European Option Pricing under GeometricLévy Processes with Proportional TransactionCosts

Haipeng Xing∗ Yang Yu

† Tiong Wee Lim‡

January 20, 2016

Abstract: Davis et al. (1993) provided a detailed discussion on the problem ofEuropean option pricing with transaction costs when the price of underlyingfollows a diffusion process, we consider a similar problem here and develop anew computational algorithm for the case that the underlying price follows ageometric lévy process. Using an approach that is based on maximization ofthe expected utility of terminal wealth, we transform the option pricing intostochastic optimal control problems, and argue that the value functions of theseproblems are the solutions of a free boundary problem, in particular, a partialintegro-differential equation, under different boundary conditions. To solvethe singular stochastic control problems associated with utility maximizationand compute the value function and no-transaction boundaries, we develop acoupled backward induction algorithm that is based on the connection of thefree boundary problem to an optimal stopping problem. Numerical examplesof option pricing under a double exponential jump diffusion model are alsoprovided.

Keywords: Jump-diffusion processes; Transaction costs; Singular stochasticcontrol

∗Department of Applied Mathematics and Statistics, State University of New York, Stony Brook, USA.E-mail:xing@ams.sunysb.edu.†Department of Applied Mathematics and Statistics, State University of New York, Stony Brook, USA.

E-mail: yyu@ams.sunysb.edu.‡Department of Statistics and Applied Probability, National University of Singapore, Republic of Sin-

gapore

1

1 Introduction

The problem of option valuation has been intensively discussed in quantitative financeever since the publication of two seminal papers by Black and Scholes (1973) and Merton(1973). In both papers, the authors considered an idealized setting where an investorincurs no transaction costs from trading in a complete market consisting of a risk-freeasset and a risky asset whose price is a geometric Brownian motion, and presented adynamic self-financing trading strategy that replicates the payoff of an option. Theythen argued that, in the absence of arbitrage, the value of an option equals the amountof initial capital required for setting up the replicating portfolio.

One important assumption in the Black-Scholes-Merton model is the geometric Brow-nian motion for the stock price process, which has been challenged by a large amountof empirical studies; see Cont (2001), Lai and Xing (2008) and reference therein. To cap-ture asymmetric, leptokurtic features in the underlying asset return distributions andvolatility smiles in the option markets, Ait-Sahalia (2002), Carr and Wu (2003), Erakeret al. (2003), Broadie et al. (2007) developed models and methods that incorporate theoccurrence of rare jumps in a price process that would otherwise follow a diffusion.Merton (1976) argued that the asset price fluctuations can be decomposed as the sum of“normal” vibrations, caused by temporary imbalance of supply and demand and othernew information that causes marginal changes in the stock value, and “abnormal” vi-brations, caused by the arrival of important new information which generates occasionaland large impact on price. Furthermore, he modeled the normal and abnormal varia-tions by a standard geometric Brownian motion and a Poisson process, respectively, andderived an option pricing formula. Subsequent work has been moving towards con-sidering other or more general Lévy process. For instance, Chan (1999) considered theproblem of pricing contingent claims on a stock whose price process follows a geometricLévy process, and Kou (2002) proposed a double exponential jump-diffusion model andderived analytical solutions for a variety of option-pricing problems.

Another important assumption in the Black-Scholes-Merton model is no transactioncosts in the replicating process. Under this and continuous trading assumptions, theoption can be perfectly replicated in such a “complete” market. However, in the presenceof proportional transaction costs in capital markets, the continuous replication policyincurs an infinite amount of transaction costs over any trading intervals, and perfecthedging becomes impossible, hence the absence of arbitrage argument is no longer valid.To deal with the problem of option pricing and hedging with transaction costs, several

2

approaches have been proposed for the case that the underlying stock price follows ageometric Brownian motion. One approach is based on super-replication in a discrete-time setting and tries to find trading strategies which produce payoffs at expirationthat are larger than or equal to the option payoff; see details in Leland (1985), Boyleand Vorst (1992), Bensaid et al. (1992), Edirisinghe et al. (1993), Soner et al. (1995), andPrimbs (2009). Another approach examines the difference between the desired payoffat maturity and the realized cash flow from a hedging strategy, and tries to achieve thebest possible tradeoff between the cost of payoff and the risk. A pioneer work along thisline is Hodges and Neuberger (1989), who used a risk-averse utility function to assessthe replication error and formulated the problem of option pricing and hedging as thatof maximizing the investor’s expected utility of terminal wealth. Their key idea behindthe utility-based approach is to make use of an indifference argument and to definethe writing price of an option as the amount of money that would make an investorindifferent, in terms of expected utility, between trading in the market with and withoutwriting the option. They interpreted the “hedging” of the option as the difference inthe two trading strategies, with and without the option. Davis et al. (1993) modifiedcertain settings of Hodges and Neuberger (1989) and developed rigorously the modelof Hodges and Neuberger (1989) for a market with proportional transaction costs. Inparticular, Davis et al. (1993) showed that the option pricing problem with transactioncosts involves solving two singular stochastic control problems formulated by Davisand Norman (1990). They also developed, for the negative exponential utility function,numerical algorithms to compute the optimal hedge and option price by making useof discrete-time dynamic programming for an approximating binomial tree for the stockprice. Further contributions to the study of the utility based option pricing approach andnumerical methods with proportional transaction costs include Clewlow and Hodges(1997), Whalley and Wilmott (1997), Andersen and Damgaard (1999), Constantinidesand Zariphopoulou (1999, 2001), Zakamouline (2006), Lai and Lim (2009), and someothers.

In this paper, we relax the assumptions of geometric Brownian motion for the stockprice process and no transaction costs in the replicating process, and consider the prob-lem of option pricing under general jump diffusion processes with proportional trans-action costs by generalizing the utility-based approach developed by Davis et al. (1993)and proposing a coupled backward induction algorithm to compute the solutions. Inparticular, under the assumption that the underlying stock price follows a geometricLévy process, we formulate the problem of option pricing with proportional transac-

3

tion costs as the maximization of the investor’s expected utility of terminal wealth, anddemonstrate that the implied singular stochastic control problem can be reduced to afree boundary problem for a partial integro-differential equation. The derived equationis in the form of backward stochastic differential equations, and can be solved by least-squares Monte Carlo regressions (Gobet et al., 2005; Bender and Steiner, 2012) or Fouriermethod (Fang and Oosterlee, 2008; Ruijter and Oosterlee, 2015). Instead of using theabove method, we consider probabilistic numerical methods to solve the derived equa-tions for the negative exponential utility function. However, our numerical method isdifferent from Davis et al. (1993), Clewlow and Hodges (1997) and Zakamouline (2006)who use discrete-time dynamic programming for an approximating binomial tree forthe stock price. Instead, we make use of the connection of the free boundary problem toan optimal stopping problem, and propose a coupled backward induction algorithm tocompute the buy-sell boundaries and value functions of the maximization problem.

The rest of the paper is organized as follows. Section 2 presents the utility based ap-proach to pricing options under a geometric Lévy process with proportional transactioncosts and its corresponding singular stochastic control problems. Section 3 introducesfor the negative exponential utility function a coupled backward induction algorithmto solve the control problems with much less computational cost. Section 4 providesintensive simulation studies that investigate the impact of jump component in the stockprice process and transaction costs on option price and the implied hedging costs. Someconcluding remarks are given in Section 5.

2 A utility-based approach to option pricing with proportional trans-action costs

2.1 Option pricing via utility maximization

Suppose that an investor is provided with an opportunity to enter into a position in aEuropean call option written on a stock with expiration date T and strike price K. Theprice of the stock is assumed to follow a geometric Lévy process

dSt = αSt−dt + σSt−dWt + St−

∫ ∞

−1ηN(dt, dη). (1)

Here the mean rate of return α > 0 and the volatility σ > 0 are constants, and Wt isa standard Brownian motion on a filtered probability space (Ω,F ,Ft, P) with W0 = 0.

4

The Poisson random measure N is Ft-centered, that is,

N(t, A) = N(t, A)− E[N(t, A)] = N(t, A)− tν(A),

in which Poisson random measure N(t, A) measures the number of jumps with ampli-tude in A ⊂ (−1, ∞) up to and including time t, N has a time-homogeneous intensity,E[N(t, A)] = tν(A), and ν is the Lévy measure associated to N. Note that to remain theprice process St > 0 for all t ≥ 0, we only allow jump sizes η > −1. We also assumethat, for technical convenience, ∫ ∞

−11∨ η2dν(η) < ∞.

In the presence of proportional transaction costs, the investor pays 0 < ζ < 1 and0 < µ < 1 of the dollar value transacted on purchase and sale of the underlying stock,respectively. Let xt denote the number of shares held in stock and yt the dollar valueof the investment in bond which pays a fixed risk-free rate r. The investor’s position(xt, yt) in stock and bond is driven by:

dxt = dLt − dMt, (2)

dyt = rytdt− aStdLt + bStdMt, (3)

with a = 1 + ζ and b = 1− µ, where Lt and Mt are nondecreasing and non-anticipatingprocesses and represent the cumulative numbers of shares of stock bought or sold, re-spectively, within the time interval [0, t], 0 ≤ t ≤ T. The process (Lt, Mt) provides usan admissible trading strategy. For convenience, we denote T (y0) the set of admissibletrading strategies that an investor starts at time zero with amount y0 of the investmentin bond and zero holdings in stock. Then equations (1), (2) and (3) compose the marketmodel in the time interval [0, T], which describes a degenerate Lévy process in R3.

Denote the terminal settlement value of the stock and option by Zi(ST, xT), wherei = 0, s and b indicates the investor’s position in the option: no call option, short call,and long call, respectively. When there is no option position, or i = 0, the liquidatedvalue of stock is given by

Z0(S, x) = xS(aIx<0 + bIx>0). (4)

If the option is cash settled, the option writer delivers (ST − K)+ in cash at T, so

Zi(S, x) = Z0(S, x)− (S− K)∆i(S), i = s, b, (5)

5

where ∆s(S) = IS>K (short call) and ∆b(S) = −IS>K (long call). If the option is assetsettled, then the writer delivers one share of stock in return for a payment of K when thebuyer exercises the option at maturity T, so

Zi(S, x) = Z0(S, x− ∆i(S)) + K∆i(S), i = s, b. (6)

Note that the cases of (4)-(6) imply the trade of 0 share of stock for i = 0, s and bwith cash settlement, and the trade of ∆i(ST) share of stock for i = s and b with assetsettlement. Using the self-financing argument of Bensaid et al. (1992) and assuming thatthe investor can choose any time in [t, T] to trade and the first trade after time t occursat time τ, we then have xu = xt for all u ∈ [t, τ), and the dollar value of the investmentat time τ is given by

yτ = yt er(τ−t) − Sτ(adLτ − bdMτ),

in which Sτ(adLτ − bdMτ) is the cost of the first trade. In general, suppose trades aftertime τ0 := t occur at times τ1 < τ2 < · · · < τn, we have

yT = yt er(T−t) −n

∑i=0

er(T−τi)Sτi(adLτi − bdMτi).

Extending the argument to continuous time and continuous states yields

yT = yt er(T−t) −Ψ(L, M; t, T) (7)

whereΨ(L, M; t, T) = a

∫[t,T)

er(T−u)SudLu − b∫[t,T)

er(T−u)SudMu (8)

is the total trading cost incurred over [t, T). In the mean while, the total hedging costincurred in [t, T] is expressed as

Ci(L, M; t, T) = Ψ(L, M; t, T)− Zi(ST, xT), i = 0, s, b. (9)

Therefore, combining (7)-(9) yields the investor’s terminal wealth in terms of the totalhedging cost

yT + Zi(ST, xT) = yt er(T−t) − Ci(L, M; t, T), i = 0, s, b,

Suppose that the writer’s utility U : R → R is a concave and increasing functionwith U(0) = 0. We assume that the investor’s goal is to maximize the expected utility ofterminal wealth under the market model (1)-(3)

Vi(t, S, x, y) = sup(Lt,Mt)∈T (y0)

E[U(yT + Zi(ST, xT))|St = S, xt = x, yt = y], (10)

6

which corresponds to no call option (i = 0), short call (i = s) and long call (i = b).With the given utility function (10), the option price can be derived from the indiffer-ence argument which is similar to utility equivalence pricing principle in economics. Inparticular, the writing price of an option is defined as the amount of money that makesthe investor indifferent, in terms of expected utility, between entering into the marketwith and without writing the option. At this reservation price, the investor is indifferentbetween selling (or buying) an option and doing nothing. Denote the reservation priceof selling (or buying) an option as the amount of cash Ps (or Pb) required initially toprovided the same expected utility as not enter into this position to the investor, Ps andPb satisfy the following equations:

Vs(0, S, x, y + Ps(S, x)) = V0(0, S, x, y) = Vb(0, S, x, y− Pb(S, x)). (11)

We denote that the set of solutions of equations (1), (2) and (3) as

TK = (St, xt, yt) ∈ R+ ×R×R|yt + Zi(xt, St) < −K,

where K is a constant that may depend on the strategy (Lt, Mt).

2.2 The Hamilton-Jacobi-Bellman equations and free boundary problems

We now derive the Hamilton-Jacobi-Bellman equations, associated with the stochasticcontrol problems, for the utility maximization problem (10). Consider a class of tradingstrategies such that Lt and Mt are absolutely continuous processes, given by

Lt =∫ t

0ludu and Mt =

∫ t

0mudu,

where lu and mu are positive and uniformly bounded by ξ < ∞. Then equations (1)-(3)consist of a vector stochastic differential equation with controlled drift, and the Bellmanequation for a value function Vi is given by

maxlt,mt

(P1Vi

)lt −

(Q1Vi

)mt

+O1Vi = 0, (12)

for (t, St, xt, yt) ∈ [0, T]×R×R×R+, in which operators P1, Q1 and O1 are defined as

P1φ :=∂φ

∂x− aSt

∂φ

∂y, Q1φ :=

∂φ

∂x− bSt

∂φ

∂y,

O1φ :=∂φ

∂t+ ry

∂φ

∂y+ αS

∂φ

∂S+

12

σ2S2 ∂2φ

∂S2

+∫ ∞

−1

[φ(t, S(1 + η), x, y)− φ(t, S, x, y)− ηSt

∂φ

∂S

]dν(η).

7

respectively. Then the optimal trading strategy is determined by the following cases(Note that all other combinations of inequalities are impossible, since the value functionsare increasing functions of x and y):

(i) buying stock at the maximum possible rate lt = ξ and mt = 0 when P1V ≥ 0 andQ1V > 0;

(ii) selling stock at the maximum possible rate mt = ξ and lt = 0 when P1V < 0 andQ1V ≥ 0;

(iii) doing nothing, that is mt = lt = 0 when P1V ≤ 0 and Q1V ≤ 0;

The above argument shows that the optimization problem (10) is a free boundary prob-lem. Besides, the state space [0, T] ×R+ ×R×R is partitioned into buy, sell and no-transaction regions, which are characterized by inequalities in (i)-(iii), respectively. Forsufficiently large ξ, the state space remains divided into a buy region B, a sell region S ,and a no-transaction region N , and the optimal trading strategy requires an immediatetransaction to the boundary of the buy region, ∂B or that of the sell region ∂S , if thestate is in region B or S . Hence the value function Vi(t, S, x, y) satisfy

Vi(t, S, x, y) = Vi(t, S, x + εx, y− aSεx) in B,

Vi(t, S, x, y) = Vi(t, S, x− εx, y + bSεx) in S ,

in which εx (the number of shares bought or sold by the investor) can take any positivevalue up to the number required to take the state to ∂B or ∂S . Instantaneous transac-tion from the interior of the buy (or sell) region to the buy (or sell) boundary suggestsequations P1Vi = 0 in B and Q1Vi = 0 in S . In the no-transaction region N , the valuefunction follows equation (12) for the trading strategies lt = mt = 0, and hence becomesO1Vi = 0. Therefore, the above derived equations can be condensed into the followingpartial integro-differential equations (PIDE):

maxP1Vi,−Q1Vi,O1Vi

= 0 (13)

for (t, St, xt, yt) ∈ [0, T]×R×R×R+. The above discussion also yields the followingfree boundary problem for the singular stochastic control value function Vi:

O1Vi = 0 in NP1Vi = 0 in BQ1Vi = 0 in S

Vi(T, S, x, y) = U(y + Zi(S, x)).

(14)

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Actually, we can further show that the value functions Vi given by (10) are weak solutionsof the variational inequality (13) in the notion of viscosity solutions.

Theorem 1. The value function Vi is a constrained viscosity solution of

maxP1φ,−Q1φ,O1φ

= 0 (15)

on [0, T]× TK.

2.3 Solutions for the negative exponential utility function

We further assume that the investor has the negative exponential utility function

U(z) = 1− e−γz, (16)

in which γ is the constant absolute risk aversion (CARA) parameter. For equation (13),this utility function can reduce much of computational effort and is simple to inter-pret. Furthermore, the option price based on the exponential utility function is a goodapproximation to the price implied by any hyperbolic absolute risk aversion utility func-tion with the same level of absolute risk aversion; see Andersen and Damgaard (1999).Since this utility function will be used for algorithm design and convergence argument inSection 3 and also simulation studies in Section 4, we show below that the value functionVi is also the unique bounded constrained viscosity solution of (15) for the exponentialutility function (16).

Theorem 2. Let u be a bounded upper semicontinuous viscosity subsolution of (15), andv a bounded from below lower semicontinuous viscosity supersolution of (15), such thatu(T, S, x, y) ≤ v(T, S, x, y) for all (S, x, y) ∈ TK. Then u ≤ v on [0, T]× TK.

Note that for the utility function (16), the definition of the value function (10) can beexpressed as

Vi(t, S, x, y) = 1− exp−γyer(T−t)Hi(t, S, x), (17)

where

Hi(t, S, x) := infL,M

E

exp[−γ(yT + Zi(ST, xT)− yter(T − t))]∣∣St = S, xt = x

= 1−Vi(t, S, x, 0).

Plugging (17) into (13) and defining the following operators for ψ(t, S, x) on [0, T]×R×R,

P2ψ :=∂ψ

∂x+ aγSer(T−t)ψ, Q2ψ :=

∂ψ

∂x+ bγSer(T−t)ψ,

9

O2ψ :=∂ψ

∂t+ αS

∂ψ

∂S+

12

σ2S2 ∂2ψ

∂S2 +∫ ∞

−1

[ψ(t, S(1 + η), x)− ψ(t, S, x)− ηS

∂ψ

∂S

]dν(η),

we obtain an variational inequality

minP2Hi,−Q2Hi,O2Hi

= 0, (18)

and the corresponding free boundary problem for Hi(t, S, x),O2Hi(t, S, x) = 0 x ∈ [Xb(t, S), Xs(t, S)],P2Hi(t, S, x) = 0 x ≤ Xb(t, S),Q2Hi(t, S, x) = 0 x ≥ Xs(t, S),

Hi(T, S, x) = exp−γZi(S, x),

(19)

in which Xb(t, S) and Xs(t, S) are the buy and sell boundaries, respectively; see Daviset al. (1993). Moreover, combining (17) with (11) yields the reservation buying and sellingprices

Pb(S, x) = −γ−1 e−rT log[

Hb(0, S, x)/H0(0, S, x)]

, (20)

Ps(S, x) = γ−1 e−rT log[

Hs(0, S, x)/H0(0, S, x)]

. (21)

In the limit as γ → ∞, Bouchard et al. (2001) have shown that the reservation priceconverges to the sum of the liquidation value of the initial endowment and the super-replication price of the option.

3 A numerical algorithm

Note that when the stock price process St follows a diffusion-only process, the freeboundary problem (19) reduces to the one in Davis et al. (1993). To solve the option pric-ing problem under diffusions, Kushner and Dupuis (1992), Davis et al. (1993) and Za-kamouline (2006) have developed a numerical scheme that involves a consistent Markovchain approximation of continuous-time price processes and then solved an appropri-ate optimization problem by discrete time dynamic programming. In particular, theirscheme is based on weak convergence of probability measures and uses Markov chainapproximation and discrete time dynamic programming algorithm. Though the exten-sion of the scheme from diffusion-only to jump diffusion processes is possible, it iscomputationally expensive as the solutions of the singular stochastic control problem(19) require the determination of both when to apply control and how much control toapply.

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In this section, we introduce a coupled backward induction algorithm that can sub-stantially reduce computational complexities. The basic idea of coupled backward induc-tion algorithm was introduced by Lai and Lim (2009). Specifically, based on a diffusion-only stock price process, Lai and Lim (2009) used utility maximization and cost con-straint argument to derive the free boundary problem for singular stochastic controlvalue functions, and then connected the free boundary problem to an optimal stoppingproblem. We employ the similar idea here and connect the free boundary problem (19)to an optimal stopping problem. However, since the underlying stock price process inour model involves both jumps and diffusions, the diffusion-based numerical scheme inLai and Lim (2009) can not be applied to problem (19). To solve this issue, we choosethe numerical scheme of Bernoulli walk with jumps in Aitsahlia and Runnemo (2007)for discretized jump-diffusion processes. In such way, we extend the idea of coupledbackward induction algorithm to geometric Lévy processes.

3.1 Change of variables

To solve the free boundary problem (19), we apply the change of variables as follows:

ρ =r

σ2 , s = σ2(t− T), z = log(S/K)− (α− β− 12)s,

θ =α

σ2 , β =1σ2

∫ ∞

−1ηdν(η),

(22)

and let hi(s, z, x) = Hi(t, S, x). Then inequality (18) becomes

minP3hi,−Q3hi,O3hi

= 0, (23)

and equation (19) can be rewritten asO3hi(s, z, x) = 0 x ∈ [Xb(s, z), Xs(s, z)]P3hi(s, z, x) = 0 x 6 Xb(s, z)Q3hi(s, z, x) = 0 x > Xs(s, z)

hi(0, z, x) = exp−γKAi(z, x),

(24)

where the operator P3,Q3 and O3 are defined as follows:

P3ψ :=∂ψ

∂x(s, z, x) + aγK ez+(θ−ρ−β−0.5)sψ(s, z, x), (25)

Q3ψ :=∂ψ

∂x(s, z, x) + bγK ez+(θ−ρ−β−0.5)sψ(s, z, x), (26)

11

O3ψ :=∂ψ

∂s+

12

∂2ψ

∂z2 +1σ2

∫ ∞

−1

[ψ(s, z + log(1 + η), x)− ψ(s, z, x)

]dν(η). (27)

Then, corresponding to definitions (4)-(6) of terminal wealth,

A0(z, x) = x ez(a Ix<0 + b Ix>0),Ai(z, x) = A0(z, x− Di(z)) + Di(z), for asset settlement i = s, b,Ai(z, x) = A0(z, x)− (ez − 1)Di(z), for cash settlement i = s, b,

with Ds(z) = Iz>0 and Db(z) = −Iz>0. Note that the operator O3 can be viewedas the infinitesimal generator of a jump diffusion process defined through the stochasticdifferential equation,

dzs = dws + dns,

where ws is a standard Brownian motion and ns is a jump process with jump size scaledby 1/σ2. This implies that while (s, z) is inside the no-transaction region, the dynamicsof hi(s, z, xs) is driven by the jump diffusion process zs, s ≤ 0. Furthermore, it followsfrom (24) that, in the buy and sell regions,

hi(s, z, x) = exp−aγK ez+(θ−ρ−β−0.5)s[x− Xb(s, z)]hi(s, z, Xb(s, z)), x ≤ Xb(s, z),

hi(s, z, x) = exp−bγK ez+(θ−ρ−β−0.5)s[x− Xs(s, z)]hi(s, z, Xs(s, z)), x ≥ Xs(s, z).(28)

We next discuss how to get a solution for the free boundary problem (24). As men-tioned in the beginning of this section, one way to solve (24) directly is to extend thenumerical scheme that Kushner and Dupuis (1992), Davis et al. (1993) and Zakamouline(2006) developed for a diffusion-only process, which involves a consistent Markov chainapproximation of continuous-time price processes and then solves an approximate op-timization problem by discrete time dynamic programming. Since our setting here in-volves both jump and diffusion processes, it is too complicated to extend the numericalscheme in Kushner and Dupuis (1992), Davis et al. (1993) and Zakamouline (2006). In-stead of solving (24) directly, we consider solving two partial integro-differential equa-tions that are much easier than the original problem (24). The first partial integro-differential equation has the same form as (24) but the boundaries are assumed to beknown. The second partial integro-differential equation is a free boundary problem withdifferential equations in the buy or sell regions are completely specified. Specifically, thesecond partial integro-differential equation is constructed as follows. Let wi = ∂hi/∂x,

12

then wi satisfies the free boundary problemO3wi(s, z, x) = 0 x ∈ [Xb(s, z), Xs(s, z)]

wi(s, z, x) = −aγK ez+(θ−ρ−β−0.5)shi(s, z, x) x ≤ Xb(s, z)wi(s, z, x) = −bγK ez+(θ−ρ−β−0.5)shi(s, z, x) x ≥ Xs(s, z)wi(0, z, x) = −γKBi(z, x)hi(0, z, x),

(29)

where O3 is defined by (25) and Bi(z, x) = ∂Ai(z, x)/∂x is given by

B0(z, x) = A0(z, x)/x,Bi(z, x) = B0(z, x− Di(z)), for asset settlement i = s, b,Bi(z, x) = B0(z, x), for cash settlement i = s, b.

In the sequel, we fix i = 0, s, b and drop the superscript i in wi and hi for notationalsimplicity. The purpose of introducing problem (29) is that, when the function hi(s, z, x)is known, (29) becomes an optimal stopping problem associated with a Dynkin game(see Lai and Lim, 2009, Section 3), which can be easily solved by backward inductionand random walk approximation (see Chernoff and Petkau, 1986; AitSahalia and Lai,1999; Lai et al., 2007).

3.2 A coupled backward induction algorithm

We now introduce a coupled backward induction algorithm that solves (24) and (29) toobtain the buy and sell boundaries Xb(s, z) and Xs(s, z) and the value function h(s, z, x)simultaneously. Since the free boundary problem (29) with known h(s, z, x) can be solvedvia random walk approximations, we consider a numerical scheme of Bernoulli walkwith jumps recently proposed by Aitsahlia and Runnemo (2007).

Given a small δ > 0, we discretize time and space as follows. Let s0 = 0 and sj =

sj−1 − δ for j ≥ 1 and set

Sδ = sn, sn − δ, . . . , δ, 0,Zδ =

√δj : j is an integer = 0,±

√δ,±2

√δ, . . . .

Note that the possibility of jump in an interval of length δ can be approximated by βδ

and hence the possibility of no jump in the interval is 1− βδ. Let Sδj be the position

of the approximating discrete process after the nth transition. On the grid Zδ, a jumpoccurs at the (j + 1) transition if Sδ

j+1 − Sδj = ±l

√δ for l ≥ 2. We then approximate the

jump distribution by partitioning the entire real line into intervals of length√

δ, whose

13

probability measure is assigned as the probability mass on the midpoints of the intervals.In particular, let F be the cumulative probability distribution function of the jump sizeY, we set

dF(l) =

F[(l + 1

2)√

δ]− F[(l − 12)√

δ], |l| ≥ 2,F[(l + 1

2)√

δ]− F[(−l − 12)√

δ], l = 0.(30)

We then assign the following probabilities

PSδj+1 − Sδ

j = ±l√

δ =

12(1− βδ) if l = 1,βδdF(l) if l 6= 1.

In practice, we need to truncate the set of l to a finite subset lmin, lmin + 1, . . . , lmax. Insuch case, we set dF(lmin) and dF(lmax) as

dF(lmin) = PY 6 (lmin + 0.5)√

δ,dF(lmax) = PY > (lmax − 0.5)

√δ.

We next use the above numerical scheme and introduce a backward induction to solve(29), provided h(s, z, x) is known. Specifically, let Tmax denote the largest expiration dateof interest and take δ and ε > 0 such that N := σ2Tmax/δ is an integer. The problem (29)with known h(s, z, x) can be solved by the following backward induction; see Lai andLim (2009). Let Xε = 0,±ε,±2ε, . . . . For i = 1, 2, . . . , N,

w(si, z, x) =

wb(si, z, x) if w(si, z, x) < wb(si, z, x),ws(si, z, x) if w(si, z, x) > ws(si, z, x),w(si, z, x) otherwise,

where x ∈ Xε and

wb(s, z, x) = −aγKez+(θ−ρ−β−1/2)h(s, z, x), (31)

ws(s, z, x) = −bγKez+(θ−ρ−β−1/2)h(s, z, x), (32)

w(si, zj, xm) =12(1− βδ)

[w(si−1, zj+1, xm) + w(si−1, zj−1, xm)

]+βδ

lmax

∑l=lmin,l 6=±1

dF(l) · w(si−1, zj+l, xm). (33)

The boundaries in (29) is determined as follows: Xb(si, z) is the largest x for whichw(si, z, x) ≤ wb(si, z, x) and Xs(si, z) is the smallest x for which w(si, z, x) ≥ ws(si, z, x).

We then solve numerically the problem (24), provided the boundaries Xb(s, z) andXs(s, z) are given. In such case, the value function in (24) with provided boundaries

14

can also be solved by backward induction. In particular, for z ∈ Zδ, define h(si, z, x) by(28) (with s replaced by si) if x ∈ Xε is outside the interval [Xb(si, z), Xs(si, z)], and leth(si, z, x) = h(si, z, x) with

h(si, zj, xq) =12(1− βδ)

[h(si−1, zj+1, xq) + h(si−1, zj−1, xq)

]+ βδ

lmax

∑l=lmin,l 6=±1

dF(l) · h(si−1, zj+l, xq),(34)

if x ∈ Xε ∩ [Xb(si, z), Xs(si, z)]. Defining wb and ws as in (31) and (32) but with hreplaced by h, it suggests the coupled backward induction algorithm described be-low to solve for Xb(si, z) and Xs(si, z), as well as to compute values of h(si, z, x) forx ∈ Xε ∩ [Xb(si, z), Xs(si, z)].

Algorithm. For i = 1, 2, . . . , N and z ∈ Zδ:

(i) Starting at x0 ∈ Xε with w(si, z, x0) < wb(si, z, x0), search for the first m ∈1, 2, . . . , (denoted by m∗ for which w(si, z, x0 + m∗ε) ≥ wb(si, z, x0 +

mε) and set Xb(si, z) = xi + m∗ε.

(ii) Let xm = Xb(si, z) + mε for m ∈ 1, 2, . . . . Compute, and store for useat si+1, w(si, z, xm) = w(si, z, xm) as defined by (33) and h(si, z, xm) =

h(si, z, xm) by (34). Search for the first m (denoted by m∗) for whichw(si, z, xm) ≥ ws(si, z, xm) and set Xs(si, z) = Xb(si, z) + m∗ε.

(iii) For x ∈ Xε outside the interval [Xb(si, z), Xs(si, z)], compute h(si, z, x) by(28) and set w(si, z, x) = wb(si, z, x) or ws(si, z, x) as defined by (31) and(32) according to whether x ≤ Xb(si, z) or x ≥ Xs(si, z).

Note that, comparing to the discrete-time dynamic programming algorithms of Daviset al. (1993), Clewlow and Hodges (1997), and Zakamouline (2006) that need to performan additional nonlinear optimization to identify the optimal trade size at each time step,the above algorithm avoids such optimization by solving the coupled problem for (w, h)instead of just for h and hence is much easier to implement. The convergence of thealgorithm can be shown by the following argument.

Theorem 3. The solution of the algorithm converges locally uniformly as δ → 0 to theunique continuous constrained viscosity solution of (24).

15

4 Numerical Study

We now perform a numerical study using the proposed algorithm. For illustration pur-pose, the price process of the underlying is assumed to follow a double exponentialjump-diffusion process (Kou (2002)) with mean rate of return α > 0 and volatility σ > 0

dSt = αSt−dt + σSt−dWt + St−d

(Nt

∑i=1

(Qi − 1)

), (35)

where α and σ are the expected return and diffusion volatility of the underlying asset,Wt; t > 0 is a standard Brownian motion with W0 = 0, Nt; t ≥ 0 is a Poisson processwith rate λ, Qi is a sequence of independent and identically distributed positive randomvariables such that the jump Y = log Q has asymmetric double exponential distributionwith density

fY(y) =

pη1 e−η1y if y ≥ 0,(1− p)η2 eη2y if y < 0.

Here 0 ≤ p ≤ 1, and p and 1 − p represent the probability of positive and negativejumps, respectively. The parameters η1 and η2 are assumed to satisfy η1 > 1 and η2 > 0.Note that under this specification, (30) becomes

dF(l) =

(1− p)(eη2(l+0.5)

√δ − eη2(l−0.5)

√δ) if l ≤ −2,

p(e−η1(l−0.5)√

δ − e−η1(l+0.5)√

δ) if l ≥ 2,(1− p)e1.5η2

√δ − pe−1.5η1

√δ if l = 0,

and the boundaries of the jump distribution is set as follows:

dF(lmin) = (1− p) eη2(lmin+0.5)√

δ,

dF(lmax) = p e−η1(lmax−0.5)√

δ.

We then study the reservation price of a short call option with K = 100 and T = 0.5under the process (35). To illustrate the algorithm, we choose α = r, which let uscharacterize the optimal hedging strategy for the utility-maximization approach by apair (rather than two pairs) of buy-sell boundaries. We further assume that α = r = 0and σ = 0.3 for the diffusion part of (35), which are same as the parameter configurationused by Clewlow and Hodges (1997). For the discretization of time, space and numberof shares of stock, we use δ = ε = 10−4 in the following study.

16

4.1 Impact of jumps

We first investigate the impact of jump components on the price of the call option andbuy and sell boundaries. In this experiment, we assume the CARA parameter γ = 1 andtransaction cost ζ = µ = 0.01. To discuss the impact of different rates and directionsof jumps, we let η1 = η2 = 25 and consider different values of Poisson rate λ and theprobability p of positive jumps.

Figure 1 shows the optimal buy (lower) and sell (upper) boundaries Xb(t, S) andXs(t, S) for the short call with p = 0.5, S0 = 90, 100, 110, and λ with asset and cashsettlements, respectively. For each pair of boundaries, the buy-region is below the buyboundary and the sell-region is above the sell boundary. The region between the buy andsell boundaries is the no-transaction region. In the asset settlement, when the call optionis at the money, (i.e., K = S = 100), the buy and sell boundaries over time are symmetricaround certain level and stretch out in the diffusion-only case (i.e., λ = 0); with the in-crease of λ, the no transaction region before expiration (i.e. T − t = 0) become narrowerand boundaries began to stretch towards the inside of the no-transaction region. Whenthe call option is in the money (i.e., K = 100, S = 110), buy and sell boundaries move up-ward to level 1 and two boundaries become 1 when the time to maturity ranges between0.02 and 0; furthermore, with the increase of λ from 0 to 10, the period during whichbuy and sell boundaries coincides become much shorter. Similar pattern is also foundwhen the call option is out of the money (i.e., K = 100, S = 90), except that buy andsell boundaries in such case move downward to level 0. In the cash settlement, with theincrease of λ, the no transaction region in the at-the-money case become much narrowerwhen the option will expire very soon. Around the expiration, the no-transaction regionof the in-the-money call option becomes wider, while that of the out-of-the-money calloption shrink to 0. This fact shows that large price movement narrows down the no-transaction region, and indicates that liquidating any excess position as soon as possibleis optimal for the option writer.

Figure 2 shows the optimal buy (lower) and sell (upper) boundaries Xb(t, S) andXs(t, S) for the short call with S0 = 90, 100, 110, and p = 0.1, 0.3, 0.5, 0.9 with asset andcash settlements, respectively. Note that with the increase of p, the trend of stock pricemovement changes from decreasing to increasing, the no-transaction regions aroundT− t = 0.5 move downward, whereas those around the expiration period move upwardin both asset and cash settlements. The configurations of no-transaction regions for thein-, at-, and out-of-the-money options are very different. In the asset settlement, whenthe call option is at-the-money, the buy and sell boundaries have downward tails, but

17

0.5 0.4 0.3 0.2 0.1 0

0

0.2

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0

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0

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0

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0

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0

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0.4

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1

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0

0.2

0.4

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1

Figure 1: Optimal buy (lower) and sell (upper) boundaries Xb(t, S) and Xs(t, S) fromCARA utility function for a short call with asset (top four panels) and cash (bottom fourpanels) settlements, respectively, where γ = 4, ζ = µ = 0.01, K = 100, S = 90 (dashedline), 100 (solid line), 110 (dot-dash line), and p = 0.5. The Poisson rates in the panels(from top to bottom and left to right) are λ = 0, 1, 5, 10, 0, 1, 5, 10, respectively.

18

0.5 0.4 0.3 0.2 0.1 00

0.2

0.4

0.6

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1

0.5 0.4 0.3 0.2 0.1 00

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0

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1

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0

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0.4

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1

Figure 2: Optimal buy (lower) and sell (upper) boundaries Xb(t, S) and Xs(t, S) fromCARA utility function for a short call with asset (top four panels) and cash (bottom fourpanels) settlements, respectively, where γ = 4, ζ = µ = 0.01, K = 100, S = 90 (dashedline), 100 (solid line), 110 (dot-dash line), and λ = 4. The rates of jumps (from top tobottom and left to right) are p = 0.1, 0.3, 0.5, 0.9, 0.1, 0.3, 0.5, 0.9, respectively.

19

Table 1: Prices of the short call option under jump-diffusion (35).

p = 0.5 λ = 0 λ = 1 λ = 5 λ = 10Price for Asset Settlement option 5.98 6.13 6.58 7.05Price for Cash Settlement option 6.31 6.46 6.92 7.39

λ = 4 p = 0.1 p = 0.3 p = 0.5 p = 0.9Price for Asset Settlement option 9.16 7.75 6.48 4.33Price for Cash Settlement option 9.59 8.13 6.82 4.59

these tails move up with the increase of p. When the option is in-the-money (out-of-the-money), it has similar pattern and more tails of the sell (buy) boundary become flatwhen p increases.

Besides the buy and sell boundaries, we also consider the impact of jumps on theprices of the short call options. Table 1 shows the prices of the short call option inFigures 1 and 2 with S0 = 100. We notice three facts from the table. First, when otherconditions are same, the call option price with cash settlement is more expensive thanthat with asset settlement, this is because the strategy in the cash settlement involveshigher hedging cost and hence yields higher option price. Second, more frequent jumpsin price (i.e., larger λ) leads to higher option price, as more frequent jumps in pricenarrow down the no-transaction region and imply higher hedging cost and higher optionprice. Third, higher probability of positive jumps leads to lower option price, as morefrequent positive jumps decreases the number of St hitting buying boundaries and hencereduce the hedging cost and option price. To have a better intuition on the impact ofjump intensity λ and positive probability p on option price, we show in Figure 3 thereservation price for the short call option with ζ = µ = 0.01, S = K = 100, T = 0.5 underasset settlement. We can see that when the probability of positive jumps p is less than0.5 (i.e., stock prices have the tendency of going down), the reservation price increaseswith the increase of λ; while if p > 0.5 (i.e., stock prices have the tendency of movingup), the reservation price decreases with the increase of λ.

20

02

46

810

00.2

0.40.6

0.81

0

2

4

6

8

10

12

14

16

18

p λ

Figure 3: Reservation price for short call option with ζ = µ = 0.01, S = K = 100,T = 0.5 and asset settlement. The probability of positive p = 0, . . . , l and the rate ofjumps λ = 0, . . . , 10.

4.2 Impact of the transaction cost

We further study the impact of transaction cost on the buy and sell boundaries by con-sidering the following parameter configuration:

K = 100, S = 100, γ = 1, λ = 5, p = 0.5, η1 = 25, η2 = 25

in asset and cash settlements, and compute the optimal buy and sell boundaries withthe transaction cost ζ = µ = 0, 0.01, . . . , 0.1. Figure 4 shows the optimal buy and sellboundaries for a short call option with asset and cash settlements. We find that the im-pact of transaction cost is always monotone for the sell boundary in both asset and cashsettlements, while such impact for the buy boundary is monotone in cash settlement butnot in asset settlement. This results in larger no-transaction regions for larger transac-tion costs, which is consistent with the fact that larger transaction costs will reduce theenthusiasm of trading. Furthermore, when transaction costs converge to 0, the buy andsell boundaries converge to the curve of hedging delta.

21

0.5 0.4 0.3 0.2 0.1 0 0

0.05

0.1

0

0.5

1

0.5 0.4 0.3 0.2 0.1 0 0

0.05

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0.05

0.1

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0.5 0.4 0.3 0.2 0.1 0 0

0.05

0.1

0.5

0.6

0.7

T−t

T−t

T−t

T−t

ζ=µζ=µ

ζ=µ ζ=µ

Figure 4: Optimal buy (left panels) and sell (right panels) boundaries for a short callwith asset (upper panels) and cash (lower panels) settlement. The transaction cost isζ = µ = 0, 0.01, . . . , 0.1.

4.3 Impact of the risk aversion parameter

We now study the impact of utility functions with different CARA parameters on the buyand sell boundaries. Figure 5 shows the optimal buy (lower) and sell (upper) boundariesXb(t, S) and Xs(t, S) for the short call with S0 = 90, 100, 110, and γ = 0.5, 1, 2, 4 with assetand cash settlements, respectively. An interesting phenomenon here is that it seems thatCARA parameters seem have little impact on buy and sell boundaries in the beginningof the trading process (i.e. T − t = 0.5) in both settlement, but the impact becomesignificant when the option is near expiration (T − t < 0.05). In particular, we noticethat the no-transaction regions for at-the-money options become much narrower when γ

gets larger, while the no-transaction regions for in- and out-of-the-money options do notchange as significantly as for at-the-money options. This indicates that when the agentbecomes more risk-averse, he tends to buy or sell stocks before the expiration so that theloss caused by changes of the option value can be avoided or mitigated.

22

0.5 0.4 0.3 0.2 0.1 0

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1

Figure 5: Optimal buy (lower) and sell (upper) boundaries Xb(t, S) and Xs(t, S) fromCARA utility function for a short call with asset (top four panels) and cash (bottom fourpanels) settlements, respectively, where ζ = µ = 0.01, K = 100, S = 90 (dashed line), 100(solid line), 110 (dot-dash line), λ = 4, and p = 0.5. The CARA parameters in the panels(from top to bottom and left to right) are γ = 0.5, 1, 2, 4, 0.5, 1, 2, 4, respectively.

23

4.4 Total hedging cost

We now study the total hedging cost of the optimal trading strategies based on the buyand sell boundaries in Section 2. In particular, we consider a short call option with strikeK = 100 and time to maturity T = 0.5. We also assume that the true stock price process isgiven by the double exponential jump diffusion processes (35) with parameters S0 = 100,η1 = η2 = 25, and λ = 1, 3, 5, 10, 12, respectively. The transaction costs are specified byζ = µ = 0.01. We then consider trading boundaries computed from a “correctly-" and an“incorrectly-" specified pricing models. The “correctly" specified model uses the jumpdiffusion process (35) with given parameters and is solved by the procedures in Sections2 and 3; the “incorrectly" specified model assumes no jump in the price process or theprice process follow a geometric Brownian motion, dSt = αStdt + σStdWt, and is solvedby the procedures in Davis et al. (1993). Then for each trading boundaries generatedfrom both models, we simulate 105 price paths according to the double exponential jumpdiffusion processes (35), and compute the total hedging cost C according to (9) and thetotal number of trades κ. We summarize the statistic on C and κ from the “correctly-"and the “incorrectly-" specified models in the second and the third columns of Table 2,respectively. Note that the correctly specified models always give larger total numberof trades but lower total hedging cost, comparing with those of the incorrectly specifiedmodels.

5 Concluding Remarks

We consider the problem of European option pricing in the presence of proportionaltransaction cost when the price of the underlying follows a jump diffusion process.Using an approach that is based on maximization of the expected utility of terminalwealth, we transform the option pricing into stochastic optimal control problems, andargue that the value functions of these problems are the solutions of a free boundaryproblem which consists of a partial integro-differential equation and different boundaryconditions. We develop a coupled backward induction algorithm to solve the singularstochastic control problems associated with utility maximization, and compute the valuefunction and no-transaction boundaries. The developed algorithm computes the valuefunctions and the trading boundaries of the stochastic control problems simultaneouslyand hence greatly reduces the computational cost.

24

Table 2: Summary statistics of simulation on total hedging cost.

Correctly-specified model Incorrectly-specified modelλ 1 3 5 10 12 1 3 5 10 12

(a) Asset settlement

C 10.255 10.551 10.838 11.517 11.774 10.262 10.558 10.851 11.544 11.801(0.004) (0.005) (0.005) (0.007) (0.007) (0.004) (0.005) (0.005) (0.006) (0.007)

κ 88.24 89.81 91.12 93.70 94.75 85.90 85.95 86.13 86.52 86.57(0.04) (0.04) (0.04) (0.05) (0.05) (0.04) (0.04) (0.04) (0.05) (0.05)

(b) Cash settlement

C 10.734 11.026 11.306 12.000 12.257 10.742 11.049 11.329 12.030 12.302(0.004) (0.005) (0.006) (0.007) (0.007) (0.004) (0.005) (0.005) (0.006) (0.007)

κ 86.13 87.72 89.16 91.85 92.83 84.00 84.14 84.22 84.51 84.66(0.05) (0.05) (0.05) (0.05) (0.05) (0.04) (0.05) (0.05) (0.05) (0.05)

Declarations of Interest

The authors report no conflicts of interest. The authors alone are responsible for thecontent and writing of the paper.

Acknowledgement

The authors thank for an associate editor and two anonymous referees for very helpfulcomment. This research is supported by the National Science Foundation under grantsDMS-0906593 and DMS-1206321 at SUNY at Stony Brook.

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Appendix of "European Option Pricing under GeometricLévy Processes with Proportional Transaction Costs"

Proofs of Theorems 1-3

Sketched Proof of theorem 1. In our discussion, the state is W = (t, w), where w = (S, x, y).Let W0 = (t0, S0, x0, y0) ∈ [0, T]× TK.

We first show that V1 is a viscosity subsolution of (15) on [0, T] × TK, or, for anysmooth function φ(W), such that V1(W)− φ(W) has a local maximum at W0 ∈ [0, T]×TK,

maxP1φ(W0),−Q1φ(W0),O1φ(W0) ≥ 0. (36)

Suppose that (36) is wrong, then for smooth function φ such that V1(W0) = φ(W0) andV1(W) ≤ φ(W) on [0, T]× TK, if P1φ(W0) < 0, Q1φ(W0) > 0, there must exist δ > 0,such that O1φ(W0) < −δ. By smoothness of φ, there exists a neighborhood R(W0) ofW0, such that for W ∈ R(W0), we have P1φ(W) < 0, Q1φ(W) > 0, and O1φ(W) < −δ.

Note that by our construction, there exists an optimal trading strategy correspondingto the processes L∗(t), M∗(t), such that W∗0 (t) = (t, B∗(t), x∗(t), y∗(t)) is the optimaltrajectory with W∗0 (t0) = W0. We show that W∗0 (t) has no jumps, almost surely, atW0 = W∗0 (t0). Denote the event that the optimal trajectory W∗0 (t) has a jump of size ε

along the direction (0, 0, 1,−aS0) by I(ω), then since V1(W) ≤ φ(W) for all W ∈ E(W0)

and V1(W0) = φ(W0), by the principle of dynamic programming,∫

I(ω) φ(t0, S0, x0 +

ε, y0 − aS0ε)dP ≥∫

I(ω) φ(t0, S0, x0, y0)dP. Therefore, letting ε → 0 and using Fatou’slemma, we have

(P1φ(W0)

)P(I) ≥ 0, and therefore P(I) = 0.

Now, define τ(ω) := inft ∈ (t0, T]|W∗0 (t) /∈ E(W0), τ(ω) is positive almost surely.Therefore,

−δE(τ) > E

∫ τ

t0

P1(W∗0 (t))dL∗(t)−E

∫ τ

t0

Q1(W∗0 (t))dM∗(t) + E

∫ τ

t0

O1(W∗0 (t))dt

= EA1 −EA2 + EA3.

Applying Itô’s formula to φ(W), we have

E(φ(W∗0 (τ))) ≤ V1(W0) + EA1 −EA2 + EA3 < V1(W0)− δE(τ),

which contradicts with the optimality of L∗(t) and M∗(t) and the dynamic program-ming principle. Therefore, at least one of items inside the maximum operator of (36) isnonnegative. Hence the value function V1 is a viscosity subsolution of (15).

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To show that V1 is a viscosity supersolution of (15), we need to show that, for allsmooth functions φ(W) such that V1(W)− φ(W) has a local minimum at W0 ∈ [0, T]×TK,

maxP1φ(W0),−Q1φ(W0),O1φ(W0) ≤ 0. (37)

Assume V1(W0) = φ(W0) and V1 ≥ φ on [0, T] × TK. Consider the trading strategyL(t) = L0 > 0, t0 ≤ t ≤ T, and M(t) = 0, t0 ≤ t ≤ T. By the dynamic programmingprinciple,

V1(t0, S0, x0, y0) ≥ V1(t0, S0, x0 + L0, y0 − aS0L0).

This inequality also holds for φ(t, S, x, y). Letting L0 → 0, we get P1φ(W0) ≤ 0. Similarly,considering the strategy L(t) = 0, t0 ≤ t ≤ T, and M(t) = M0 > 0, t0 ≤ t ≤ T, wecan argue that −Q1φ(W0) ≤ 0. Then for the case where no trading occurs. By thedynamic programming principle E(V1(Wd

0 (t))) ≤ V1(t0, S0, x0, y0), where Wd0 (t) is the

state trajectory when L(t) = M(t) = 0, t0 ≤ t ≤ T and Wd0 (t) ∈ E(W0). Applying

Itô’s formula on φ(t, S, x, y), we get E∫ t

t0O1Wd

0 (ξ)dξ ≤ 0. Let t → t0, we then haveO1φ(W0) ≤ 0. This completes the proof.

Sketched Proof of theorem 2. The proofs follows the arguments used in Section 5 of Ishiiand Lions (1990) and Theorem 3 of Davis et al. (1993), so we only present the main steps.First, we can construct a positive strict supersolution h of (15) on [0, T]×TK, by replacingthe diffusion component with the corresponding jump-diffusion part in the proof ofTheorem 3 of Davis et al. (1993). Then we define a linear combination of supersolutionsv and h, vθ = θv + (1− θ)h, where θ ∈ (0, 1), which is a viscosity supersolution of theequation H(W, vt, Dv, D2v)− (1− θ) f . Then based on arguments similar to Lemma 2 ofDavis et al. (1993), we get u ≤ vθ. Letting θ → 1 yields u ≤ v.

Sketched Proof of theorem 3. Denote the solution derived in Section 3.2 as hi,δ. For conve-nience, we omit the superscript i of hi and hi,δ in the sequel. Define hδ(s, z, x) such thathδ(s, z, x) = hδ(si, zj, xq), for s ∈ (si, si−1], z ∈ [zj − 1

2

√δ, zj +

12

√δ), x ∈ [xq − 1

2 ε, xq +12 ε),

and hδ(s, z, x) = exp−γKA(z, x), for s = 0. Let

h(s, z, x) = lim infδ,ε→0

hδ(si, zj, xq), h(s, z, x) = lim supδ,ε→0

hδ(si, zj, xq).

We first show h is a viscosity supersolution of (23), that is, minP3φ,−Q3φ,O3φ

≥

0. Suppose that w0 is a local minimum of h− φh on the domain for a smooth function φh.That is, h(v0) = φ(v0), φ is bounded outside a neighborhood of v0 such that h ≥ φ(v).By definition, there exist sequences δn ∈ R+ and vn := (sδn

i , zδnj , xδn

q ), such that δn → 0,

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vn → v0 := (s, z, x), hδn(vn) → h(v0), vn is a global minimum point of ψh := hδn − φ.Then ψh → 0 and hδn(v) ≥ φ(v) + ψh(v) for any v.

For the first operator P3, the boundary constraint (31) implies that hδn(wδn) = hδn(wδn),where wδn are the points on the buy boundary. Then we have hδn(wδn) ≤ φ(wδn) +

hδn(wδn)− φ(wδn), which further implies that P3φ ≥ 0. Similar arguments can show that−Q3φ ≥ 0. For the third operator O3, we notice that numerical scheme (34) implies thathδn(wδn) = Ehδn(wδn+1). Then hδn(wδn) ≤ Eφ(wδn+1)+ hδn(wδn)− φ(wδn+1), togetherwith Itô’s formula, suggests that O3φ ≥ 0.

Similar arguments can be applied to show that h is a viscosity subsolution of equation(23). Then the uniqueness of the solution indicates that h ≥ h. By the definition of h andh, same argument implies that h ≤ h. Finally, it suggests that h = h = h and the uniformconvergence of hδ to h.

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