predictable

igma–algebra on [0, T] × , which is the sigma–algebra of all left

ontinuous functions with right limits on [0, T] × . The optimal

ortfolio for the informed or uninformed investor is π ∗,i ∈ Ai(x) such

hat(log V i, π ∗,i

T

) = maxπ∈Ai(x)

E log V i, πT .

hat is, π ∗,i = arg maxπ∈Ai(x) E log V i, πT . Since θt ≡ θ i

t is independent

f πt , observe that E log V i, πT is maximized iff E

∫ T0 f (i)

U,S(πit)dt is max-

mized. That is, if f (i)U,S(π

i) defined by (31) below is maximized on

he admissible set Ai(x). This approach is similar to the optimization

ethod used by Amendinger et al. (1998), Ankirchner et al. (2006),

nd Liu, Longstaff, and Pan (2003).

.3. The objective function G

In order to obtain optimal portfolio strategies, we need the as-

istance of an objective function G. For any Lévy measure va(·),∈ {U, S}, define the (partial) objective function Ga by the prescrip-

ion Ga : [0, 1] −→ R,

a(α ; s)�=

∫R

log(1 + α(es x − 1))va(dx), α ∈ [0, 1],

s ∈ Iσ�= (0, σmax] ∪ {1}, (25)

here s is a non-negative parameter or function representing volatil-

ty and σmax is positive. Note that α, and hence π , is restricted to the

omain [0, 1].

tanding assumption. We assume that an integer k ≥ 2 exists such

hat

R

(e±s x − 1

)kva(dx) < ∞, where a ∈ {U, S}, s ∈ Iσ .

efine the objective function G : [0, 1] −→ R by

(α ; s)�= GU(α ; s)+ GS(α ; 1). (26)

emma 3. Let s ∈ Iσ . If α ∈ [0, 1], then G′′(α ; s) < 0.

his result proves that G is strictly concave on [0, 1] for each fixed

olatility level s, and hence, will admit unique optimal portfolios. We

ow give a major result for optimal portfolios strategies of investors.

heorem 3. Let i ∈ {0, 1} and q ∈ [0, 1]. For the ith investor, the op-

imal portfolio π i, that maximizes the expected logarithmic utility from

erminal wealth over the investment period [0, T], is given by

it = μi

t − rt + G′(π it ; qσt

)σ 2

t

≡ μit − rt + G′

U

(π i

t ; qσt

) + G′S

(π i

t ; 1)

σ 2t

.

(27)

n terms of the stock’s total return bit, the optimal portfolio is

it = bi

t − rt − KU(qσt)− KS(1)+ G′(π it ; qσt

)σ 2

t

, (28)

here

a(s) =∫

(es x − 1)va(dx), a ∈ {U, S}, s ≥ 0. (29)


T

x

u

w

G

R

R

πl

n

w

3

s

s

s

d

A

I

V

w

V

T

π

w

G

O

p

a

m

d

a

3

c

s

t

t

i

M

w

K

F

n

σ

M

w

K

f

αn

t

G∫p

G

W

L

he maximum expected logarithmic utility from terminal wealth, with

> 0 in initial capital, is

i(x) = log x + 1

2E

∫ T

0

(θ i

t

)2dt + E

∫ T

0

f (i)U,S

(π i

t

)dt, (30)

here

f (i)U,S(π

i) = G(π i ; qσ)− 1

2(π iσ − θ i)2, (31)

(π i ; qσ) = GU(π i ; qσ)+ GS(πi ; 1),

μit = μt + υ i

tσt − qσt

∫R

xυU(dx),

υ it = υ i,B

t + υZt ,

υ1,Bt = −λ q UB

t , υ0,Bt = −λ

∫ t

0

e−λ(t−s)(1 + γ (s))dB0s ,

υZt = −λ q UZ

t ,

bit = μi

t + KU(qσt)+ KS(1). (32)

emark 3.

(a) The optimal portfolio strategy for each investor consists of

three components. First, there is a continuous component

which is the Merton (1971) optimal for that investor. The sec-

ond component is due to jumps in the mispricing process, level

of mispricing q and volatility. The third component is due to

jumps in the asset price only. One may also present the optimal

portfolio strategy of each investor through it total expected

return bt , the derivative of objective function G and kernel,

Ka.

(b) Let uiT,c(x) = log x + 1

2 E∫ T

0 (θ it)

2dt and uiT,d

(x) = E∫ T

0 f (i)U,S(π

it)dt.

Then ui(x) ≡ uiT(x) = ui

T,c(x)+ uiT,d

(x), where uiT,c(x) is the con-

tinuous component representing the maximum expected loga-

rithmic utility from terminal wealth for the purely continuous

Merton case, and uiT,d

(x) is the discontinuous component which

is due to jumps in the mispricing and the asset prices.

emark 4. Because of the intractability of the integrals∫

R log(1 +(eσ x − 1))va(dx), where a ∈ {U, S}, we compute the optimal portfo-

io π i, using approximation techniques. This is done via the instanta-

eous centralized moments of return for both Lévy processes X and Z,

ith Lévy measures vS and vU , respectively.

.4. The case NU = NS = N

We now study the case where X and Z, given by (5) and (6), re-

pectively, have the same Poisson random measure N, and hence the

ame Lévy measure v = vU = vS. As already demonstrated by (20), the

tock has log-return dynamics

(log St) = μ∗,it dt + σt dBi

t +∫

R

x (qσt + 1)N(dt, dx).

pplication of Itô’s formula yields

dSt

St−= μi

t dt + σt dBit +

∫R

(e(qσt+1)x − 1)N(dt, dx).

t follows from Theorem 1, that the dynamics of the wealth processi is

dVit

Vit−

= (rt + πt σt θ

it

)dt + πt σt dBi

t +∫

R

πt(e(qσt+1)x − 1)N(dt, dx),

ith explicit solution

it = Vi

0 exp

(∫ t

0

rs ds +∫ t

0

(πsσsθ

is − 1

2π2

s σ 2s

)ds +

∫ t

0

πsσsdBis

)× �0≤s≤t(1 + πs(e

(qσs+1) Xs − 1)).
a
he optimal portfolio π i is given by the equation

i = σθ i + G′(π i; qσ)

σ 2= μi − r + G′(π i; qσ)

σ 2,

here

(α ; σ) =∫

R

log(1 + α(e(σ+1)x − 1))v(dx).

bserve that the optimal portfolio strategy for each investor now de-

ends on two components: a Merton (1971) type optimal component

nd a jump component that depends on the jump intensity, level of

ispricing, and the volatility of the diffusive component of the price

ynamic. The budget constraint also has a similar dependence, in

ddition to having a diffusive component.

.5. Approximation of optimal portfolios using instantaneous

entralized moments

We now develop the tools required to estimate optimal portfolio

trategies of investors. We adopt the definition of instantaneous cen-

ralized moments of return as presented in Cvitanic et al. (2008). For

he risky asset with price S and positive integer k, we define the kth

nstantaneous centralized moment by the prescription

S(k)�=

∫R

(ex − 1)kvS(dx) =k∑

j=1

(−1)k−j

(k

j

)KS(j),

here the kernel KS is defined by

S(s)�=

∫R

(es x − 1)vS(dx), s ≥ 0.

or the jump component of the mispricing process U, the instanta-

eous centralized moments of return are dependent on the volatility

t , and are given by:

U(k ; σ)�=

∫R

(eσ z − 1)kvU(dz) =k∑

j=1

(−1)k−j

(k

j

)KU(j ; σ),

here the kernels are

U(j ; σ)�=

∫R

(ej σ z − 1)vU(dz).

We use approximation methods to compute the optimal port-

olios since the partial objective functions Ga(α ; s)�= ∫

R log(1 +(es x − 1)va(dx), a ∈ {U, S} , s ∈ Iσ ≡ (0, σmax] ∪ {1}, is in general

ot tractable. We assume, as before, that∫

R(e±s x − 1)2va(dx) < ∞o ensure that the optimal portfolios exist (i.e.; G′′(α) < 0). Since

(α ; s) = GU(α ; s)+ GS(α ; 1), where α ∈ [0, 1] and s ∈ Iσ , then if

R(e± sx − 1)kva(dx) < ∞, we can approximate G(α ; s)by a kth degree

olynomial Gk(α ; s), where

k(α ; s)�=

k∑j=1

(−1)j−1M(j ; s)αj

j!, (33)

M(j ; s)�= MU(j ; s)+ MS(j ; 1), j = 1, 2, · · · , k.,

Ma(j ; s) =∫

R

(es x − 1)jva(dx) =j∑

r=1

(−1)r−1

(j

r

)Ka(r ; s),

Ka(j ; s) =∫

R

(ej s x − 1)va(dx), a ∈ {U, S}.e have the following lemma.

emma 4. Let k ∈ N. If∫

R(e±s x − 1)kva(dx) < ∞, where s ∈ Iσ , and

∈ {U, S}, then the kth instantaneous centralized moment exists and is


T

c

l

γ

T

i

i

c

t

j

t

s

t

a

c

m

d

t

T

s

a

w

t

H

h

i

t

t

W

a

T

i

w

given by

M(k ; s) =k∑

j=1

(−1)j−1

(k

j

)K(j ; s) = MU(k ; s)+ MS(k ; 1),

where

K(j ; s) = KU(j s)+ KS(j), Ka(γ ) =∫

R

(eγ x − 1)va(dx), a ∈ {U, S}.Note that the kernel Ka, where a ∈ {U, S}, is used to calculate the in-

stantaneous centralized moments. Because of the intractable nature

of the integrals involved in computing the optimal portfolio strate-

gies, we resort to approximations of these portfolios using the instan-

taneous centralized moments. The following approximation result is

a direct consequence of Theorem 3.

Theorem 4. Let i ∈ {0, 1} and π i be the optimal portfolio that maximizes

the expected logarithmic utility from terminal wealth over the investment

period [0, T]. Then π(k)i, the approximation of the optimal portfolio

based on a kth degree polynomial approximation (33) of the objective

function G, is given by

πt(k)i = μi

t − rt + G′k(πt(k)i ; qσt)

σ 2t

. (34)

The maximum expected logarithmic utility from terminal wealth, ui(x),with x > 0 as initial capital, is approximated by

u(k)i(x) = log x + 1

2E

∫ T

0

(θ it)

2dt + E

∫ T

0

f (i)U,S(πt(k)

i)dt, (35)

where f (i)U,S(π

i) is defined by (31).

Remark 5. Note from Eq. (34) that the excess stock holding overμi

t−rt

σ 2t

,

the Merton (1971) optimal, isG′

k(πt(k)

i ; qσt)

σ 2t

, which is dependent on

M(j ; s) = MU(j ; s)+ MS(j ; 1), j = 1, 2, · · · , k, the instantaneous cen-

tralized moments of return contained in Eq. (33). Each moment is the

sum of the instantaneous centralized moments for the mispricing and

asset price, respectively.

4. Asymptotic utilities

In this section we discuss the asymptotic results of the optimal

expected utilities for both informed and uninformed investors. We

set the interest rate r = 0 without loss of generality.

As before, we assume that the objective function G(α) ≡ G(α ; s) =GU(α ; s)+ GS(α ; 1) is restricted to the domain [0, 1], where s ∈ Iσ ,

and that∫

R(e±s x − 1)2va(dx) < ∞, for each a ∈ {U, S}. That is, we have

at least a quadratic approximation of G(α). Let Ma(j ; s) = ∫R(es x −

1)jva(dx), a ∈ {U, S}. Define

M(j ; σt)�= MU(j ; σt)+ MS(j ; 1), j = 1, 2, · · · , k, (36)

At�= −M(2 ; qσt)

2(σ 2t + M(2 ; qσt))

,

Bt�= 2M(1 ; qσt)

2(σ 2t + M(2 ; qσt))

,

Ct�= M2(1 ; qσt)

2(σ 2t + M(2 ; qσt))

.

Note that A, B, and C are functions of the diffusive volatility σt . Let

Q(θ) be defined by

Q(θ)�= Aθ2 + Bθ + C, (37)

where θ = μσ represents Sharpe ratio. Let

γ 2t = σ 2

t

σ 2t + M(2 ; qσt)

. (38)

hen γt is the proportion of total volatility due to the diffusive

omponent of the risky asset with dynamics (1). We assume that

imt→∞σt = σ∞ = σ > 0, and set

2 = limt→∞

γ 2t = σ 2

∞σ 2∞ + M(2 ; qσ∞)

. (39)

heorem 5. Assume that∫

R(e±s x − 1)kv(dx) < ∞, s ∈ Iσ , k ≥ 2 and G

s restricted to [0, 1]. Let x > 0 be the initial wealth of investors and

∈ {0, 1}.

(a) As T → ∞, the asymptotic optimal expected utility for the ith

investor due to jumps is

uiT, d(x) ∼

∫ T

0

Q

(μt

σt: σt, M(1; qσt)M(2; qσt)

)dt

+ λ

2A∞(1 − p)(1 + (−1)i+1p)

[1 +

∫R

z2 vU(dz)

]T,

(40)

where

A∞ = limt→∞

At = − M(2; qσ∞)

2(σ 2∞ + M(2; qσ∞)

) . (41)

(b) The asymptotic excess optimal expected utility of the uninformed

investor relative to the informed due to jumps is strictly positive

and given by

u0T, d(x)− u1

T, d(x) ∼ −λ A∞ p (1 − p)

[1 +

∫R

z2 vU(dz)

]T.

(42)

Observe that the optimal expected utility is the sum of two disjoint

omponents—one is continuous, while the other is discrete. uiT, d

(x) is

he discrete component of optimal expected utility. It is driven by the

ump processes only. Part (a) shows that the jump component of op-

imal expected utility depends on the diffusive volatility σt , first and

econd instantaneous centralized moments, M(1; qσt) and M(2; qσt),he level of mispricing q, the jump intensity of the mispricing process,

nd the long-run proportion of total volatility due to the diffusive

omponent of the asset price dynamics. Part (b) gives the excess opti-

al expected utility of the uninformed investor due to jumps only. It

epends on the proportion of mispricing, the mean-reversion speed,

he variance of the mispricing process, and a long-run variance ratio.

his excess is positive because A∞, the long-run proportion of diffu-

ive variance to total variance of the percentage return of the risky

sset is always negative, unless there is no diffusive component, in

hich case it is zero. This is an unexpected result, which suggests

hat jumps favour the uninformed investor from a utility standpoint.

owever, as the next major result shows, the informed investor still

as positive excess utility relative to the uninformed investor. This

mplies that the informed investor gains most of his/her utility from

he diffusive component of the asset price dynamics, and that it pays

o be more informed unless there is too little or too much mispricing.

e have the following major result as a consequence of the quadratic

pproximation of G(α ; σ).

heorem 6. Assume that the conditions of Theorem 5 hold. Let the

nvestment horizon T −→ ∞. Under quadratic approximation of G(α ; s),e have

(a) The asymptotic maximum expected logarithmic utility from ter-

minal wealth for the ith investor, with initial capital x > 0, is

uiT(x) ≈ log x + 1

2

∫ T

0

(μt

σt

)2

dt +∫ T

0

Q

(μt

σt

)dt

+ λ

4(1 − p)(1 + (−1)i+1p)

[1 +

∫R

z2 vU(dz)

]T.


l

c

l

m

l

i

o

o

a

h

I

e

i

i

(

m

e

a

s

d

j

λt

O

c

o

b

u

p

c

t

i

t

t

m

2

m

o

i

m

a

c

t

c

5

i

w

a

t

r

p

s

i

m

p

o

t

v

i

c

u

v

g

c

t

n

w

(

m

o

r

i

t

l

t

p

t

g

t

c

n

i

u

l

c

p

t

p

o

t

i

i

t

m

e

w

(b) The asymptotic excess expected logarithmic utility of the informed

investor is

u1T(x)− u0

T(x) ≈ λ

2p (1 − p)

[1 +

∫R

z2 vU(dz)

]T,

where λ = λγ 2 is the long-run adjusted mean-reversion speed

given.

(c) Let λ be the speed of the original mispricing process. There exists

an adjusted continuous O–U mispricing process with speed λ =[1+∫

R z2 vU(dz)]σ 2∞σ 2∞+M(2 ; qσ∞)

, and a continuous diffusion process linked to the

original asset price process with volatility σt , such that the excess

optimal expected utility of the informed investor is

u1T(x)− u0

T(x) ≈ λ

2p (1 − p)T.

Moreover, the excess utility is greater (less) than its continuous ge-

ometric Brownian motion (GBM) counterpart if the product of σ 2∞and the quadratic variation

∫R z2 vU(dz) of the jump component

of U is greater (less) than M(2 ; qσ∞), the second instantaneous

centralized moment of return of the asset price jump process in

the long-run.

Observe from Part (a) that under quadratic approximation, the

ong-run optimal expected utility of each investor is the sum of four

omponents. The first two components are continuous, while the

ast two components depend on the jumps in both asset price and

ispricing processes through their respective Lévy measures. The

evel of mispricing q, its adjusted speed λ, diffusive volatility σt , and

nvestment period T are also important ingredients that influence

ptimal expected utility.

Part (b) presents a version of the excess optimal expected utility

f the informed investor over the uninformed that depends on the

djusted mean-reversion speed λ of the mispricing process, which

as been dampened by the total variance of the asset price process.

t also depends on the level of mispricing and the Lévy measure gov-

rning the mispricing process. Our model shows that the informed

nvestor has more long-run utility, and therefore, it pays to be more

nformed in the long-run, unless there is too little (q ≈ 0) or too much

q ≈ 1) mispricing in the market. Excess optimal expected utility is

aximized when the percentage of mispricing reaches 75%.

Part (c) is another representation of the long-run excess optimal

xpected utility that is identical in structure to Buckley et al. (2014)

nd Guasoni (2006), and indeed nests the mean reversion speeds pre-

ented therein. In this case, the model is equivalent to a continuous

iffusion with a continuous O–U mispricing process having a new ad-

usted mean-reversion speed λ, that is a function of the original speed

, the diffusive volatility, and variances which are constructed from

he Lévy measures of both asset price and mispricing jump processes.

bserve that if there are no jumps, the model reduces to the purely

ontinuous model of Guasoni (2006).

In our more complete framework, the excess utility can be more

r less than its continuous GBM counterpart and is entirely governed

y the jump intensities. In particular, the excess optimal expected

tility is greater (less) than its continuous GBM counterpart if the

roduct of σ 2∞ and the quadratic variation∫

R z2 vU(dz) of the jump

omponent of U is greater (less) than M(2 ; qσ∞), the second instan-

aneous centralized moment of return of the asset price jump process

n the long-run. In the case where there are no jumps in mispricing,

he quadratic variation due to mispricing is zero and consequently

he excess utility is always less than the continuous GBM bench-

ark. As in prior studies by Guasoni (2006) and Buckley et al. (2012,

014), we obtain maximum excess optimal expected utility when

ispricing in the market is at 75%. It is worth noting that whether

r not jumps exist, there is no utility advantage for the informed

nvestor if the market is totally free of mispricing or completely

ispriced.

Our model could be applied to options pricing that incorporates

symmetric information and mispricing of the underlying asset. It

ould be further generalized to a multi-market setting with at least

wo correlated risky assets, and their respective mispricings pro-

esses. Preferences for investors could be general CRRA type.

. Conclusion

We extend the theory of mispricing models under asymmetric

nformation and logarithmic preferences to the discontinuous frame-

ork where jumps in both the asset price and mispricing process

re incorporated. We obtain analogous but more general results than

hose presented in Buckley et al. (2014) and Guasoni (2006). We de-

ive explicit formulas for the optimal portfolios and maximum ex-

ected logarithmic utilities for both investors under reasonable as-

umptions. In particular, we show that the optimal portfolio of each

nvestor contains excess stock holdings which depend on the Lévy

easure of the jump processes driving both the stock price and mis-

ricing processes, through their instantaneous centralized moments

f return.

Under quadratic approximation of the optimal portfolios, we find

hat the optimal expected utility from terminal wealth for each in-

estor is increased by the presence of jumps in the mispricing. Thus,

nvestors are better off when mispricing jumps than when it changes

ontinuously. We also show that the uninformed investor has more

tility resulting from jumps in mispricing. However, the informed in-

estor still has positive excess utility over the uninformed, which sug-

ests that the informed investor gets more utility from the diffusive

omponent driving the asset price process. Furthermore, we show

hat the asymptotic excess maximum expected utility from termi-

al wealth of the informed investor is λ2 p (1 − p)

[1 + ∫

R x2 vU(dx)]

T ,

hich is similar to that presented in Buckley et al. (2014) and Guasoni

2006), but increased by a factor directly attributed to the jumps in

ispricing and asset price. As in the case of jumps in asset price

nly, the mean-reversion speed λ is replaced by an adjusted mean-

eversion speed λ, which is a scalar multiple of the original speed used

n the Brownian motion case and dampened by the total volatility of

he stock including the component produced by the jumps. Equiva-

ently, the excess optimal expected utility is λ2 p (1 − p)T , where λ is

he mean-reversion speed of an adjusted continuous O–U mispricing

rocess which depends on the volatility of the long-run asset price. In

his framework, the excess optimal utility of the informed investor is

reater (less) than its continuous Merton (1971) GBM counterpart if

he product of σ 2T and the quadratic variation

∫R z2 vU(dz)of the jump

omponent of U is greater (less) than M(2 ; qσT), the second instanta-

eous centralized moment of return of the asset price jump process

n the long-run.

We also show that the uninformed investor has more expected

tility resulting from jumps in mispricing in the long-run, which is

arger than what obtained in the Brownian motion case. Despite this

ounter-intuitive and surprising result, the informed investor still has

ositive excess utility over the uninformed, which clearly suggests

hat the informed investor gets more utility from the diffusive com-

onent of the asset price process. Thus, notwithstanding the presence

f asymmetric information, mispricing and jumps, our results show

hat it still pays to be more informed in the long-run, unless there

s too little or too much mispricing in the market. In this case, the

nformed investor has no utility advantage.

Our study can be extended as follows. First, it could be applied

o specific asset classes such as currencies or indices. Second, our

odel assumes that volatility is constant or deterministic. It is well-

stablished empirically that volatility is stochastic, and hence, it

ould be natural to extend the model to the stochastic volatility


S

q

V

T

V

B

(

(

A

P

i

d

w

a

f

d

T

d

case. Third, it could also be generalized to multiple risky as-

sets under constant or stochastic volatility, where investors have

CRRA preferences. Fourth, the pricing of options under asymmet-

ric information and mispricing has not received much attention in

the literature. This could be another fruitful direction for future

research.

Appendix A. Proofs

A.1. Proof of Proposition 1

Proof. By the martingale property of Z, E(UZt ) = E

∫ t0 e−λ(t−s)dZs = 0.

By Itô-isometry, and the fact that t∫

R z2 vU(dz) = E(Z2t ) < ∞, we have

Var(UZt ) = E(UZ

t )2 = E

(∫ t

0

e−λ(t−s)dZs

)2

= E

∫ t

0

e−2λ(t−s)d[Z, Z]s

=∫ t

0

e−2λ(t−s)

∫R

z2 vU(dz)ds

=∫ t

0

e−2λ(t−s)ds

∫R

z2 vU(dx)

= 1 − e−2λ t

2λ

∫R

z2 vU(dz).

The last result follows directly by letting t → ∞.


Proof. By Eq. (8), Ut�= UB

t + UZt , whence E(Ut) = E(UB

t )+ E(UZt ) = 0.

Since UB and UZ are independent processes, then it follows from

Proposition 1 that

E(U2t ) = Var(Ut) = Var(UB

t )+ Var(UZt )

= 1 − e−2λ t

2λ+

(1 − e−2λ t

2λ

) ∫R

z2 vU(dz)

=(

1 − e−2λ t

2λ

) (1 +

∫R

z2 vU(dz)

).

The last result follows directly by letting t → ∞.

A.3. Proof of Lemma 1

Proof. Since Yct

�= Yt − q UZt = p Wt + q UB

t is a continuous Gaussian

process (being the sum of two such independent processes), then by

Hitsuda (1968) there exists an F0-Brownian motion B0 and a pro-

cess φ0, Bt adapted to F0

t , such that B0t = Yc

t + ∫ t0 φ0, B

s ds, where φ0, Bs =

−υ0, Bs = λ

∫ t0 e−λ(t−s)(1 + γ (s))dB0

s . Since K0t = F0

t ∨ σ(Zu, Xu : u ≤ t)

then F0t ⊂ K0

t , and since B0t and φ0, B

t are F0t -adapted, they are K0

t -

adapted. Thus, B0t = Yt − q UZ

t + ∫ t0 φ0, B

s ds = Yt − q UZt − ∫ t

0 υ0, Bs ds.


Proof. We prove the case for i = 1. From Eq. (12),

E(υ1

t

) = E(υ1,B

t

) + E(υZ

t

) = −λ qE(UB

t

) − λ qE(UZ

t

) = 0.

ince UB and UZ are independent O–U processes, then with2 = 1 − p2, we have

ar(υ1

t

) = Var(υ1,B

t

) + Var(υZ

t

) = λ2q2 Var(UB

t

) + λ2 q2 Var(UZ

t

)= λ2q2

(1 − e−2λ t

2λ

)+ λ2 q2

(1 − e−2λ t

2λ

) ∫R

z2 vU(dz)

= λ

2(1 − e−2λ t)q2

(1 +

∫R

z2 vU(dz)

)= λ

2(1 − e−2λ t)(1 − p)(1 + (−1)1+1p)

(1 +

∫R

z2 vU(dz)

)= λ

2(1 − e−2λ t)(1 + p)(1 + (−1)1p)

(1 +

∫R

z2 vU(dz)

).

he proof for i = 0 follows similarly. Specifically, by Itô isometry,

ar(υ0

t

) = Var(υ0,B

t

) + Var(υZ

t

)= λ2Var

∫ t

0

e−λ(t−s)(1 + γ (s))dB0s + λ2 q2 Var(UZ

t )

= λ2

∫ t

0

e−2λ(t−s)(1 + γ (s))2ds

+λ2 q2

(1 − e−2λ t

2λ

) ∫R

z2 vU(dz)

y Part 3 of Proposition 3, Varυ i,Bt = E[υ i,B

t ]2 → λ2 (1 − p)(1 +

−1)i+1p) as t → ∞ and hence Varυ it = E[υ i

t]2 → λ

2 (1 + p)(1 +−1)ip)(1 + ∫

R z2 vU(dz)).

.5. Proof of Theorem 1

roof. Let i = 1. By Eq. (18), the dynamics for the informed investor

s equivalently given by

(I(t)) = (μ∗,1t dt + σt dB1

t +∫

R

qσtz NU(dt, dz)+∫

R

x NS(dt, dx),

(A.1)

here I(t) = log St ⇐⇒ St = elog St = eI(t). Let f (x) = ex. Then f ∈ C2(R)nd by Itô’s formula (cf. Theorem 4.4.7, Applebaum (2004)), we have

rom (A.1) that

(f (It)) = df

dx(It−)+ 1

2

d2f

dx2(It−)d[Ic, Ic]t

+∫

R

(f (It− + q x σt)− f (It−)

)NU(dt, dx)

+∫

R

(f (It− + x)− f (It−)

)NS(dt, dx)

= eIt−

[μ∗,1

t dt + σt dB1t + 1

2σ 2

t dt

]+

∫R

eIt−(eqσtx − 1)NU(dt, dx)

+∫

R

eIt−(ex − 1)NS(dt, dx)

= eIt−

[μ1

t dt + σtdB1t +

∫R

(eqσtx − 1)NU(dt, dx)

+∫

R

(ex − 1)NS(dt, dx)

].

hus, we get

(elog St) = St−

[μ1

t dt + σtdB1t +

∫R


+∫

(ex − 1)NS(dt, dx)

]
R


w

T

A

P

c

N

P

T

T

D

o

V

w

A

P

1

s

G

A

P

s

t

w

l

T

E

T

E

w

f

T

u

T

c

1

w

h

f

w

f

A

P

K

M

A

P

a

E

hich is equivalent to the percentage return equation:

dSt

St−= μ1

t dt + σt dB1t +

∫R


+∫

R

(ex − 1)NS(dt, dx). (A.2)

he proof for the uninformed investor is identical.


roof. We provide the proof for the informed investor. The crucial

omponent of this proof is Lemma 2, which states that NU(t, A) and

S(t, A) do not jump at the same time, because they are independent

oisson processes on each bounded Borel set A ∈ B(R − {0}). From

heorem 1, the percentage return of V1 is

dV1t

V1t−

= (1 − πt)rtdt + πtdSt

St

= (1 − πt)rt dt + πt μ1t dt + πt σt dB1

t

+∫

R

πt(eqσtz − 1)NU(dt, dz)+

∫R

πt(ex − 1)NS(dt, dx).

= (rt + πt σt θ

1t

)dt + πt σt dB1

t +∫

R

πt(eqσtz − 1)NU(dt, dz)

+∫

R

πt(ex − 1)NS(dt, dx).

his proves Eq. (23). By applying the result on Doleans–

ade/Stochastic exponentials, and using the fact of the independence

f jumps of NU and NS, Eq. (23) yields the solution

1t = V1

0 exp

(∫ t

0

rsds +∫ t

0

(πsσsθ

1s − 1

2π2

s σ 2s

)ds +

∫ t

0

πsσsdB1s

)× �0≤u≤t(1 + πu(e

qσu Zu − 1))× �0≤s≤t(1 + πs(e Xs − 1)),

ith discounted wealth given by (24).

.7. Proof of Lemma 3

roof. Let a ∈ {U, S} and s ∈ Iσ . By the standing assumption,∫

R(e± sx −)2va(dx) < ∞. Thus GU, GS ∈ C2[0, 1], and for all α ∈ [0, 1] and

∈ Iσ , G′′a(α ; s) = − ∫

R(es x−1)2va(dx)(1+α(es x−1))2 < 0. Thus, G′′(α ; s) = G′′

U(α ; s)+′′S(α ; 1) < 0.


roof. We provide the proof for the informed investor (i = 1). It is

imilar for the uninformed investor (i = 0). Since utility is logarithmic,

hen using Poisson integration, the utility from terminal (discounted)

ealth for the informed investor is:

og V1T = log x +

∫ T

0

(πsσsθ

1s − 1

2π2

s σ 2s

)ds +

∫ T

0

πsσsdB1s

+∑

0≤u≤T

log(1 + πu(eqσu Zu − 1))

+∑

0≤s≤T

log(1 + πs(e Xs − 1))

= log x +∫ T

0

(πsσsθ

1s − 1

2π2

s σ 2s

)ds +

∫ T

0

πsσsdB1s

+∫ T

0

∫R

log(1 + πu(eqσuz − 1))NU(du, dz)

+∫ T ∫

log(1 + πs(ex − 1))NS(ds, dx)

0 R

aking expectation yields

log V1T = log x + 1

2E

∫ T

0

(θ1

s

)2ds − 1

2E

∫ T

0

(πsσs − θ1

s

)2ds

+∫ T

0

∫R

log(1 + πu(eqσuz − 1))vU(dz)ds

+∫ T

0

∫R

log(1 + πs(ex − 1))vS(dx)ds

hus

log V1T = log x + 1

2E

∫ T

0

(θ1

t

)2dt + E

∫ T

0

f (1)U,S(πt)dt, (A.3)

here

(1)U,S(πt) = −1

2

(πtσt − θ1

t

)2 +∫

R

log(1 + πt(eqσtz − 1))vU(dz)

+∫

R

log(1 + πt(ex − 1))vS(dx)

= GU(πt; qσt)+ GS(πt; 1)− 1

2

(πtσt − θ1

t

)2

= G(πt; qσt)− 1

2

(πtσt − θ1

t

)2.

he value function is

1(x) = maxπ∈A1(x)

E log V1,πT = log x + 1

2E

∫ T

0

(θ1

t

)2dt

+ maxπ

E

∫ T

0

f (1)U,S(πs)ds

= log x + 1

2E

∫ T

0

(θ1

t

)2dt + E

∫ T

0

maxπ

f (1)U,S(πs)ds. (A.4)

hus, the objective function f (1)U,S(π) given by (31) is strictly con-

ave by Lemma 3 if π ∈ [0, 1],∫

R(e±σ x − 1)2va(dx) < ∞ and∫

R(e± x −)2va(dx) < ∞, where a ∈ {U, S}. Thus, f (1)

U,S(π) has a maximum at π1

here ddπ

(f (1)U,S(π))|π=π1 = 0. Dropping the subscripts “U” and “S”, we

ave

′(π) = G′(π ; qσ)− σ (πσ − θ1)

= G′U(π ; qσ)+ G′

S(π ; 1)− σ(πσ − θ1) = 0,

hence Eq. (27) holds. The maximum expected utility u1(x) follows

rom the value function (A.4) with maxπ f (1)U,S(π) = f (1)

U,S(π1).

.9. Proof of Lemma 4

roof. Obviously, Ka(j ; s)= ∫R(ej s x − 1)va(dx)=Ka(j s). Thus K(j ; s) =

U(j ; s)+ KS(j ; 1) = KU(j s)+ KS(j), whence

(k ; s) =k∑

j=1

(−1)j−1

(k

j

)K(j ; s)

=k∑

j=1

(−1)j−1

(k

j

)(KU(j ; s)+ KS(j ; 1))

=k∑

j=1

(−1)j−1

(k

j

)KU(j ; s)+

k∑j=1

(−1)j−1

(k

j

)KS(j ; 1)

= MU(k ; s)+ MS(k ; 1).


roof. Part (a): Assume that∫

R(e±s x − 1)2v(dx) < ∞, where s ∈ Iσnd G is restricted to [0, 1]. By Proposition 4, as t → ∞(υ i

t

)2 −→ λ

2(1 − p)(1 + (−1)i+1p)

[1 +

∫z2 vU(dz)

], i ∈ {0, 1}.

R


B

B

B

B

B

C

C

C

C

C

F

G

G

G

G

G

H

H

H

H

K

K

K

K

L

L

L

M

M

M

Similar to the proof of Theorem 3 in Buckley et al. (2014), as T → ∞,

we have

uiT, d(x) ∼ E

∫ T

0

Q

(μt

σt: σt, M(1; qσt), M(2; qσt)

)dt +

∫ T

0

AtE(υ i

t

)2dt

∼ E

∫ T

0

Q

(μt

σt

)dt + T lim

t→∞AtE

(υ i

t

)2

= E

∫ T

0

Q

(μt

σt

)dt + T A∞ lim

t→∞E(υ i

t

)2

= E

∫ T

0

Q

(μt

σt

)dt

+ λ

2A∞(1 − p)(1 + (−1)i+1p)

[1 +

∫R

z2 vU(dz)

]T,

where A∞ = limt→∞ At = limt→∞ − M(2;qσt)

2(σ 2t +M(2;qσt))

= − M(2;qσ∞)

2(σ 2∞+M(2;qσ∞)).

Part (b): From part(a), since λ2 A∞

∫R z2 vU(dz)T is common for both

investors, it follows that the excess optimal utility of the uninformed

investor due to the jumps is

u0T, d(x)− u1

T, d(x) ≈ −λ

2A∞[(1 − p)(1 + p)− (1 − p)(1 − p)]

×[

1 +∫

R

z2 vU(dx)

]T

= −λ A∞ p (1 − p)

[1 +

∫R

z2 vU(dz)

]T,

which is non-negative.

A.11. Proof of Theorem 6

Proof. (a) The total optimal asymptotic utility of the ith investor

is uiT(x) = ui

T, c(x)+ uiT, d

(x), where the continuous component

is

uiT, c(x) ∼ log x + 1

2

∫ T

0

(μt

σt

)2

dt + λ T

4(1 − p)(1 + (−1)i+1p)

×(

1 +∫

R

z2 vU(dz)

),

and the discontinuous component is given by Theorem 5 as

uiT, d(x) ∼

∫ T

0

Q

(μt

σt

)dt + λ

2A∞(1 − p)(1 + (−1)i+1p)

×(

1 +∫

R

z2 vU(dz)

)T,

where Q(·) also depends on σt , M(1; qσt) and M(2; qσt).(b) This follows by taking the difference of the optimal utilities for

both investors.

(c) This follows immediately from (b) by replacing

λ[1 + ∫R z2 vU(dz)] with λ.

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a discontinuous mispricing model under asymmetric information

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