Forward Indifference Valuation of American Options

Tim Leung∗ Ronnie Sircar† Thaleia Zariphopoulou‡

April 9, 2010

Abstract

We analyze the American option valuation problem with the forward performancecriterion introduced by Musiela and Zariphopoulou (2008). In this framework, utilityevolves forward in time without reference to a specific future time horizon. Moreover,risk preferences change with changing market conditions, which is natural as investorsare clearly more risk averse in economic downtimes. We examine two applications: thevaluation of American options with stochastic volatility and the modeling of early ex-ercises of American-style employee stock options. Finally, we introduce the marginalforward performance price. Among others, we find that, for arbitrary preferences, themarginal forward indifference price is always independent of the investor’s wealth and isrepresented as the claim’s risk-neutral price under the minimal martingale measure.

1 Introduction

Utility maximization theory has been central to quantifying rational investment decisionsand risk-averse valuations of assets at least since the work of von Neumann and Morgensternin the 1940s. In the Merton (1969) problem of dynamic portfolio optimization, utility isdefined at some time horizon in the future, when investment decisions are assessed in termsof expected utility of wealth. For portfolios involving derivatives, and associated utilityindifference pricing problems, derivatives payoffs or random endowments may be realizedat random times, which requires the specification of utility at other times, not just at asingle terminal time. This consideration is particularly important for investment and pricingproblems involving defaultable securities or American options.

One way to view this issue is to consider the definition of utility at the time of a randomcash flow as analogous to specifying what the investor does with the endowment thereafter.Any answer to the latter question necessarily involves details of the market in which hemight invest, and utilities and markets are inextricably linked. Some examples include Ober-man and Zariphopoulou (2003), Leung and Sircar (2009a) for utility indifference pricing ofAmerican options, and Sircar and Zariphopoulou (2010), Leung et al. (2008), and Jaimungaland Sigloch (2010) for defaultable securities. This approach allows for comparing utilities ofwealth at different times. However, as is common in classical utility indifference pricing, theinvestor’s risk preferences at intermediate times and the optimal investment decisions stilldirectly depend on an a priori chosen investment horizon.

∗Department of Applied Mathematics & Statistics, Johns Hopkins University, Baltimore MD 21218;

leung@jhu.edu. Work supported by NSF grant DMS-0908295 and the Johns Hopkins Duncan Travel Fund.†ORFE Department, Princeton University, E-Quad, Princeton NJ 08544; sircar@princeton.edu. Work

supported by NSF grant DMS-0807440.‡The Oxford-Man Institute of Quantitative Finance, University of Oxford, Oxford, UK; Departments of

Mathematics, and Information, Risk and Operations Management, University of Texas at Austin, Austin, TX

78712; thaleia.zariphopoulou@oxford-man.ox.ac.uk. Work partially supported by NSF grant DMS-0636586.

1

This issue has been addressed by Zariphopoulou and co-authors through the constructionof the forward performance criterion (see, for example, Musiela and Zariphopoulou (2008)).In this approach, the investor’s utility is specified at an initial time, and his risk preferences atsubsequent times evolve forward without reference to any specific ultimate time horizon. Thisresults in a stochastic utility process, called the forward performance process, which satisfiescertain desirable axiomatic properties so that it evolves consistently with the random marketconditions. Hence, this approach necessarily blends market models and risk preferences.The risk profile of a given investor is no longer considered separately from his investmentopportunities and the market. This is entirely natural: the current economic crisis has clearlyshown increased risk aversion in investors as the market has fallen.

In this paper, we develop an indifference valuation methodology based on the forwardperformance criterion. Specifically, we study the valuation of a long position in an Americanoption in an incomplete diffusion market model. This leads us to a combined stochasticcontrol and optimal stopping problem. In Section 2, our main objective is to analyze theoptimal trading and exercising strategies that maximize the option holder’s forward perfor-mance from the trading wealth together with the option payoff upon exercise. We also studythe holder’s forward indifference price for the American option, defined by comparing hisoptimal expected forward performance with and without the derivative (see Definition 2).

In Section 3, we discuss the exponential forward indifference valuation of an Americanoption in a stochastic volatility model. Due to the analytical properties of the exponentialforward performance, we show that the forward indifference price is wealth-independent,and derive its associated variational inequality as well as a dual representation. We alsoprovide a comparative analysis between the forward and classical exponential indifferenceprices. For instance, we show that the forward indifference price representation involves arelative entropy minimization (up to a stopping time) with respect to the minimal martingalemeasure, as opposed to the minimal entropy martingale measure that arises in the classicalexponential utility indifference price (see, among others, Rouge and El Karoui (2000) andDelbaen et al. (2002) for European claims, and Leung and Sircar (2009b) for Americanclaims). This contrasting difference is also reflected in the asymptotics results of indifferenceprices discussed here.

Another application is the modeling of early exercises of employee stock options (ESOs),which are American-style call options written on the firm’s stock granted to the employeeas a form of compensation. In Section 4, we assume a forward performance criterion for theemployee and investigate the impacts of various factors, such as wealth and risk tolerancefunction, on the employee’s exercise timing. In particular, we find that the employee tendsto exercise the ESO earlier when his wealth approaches zero.

Lastly, in Section 5, we introduce an alternative valuation mechanism for American op-tions based on the marginal forward performance. As is well known in the classical utilityframework, the marginal utility price represents the per-unit price that a risk-averse investoris willing to pay for an infinitesimal position in a contingent claim. In general, the marginalutility price is closely linked to the investor’s utility function and the market setup, and itonly becomes wealth independent under very special circumstances (see Kramkov and Sirbu(2006) for details). We adapt the classical definition to our forward performance frameworkand give a definition of the marginal forward indifference price. In contrast to the classicalmarginal utility price, the marginal forward indifference price turns out to be independentof the holder’s wealth and forward performance criterion, and it is simply the risk-neutralprice under the minimal martingale measure. Section 6 concludes the paper and discussesextensions for future investigation.

2

2 Forward Investment Performance Measurement and Indif-ference Valuation

We fix a filtered probability space (Ω,F ,P), with a filtration (Ft)t≥0 that satisfies the usualconditions of right continuity and completeness. The financial market consists of two liquidlytraded assets, namely, a riskless money market account and a stock. The money marketaccount has the price process B that satisfies

dBt = rtBtdt, (1)

with B0 = 1 ,where (rt)t≥0 is a non-negative Ft-adapted interest rate process. We shall workwith discounted cash flows throughout.

The discounted stock price S is modeled as a continuous Itô process satisfying

dSt = Stσt (λt dt+ dWt) , (2)

with S0 > 0, where (Wt)t≥0 is an Ft-adapted standard Brownian motion. The Sharpe ratio(λt)t≥0 and volatility coefficient (σt)t≥0 are taken to be bounded Ft-adapted processes sothat a strong solution exists for the SDE (2).

Starting with initial endowment x ∈ R, the investor dynamically rebalances his portfolioallocations between the stock S and the money market account B. Under the self-financingtrading condition, the discounted wealth satisfies

dXπt = πtσt(λt dt+ dWt), (3)

where (πt)t≥0 represents the discounted cash amount invested in S. As is standard (see,for example, Karatzas and Shreve (1998)), the set of admissible strategies Z consists of allself-financing Ft-adapted processes (πt)t≥0 such that

∫ u

0 σ2t π

2t dt < ∞ almost surely for each

u > 0. For 0 ≤ s ≤ t, we denote by Zs,t the set of admissible strategies over the period [s, t].Recently, Musiela and Zariphopoulou introduced the forward investment performance

criterion for portfolio selection. In the standard Merton portfolio optimization problem, riskaversion is modeled by a deterministic utility function Û(x) defined at some fixed terminaltime T and the investor’s risk preferences (or indirect utility) at intermediate times areinferred backwards. Starting at time t ≤ T with Ft-measurable wealth Xt, the Merton valueprocess is

Mt(Xt) = ess supπ∈Zt,T

IE{

Û(XπT )| Ft}

. (4)

When a dynamic programming principle holds, namely

Mt(Xt) = ess supπ∈Zt,s

IE {Ms(Xπs )| Ft} , 0 ≤ t ≤ s ≤ T, (5)

it follows that, under suitable technical conditions, the value process (Mt(Xπt ))0≤t≤T is a

supermartingale for any arbitrary admissible strategy π, and it is a martingale under sometrading strategy π̂∗. These are adopted as defining characteristics of the forward performancecriterion. Some well-known Merton problems where these conditions hold are i) markets withMarkovian dynamics where the optimal portfolio allocation can be found by solving a Bell-man PDE; ii) when the utility is of exponential type, in which case (5) holds under quitegeneral semimartingale models (see Mania and Schweizer (2005); Leung and Sircar (2009b));and iii) when expected utility is replaced by a dynamic time-consistent concave utility func-tional, defined, for instance, from a BSDE in Itô markets (see Klöppel and Schweizer (2006);Cheridito and Kupper (2009)).

3

However, in the forward performance framework, the investor’s utility function u0(x)is defined at time 0, and his performance criterion evolves forward in time. We recall thedefinition of the forward performance process given in Musiela and Zariphopoulou (2008):

Definition 1 An Ft-adapted process (Ut(x))t≥0 is a forward performance process if:1. for each t ≥ 0, the mapping x 7→ Ut(x) is strictly increasing and concave in x ∈ R,

2. for each t ≥ 0 and each admissible strategy π ∈ Z, IE{(Ut(Xπt ))+} < ∞ ,

3. for each π ∈ Z,IE{Us(Xπs )| Ft} ≤ Ut(Xπt ), for s ≥ t ,

4. there exists a strategy π∗ ∈ Z such thatIE{Us(Xπ

∗

s )| Ft} = Ut(Xπ∗

t ), for s ≥ t . (6)and

5. it satisfies the initial datum U0(x) = u0(x), x ∈ R, where u0 : R 7→ R is a strictlyconcave and increasing function of x.

In this definition, conditions 3 and 4 indicate that the forward performance process(Ut(X

πt ))t≥0 is a (P,Ft) supermartingale for any strategy π, and a martingale for some

strategy π∗. Condition 4 is also called the horizon-unbiased condition in Henderson andHobson (2007).

2.1 Forward Indifference Price

We introduce forward indifference valuation from the perspective of the holder of an Americanoption. The option payoff is characterized by an Ft-adapted bounded process (gt)0≤t≤T witha finite expiration date T . The collection of admissible exercise times is the set of stoppingtimes with respect to F0,T = (Ft)0≤t≤T taking values in [0, T ]. For 0 ≤ s ≤ t ≤ T , we denoteby Ts,t the set of Fs,t stopping times.

The option holder chooses his dynamic trading strategy π and exercise time τ so asto maximize his expected forward performance from investing in S and the money marketaccount and from receiving the option payoff. This leads to a combined stochastic controland optimal stopping problem. We define

Vt(Xt) = ess supτ∈Tt,T

ess supπ∈Zt,τ

IE {Uτ (Xπτ + gτ ) | Ft} , t ∈ [0, T ], (7)

which is the holder’s value process starting at time t with wealth Xt.In the classical case with a terminal utility function Û , the holder’s investment problem

isess supτ∈Tt,T

ess supπ∈Zt,τ

IE {Mτ (Xπτ + gτ ) | Ft} ,

where M is the solution to the Merton problem defined in (4). Here M plays the role ofintermediate utility at times τ ≤ T , and corresponds to specifying that option proceedsreceived at exercise time τ are re-invested in the Merton optimal strategy, up till time T . Bycontrast, the forward performance process U specifies utilities at all times, without referenceto any specific horizon.

The holder’s forward indifference price pt for the American option g is defined as the cashamount such that the option holder is indifferent between two positions: optimal investmentwith the American option, and optimal investment without the American option but withextra wealth pt.

4

Definition 2 The holder’s forward indifference price process (pt)0≤t≤T for the Americanoption is defined by the equation

Vt(Xt) = Ut(Xt + pt), t ∈ [0, T ]. (8)

The forward indifference price is useful for characterizing the option holder’s optimalexercise time τ∗. Under quite general conditions, the optimal stopping time is recovered asusual as the first time the value process reaches the reward process. From (7) and (8),

τ∗t = inf {t ≤ s ≤ T : Vs(Xs) = Us(Xs + gs)}= inf {t ≤ s ≤ T : Us(Xs + ps) = Us(Xs + gs)}= inf {t ≤ s ≤ T : ps = gs} . (9)

The representation (9) implies that the option holder will exercise the American option assoon as the forward indifference price reaches (from above) the option payoff. It allows us toanalyze the holder’s optimal exercise policy through his forward indifference price.

In Sections 3 and 4, we will focus our study on two specific financial applications: i)the valuation of an American option written on a stock S with stochastic volatility underforward performance criterion of exponential type (to be defined in (23)), and ii) modelingearly exercises of employee stock options (ESOs), where we go beyond the exponential forwardperformance criterion.

2.2 Forward Performance of Generalized CARA/CRRA Type

In this paper, we will use a special class of forward performance processes introduced byMusiela and Zariphopoulou (2008). They are represented by the compilation of a determin-istic function u(x, t) which models the investor’s dynamic risk preference, and an increasingprocess At that solely depends on the market. Recently, Berrier et al. (2009) and Musielaand Zariphopoulou (2010) studied various properties of this family of forward performance,especially its time-monotonicity, and gave a dual representation of the forward performance.

Theorem 3 (Proposition 3 of Musiela and Zariphopoulou (2008)) Define the stochastic pro-cess

At =

∫ t

0λ2s ds, t ≥ 0. (10)

Let the function u : R × R+ 7→ R be C3,1, strictly concave and increasing in the spatialargument. Also, assume that it satisfies the partial differential equation

ut =1

2

u2xuxx

, (11)

with initial condition u(x, 0) = u0(x) , where u0 ∈ C3(R). Then, the process Ut(x) defined by

Ut(x) = u (x,At) , t ≥ 0, (12)

is a forward performance process. Moreover, the martingale-generating strategy π∗ in (6) isgiven by

π∗t = −λtσt

ux(X∗t , At)

uxx(X∗t , At), (13)

where (X∗t )t≥0 is the associated wealth process X∗ = Xπ

∗following (3).

5

A quantity that plays a crucial role in the description of the wealth and portfolio processes(X∗, π∗) is the so-called local risk tolerance function R : R× R+ 7→ R+ defined by

R(x, t) = − ux(x, t)uxx(x, t)

. (14)

With this notation, the dynamics of X∗ is

dX∗t = R(X∗t , At)λt (λt dt+ dWt) . (15)

Given enough regularity, if u satisfies (11), then, by differentiation, R is solution of the fastdiffusion equation, namely

Rt +1

2R2Rxx = 0. (16)

One way to construct forward performance criteria is to look for solutions of this PDE.In Zariphopoulou and Zhou (2008), the following two-parameter family of risk tolerancefunctions was introduced:

R(x, t;α, β) =√

αx2 + βe−αt, α, β > 0. (17)

We illustrate an example of this risk tolerance in Figure 1.

Figure 1: The risk tolerance function R(x, t;α, β) with α = 4, and β = 0.25. For any fixed wealthx, the risk tolerance decreases over time. At any fixed time, the risk tolerance increases as wealthdecreases or increases away from zero.

There are several reasons to work with this family. First, it yields, in the limit, risktolerance functions that resemble those related to the three most popular cases, specifically,the exponential, power and logarithmic. Indeed,

limα→0

R(x, t;α, β) =√

β, (exponential) (18)

limβ→0

R(x, t;α, β) = αx, x ≥ 0, (power)

limβ→0

R(x, t; 1, β) = x, x > 0. (logarithmic)

6

We remark that R(x, t;α, β) is well-defined for wealth x ∈ (−∞,∞), except in the limit caseβ ↓ 0. In Figure 2, we illustrate the limit in (18) where the risk tolerance function converges tothe constant

√β as α ↓ 0. As we will see in Section 3, the constant risk tolerance corresponds

to the forward performance measure of exponential type.

−1 −0.5 0 0.5 10

0.5

1

1.5

2

2.5

wealth (x)

R(x

,t;α,

β)

Risk Tolerance Function

α=4

α=1

α=0

α=0.5

Figure 2: As α decreases from 4 to 0, with β = 0.25 and t = 1, the risk tolerance function R(x, t;α, β)converges to the constant level

√β = 0.5, as predicted by the limit in (18).

Zariphopoulou and Zhou (2008) compute the corresponding dynamic risk preference func-tion u(x, t):

Proposition 4 (Proposition 3.2 of Zariphopoulou and Zhou (2008)) The dynamic risk pref-erence function u(x, t;α, β) associated with R(x, t;α, β) in (17) is given by

u(x, t;α, β) = mκ1+

1

κ

α− 1e1−κ2

tβκe−αt + (1 + κ)x(κx+

√

αx2 + βe−αt)

(κx+√

αx2 + βe−αt)1+1

κ

+ n, α 6= 1, (19)

u(x, t; 1, β) =m

2

(

log(x+√

x2 + βe−t)− e−t

βx(

x−√

x2 + βe−t)

− t2

)

+ n, α = 1,

where κ =√α, and m > 0, n ∈ R are constants of integration.

As mentioned earlier, in the context of the domain of the local risk tolerance, the functionu(x, t;α, β) is also well-defined for all x ∈ R, except in the limit case β ↓ 0. This propertyis particularly useful in indifference valuation for it eliminates the nonnegativity constraintson the investor’s wealth (with and without the claim in hand).

3 American Options under Stochastic Volatility

In this section, we study the forward indifference valuation of an American option in astochastic volatility model. We work with the exponential forward performance, which, asmentioned in the previous section, corresponds to the parameter choice α = 0. A comparativeanalysis with the classical exponential utility indifference pricing is provided in Section 3.4.

7

The underlying stock price S, discounted by the money market account, is modeled as adiffusion process satisfying

dSt = µ(Yt)St dt+ σ(Yt)St dWt. (20)

The drift and volatility coefficients µ(Yt) and σ(Yt) are driven by a non-traded stochasticfactor Y which evolves according to

dYt = b(Yt) dt+ c(Yt) (ρdWt +√

1− ρ2dŴt), (21)

with correlation coefficient ρ ∈ (−1, 1). The processes W and Ŵ are two independent stan-dard Brownian motions defined on (Ω,F , (Ft)t≥0,P), where Ft is taken to be the augmentedσ-algebra generated by ((Wu, Ŵu); 0 ≤ u ≤ t). The volatility function σ(·) and the diffusioncoefficient c(·) are smooth, positive and bounded. The drift coefficient µ(·) is bounded con-tinuous, and b(·) is Lipschitz continuous on R. The American option yields payoff g(Sτ , Yτ , τ)at exercise time τ ∈ [0, T ], where g(·, ·, ·) is smooth and bounded.

3.1 Exponential Forward Indifference Price

We model the American option holder’s risk preferences by the exponential forward perfor-mance process. By setting the parameter α = 0 in (17), the risk tolerance becomes a constant√β (see (18) and Figure 2). In turn, (11) and (14) yield the corresponding exponential risk

preference function u(x, t):

u(x, t) = −e−γx+ t2 , (22)with γ = 1/

√β. The parameter γ can be considered as the investor’s local risk aversion.

Applying Theorem 3 yields the exponential forward performance process:

U et (x) = −e−γx+1

2

∫ t0λ(Ys)2ds, (23)

where λ(y) = µ(y)/σ(y).As defined in (7), the holder’s forward performance value process is given by

V et (Xt) = ess supτ∈Tt,T

ess supπ∈Zt,τ

IE{

−e−γ(Xπτ +g(Sτ ,Yτ ,τ))e 12∫ τ0λ(Ys)2ds | Ft

}

= e1

2

∫ t0λ(Ys)2ds ess sup

τ∈Tt,T

ess supπ∈Zt,τ

IE{

−e−γ(Xπτ +g(Sτ ,Yτ ,τ))e 12∫ τtλ(Ys)2ds | Ft

}

.

Restricting to Markovian controls, we look for a solution the form

V et (Xt) = e1

2

∫ t0λ(Ys)2ds V (Xt, St, Yt, t), (24)

where the function V : R×R+×R×R+ 7→ R− solves

V (x, s, y, t) = supτ∈Tt,T

supπ∈Zt,τ

IE{

−e−γ(Xπτ +g(Sτ ,Yτ ,τ))e 12∫ τtλ(Ys)2ds |Xt = x, St = s, Yt = y

}

.

(25)We proceed with a variational inequality that V is expected to satisfy. To facilitate

notation, we introduce the following differential operators and Hamiltonian:

LSY v =1

2σ(y)2s2vss + ρc(y)σ(y)svsy +

1

2c(y)2vyy (26)

+ λ(y)σ(y)svs + b(y)vy,

L0SY v = LSY v − ρc(y)λ(y)vy − λ(y)σ(y)svs

=1

2σ(y)2s2vss + ρc(y)σ(y)svsy +

1

2c(y)2vyy + (b(y)− ρc(y)λ(y))vy, (27)

8

and

H(vxx, vxy, vxs, vx) = maxπ

(

π2σ(y)2

2vxx + π

(

ρσ(y)c(y)vxy + σ(y)2svxs + λ(y)σ(y)vx

)

)

.

Note that LSY and L0SY are, respectively, the infinitesimal generators of the Markov process(St, Yt)t≥0 under the historical measure P and the minimal martingale measure Q

0 (to bedefined in (34)).

As is usual, given sufficient regularity, the solution V of (25) satisfies the variationalinequality

Vt + LSY V +H(Vxx, Vxy, Vxs, Vx) +λ(y)2

2V ≤ 0,

V (x, s, y, t) ≥ −e−γ(x+g(s,y,t)),(

Vt + LSY V +H(Vxx, Vxy, Vxs, Vx) +λ(y)2

2V)

·(

− e−γ(x+g(s,y,t)) − V (x, s, y, t))

= 0,

V (x, s, y, T ) = −e−γ(x+g(s,y,T )),(28)

for (x, s, y, t) ∈R×R+×R+× [0, T ]. The existence of a smooth solution to this variationalinequality is an open question, and we do not address it here.

In view of Definition 2 and (24), the option holder’s exponential forward indifference priceprocess (pt)0≤t≤T is given by

pt = −1

γlog

(

− V (Xt, St, Yt, t))

−Xt. (29)

Next, applying the transformation

V (x, s, y, t) = −e−γ(x+p(s,y,t)) (30)

to (28) yields

pt + L0SY p−1

2γ(1− ρ2)c(y)2p2y ≤ 0,

p(s, y, t) ≥ g(s, y, t),(

pt + L0SY p−1

2γ(1− ρ2)c(y)2p2y

)

· (g(s, y, t)− p(s, y, t)) = 0,

p(s, y, T ) = g(s, y, T ),

(31)

for (s, y, t) ∈ R+×R+× [0, T ]. Substituting (30) into (29), we verify that the exponentialforward indifference price is given by pt = p(St, Yt, t). Therefore, the indifference price solvesthe free boundary problem (31), and it is wealth independent.

By the first-order condition in (28) and the transformation (30), the optimal hedgingstrategy (π̃∗t )0≤t≤T can be expressed in terms of the indifference price, namely,

π̃∗t =λ(Yt)

γσ(Yt)+

Stγps(St, Yt, t) +

ρc(Yt)

γσ(Yt)py(St, Yt, t).

The first term in this expression is the martingale-generating strategy in (13), while thesecond and third terms correspond to the extra demand in stock S due to changes in S andthe stochastic factor Y respectively.

9

The optimal exercise time is the first time that the indifference price reaches the optionpayoff:

τ∗t = inf{t ≤ u ≤ T : p(Su, Yu, u) = g(Su, Yu, u)} . (32)In practice, one can numerically solve the variational inequality (31) to obtain the optimalexercise boundary which represents the critical levels of S and Y at which the option shouldbe exercised. We remark that the indifference price, the optimal hedging and exercisingstrategies are all wealth independent. The same occurs in the classical indifference valuationwith exponential utility.

3.2 Dual Representation

The option holder’s forward performance maximization in (24) can be considered as theprimal optimization problem, and it yields the first expression for the forward indifferenceprice in (29). Our objective is to derive a dual representation for the forward indifferenceprice, which turns out to be related to pricing the American option with entropic penalty.This result will allow us to express the price in a way analogous to the classical exponentialindifference price. We carry out this comparison in Section 3.4.

We first define the set of equivalent local martingale measures considered in this stochasticvolatility model.

Definition 5 Define a probability measure Qφ by

dQφ

dP= exp

(

−12

∫ T

0λ(Ys)

2 + φ2s ds−∫ T

0λ(Ys) dWs −

∫ T

0φs dŴs

)

,

where (φt)0≤t≤T is an Ft-adapted process such that

IEQφ

{∫ T

0φ2tdt

}

< ∞. (33)

Its density process is denoted by Zφt = IE{dQφ

dP|Ft}, 0 ≤ t ≤ T .

By Girsanov’s Theorem, S becomes a (local) martingale under Qφ, and W φt = Wt +∫ t

0 λ(Ys)ds and Ŵφt = Ŵt +

∫ t

0 φs ds are independent Qφ-Brownian motions. The process φ

is commonly referred to as the risk premium for the second Brownian motion Ŵ .When φ = 0, we obtain the minimal martingale measure Q0, given by

dQ0

dP= exp

(

−12

∫ T

0λ(Ys)

2 ds−∫ T

0λ(Ys) dWs

)

. (34)

In fact, we can express Qφ in terms of Q0, namely,

dQφ

dQ0=

dQφ

dP

/

dQ0

dP= exp

(

−12

∫ T

0φ2s ds−

∫ T

0φs dŴ

0s

)

. (35)

We also denote the density process of Qφ with respect to Q0 by Zφ,0t = IEQ0{dQφ

dQ0|Ft}.

We can think of Q0 as the prior risk-neutral measure, and define the conditional relativeentropy Hτt (Q

φ|Q0) of Qφ with respect to Q0 over the interval [t, τ ] as

Hτt (Qφ|Q0) = IEQφ

{

logZφ,0τ

Zφ,0t|Ft

}

.

10

Direct computation from (35) shows that this relative entropy is, in fact, a quadratic penal-ization on the risk premium φ. In other words,

Hτt (Qφ|Q0) = 1

2IEQ

φ

{∫ τ

t

φ2s ds|Ft}

. (36)

Note that condition (33) guarantees the finiteness of the relative entropy of Qφ with respectto Q0. Also, we denote Mf to be the set of Qφ in Definition 5.

Next, we apply these results to obtain a dual representation of the exponential forwardindifference price.

Proposition 6 The exponential forward indifference price can be represented as the solutionof the combined stochastic control and optimal stopping problem

p(St, Yt, t) = ess supτ∈Tt,T

ess infQφ∈Mf

(

IEQφ {g(Sτ , Yτ , τ)|Ft}+

1

γHτt (Q

φ|Q0))

. (37)

The optimal measure is given by Qφ∗, which corresponds to the risk premium (see (35))

φ∗t = −γc(Yt)√

1− ρ2 py(St, Yt, t), (38)

and the optimal stopping time is given by (32).

Proof. Motivated by the right-hand side of (37) and equation (36), we consider the candidatefunction

w(s, y, t) := supτ∈Tt,T

infQφ∈Mf

IEQφ

{

g(Sτ , Yτ , τ) +1

2γ

∫ τ

t

φ2s ds|St = s, Yt = y}

. (39)

Note that the infinitesimal generator of (S, Y ) under Qφ is given by

LφSY w = L0SY w + φc(y)√

1− ρ2wy,

where L0SY is given in (27). It follows that the associated variational inequality for w(s, y, t)is

wt + L0SY w + infφ(φc(y)

√

1− ρ2wy +φ2

2γ) ≤ 0,

w(s, y, t) ≥ g(s, y, t),(

wt + L0SY w + infφ(φc(y)

√

1− ρ2wy +φ2

2γ)

)

· (g(s, y, t)− w(s, y, t)) = 0,

w(s, y, T ) = g(s, y, T ),

(40)

for (s, y, t) ∈ R+×R+×[0, T ].After the minimization over φ above, we obtain the optimal control process

φ∗t = −γc(Yt)√

1− ρ2wy(St, Yt, t).

In turn, the first inequality in (40) becomes

wt + L0SY w −1

2γ(1− ρ2)c(y)2w2y ≤ 0.

11

Comparing it and the variational inequality (31) for p(s, y, t) yields w = p.

The dual representation (37) provides an alternative interpretation of the forward indif-ference price. In essence, the holder tries to value the American option over a set of equivalentlocal martingale measures {Qφ}, and his selection criterion for the optimal pricing measureis based on relative entropic penalization. Indeed, the second term in (37) is the relativeentropy of a candidate measure Qφ with respect to the minimal martingale Q0 up to theexercise time. This leads the holder to assign the corresponding optimal risk premium φ∗ in(38). We will compare this result to its classical analogue in Section 3.4.

3.3 Risk Aversion and Volume Asymptotics

Proposition 6 provides a convenient representation for analyzing the indifference price’s sen-sitivity with respect to risk aversion and the number of options held. First, let us considera risk-averse investor with local risk aversion γ (see 22) who holds a > 0 units of Americanoptions, and suppose that all a units are constrained to be exercised simultaneously. In thiscase, the holder’s indifference price p(s, y, t; γ, a) is again given by (37) but with the payoffg(Sτ , Yτ , τ) replaced by ag(Sτ , Yτ , τ). The optimal exercise time τ

∗(a, γ) is the first time thatthe forward indifference price reaches the payoff from exercising all a units:

τ∗(a, γ) = inf{t ≤ u ≤ T : p(Su, Yu, u; γ, a) = ag(Su, Yu, u)}. (41)

Proposition 7 Fix a > 0 and t ∈ [0, T ]. If γ2 ≥ γ1 > 0, then

p(s, y, t; γ2, a) ≤ p(s, y, t; γ1, a)

andτ∗(a, γ2) ≤ τ∗(a, γ1), almost surely.

Proof. For γ2 ≥ γ1 > 0, it follows from (37) that p(s, y, t; γ2, a) ≤ p(s, y, t; γ1, a). Therefore,as γ increases, p(s, y, t; γ, a) decreases, while the payoff ag(s, y, t) does not depend on γ. By(41), this leads to a shorter exercise time (almost surely).

Furthermore, we observe that, as γ increases to infinity, the penalty term in the indiffer-ence price representation (37) vanishes. Consequently, we deduce the following limit:

limγ→∞

p(s, y, t; γ, a) = a · supτ∈Tt,T

infQφ∈Mf

IEQφ {g(Sτ , Yτ , τ)|St = s, Yt = y} . (42)

This limiting price is commonly referred to as the sub-hedging price of the American options(see, for example, Karatzas and Kou (1998)). On the other hand, as the holder’s risk aversionγ decreases to zero, one can deduce from (37) that it is optimal not to deviate from the priormeasure Q0 (i.e. φ = 0), yielding zero entropic penalty. This leads to valuing the Americanoptions under the minimal martingale measure Q0, namely,

limγ→0

p(s, y, t; γ, a) = a · supτ∈Tt,T

IEQ0 {g(Sτ , Yτ , τ)|St = s, Yt = y} . (43)

We have provided the formal arguments for these risk-aversion limits. For more technicaldetails, we refer the reader to Leung and Sircar (2009b) who have shown these asymptoticresults for the traditional exponential indifference price of American options in a generalsemimartingale framework, and their proofs can be easily adapted here.

12

Finally, we observe from (37) the volume-scaling property:

p(s, y, t; γ, a)/a = p(s, y, t; aγ, 1).

Hence, as the number of options held increases, the holder’s average indifference pricep(s, y, t; γ, a)/a decreases, and by (41) the options will be exercised earlier. Moreover, theindifference price limits in (42)-(43) lead to the following limits:

lima→∞

p(s, y, t; γ, a)

a= sup

τ∈Tt,T

infQφ∈Mf

IEQφ {g(Sτ , Yτ , τ)|St = s, Yt = y}

and

lima→0

p(s, y, t; γ, a)

a= sup

τ∈Tt,T

IEQ0 {g(Sτ , Yτ , τ)|St = s, Yt = y} .

3.4 Comparison with the Classical Exponential Utility Indifference Price

In this section, we provide a comparative analysis between the classical and forward indif-ference valuation. We start with a brief review of the classical indifference pricing withexponential utility under stochastic volatility models. We refer the reader to, for example,Sircar and Zariphopoulou (2004) and Benth and Karlsen (2005) for European options, andOberman and Zariphopoulou (2003) for American options.

In the traditional setting, the investor’s risk preferences at time T are modeled by theexponential utility function −e−γx, γ > 0. The value function of the Merton problem is

M(x, y, t) = supπ∈Zt,T

IE{

−e−γXπT |Xt = x, Yt = y}

, (44)

with XπT given by (3). As is well known, the function M admits a separation of variables dueto the exponential utility.

Proposition 8 The value function M(x, y, t) is given by

M(x, y, t) = −e−γxf(y, t)1

1−ρ2 , (45)

where f solves

ft + L0Y f =1

2(1− ρ2)λ(y)2f, (46)

for (x, t) ∈ R × [0, T ), with f(y, T ) = 1, for y ∈ R. The operator L0Y is the infinitesimalgenerator of Y under the minimal martingale measure Q0, and is given by

L0Y f = (b(y)− ρc(y)λ(y)) fy +1

2c(y)2fyy.

Details can be found, for example, in Theorem 2.2 of Sircar and Zariphopoulou (2004).If the American option g is held, then the investor seeks the optimal trading strategy and

exercise time to maximize the expected utility from trading wealth plus the option’s payoff.Upon exercise of the option, the investor will reinvest the contract proceeds, if any, to histrading portfolio, and continue to trade up to time T . As a consequence, the holder facesthe optimization problem

V̂ (x, s, y, t) = supτ∈Tt,T

supπ∈Zt,τ

IE {M (Xπτ + g(Sτ , Yτ , τ), Yτ , τ) |Xt = x, St = s, Yt = y } , (47)

13

where M is defined in (44).The indifference price of the American option p̂ is found from the equation

M(x, y, t) = V̂ (x− p̂(x, s, y, t), s, y, t). (48)

Using (45) and (48), we obtain the formula

V̂ (x, s, y, t) = −e−γ(x+p̂(x,s,y,t))f(y, t)1

1−ρ2 . (49)

To derive the variational inequality for the indifference price, one can use the variationalinequality for V and then apply the transformation (49). Again, the choice of exponentialutility yields wealth-independent indifference prices, i.e. p̂(x, s, y, t) = p̂(s, y, t). We obtain

p̂t + LESY p̂−1

2γ(1− ρ2)c(y)2p̂2y ≤ 0,

p̂(s, y, t) ≥ g(s, y, t),(

p̂t + LESY p̂−1

2γ(1− ρ2)c(y)2p̂2y

)

· (g(s, y, t)− p̂(s, y, t)) = 0,

p̂(s, y, T ) = g(s, y, T ),

(50)

for (s, y, t) ∈ R+×R+×[0, T ]. Herein,

LESY w = L0SY w + l(y, t)c(y)√

1− ρ2wy,

where

l(y, t) =1

√

1− ρ2c(y)

fy(y, t)

f(y, t).

In fact, LESY is the infinitesimal generator of (S, Y ) under the minimal entropy martingalemeasure (MEMM) QE , which is given by

dQE

dP= exp

(

−12

∫ T

0(λ(Ys)

2 + l(Ys, s)2) ds−

∫ T

0λ(Ys) dWs +

∫ T

0l(Ys, s) dŴs

)

. (51)

For a detailed study on the application of MEMM in indifference pricing, we refer the readerto Fritelli (2000) and Delbaen et al. (2002).

It is important to notice that the fundamental difference between variational inequalities(31) and (50) lies in the operators L0SY and LESY . Indeed, these variational inequalities reflectthe special roles of the minimal martingale measure in the forward indifference setting andthe minimal entropy martingale measure in the classical model. The MEMM also arises inthe dual representation of p̂. Namely, one can show that (see Leung and Sircar (2009b))

p̂(St, Yt, t) = ess supτ∈Tt,T

ess infQφ∈Mf

(

IEQφ {g(Sτ , Yτ , τ)|Ft}+

1

γHτt (Q

φ|QE))

. (52)

This duality result bears a striking resemblance to (37), except here the relative entropy termis computed with respect to the minimal entropy martingale measure QE rather than withrespect to Q0.

In the classical setting, the computation of the indifference price involves two steps,namely, first solving PDE (46) followed by variational inequality (50). However, in theforward indifference valuation, the indifference price can be obtained by solving only onevariational inequality (31). Hence, with exponential performance, the forward indifferenceformulation allows for more efficient computation than in the classical framework.

14

Remark 9 If the claim is written on Y only, say with payoff function g(y, t), then the indif-ference price does not depend on S. Applying a logarithmic transformation to the variationalinequality (31), the nonlinear variational inequality can be linearized. As a result, in additionto (37), the forward indifference price admits the representation:

p(y, t) = − 1γ(1− ρ2) log infτ∈Tt,T

IEQ0{

e−γ(1−ρ2)g(Yτ ,τ) |Yt = y

}

.

In contrast, the classical exponential utility indifference price of an American option with thesame payoff function g(y, t) can be found in Oberman and Zariphopoulou (2003), and it isgiven by

p̂(y, t) = − 1γ(1− ρ2) log infτ∈Tt,T

IEQE{

e−γ(1−ρ2)g(Yτ ,τ) |Yt = y

}

,

where QE is the minimal entropy martingale measure defined in (51). Again, we see that Q0

in the forward performance framework plays a similar role as QE in the classical setting.

In summary, we have studied the exponential forward indifference price through its as-sociated variational inequality and dual representation, which have very desirable structuresand interpretations due to the nice analytic properties of the exponential forward perfor-mance. We have also seen in the analysis that higher risk aversion leads to earlier exercisetimes.

4 Modeling Early Exercises of Employee Stock Options

Now, we consider the problem of early exercises of employee stock options (ESOs). Theseare American call options granted by a company to its employees as a form of compensation.A typical ESO contract prohibits the employee from selling the option and from hedging byshort selling the firm’s stock. The sale and hedging restrictions may induce the employee toexercise the ESO early and invest the option proceeds elsewhere. In fact, empirical studies(for example, Bettis et al. (2005)) show that employees tend to exercise their ESOs very early.Recent studies, for example, Henderson (2005) and Leung and Sircar (2009a), apply classicalindifference pricing to ESO valuation. In those papers, the employee was assumed to have aclassical exponential utility specified at the expiration date T of the options. Here, we assumea forward performance criterion for the employee, which is not anchored to a specific futuretime, and then compute the optimal exercise strategy. Modeling the employee’s exercisetiming is crucial to the accurate valuation of ESOs.

We assume that the employee dynamically trades in a liquid correlated market indexand a riskless money market account in order to partially hedge against his ESO position.Alternative hedging strategies for ESOs have also been proposed. For instance, Leung andSircar (2009b) considered combining static hedges with market-traded European or Americanputs with the dynamic investment in the market index.

We focus our study on the case of a single ESO. Typically, ESOs have a vesting periodduring which they cannot be exercised early. The incorporation of a vesting period amountsto lifting the employee’s pre-vesting exercise boundary to infinity to prevent exercise, butleaving the post-vesting policy unchanged. The case with multiple ESOs can be studied asa straightforward extension to our model though the numerical computations will be morecomplex and time-consuming; see Grasselli and Henderson (2009) for the case of multipleperpetual ESOs with exponential utility. Our main objective is to examine the non-trivialeffects of forward investment performance criterion on the employee’s optimal exercise timing.

15

4.1 The Employee’s Optimal Forward Performance with an ESO

We assume that the money market account yields a constant interest rate r ≥ 0. Thediscounted prices of the market index and the firm’s stock are modeled as correlated lognormalprocesses, namely,

dSt = µSt dt+ σSt dWt, (traded) (53)

dYt = bYt dt+ cYt (ρdWt +√

1− ρ2 dŴt) , (non-traded) (54)

where µ, σ, b, c are constant parameters. The discounted ESO payoff, with expiration dateT , is given by

g(Yτ , τ) = (Yτ −Ke−rτ )+, for τ ∈ T0,T .Note that the Sharpe ratio of S is now a constant λ = µ/σ, and the option payoff is indepen-dent of S. The employee dynamically trades in the index Y and the money market account,so his discounted wealth process satisfies

dXπt = πtσ(λ dt+ dWt). (55)

We proceed with the employee’s forward performance criterion Ut(x). First, we adoptthe risk tolerance function in (17), namely, R(x, t) =

√

αx2 + βe−αt, and the correspondingdynamic risk preference function u(x, t) given in Proposition 4. Then, we apply Theorem3 to obtain the employee’s forward performance Ut(x) = u(x, λ

2t). In turn, the employee’smaximal forward performance in the presence of the ESO is given by

V (x, y, t) = supτ∈Tt,T

supπ∈Zt,τ

IE{

u(Xπτ + g(Yτ , τ), λ2τ) |Xt = x, Yt = y

}

. (56)

In contrast to the stochastic volatility problem in Section 3, the state variable S is no longerneeded, but we work with a more general forward performance criterion than the exponentialone.

The value function is conjectured to satisfy the following variational inequality:

Vt + LY V −(ρcyVxy + λVx)

2

2Vxx≤ 0,

V (x, y, t) ≥ u(x+ g(y, t), λ2t),(

Vt + LY V −(ρcyVxy + λVx)

2

2Vxx

)

·(

u(x+ g(y, t), λ2t)− V (x, y, t))

= 0,

V (x, y, T ) = u(x+ g(y, T ), λ2T ),

(57)

for (x, y, t) ∈ R×R+×[0, T ]. We remark that the variational inequality (57) is highly nonlinear,and it can be simplified only for very special local utility functions (e.g. the exponentialutility). Here, we do not attempt to address the related existence, uniqueness, and regularityquestions.

4.2 Numerical Solutions

We apply a fully explicit finite-difference scheme to numerically solve (57) for the employee’soptimal exercising strategy. First, we restrict the domain R×R+× [0, T ] to a finite domainD = {(x, y, t) : −L1 ≤ x ≤ L2, 0 ≤ y ≤ L3, 0 ≤ t ≤ T}, where Lk, k = 1, 2, 3, are chosen tobe sufficiently large to preserve the accuracy of the numerical solutions. The numerical

16

scheme also requires a number of boundary conditions. For y = 0, we have Yt = 0, t ≥ 0,and thus the ESO becomes worthless. Therefore, we set

V (x, 0, t) = u(x, λ2t).

When Y hits the high level L3, we assume that the employee will exercise the ESO there,implying the condition

V (x, L3, t) = u(x+ g(L3, t), λ2t).

We also need to set appropriate boundary conditions along x = −L1 and x = L2. To thisend, we adopt the Dirichlet boundary conditions

V (−L1, y, t) = u(−L1 + g(y, t), λ2t), V (L2, y, t) = u(L2 + g(y, t), λ2t),

which imply that the employee will exercise the ESO at these boundaries.Then, a uniform grid is applied onD with nodes {(xi, yj , tn) : i = 0, ..., I; j = 0, ..., J ; n =

0, ..., N}, with ∆x = (L1+L2)/I, ∆y = L3/J , and ∆t = T/N being the grid spacings. Next,we use the discrete approximations V ni,j ≈ V (xi, yj , tn) where xi = i∆x, yj = j∆y, andtn = n∆t. Also, we discretize the partial differential inequality in (57) by approximating thex and y derivatives with central differences:

∂V

∂x(xi, yj , tn) ≈

V ni+1,j − V ni−1,j2∆x

,∂2V

∂x2(xi, yj , tn) ≈

V ni+1,j − 2V ni,j + V ni−1,j∆x2

, (58)

∂V

∂y(xi, yj , tn) ≈

V ni,j+1 − V ni,j−12∆y

,∂2V

∂y2(xi, yj , tn) ≈

V ni,j+1 − 2V ni,j + V ni,j−1∆y2

, (59)

and∂2V

∂x∂y(xi, yj , tn) ≈

V ni+1,j+1 − V ni+1,j−1 + V ni−1,j−1 − V ni−1,j+14∆x∆y

.

The t-derivative is approximated by the backward difference

∂V

∂t(xi, yj , tn) ≈

V n+1i,j − V ni,j∆t

.

This results in an explicit finite-difference scheme which solves for all V ni,j iteratively backwardin time starting at tN = T .

During each time step, the inequality constraint V (x, y, t) ≥ u(x+g(y, t), λ2t) is enforced.By comparing the value function and the obstacle term, we identify the continuation regionC where the ESO is not exercised, and the exercise region E where the ESO is exercised,namely

C = {(x, y, t) ∈ R×R+×[0, T ] : V (x, y, t) > u(x+ g(y, t), λ2t)}, (60)E = {(x, y, t) ∈ R×R+×[0, T ] : V (x, y, t) = u(x+ g(y, t), λ2t)}. (61)

From the numerical example in Figure 3, we observe that the value function dominatesthe obstacle term. At any time t and wealth x, we locate the optimal stock price level y∗(x, t)that separates the two regions C and E . As a result, the employee will exercise the ESO assoon as Yt hits the threshold y

∗(Xt, t):

τ∗ = inf{0 ≤ t ≤ T : Yt = y∗(Xt, t)}. (62)

In the case of call options, the boundary lies above the strike K. Figure 4 shows an exampleof the optimal exercise boundary for the ESO.

17

0 0.5 1 1.5 20.2

0.4

0.6

0.8

1

1.2

1.4

1.6

y

Val

ue F

unct

ion

and

Obs

tacl

e

V(0,y,0)

u((y−K)+,0)

Figure 3: The value function V (x, y, t) dominates the obstacle term u(x+g(y, t), λ2t). The parametersare µ = 11.5%, σ = 35%, b = 6%, c = 40%, ρ = 50%, r = 1%, K = 1, T = 1, α = 4, β = 0.25.At t = 0 and x = 0, the critical stock price y∗(0, 0) = 1.58 is the point at which the value functiontouches the obstacle term (above the strike).

Figure 4: The optimal exercise policy is characterized by the critical stock price y∗(x, t) as a functionof wealth x and time t. It decreases as time approaches maturity. In addition, it tends to shift loweras wealth is near zero.

18

4.3 Behavior of the Optimal Exercise Policy

We illustrate the employee’s optimal exercise boundary in Figure 4. Not surprisingly, theexercise boundary y∗(x, t) decreases with respect to time, which implies that the employeeis willing to exercise the ESO at a lower stock price as it gets closer to expiry.

From Figure 5, we observe that the exercise boundary is wealth dependent. The employeetends to delay exercising the ESO when his wealth deviates away from zero. We can gainsome intuition from our choice of risk tolerance function R(x, t;α, β). As wealth approacheszero, the employee’s risk tolerance decreases (recall Figure 1), or equivalently, risk aversionincreases. Higher risk aversion influences the employee to exercise earlier to secure smallgains rather than waiting for future uncertain payoffs.

Finally, we show in Figure 6 that the exercise boundary tends to shift upward for highervalues of α and β, given the initial wealth x = 0. The effect of β is intuitive because the risktolerance function is increasing with respect to β. Therefore, the option holder with a higherβ is effectively less risk averse and may be willing to hold on to the ESO longer.

0 0.2 0.4 0.6 0.8 11

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

Sto

ck p

rice

leve

l

Time

x=1.5x=1x=0.5x=0

0 0.2 0.4 0.6 0.8 11

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

Sto

ck p

rice

leve

l

Time

x=−1.5x=−1x=0x=−0.3

Figure 5: (Left): The exercise boundary shifts upward as wealth increases from zero. (Right): Theexercise boundary is the lowest when wealth x = −0.3. As wealth further decreases from −0.3 to −1.5,the exercise boundary rises again. The parameters here are the same as in Figure 3.

0 0.2 0.4 0.6 0.8 11

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

Sto

ck p

rice

leve

l

Time

β=1β=0.25β=1/9

0 0.2 0.4 0.6 0.8 11

1.1

1.2

1.3

1.4

1.5

1.6

1.7

Sto

ck p

rice

leve

l

Time

α=4α=1.5α=0.5

Figure 6: (Left): A higher value of β leads to a higher exercise boundary (at initial wealth x = 0).(Right): A higher value of α shifts the exercise boundary upward (initial wealth x = 0). The parametershere are taken to be same as those in Figure 3, except for α and β specified in the figures above.

19

5 Marginal Forward Indifference Price of American Options

In this section, we introduce the marginal forward indifference price of American options.A related concept in the classical utility framework is the marginal utility price, which isuseful as an approximation for pricing a small number of claims. For completeness and theupcoming comparison with the forward analogue, we provide a brief review of the marginalutility price in the diffusion market.

5.1 Review of the Classical Marginal Utility Price

In traditional utility maximization, the investor’s risk aversion is modeled by a deterministicutility function Û(x) defined at time T . In the Itô diffusion market introduced in Section2, he dynamically invests in the money market account and stock S, and his wealth processfollows (3). The corresponding Merton portfolio optimization problem was given in (4).

Next, suppose that the investor decides to buy δ units of a European claim, each offeringpayoff CT ∈ FT . The marginal utility price is the per-unit price that the investor is willingto pay for an infinitesimal position (δ ≈ 0) in the claim. This problem has been studied indetail by Davis (1997, 2001), who shows by a formal small δ expansion that the investor’smarginal utility price at time t is given by

ĥt =IE

{

Û ′(X̂∗T )CT | Ft}

M ′t(Xt), t ∈ [0, T ], (63)

where X̂∗T is the optimal terminal wealth in the Merton value process Mt(Xt) given in (4).Kramkov and Sirbu (2006) adopt (63) as the definition of the marginal utility price, whichwe adapt to the case of American options.

Definition 10 The marginal utility price process (ht)0≤t≤T for an American option withpayoff process (gt)0≤t≤T is defined as

ht =ess supτ∈Tt,T IE

{

M ′τ (X̂∗τ ) gτ | Ft

}

M ′t(Xt). (64)

Among others, one important question is under what conditions does the marginal utilityprice become independent of the investor’s wealth. In the classical setting for options withoutearly exercise, wealth-independence of marginal utility prices is very rare. In fact, Kramkovand Sirbu (2006) show that only exponential and power utilities yield wealth-independentmarginal utility prices for any payoff and in any financial market. Herein, we complementtheir findings by giving a condition for wealth-independent marginal utility prices for Amer-ican options in the stochastic volatility model considered in Section 3.

Proposition 11 Suppose S and Y follow SDEs (20)-(21), and consider the value functionof the Merton problem

M(x, y, t) = supπ∈Zt,T

IE{

Û(XπT )|Xt = x, Yt = y}

, (65)

for (x, y, t) ∈R×R+×[0, T ]. If M satisfies

Mxy(x, y, t) = Mx(x, y, t)L(y, t), (66)

20

where L : R+×[0, T ] 7→ R is a C1(R+×[0, T ]) function, then the marginal utility price for anAmerican option with payoff function g(s, y, t) is given by

h(s, y, t) = supτ∈Tt,T

IEQϕ {g(Sτ , Yτ , τ) |St = s, Yt = y} , (67)

where Qϕ is an equivalent martingale measure defined by

dQϕ

dP= exp

(

−12

∫ T

0(λ(Ys)

2 + ϕ(Ys, s)2) ds−

∫ T

0λ(Ys) dWs +

∫ T

0ϕ(Ys, s) dŴs

)

, (68)

withϕ(y, t) =

√

1− ρ2c(y)L(y, t). (69)

A proof is provided in Appendix A.1.

Remark 12 If M admits the separation of variables

M(x, y, t) = k(x, t)w(y, t),

for some suitably differentiable functions k(x, t) and w(y, t), then assumption (66) is satisfied.Examples include the exponential or power utilities (also see Corollary 13 for the exponentialcase).

We observe from the Proposition 11 that the marginal utility price is wealth-independent.Nevertheless, the associated pricing measure Qϕ depends on the factor L, and therefore onthe utility function. As we discuss next, in the special example with exponential utility, themeasure Qϕ turns out to be the minimal entropy martingale measure and is independentof both wealth and utility function. Notice, however, that it is not independent of thehorizon choice. The following proposition is a straightforward generalization of this well-known result to the case of American options; see, for example, Becherer (2001) for the caseof claims without early exercise opportunities.

Proposition 13 Assume Û(x) = −e−γx. Then, the marginal utility price is given by

h(s, y, t) = supτ∈Tt,T

IEQE {g(Sτ , Yτ , τ) |St = s, Yt = y} , (70)

where QE is the minimal entropy martingale measure given in (51).

Proof. From the Merton solution (45) in Proposition 8, it is immediate that assumption(66) is satisfied, and that the factor L is given by

L(y, t) =1

(1− ρ2)fy(y, t)

f(y, t).

The pricing formula (70) and the associated measure then follow from Proposition 11.

Next, we consider a special situation where the coefficients µ(·) and σ(·) of S are constant.In this case, Y does not affect the dynamics of S, but plays the role of a non-traded asseton which the claim g is written. Therefore, the market is still incomplete. We show that inthis case every utility function will yield the same wealth-independent marginal utility price,given as an expectation under the minimal martingale measure (see Definition 34).

21

Proposition 14 If the drift coefficient µ(·) and volatility coefficient σ(·) of S are constant,then the marginal utility price is independent of wealth and utility, and it is given by

h(s, y, t) = supτ∈Tt,T

IEQ0 {g(Sτ , Yτ , τ) |St = s, Yt = y} , (71)

where Q0 is the minimal martingale measure.

A similar result for European claims has been shown by Kramkov and Sirbu (2006). Here,we study American options and provide an alternative proof in Appendix A.2.

5.2 The Marginal Forward Indifference Price Formula

Following the classical formulation, we introduce for our model the following definition of themarginal forward indifference price.

Definition 15 Let Ut(x) = u(x,At) (see Theorem 3) be the investor’s forward performanceprocess. The marginal forward indifference price process (p̃t)0≤t≤T for an American optionwith an Ft-adapted bounded payoff process (gt)0≤t≤T is defined as

p̃t =ess supτ∈Tt,T IE {ux (X∗τ , Aτ ) gτ | Ft}

ux(Xt, At), (72)

where At =∫ t

0 λ2sds and X

∗ evolves according to (15).

At first glance, the marginal forward indifference price in (72) might depend on theholder’s risk preferences and wealth. Nevertheless, as the next proposition shows, themarginal forward indifference price is independent of both of these elements, and is simplygiven as the expected discounted payoff under the minimal martingale measure, regardlessof the investor’s forward performance criterion.

Proposition 16 The marginal forward indifference price of an American option with payoffprocess (gt)0≤t≤T is given by

p̃t = ess supτ∈Tt,T

IEQ0{gτ | Ft}, (73)

where Q0 is the minimal martingale measure. Consequently, p̃t is independent of both theholder’s wealth and his forward performance.

Proof. Comparing (72) and (73), we observe that it is sufficient to show that, for anyτ ∈ Tt,T ,

ux (X∗τ , Aτ )

ux (X∗t , At)= exp

(

−12

∫ τ

t

λ2s ds−∫ τ

t

λs dWs

)

. (74)

Indeed, this leads to the desired measure change from the historical measure P to the minimalmartingale measure Q0.

Applying Itô’s formula to ux(X∗t , At) and using the SDE (15) for X

∗ gives

dux (X∗t , At)

= λ2t

(

uxt(X∗t , At) +R(X

∗t , At)uxx(X

∗t , At) +

R(X∗t , At)2

2uxxx(X

∗t , At)

)

dt

+ λtR(X∗t , At)uxx(X

∗t , At) dWt. (75)

22

Next, we show that the drift vanishes. Indeed, using that u(x, t) satisfies (11), it follows fromdifferentiation that

uxt = ux −u2xuxxx2u2xx

. (76)

We substitute (76) into the drift of SDE (75) and use the fact thatR(x, t) = −ux(x, t)/uxx(x, t).Then the drift term vanishes, yielding

dux (X∗t , At) = λtR(X

∗t , At)uxx(X

∗t , At) dWt

= λtux(X∗t , At) dWt.

This implies that the process (ux (X∗t , At))t≥0 is given by the stochastic exponential repre-

sentation in (74). We easily conclude.

Corollary 17 Assume that S and Y follow the SDEs (20)-(21). The marginal forwardindifference price for an American option with payoff function g(s, y, t) is given by:

p̃(s, y, t) = supτ∈Tt,T

IEQ0 {g(Sτ , Yτ , τ) |St = s, Yt = y} , (77)

where Q0 is the minimal martingale measure defined in (34). Consequently, the marginalforward indifference price is independent of the holder’s wealth and forward performancecriterion.

Proposition 16 illustrates a crucial feature of the forward indifference pricing mechanism.If we consider that, in a general Itô diffusion market, different investors adopt differentforward performances according to Theorem 3, then their marginal forward indifference pricesfor an American claim will necessarily be the same, regardless of their wealth and choicesof forward performance. However, in the traditional setting, the marginal utility price for ageneral utility function is both wealth and utility independent only in a correlated geometricBrownian motions model (see Kramkov and Sirbu (2006) and Proposition 14 above). Dueto the simple structure of this special market model, the marginal forward indifference priceand the marginal utility price have the same form (see Proposition 14 and Corollary 17) eventhough they are derived from very different mechanisms.

While the forward indifference valuation yields a nonlinear pricing rule, the marginalpricing rule is linear with respect to the quantity of options. In contrast to the nonlinearvariational inequalities in (28) and (57) in Markovian models such as those in Sections 3 and4 , the marginal forward utility price yields a linear free boundary problem with reduceddimension due to its wealth-independence. Therefore, the marginal formulation is moreamenable for computations, and can potentially be used as an efficient approximation forvaluing and hedging small derivatives position.

6 Conclusions and Extensions

In summary, we have discussed the forward indifference valuation for American options inan incomplete model with a stochastic factor. We have applied it to value American optionswith an exponential forward performance criterion and to model the early exercises of ESOs.The option holder’s optimal hedging and exercising strategies are found from solving theunderlying variational inequalities.

The forward indifference valuation mechanism is profoundly different from the one in theclassical approach. This is best illustrated in Section 3, in which the exponential forward

23

indifference price is expressed in terms of relative entropy minimization with respect to theminimal martingale measure, rather than the minimal entropy martingale measure as is thecase in the traditional setting. The minimal martingale measure also plays a crucial role asthe pricing measure for the marginal forward indifference price. In contrast to the classicalmarginal utility price, the marginal forward indifference price is independent of both theinvestor’s wealth and risk preferences.

Several major challenges and interesting problems remain for future investigation. Theseinclude the existence and regularity results for the variational inequalities associated with theoptimal forward performance and the forward indifference price. The nonlinearity of the vari-ational inequalities also requires the development of efficient numerical schemes. Moreover,even though we have focused on the valuation of American options, it is important to examineits impact in the host of other applications where traditional utility valuation has been used,for example, credit derivatives (Sircar and Zariphopoulou (2010); Jaimungal and Sigloch(2010)), volatility derivatives (Grasselli and Hurd (2008)), insurance products (Bayraktarand Ludkovski (2009)), and order book modeling (Avellaneda and Stoikov (2008)). In allof these, exponential utility is chosen for its convenient analytic properties. Forward perfor-mance provides a convenient tool to i) move away from exponential utility, and ii) removethe horizon dependence.

A Appendix

A.1 Proof of Proposition 11

First, we recall the associated HJB equation of the Merton value function, namely,

Mt =(ρσ(y)c(y)Mxy + λ(y)σ(y)Mx)

2

2σ(y)2Mxx− b(y)My −

c(y)2

2Myy, (78)

with M(x, y, T ) = Û(x). The optimal strategy is given in the feedback form

π̂∗t = −ρc(Yt)Mxy(X̂

∗t , Yt, t) + λ(Yt)Mx(X̂

∗t , Yt, t)

σ(Yt)Mxx(X̂∗t , Yt, t). (79)

Next, we look at the SDE that the process Mx

(

X̂∗t , Yt, t)

satisfies. By Ito’s formula, we get

dMx(X̂∗t , Yt, t) =

(

Mxt +π̂∗

2

t σ2(Yt)

2Mxxx + π̂

∗t σ(Yt)λ(Yt)Mxx

+ b(Yt)Mxy +c2(Yt)

2Mxyy + π̂

∗t ρσ(Yt)c(Yt)Mxxy

)

dt (80)

+ π̂∗t σ(Yt)Mxx dWt + c(Yt)Mxy

(

ρdWt +√

1− ρ2dŴt)

.

Next, we show that the drift above vanishes. Indeed, differentiating (78) with respect tox yields

Mxt = ρ2c(y)2

MxyMxxyMxx

− ρ2c(y)2

2

M2xyMxxx

M2xx+ λ(y)2Mx −

λ(y)2

2

M2xMxxxM2xx

+ ρλ(y)c(y)

(

Mxy +MxxyMxMxx

− MxyMxMxxxM2xx

)

− b(y)Mxy −c(y)2

2Mxyy,

which, combined with (80), eliminates the drift.

24

The SDE for Mx

(

X̂∗t , Yt, t)

simplifies to

dMx(X̂∗t , Yt, t) = (π̂

∗t σ(Yt)Mxx + ρc(Yt)Mxy) dWt +

√

1− ρ2c(Yt)Mxy dŴt= −Mx

(

λ(Yt)dWt −√

1− ρ2c(Yt)L(Yt, t) dŴt)

,

where the second equality follows from (79) and assumption (66).This implies that the process (Mx(X̂

∗t , Yt, t))t≥0 is given by the stochastic exponential

Mx(X̂∗τ , Yτ , τ)

Mx(X̂∗t , Yt, t)= exp

(

−12

∫ τ

t

(λ(Ys)2 + ϕ(Ys, s)

2) ds−∫ τ

t

λ(Ys) dWs +

∫ τ

t

ϕ(Ys, s) dŴs

)

,

(81)with ϕ(y, t) =

√

1− ρ2c(y)L(y, t). Comparing (63) and (67), we observe that (81) will leadto the necessary change of measure from P to Qϕ, and (67) follows.

A.2 Proof of Proposition 14

With σ(·) being constant, the investor’s wealth Xπ does not depend on Y . The HJB equation(78) then reduces to

Mt =λ2

2

M2xMxx

(82)

with M(x, T ) = Û(x). In turn, the optimal feedback strategy simplifies to

π̂∗t = −λ

σ

Mx(X̂∗t , t)

Mxx(X̂∗t , t). (83)

By Ito’s formula, we then get

dMx(X̂∗t , t) =

(

Mxt + π̂∗t µMxx +

π̂∗2

t σ2

2Mxxx

)

dt+ π̂∗t σMxx dWt

= −λMx(X̂∗t , t) dWt,

where the second equality follows from (83) and the fact that

Mxt = λ2Mx −

λ2

2

M2xMxxxM2xx

by differentiating PDE (82). As a result, (Mx(X̂∗t , t))t≥0 is given by the stochastic exponential

Mx(

X̂∗τ , τ)

Mx(

X̂∗t , t) = exp

(

−λ2

2(τ − t)− λ(Wτ −Wt)

)

, (84)

for τ ∈ Tt,T . Comparing (63) and (71), we observe that (84) yields the necessary change ofmeasure from P to the minimal martingale measure Q0 (see (34)), and formula (71) follows.

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