Factorization of European and American Option Prices

under Complete and Incomplete Markets�

Alfredo Ibáñezy

Quantitative Development, Capital Markets

Caja Madrid

This draft: January 6, 2007

Abstract

In a standard option-pricing model, with continuous-trading and di¤usion processes, this paper

shows that the price of one European�style option can be factorized into two intuitive components:

One robust, X0, which is priced by arbitrage, and a second, �0, which depends on a risk orthogonal

to the traded securities. This result implies the following: 1) In an incomplete market, these parts

represent the price of a hedging portfolio, which is unique, and a premium, which depends only on

the risk premiums associated with the residual risk, respectively. 2) In a complete market, it allows

factoring the contribution of the di¤erent sources of risk to the �nal option price. For example,

in a stochastic volatility model, we can quantify the impact on the option price of volatility risk

relative to market risk, �0 and X0, respectively. Hence, certain misspricings in option markets can

be directly related to the premium, �0. 3) Moreover, these results extend to American securities,

which have a third component �an additional early-exercise premium.

Key words: Option-pricing; Incomplete Markets; Dynamic Hedging; Option Price Factoriza-

tion; Stochastic Volatility; American Options.

JEL codes: G12, G13

�Previous versions of this paper were entitled, �Option-Pricing in Incomplete Markets: The Hedging Portfolio plus a

Risk Premium-Based Recursive Approach.�I am grateful to seminar participants at Universidad Carlos III and CEMFI

(Madrid), Nova (Lisboa), USC and UCLA (Los Angeles), the 14th Derivative Securities Conference (New York), the

Society for Computational Economics (Washington D.C.), and to Julio Cacho-Díaz, Ming Huang, Francis Longsta¤,

Eduardo Schwartz, and, especially, two anonymous referees for helpful comments. Any remaining errors are of course

my own. Part of this research was done when I was a¢ liated to Instituto Tecnológico Autónomo de México (ITAM)

and to Dep. de Economía de la Empresa, Universidad Carlos III de Madrid, and I was a Visiting Scholar in the Finance

Department, the Anderson School at UCLA.yTorre Caja Madrid. Po de la Castellana, 189, 3a planta. 28046 Madrid, Spain. Phone: + 34-91-4239296. Fax: +

34-91-4239735. e-mail: aibanero@cajamadrid.es

1

1 Introduction

In an incomplete market there is not a replicating portfolio for those securities that are not spanned,

and thus, one cannot apply the �law of one price�and obtain a unique solution. On the contrary,

there are upper and lower arbitrage bounds, which contain the non arbitrage prices (Merton (1973)).

One must make further assumptions to select one of these prices or to constrain the arbitrage bounds.1

This paper shows that the price of a European�style security can be decomposed into the price of

a hedging portfolio plus a premium. More precisely, for any non-spanned security, we do not provide

a speci�c price, but show that any arbitrage-free price C0 can be factored into two components:

C0 = X0 + �0. The �rst component, X0, is the price of a hedging portfolio and is unique, as in a

complete market, and, therefore, does not depend on C0. The second one, �0, is the premium, which

depends only on the stream of risk premiums associated with the residual risk, and is implicit in C0.

The price factorization implies that we can see an option price as the sum of two separate parts,

focusing on the premium �0. These results are simple and intuitive, but have not been explicitly

written out as we do here. Moreover, these results can be applied to option prices in a complete

market and extend to American-style securities as well. The decomposition holds in the standard

asset-pricing model, which assumes continuous-trading and di¤usion processes. These results follow

from the fact that the option payo¤ can be divided into two orthogonal components: one spanned

by the traded securities and a second orthogonal to them. Let us illustrate these results in a simple

one-period model, which also clari�es the assumptions underlying them.

A One�period Example: We price a security C1. There are two traded securities, a riskless

bond and a risky asset with prices 1 and x0 = 1 and payo¤s R > 0 and x1, respectively. We assume

that C1 � N (�c; �2c) and x1 � N (�x; �2x) are Gaussian (� is the mean and �2 is the variance), sothat minimizing the variance is the optimal and unique hedging criterion. Let �, with j�j < 1, be thecorrelation between C1 and x1. Then, the beta portfolio h�1 = �

�1x �c� minimizes the variance of Y1 =

h0R + h1x1 � C1, where h�0 = 1R (�c � h

�1�x). The residual risk veri�es that Y1 � N (0;

�1� �2

��2c)

and E [Y1x1] = 0. We denote by X0 = h�0 + h�1 the price of this hedging portfolio.

Let C0 be the price of the security C1. First, C1 � Y1 is the spanned payo¤ of C1, since thehedging criterion is unique, and X0 is its corresponding price. Therefore, C0�X0 is the risk premiumassociated with the unspanned payo¤ or residual risk Y1, and holds for every price C0. Second, it is

a separate matter to determine the magnitude of this premium or how it is invested (i.e., C0 �X0and �, respectively), where the �nal residual risk is given by Y1 + (C0 �X0) (R+ �(x1 �R)). �

1 Including the use of utility maximization arguments (Merton (1976) and Rubinstein (1976)); to consider prices of

risk associated with non-traded state variables (Heston (1993)); to compute an optimal hedging portfolio, whose price

is the desired incomplete market price (Merton (1998)); to use a risk/reward criterion, such as the gains-to-losses ratio

(Bernardo and Olivier (2000)) or the Sharpe ratio (Cochrane and Saá-Requejo (2000)). See also Carr et al. (2001) and

Cerný (2003). See Broadie and Detemple (2004, 1151) for a recent survey and more references to the literature.

2

The arguments are similar in a dynamic model. Consider, for example, Heston�s (1993) model,

which depends on a non-traded variable, stochastic volatility. Assume that the only traded securities

are the market portfolio and a bank account. Hence, the market is incomplete. By using dynamic

spanning, it is possible to decompose the option payo¤ into two components: One that depends

on market risk only, and that, therefore, can be hedged and priced by arbitrage. And a second

component, which is orthogonal to the market and depends on stochastic volatility. We show that

the price of this second component depends only on the stream of volatility risk premiums.

The option price factorization, C0 = X0 + �0, has interesting applications for option�pricing,

which we describe next. First, it is easy to de�ne an upper and a lower bound in an incomplete

market, because the price of the hedging portfolio is the same for both bounds. For example, if one

considers positive (negative) risk premiums for the upper (lower) bound, the term �0 > 0 (�0 < 0).

Therefore, it is also easy to constrain the arbitrage bounds by constraining the risk premiums.

Second, the factorization is derived by using the risk-neutral measure that assigns zero risk

premiums to the orthogonal risk, and applying Feynman-Kac theorem. Under this measure, the

discounted upper (lower) bound is a super-martingale (sub-martingale) if the risk premiums are

positive (negative). The discounted price of the hedging portfolio is the martingale component.

These results are related to Ross (1978) and Harrison and Kreps (1979), but in incomplete markets.

As another application, consider the problem of pricing a portfolio of N securities. The price of

this portfolio and the sum of the N individual prices can only di¤er in an incomplete market. The

factorization implies that this di¤erence is due to the valuation of the residual risk, since the price

of the hedging portfolio is the same in both cases. If there is some diversi�cation in the portfolio of

the N securities (e.g., from having o¤setting positions), this portfolio can be cheaper.

The decomposition can be applied to a complete market: to factor the contribution of the di¤erent

sources of risk to the �nal option price. For example, in a stochastic volatility model, we can quantify

the price impact of stochastic volatility relative to market risk, �0 and X0, respectively. Note that

�0 can be explicitly computed as the di¤erence of two option prices; i.e., C0 �X0. Accordingly, thepercent option premium �0=X0 is a simple measure of mispricing (see Ibáñez (2006)).

The decomposition also applies to American�style securities. Extending previous results of com-

plete markets (e.g., Kim (1990), Carr, Jarrow, and Myneni (1992), and Broadie and Detemple (2004)),

an American option is divided into three components: a risk premium, the price of the hedging port-

folio of the equivalent European option, and an extra early�exercise premium. We indeed show two

di¤erent factorizations. These are novel results as, unlike European options, American options in

incomplete markets have received little attention.

As noted above, the premium, �0, depends on the risk premiums associated with the residual

risk. However, di¤ering from a complete market, these risk premiums do not need to depend on

�prices of risk� to avoid arbitrage. This is a �exibility which could be used to better �t volatility

3

smiles.2 If one associates a price of risk with every non-traded variable, the approach reduces to risk-

neutral pricing. One can show that Merton (1998) and Cochrane and Saá-Requejo (2000) present

two approaches where the risk premiums are equal to zero and are proportional to the residual risk

volatility, respectively.3 This paper is related to Naik and Lee (1990) who also focus on and quantify

the extra option (equilibrium) price in a model where volatility is constant but with jump risk.

The rest of the paper is as follows. Sections 2 and 3 give the price decomposition in discrete-time

one-period and multi-period models, respectively. Section 4 considers the continuous-time model for

di¤usion processes. Section 5 studies an example of Basis Risk. Section 6 concludes.

2 The One-period Model

Assume a one-period model. Let t and t+1 be the �rst and the second periods, respectively. LetK be

the number of states, = f!1; !2; :::; !Kg the state space, and Pt the true probability measure, withPt(!) > 0 for all ! 2 . There exists a riskfree asset with prices S0t = 1 and S0t+1 = 1+r (where r > 0is the one-period riskless rate), and N risky assets with initial prices St = fS1t ; S2t ; :::; SNt g and �nalprices St+1 = fS1t+1; S2t+1; :::; SNt+1g de�ned on . Assume that there are no arbitrage opportunities.

Let H be the portfolio�s or trading strategy�s space. In particular, there are no constraints on

this space (i.e., H = RN+1). Let ht+1 = (h0t+1; h1t+1; :::; hNt+1); for ht+1 2 H and ht+1 chosen at time

t; be a (hedging) portfolio with value process Xh = fXht ; X

ht+1g; i.e., Xh

t = h0t+1+

PNn=1 h

nt+1S

nt and

Xht+1 = h

0t+1(1 + r) +

PNn=1 h

nt+1S

nt+1. Assume that this market is incomplete (i.e., K > N + 1) and

that a contingent claim with payo¤ Ct+1 is not replicable. That is, there does not exist a portfolio

ht+1 such that Xht+1(!) = Ct+1(!) for every ! 2 . Let C�t and C+t be the two arbitrage bounds of

this security, which solve two linear programs (see Ingersoll (1987) or Pliska (1997)). Then, its price

Ct must satisfy that C�t < Ct < C+t to avoid arbitrage opportunities.

2.1 A Hedging Portfolio plus a Premium Associated with the Residual Risk

Let Y ht+1, the hedging error or residual risk produced by the portfolio ht+1, be de�ned as

Y ht+1 = Xht+1 � Ct+1: (1)

Let bht+1 be an optimum portfolio associated with a hedging criterion f(Y ht+1); i.e.,

bht+1 = arg minfht+1g

f(Y ht+1); (2)

and Xbht its price. We do not specify the function f().

2E.g., Du¤ee (2002) and Duarte (2004) specify �exible prices of risk for (complete markets) term structure models.3Merton (1998, 333) argues that the risk premiums should be zero, as the residual risk is orthogonal with the traded

assets and hence with the equilibrium market portfolio. I thank Robert L. McDonald for pointing out this reference.

4

It is convenient to assume (as we do in the next subsection) that bht+1 satis�es C�t < Xbht < C

+t .

For example, a hedging portfolio with a zero expected hedging error under the objective probability

P (i.e., EPt�Y ht+1

�= 0, where EPt [:] denotes the expectation under P) satis�es this constraint since

this portfolio has both positive and negative errors.

Let yt be a risk premium associated with the residual risk Y bht+1. Then, yt does not introducearbitrage opportunities if C�t �X

bht < yt < C

+t �X

bht . Equivalently, C

�t < X

bht + yt < C

+t , and thus,

Xbht + yt is an arbitrage free price. In particular, yt can be zero if this residual risk is not priced.

Therefore, we assume that C�t < Xbht + yt < C

+t . This paper de�nes the incomplete market price

Ct as the price of the hedging portfolio, Xbht ; plus the risk premium, yt; i.e.,

Ct = Xbht + yt: (3)

First, Xbht is, intuitively, the application of the law of one price if we assume that the residual risk is

zero. Second, we add a risk premium yt to compensate the residual risk Ybht+1.

We assume that the risk premium yt is invested in riskless bonds and rede�ne the hedging strategy

as bh0+yt bonds and bhn risky assets for n = 1; 2; :::; N: Therefore, the total risk assumed by the writerof this security is

Xbht+1 � Ct+1 + yt(1 + r) = Y

bht+1 + yt(1 + r): (4)

Upper and lower bounds In an incomplete market, one is interested in de�ning two bounds,

an upper (lower) bound, Cst (Clt), obtained when hedging the short (long) position; i.e., �Ct+1

(Ct+1). Moreover, these bounds should satisfy Cst � C lt to make economic sense. Consider two

optimal hedging portfolios, bht+1(s) and bht+1(l), and two risk premiums, yst and ylt, associated withthe hedging errors Y

bh(s)t+1 =

�Xbh(s)t+1 � Ct+1

�and Y

bh(l)t+1 = �

�Xbh(l)t+1 � Ct+1

�for the short and the long

position�s, respectively. Then these two bounds can be de�ned as in equation (3); i.e.,

Cst = Xbh(s)t + yst and C

lt = X

bh(l)t � ylt: (5)

In particular, Xbh(s)t � X

bh(l)t and yst � �ylt are su¢ cient conditions for Cst � C lt. For example,bht+1 = bht+1(s) = bht+1(l) and yt = yst = ylt � 0, i.e., the same portfolio and the same nonnegative

risk premium, imply that Cst = Xbht + yt � C lt = X

bht � yt.

2.2 A Risk-Neutral Formulation: The Price Decomposition

In standard frictionless markets, for both complete and incomplete markets, an arbitrage free price

can be expressed as an expectation under a risk-neutral probability measure (henceforth, RNP mea-

sure). By using RNP measures, we are going to derive a related but novel result.

Let Qt be a RNP measure, and EQt [:] be the conditional expectation operator. Qt satis�es

Snt = EQt

�Snt+11 + r

�; n = 1; 2; :::; N: (6)

5

Let us recall the implications of a RNP measure, Qt > 0. The existence of Qt is equivalent to

nonarbitrage and the uniqueness of Qt is equivalent to market completeness. Let C = fCt; Ct+1g bethe value process of an arbitrary security. If Ct = E

Qt

hCt+11+r

i, Ct is an arbitrage-free price. Therefore,

Qt is a tool that allows us to compute the price Ct as a simple risk-neutral expectation. This last

point is the one that is important in the present nonarbitrage, incomplete market context.

Recall that portfolio bht+1 satis�es C�t < Xbht < C

+t , then there is a RNP measure bQt such that

Xbht = E

bQt

�Ct+11 + r

�; (7)

where the notation bQt highlights the dependence on portfolio bh. That is, bQt allows us to computethe price of the hedging portfolio Xbh

t by the risk-neutral expectation of the discounted payo¤ Ct+1:

Consequently, from equation (3), the incomplete market price of Ct can also be expressed as

Ct = EbQt

�Ct+11 + r

�+ yt: (8)

In sum, from equations (5) and (7),

Ct = Xbht + ayt = E

bQt

�Ct+11 + r

�+ ayt; (9)

where a = +1 (�1) for the short (long) position and upper (lower) bound.On the other hand, if yt 6= 0, there exists a di¤erent RNP measure bQyt such that

Ct = Xbht + yt = E

bQyt

�Ct+11 + r

�; (10)

which, as it is well known, can be used for pricing and to prove that Ct is arbitrage free.

Remark 1. The de�nition of the price of an arbitrary security in incomplete markets, given in

equation (3), where this price is equal to the price of a hedging portfolio plus a risk premium, is the

main result of the one-period model. Then, equation (8) and the measure bQt will allow us to extendthis important result to the case of multiperiod markets.

Remark 2. Although we have assumed a frictionless market, the de�nition of equation (3) is

independent of market frictions such as portfolio constraints or transaction costs. For equation (7)

to hold, it is necessary to �nd a probability measure which allows us to compute the price Xbht as

the discounted expectation of Ct+1. For a frictionless market, bQt is a RNP measure. For a frictionmarket, this problem is left for future research.

3 The Multiperiod Model

Consider a discrete-time multiperiod model with initial time 0, �nal time T , and M trading dates

such that t = 0; 1; :::;M�1; and �t = TM . This multiperiod model is de�ned over a probability space

6

(;F ; P; fFtg), with = f!1; !2; :::; !Kg �nite. The stochastic processes Snt are adapted and thehedging strategies hnt+1 are predictable with regard to the �ltration Ft, for t = 0; 1; :::;M and n =

0; 1; :::; N . There is a �bank account�with value process S0 = fS00 ; S01 ; :::; S0Mg = f1; er�t; :::; erT g.For simplicity, the short rate r is constant, but we can consider that r is predictable (i.e., rt+1 is

Ft�measurable). We assume that the model is arbitrage free and incomplete. The objective is toprice a contingent claim C0; whose payo¤ CM occurs in the last period and is not replicable.

Let h = fh1; h2; :::; hMg be a self-�nancing dynamic portfolio with value process Xh = fXh0 ; X

h1 ;

:::; XhMg, where Xh

0 =PNn=0 h

n1S

n0 and X

ht =

PNn=0 h

nt S

nt for t = 1; 2; :::;M: The asterisk denotes

discounted values. It is well known that a portfolio h is self-�nancing if it holds that

Xh�M = Xh�

0 +

M�1Xt=0

�Xh�t+1; (11)

where Xh�t = e�rt�tXh

t , �Xh�t+1 =

PNn=1 h

nt+1�S

n�t+1, and �S

n�t+1 = e

�r(t+1)�t �Snt+1 � er�tSnt � is thediscounted gain process for every risky asset n = 1; 2; :::; N . See Pliska (1997, chapter 3) for details.

3.1 A Hedging Portfolio plus a Risk Premium-Based Approach

For a self-�nancing portfolio h, the hedging error is de�ned by Y hT = a�XhM � CM

�, where a = +1

(a = �1) is the short (long) position. We rewrite this hedging error,

Y h�T = a�Xh�M � C�M

�= a

�Xh�0 � C�0

�+M�1Xt=0

a��Xh�

t+1 ��C�t+1�=

MXt=0

�Zh�t ; (12)

where Y h�T = e�rTY hT and C = fC0; C1; :::; CMg is a Ft�adapted stochastic process. Moreover, wede�ne �C�t = C

�t � C�t�1; C�t = e�rt�tCt, and �Zh�t+1 = e�r(t+1)�t�Zht+1 = a

��Xh�

t+1 ��C�t+1�for

t = 0; 1; :::;M � 1 and �Zh0 = a�Xh0 � C0

�. Consequently, the total hedging error, Y h�T ; is equal to

the sum of the one-period replication errors �Zh�t , t = 0; 1; :::;M:

The Option Price De�nition As in the one-period model (see equation (3)), we de�ne the

option price as the price of a hedging portfolio plus an extra risk premium. That is, for every period

t = 0; 1; :::;M � 1, we divide the option price into two parts,

Ct = Xbht+1t + ayt�t; (13)

and assume C�t < Xbht+1t < C+t (i.e., C

�t � X

bht+1t � C+t if C�t = C+t ) and C�t < X

bht+1t +ayt�t < C

+t .

We can understand yt�t as a risk premium associated with �Zbht+1. This is because of the factthat if C�t = C

+t , then C

�t = X

bht = C

+t and therefore yt = 0. In this case, the option value can be

replicated at time t+ 1 (i.e., there is bht+1 such that Xbht+1t+1 = Ct+1, and then, �Z

bht+1 = 0).

Option prices are well-de�ned since CM is known at maturity and since Ct is Ft�measurable.These prices (i.e., the value process C = fC0; C1; :::; CM�1g) are arbitrage free if and only if everyone-period price is arbitrage free (equivalently, it does exist a RNP measure, see Pliska (1997)).

7

The Multiperiod Hedging Error The hedging portfolio bh, which appears in the option pricede�nition (equation (13)), is not self-�nancing if either the risk premium yt+1 6= 0 or the hedging

error �Zbht+1 6= 0; since bht+1 and bht+2 are chosen in two independent steps. Accordingly, we introducea new self-�nancing portfolio, denoted by eh and with value process Xeh, and which �nances/investsthe hedging errors and risk premiums at the riskless rate. We also rewrite the hedging errors.

The new notation Xbht+1t (in equation (13), instead of Xbh

t =PNn=0

bhnt Snt ) is to distinguish betweenXbht+1t+1 =

PNn=0

bhnt+1Snt+1 and Xbht+2t+1 =

PNn=0

bhnt+2Snt+1; and applies only to this non self-�nancingportfolio. Recall that Ct = X

bht+1t + ayt�t and Xh�

t = e�rt�tXht . We rede�ne the hedging error as

�Ybht+1t+1 = a

�Xbht+1t+1 � Ct+1

�for t = 0; 1; :::;M � 1, where a�Y

bht+1t+1 =

�Xbht+1t+1 � Ct+1

�since a2 = 1.

That is, at the initial time t = 0;

Xeh�0 = C�0 = X

bh1�0 + ay�0�t, andehn1 = bhn1 for n = 1; 2; :::; N and eh01 = bh01 + ay�0�t:

At time t = 1;

Xeh�1 = X

bh1�1 + ay�0�t = X

bh1�1 + ay�0�t� C�1 +

�Xbh2�1 + ay�1�t

�= X

bh2�1 + a

��Y

bh�1 + y�0�t+ y

�1�t

�, and

ehn2 = bhn2 for n = 1; 2; :::; N and eh02 = bh02 + a��Y bh�1 + y�0�t+ y�1�t

�:

Then, for any time t = 0; 1; :::;M � 1,

Xeh�t = X

bht+1�t +

t�1Xi=0

a��Y

bh�i+1 + y

�i�t

�+ ay�t�t, and (14)

ehnt+1 = bhnt+1 for n = 1; 2; :::; N and eh0t+1 = bh0t+1 + t�1Xi=0

a��Y

bh�i+1 + y

�i�t

�+ ay�t�t: (15)

Xeh�M = X

bhM�M +

M�2Xt=0

a��Y

bh�t+1 + y

�t�t

�+ay�M�1�t�C�M+C�M = C�M+

M�1Xt=0

a��Y

bh�t+1 + y

�t�t

�: (16)

The multiperiod hedging error of the self-�nancing strategy eh is equal to the sum of the one-period

hedging errors plus the associated risk premiums �nanced/invested at the riskless rate r; i.e.,

YehT = a

�XehM � CM

�= a

M�1Xt=0

a��Y

bh�t+1 + y

�t�t

�erT =

M�1Xt=0

��Y

bh�t+1 + y

�t�t

�erT : (17)

What follows is a key result of the paper and is mathematically simple. Therefore, we want to

properly motivate this result. The procedure described above allows option�pricing by providing a

hedging portfolio and the risk premiums (i.e., equation (13)), period by period and state by state.

Imagine a sensible hedging criterion and fair risk premiums, bh and y, respectively. We may wonderwhat the initial option price C0 looks like and/or if has some structure. We show that C0 can be

factorized into two intuitive components, the price of a hedging portfolio, which is unique, plus a

term which depends only on the stream of risk premiums associated with the residual risk.

8

3.2 A Risk-Neutral Formulation: The Price Decomposition

Similar to equation (8) in the one-period model, the option price can be written as (where bh is theprevious non-self-�nancing portfolio in equation (13) and X

bht+1t = E

bQt

�e�r�tCt+1

�),

Ct = EbQt

�Ct+1er�t

�+ ayt�t; (18)

t = 0; 1; :::;M � 1; and a = +1 (a = �1) de�nes the upper (lower) price bound.Then, at maturity, the price is equal to CM : One period before maturity M � 1,

CM�1 = EbQM�1

�CMer�t

�+ ayM�1�t: (19)

Two periods before maturity M � 2,

CM�2 = EbQM�2

�CM�1er�t

�+ ayM�2�t = E

bQM�2

�CMer2�t

�+ a

�yM�2�t+ E

bQM�2

hyM�1er�t

�ti�; (20)

by using the law of the iterated expectation�i.e., E

bQM�2 [CM ] = E

bQM�2

hEbQM�1 [CM ]

i�. And, recur-

sively, at the initial period 0; we have the following result.

Proposition 1 Assume equation (18). Then, C0 is as follows,

C0 = EbQ0

�CMerT

�+ aE

bQ0

"M�1Xt=0

ytert�t

�t

#: � (21)

Like the one-period model, the option price C0 is divided into two parts: EbQ0

�e�rTCM

�and

aEbQ0

hPM�1t=0 e�rt�tyt�t

i, which depends only on the risk premiums y. To see the term E

bQ0

�e�rTCM

�we need an additional assumption. Denote by bht+1(y = 0) the hedging portfolio if every risk premiumis equal to zero in equation (13) (i.e., not only yt = 0 but also yt+1 = yt+2 = ::: = yM�1 = 0) and bybQ(y=0)t the corresponding RNP measure, for every period t. Then, E

bQ(y=0)0

hCMerT

i= X

bh(y=0)0 follows

from equations (13) and (21).

The assumption EbQ0

hCMerT

i= E

bQ(y=0)0

hCMerT

i(e.g., if bQ = bQ(y=0)) implies that E bQ

0

hCMerT

i= X

bh(y=0)0 .

That is, EbQ0

hCMerT

iis equal to the price of the hedging portfolio which assumes that all risk premiums

are equal to zero zero. Therefore, if �EbQ0

hCMerT

i= E

bQ(y=0)0

hCMerT

i,�

C0 = Xbh(y=0)0 + aE

bQ0

"M�1Xt=0

ytert�t

�t

#= E

bQ0

�CMerT

�+ aE

bQ0

"M�1Xt=0

ytert�t

�t

#; (22)

which is the multiperiod extension of equation (9) in the one-period model.

Remark 3. Proposition 1 holds for any given arbitrage-free price process C = fC0; C1; :::; CMgand any risk�neutral measure bQ, where then yt is de�ned through equation (18). And this result doesnot depend on whether the risk premiums and hedging errors are �nanced/invested at the riskless

9

rate or not. To give economic content to Proposition 1, we de�ne Ct as the price of a hedging portfolio

plus a risk premium in equation (13).

Remark 4. Proposition 1 and equation (22) give a novel decomposition of option prices in an

incomplete market. If �EbQ0

hCMerT

i= E

bQ(y=0)0

hCMerT

i,� the price of an European�style security, C0,

is equal to the price of a hedging portfolio plus a multiperiod premium. In the next section, we

show that this assumption holds in the continuous-time model for di¤usion processes. We also show

that the option payo¤ can be factorized into two orthogonal components, the one spanned by the

traded securities and a second orthogonal to them. Thus, equation (22) is not only a mathematical

decomposition but economic meaningful.

As equation (10) in the one-period model, there exists a di¤erent RNP measure bQy such thatCt = X

bht + ayt = E

bQyt

�e�r�tCt+1

�for t = 0; 1; :::;M � 1 (since C�t < X

bht + ayt < C

+t ). Therefore,

from the law of the iterated expectation,

C0 = EbQy0

�CMerT

�: (23)

Consequently, the multiperiod risk premium can be computed as, from equations (22) and (23),4

aEbQ0

"M�1Xt=0

ytert�t

�t

#= E

bQy0

�CMerT

�� E bQ

0

�CMerT

�: (24)

Finally, if all the risk premiums are zero because, for example, the market is complete (and bh isthe replicating portfolio), then in all the equations above we obtain the very well-known result,

if y0 = y1 = ::: = yM�1 = 0, then C0 = EbQ0

�CMerT

�= X

bh0 : (25)

3.3 The Decomposition of American�style Securities in Incomplete Markets

American�style securities in incomplete markets have received little attention (an exception is Detem-

ple and Sundaresan (1999)). Like European-style securities, we do not give a speci�c price but show

a factorization of the American option price into three components: a multiperiod risk premium,

the price of the hedging portfolio of the equivalent European option, and an extra early-exercise

premium. We, indeed, provide two di¤erent factorizations.

Let It = I (St; E) be the intrinsic payo¤ where E is the strike price. Let C and CA denote the

price of the European and the American option, respectively. At maturity, CM = CAM = I (SM ; E).

To relate the American option to its equivalent European option, we consider the two RNP mea-

sures associated with the equivalent European option ( bQ and bQy). Then, the price of the American4For clarity, we use three sets of RNP measures, bQ; bQy; and bQ(y=0)t . If y 6= 0, bQ 6= bQy. If y 6= 0, bQ and bQ(y=0)t can

be related; e.g., they are related in continuous time. If y = 0, then bQ; bQy; and bQ(y=0)t are the same.

10

option CA veri�es the two following equations, for t = 0; 1; :::;M � 1,

CAt = ayAt �t+ EbQt

�e�r�tmax

�It+1; C

At+1

�= ayAt �t+ E

bQt

�e�r�t

�"+t+1 + C

At+1

��; (26)

CAt = EbQyt

�e�r�tmax

�It+1; C

At+1

�= E

bQyt

�e�r�t

�"+t+1 + C

At+1

��; (27)

where max�It+1; C

At+1

= fIt+1 � CAt+1g+ + CAt+1 = "+t+1 + CAt+1 and "t+1 = It+1 � CAt+1.

Equation (26) shows that the American option provides two premiums from t to t + 1, a risk

premium and an early exercise premium. From equations (26) and (27), yAt �t veri�es that

ayAt �t = EbQyt

�e�r�tmax

�It+1; C

At+1

�� E bQ

t

�e�r�tmax

�It+1; C

At+1

�: (28)

Factorization 1 From equation (26), we prove that the American option veri�es the following,

CA0 = ayA0 �t+ EbQ0

�e�r�t"+1

�+ E

bQ0

�e�r�tCA1

�= ayA0 �t+ E

bQ0

�e�r�t"+1

�+ E

bQ0

he�r�tayA1 �t+ E

bQ1

�e�r2�t"+2

�+ E

bQ1

�e�r2�tCA2

�i= aE

bQ0

�yA0 �t+ e

�r�tyA1 �t�+ E

bQ0

�e�r�t"+1 + e

�r2�t"+2�+ E

bQ0

�e�r2�tCA2

�= ::: = E

bQ0

"MXt=1

"+tert�t

#| {z }

(A1)

+ aEbQ0

"M�1Xt=0

yAtert�t

�t

#| {z }

(B1)

+EbQ0

�CMerT

�| {z }

(C1)

(29)

The American option is factorized into three parts: An early-exercise premium (A1), a risk premium

(B1), and (C1) which is related to the price of the hedging portfolio of an equivalent European option.

To link pricing with hedging, we de�ne

XbhAt+1t = E

bQt

�e�r�tmax

�It+1; C

At+1

�; (30)

where bhAt+1 is the hedging portfolio of the payo¤max�I (St+1; E) ; CAt+1; i.e., the hedging error is�Y

bhAt+1t+1 = a

�XbhAt+1t+1 �max

�It+1; C

At+1

�: (31)

Consequently (extending equation (17)), the total residual risk is given byPb��1t=0

��Y

bhA�t+1 + y

A�t �t

�erb� ,

where b� 2 f1; 2; :::;Mg is the optimal stopping-time de�ned by the �rst �(t) such that I (S� ; E) � CA� .Factorization 2 From equation (27), we show the following factorization,

CA0 = EbQy0

�e�r�t"+1

�+ E

bQy0

�e�r�tCA1

�;

= EbQy0

�e�r�t"+1

�+ E

bQy0

�EQ

bh;y1

�e�r2�t"+2

�+ E

bQy1

�e�r2�tCA2

��= E

bQy0

�e�r�t"+1 + e

�r2�t"+2�+ E

bQy0

�e�r2�tCA2

�= ::: = E

bQy0

"MXt=1

"+tert�t

#+ E

bQy0

�CMerT

�: (32)

11

The American option is factorized into two parts, and early exercise premium plus the price of an

equivalent European option. Next, the European option is equal to a multiperiod risk premium

(which depends on y instead of yA) plus the price of a hedging portfolio, from Proposition 1,

EbQy0

�CMerT

�= aE

bQ0

"M�1Xt=0

ytert�t

�t

#| {z }

(B2)

+EbQ0

�CMerT

�| {z }

(C2)

; (33)

and therefore,

CA0 = EbQy0

"MXt=1

"+tert�t

#| {z }

(A2)

+ aEbQ0

"M�1Xt=0

ytert�t

�t

#| {z }

(B2)

+EbQ0

�CMerT

�| {z }

(C2)

: (34)

Remark 5. 1) The terms (C1) and (C2) are the same; however, (A1) and (B1) are di¤erent from

(A2) and (B2), respectively. 2) The term (B2) can be computed as the di¤erence of two European

options, from equation (33); however, (B1) cannot be computed as the di¤erence of two American

options (under bQy and bQ, respectively), since the early exercise premium ("+ in (A1)) changes and

is not the same in both cases. 3) Like for European options, if �EbQ0

hCMerT

i= E

bQ(y=0)0

hCMerT

i, then (C1)

is equal to the price of the hedging portfolio which assumes all zero risk premiums. Finally, 4) if it

is never optimal to exercise (i.e., "+ = 0 and yA = y), then (A1) = (A2) = 0 and (B1) = (B2).

The factorization in equation (29) depends on the true risk premiums yA, but equation (34) de-

pends on the risk premiums associated with the equivalent European option. Both decompositions

can be used to reduce the valuation of American options to the European case. These two factoriza-

tions are also obtained in continuous-time in Section 4, and we explicitly obtain the early-exercise

and risk premiums of an American put on a non-traded security in Section 5.

4 The Continuous Time Model

Assume a vector of J state variables, S(t) = (S1(t); S2(t); :::; SJ(t)), which follows the following

di¤usion process under the objective probability measure, P,

dS(t) = � (t; S(t)) dt+�(t; S(t)) dzt; (35)

where � is a (J � 1) vector, � is a (J �K) matrix, and z is a (K � 1) vector of independent Wienerprocesses. We assume that � (t; S(t)) and � (t; S(t)) satisfy growth and regularity conditions such

that the process dS is well de�ned and has a unique solution. We remark that this continuous-time

model requires technical conditions similar to those of complete markets, we refer to Du¢ e (2001).

Let r(t) be the instantaneous short rate, and r is constant to save notation. We assume that

only the �rst N state variables S1(t); S2(t); :::; SN (t), with 0 � N � K, are tradable. For example,SN+1(t); SN+2(t); :::; SJ(t) correspond with illiquid assets, stochastic volatility, etc.

12

We consider the partition of the volatility matrix �0 = [A0 B0] ; where A and B contain the �rst

N and the last J � N rows of �, respectively. We assume that the rank of the matrix A is equal

to N (almost sure), i.e., there are no redundant tradable assets. This implies that the model is

arbitrage free, and hence, there exist one (multiple if N < K) risk-neutral probability measures for

the N tradable assets (under technical conditions, see Du¢ e (2001)). The market is incomplete if

N < K. Two special cases are N = 0, then the only hedging instrument is the risk-free asset (we

de�ne � = B). And N = K, the market is complete and we obtain the arbitrage-free price (and the

matrix A is invertible). We assume no portfolio constraints.

4.1 The Hedging Strategy

Let C(t; S(t)) and Xh(S(t)) be the price of a derivative security and the price of a hedging portfolio h,

respectively, where Xh(S(t)) =PNn=1 hn(t)Sn(t), h0(t) = 0, and C(T; S(T )) is the European option

payo¤ at maturity (with the notation slightly changed). By Ito�s lemma, dC and dXh satisfy

dC = �cdt+ C0S�dz; (36)

dXh = �hdt+ h0Adz; (37)

where

�c = Ct +1

2

JXi=1

JXj=1

CSS(i;j)

KXk=1

�i;k�j;k

!+ �0CS and �h = �

0(1:N)h; (38)

and h(t) = (h1(t); h2(t); :::; hN (t)). CS is the (K � 1) vector of �rst derivatives, CSS is the (K �K)matrix of second derivatives, and we have suppressed the dependence of all variables on t and S(t).

De�ne the hedging error

dY ht = a�h0A� C 0S�

�dzt = a

�h0A� C 0S

�A0 B0

�0�dzt = a

�(h� CS(1:N))0A� C 0S(N+1:J)B

�dzt:

(39)

Because dC and dX follow di¤usion processes and because of the fact that continuous trading is

allowed, the in�nitesimal hedging errors are (conditionally) normally distributed and consequently

the appropriate, and unique, hedging criterion is to minimize the instantaneous variance.5 Therefore,

the hedging criterion f(dYt) is given by minimizing

f(dY ht ) =1

dtEPt

hdY ht

i2= (h� CS(1:N))0A� C 0S(N+1:J)B 2 ; (40)

where k:k2 is the Euclidean norm. Let denote g = h�CS(1:N): Then, the N orthogonality conditions

(i.e., EPthdSndY

bht

i= 0; n = 1; 2; :::; N) for this problem imply that

bg = �AA0��1AB0CS(N+1:J) and bh = CS(1:N) + bg; (41)

5Another approach in incomplete markets, which focuses more on hedging than pricing, studies optimal dynamic

portfolios. See, e.g., Du¢ e and Richardson (1991), Schweizer (1992), Heath et al. (2001), and Bertsimas et al. (2001).

13

where bh is the optimal minimum variance portfolio and the matrix AA0 is invertible since the rank

of A is equal to N . Then, dXbh = �bhdt+bh0Adz is the dynamics of the optimal hedging portfolio, anddY

bht = a

�bg0A� C 0S(N+1:J)B� dzt = aC 0S(N+1:J)B �A0 �AA0��1A� I� dzt (42)

is the remaining residual risk.

The residual risk, dY bht , depends on three terms: the option Deltas with regard to the non-tradedstate variables, CS(N+1:J); the volatility matrix of these non-traded variables, B; and the matrix

(A0 (AA0)�1A � I) which is related to the market incompleteness (if the market is complete, A is

invertible and this term is zero). Then, B(A0 (AA0)�1A� I)dzt is the risk which cannot be spannedby S(1:N).6 The Deltas, CS(N+1:J), can be used for risk management. In particular, dY

bht can be equal

to zero if C(t; S(t)) is replicable at time t in state S(t).

By denoting �Yk =hC 0S(N+1:J)B

�A0 (AA0)�1A� I

�i(k), k = 1; 2; ::;K, the instantaneous volatil-

ity of the residual risk is given byqPK

k=1

��Yk�2.

4.2 A Partial Di¤erential Equation: the Law of One Price plus a Risk Premium

First, if we forget the residual risk dY bht , the law of one price implies that, similar to the Black-

Scholes-Merton model, the return of a riskless portfolio must be equal to the riskless rate; i.e.,

a��bh � �c� = a �Xbh � C� r; (43)

Second, if dY bht 6= 0, we add a risk premium yt to compensate the residual risk; i.e.,

a��bh � �c� = a �Xbh � C� r + yt; (44)

which is equation (13) but in continuous-time. If dY bht = 0, yt = 0 and we have the standard

application of non arbitrage arguments with complete markets. The intuition is that the investor

obtains an extra premium ytdt for carrying extra risk on dYbht . It means that the risk-return trade-o¤

of N�ytdt;

PKk=1

��Yk�2dt�is attractive for the writer (or buyer) of the option.

The orthogonality conditions, EPthdSndY

bht

i= 0, imply that the risk premium yt can be spec-

i�ed independently of the tradable assets dSn. From the PDE equation (44), ��a (�c � rC) =�a��bh � rXbh� + yt�is the option risk premium and yt is the risk premium if the option is hedged

by portfolio bh, (recall that a = +1 (a = �1) for a short (long) option position).Equivalently, since a2 = 1, the latter PDE equation can be rewritten as

�c ���bh � rXbh� = rC � ayt; (45)

6Let us show two examples for N = 1 and K = 2. If A = (1; 0), then (A0 (AA0)�1A � I)dz = (0; dz2)0 which is

orthogonal to dz1. If A = (1; 1), (A0 (AA0)�1A� I)dz = 1

2(�dz1 + dz2; dz1 � dz2)0 which is orthogonal to dz1 + dz2.

14

We are interested in two bounds Cs and C l, with Cs � C l: A su¢ cient condition is given by �yst � ylt(e.g., yst = y

lt � 0), where yst (ylt) is the risk premium associated with Cs (C l). It is the same condition

in equation (5) in the one-period model. Intuitively, a lower (larger) term �ayt in the PDE equation(45), i.e., a lower (larger) risk-neutral drift, implies a larger (lower) option price, since at maturity

C = Cs = C l. This is formally proved below.

Finally, noting that

�0CS ���bh � rXbh� = �0CS �

��0(1:N)

bh� rXbh�= �0CS � (�� rS)0(1:N)

�bg + CS(1:N)�= rS0(1:N)CS(1:N) + �

0(N+1:J)CS(N+1:J) � (�� rS)

0(1:N) bg

= rS0(1:N)CS(1:N) +��0(N+1:J) � (�� rS)

0(1:N)

�AA0

��1AB0

�CS(N+1:J)(46)

and substituting �c and bh, equation (45) is given explicitly byCt +

1

2

JXi=1

JXj=1

CSS(i;j)

KXk=1

�i;k�j;k

!+

rS0(1:N)CS(1:N) +��0(N+1:J) � (�� rS)

0(1:N)

�AA0

��1AB0

�CS(N+1:J) = rC � ayt: (47)

4.3 The RNP Measures bQ and bQyFrom equations (35) and (43) we extract the RNP measure bQ. This is one of the innovations ofthe paper. The literature extracts the RNP measure bQy from equations (35) and (45). bQ allows to

separate the option price in the price of a hedging portfolio plus a premium, which has a natural

interpretation in incomplete markets. On the other hand, bQy is best used for pricing purposes andto prove no arbitrage (Harrison and Kreps (1979)).

Let Q be a RNP measure. Q can be characterized through the Radon-Nikodyn derivative, �; i.e.,

dQ

dP = �T , and �0 = 1 andd�t�t= ��0tdzt for t 2 [0; T ];

where e�rT �T is also the state price density and where �t is a vector of prices of risk associated

with dz (and the Novikov�s condition holds, E0hexp

�12

R T0 �

0t�tdt

�i< 1). By Girsanov�s theorem,

dzQt = dzt + �tdt are Wiener processes under Q.

In what follows, we assume a standard result of pricing by arbitrage in frictionless markets (see

Du¢ e (2001, 111-114)), so we can indistinctly use any of these three properties. Under technical

conditions, the following three properties are equivalent, (a) the existence of a market prices of risk

process �, (b) the existence of a risk-neutral probability measure Q, and (c) non arbitrage. It holds

for both complete and incomplete markets.

15

Formally, a Feynman-Kac theorem supports that a PDE has a probabilistic solution (Du¢ e (2001,

343)). For the N tradable assets, the risk-neutral drift must be equal to the riskless rate r; i.e.,

�(1:N) �A� = rS(1:N): (48)

For the rest of nontradable assets, its risk-neutral drift is implicit in the PDE pricing equation (47).

From the loadings of the vector CS and the risk-premium yt, we consider the term

rS0(1:N)CS(1:N) +��0(N+1:J) � (�� rS)

0(1:N)

�AA0

��1AB0

�CS(N+1:J) + ayt; (49)

Since �0 = [A0B0] and �0CS � �0�0CS = �0CS � �0 [A0B0]CS ,

�0CS � �0�0CS =��0(1:N) � �

0A0�CS(1:N) +

��0(N+1:J) � �

0B0�CS(N+1:J)

= rS0(1:N)CS(1:N) +��0(N+1:J) � �

0B0�CS(N+1:J): (50)

Then, from the de�nition of market price of risk, equations (49) and (50) must be equal; i.e.,��0(N+1:J) � �

0B0�CS(N+1:J) =

��0(N+1:J) � (�� rS)

0(1:N)

�AA0

��1AB0

�CS(N+1:J) + ayt; (51)

and from equation (48),

��0B0CS(N+1:J) = ��0A0�AA0

��1AB0CS(N+1:J) + ayt: (52)

Consequently, from equations (48) and (52), the vector of market prices of risk, �, satis�es24 A

C 0S(N+1:J)B�I �A0 (AA0)�1A

� 35� =24 (�� rS)(1:N)

�ayt

35 : (53)

Note that � does not depend on the drift of the nontradable state variables, �(N+1:J). If the market

is complete, A is invertible. Therefore, � is unique from equation (48) and equation (53) holds if

yt = 0 (and the option price is arbitrage free). On the other hand, if the market is incomplete, � is

obtained from equation (53) and there can be multiple solutions if K > N + 1.

Finally, if yt = 0, t 2 [0; T ], then24 A

C 0S(N+1:J)B�I �A0 (AA0)�1A

� 35� =24 (�� rS)(1:N)

0

35 ; (54)

where the option price C veri�es equation (47) with y = 0. Now, we can show that there is a solution

of the system (54) which does not depend on C 0S(N+1:J). That is, the new system of J equations24 A

B�I �A0 (AA0)�1A

� 35� =24 (�� rS)(1:N)

0(1:J�N)

35 (55)

16

always has a solution.7 Note that if � = �� solves (55), it solves (54). For example, for J = N + 1,

equations (54) and (55) are equivalent (if C 0S(N+1) 6= 0). Consequently, the expectation EbQt

hC(T )

er(T�t)

idoes not depend on the option price, and � can be directly obtained from equation (55).

4.4 The Price Decomposition

First, the total hedging error is simply the sum of the one-period hedging errors plus the one-period

risk premiums, �nanced/invested at the riskless rate r. The dynamic of a self-�nancing portfolio bhsatis�es X�bh(T ) = X�bh(0) + R T0 dX�bh(t). And C�(T; S(T )) = C�(0; S(0)) + R T0 dC�(t; S(t)). We de�neX�bh(0) = C�(0; S(0)). Therefore,

Ybh�T = a

�X�bh(T )� C�(T; S(T ))

�= a

Z T

0dX�bh(t)� a

Z T

0dC�(t; S(t))

= a

Z T

0e�rt

�(�bh � rX�bh)dt+ bh0Adzt�� aZ T

0e�rt

�(�c � rC))dt+ C 0S�dzt

�= a

Z T

0e�rt

�(�bh � rX�bh)� (�c � rC)

�dt+ a

Z T

0e�rtdY

bht

= e�rTZ T

0er(T�t)ytdt� ae�rT

Z T

0er(T�t)C 0S(N+1:J)B

�A0�AA0

��1A� I

�dzt; (56)

where the second line is from Itô�s Lemma, and the four line is from the pricing PDE equation (45)

and from the hedging error in equation (42).

That is, aXbh(T ) = aC(T; S(T )) + Y bhT . The hedging portfolio tracks the option payo¤ except fora residual risk Y bhT , which contains two parts. The �rst part depends on the risk premium y. The

second part is orthogonal to the traded assets�EPt

hdS(t)dY

bht

i= 0�. Therefore, the spanned option

payo¤ does not depend on y (except for the loadings CS(N+1:J)), and it makes economic sense to

decompose the option price in the price of a hedging portfolio plus a premium.

Second, under the risk-neutral dynamics bQ we have (dz bQt are Wiener processes under bQ),8C(T ) = C(0) +

Z T

0(rC � ayt) dt+

Z T

0C 0S�dz

bQt ; (57)

7Recall that we assume that AA0 is invertible. Equation (55) can be rewriten as24 A

B

35� =24 IN�N

BA0 (AA0)�1

35 (�� rS)(1:N) :Without lost of generality, let � = �� be a solution of this system obtained from a system of independent rows of �.

Now, consider that the volatility vector of a new non-traded variable is a linear combination of the others, i.e., b0 = 0�.

Then, we prove that �� is also a solution if � has an additional linear dependent row; i.e., since �0 = [A0 B0],

b0A0�AA0

��1(�� rS)(1:N) =

0

24 A

B

35A0 �AA0��1 (�� rS)(1:N) = 024 IN�N

BA0 (AA0)�1

35 (�� rS)(1:N) = 0��� = b0��:8This follows from Feynman-Kac Theorem, and implies that, di¤erent to the discrete-time model, we do not need

to de�ne the risk-neutral measure bQ(y=0). I am indebted to an anonymous referee for remarking this point.

17

from equation (45), and in discounted prices

C�(T ) = C�(0)� aZ T

0e�rtytdt+

Z T

0e�rtC 0S�dz

bQt : (58)

Therefore, taking risk-neutral expectations under bQ; and given C(0) = C�(0); we see thatEbQ0 [C

�(T )] = C(0)� aE bQ0

�Z T

0y�t dt

�(59)

and consequently,

C(0) = EbQ0

�C(T )

erT

�+ aE

bQ0

�Z T

0

ytertdt

�: (60)

Denote by bht(y = 0) (by Cbh(y=0)(0)) the optimal portfolio (the option price) when all risk premi-ums are zero (i.e., ys = 0 for s 2 (t; T ]). The portfolio bht(y = 0) satis�es thatX�bh(y=0)(0) = C�bh(y=0)(0),and from equation (60), X�bh(y=0)(0) = E bQ

0

hC(T )erT

i. Consequently,

C(0) = Xbh(y=0)(0) + aE bQ0

�Z T

0

ytertdt

�; (61)

the option price is equal to the price of a hedging portfolio plus a multiperiod risk premium. Equations

(56), (60) and (61) are just equations (17), (21) and (22), respectively, but in continuous time.

Let us assume that yt � 0 (almost sure), for all t. Then, from equation (60), under the bQprobability measure, the discounted upper (lower) bound is a super-martingale (sub-martingale),

and the discounted price of the hedging portfolio, e�rtXbh(y=0)(t) = E bQt

�e�rTCT

�, is the martingale

component. This proves that the upper bound (a = +1) is larger than the lower bound (a = �1).Finally, since

C(0) = EbQy0

�C(T )

erT

�; (62)

then the multiperiod risk premium is given by

aEbQ0

�Z T

0e�rtytdt

�= E

bQy0

�C(T )

erT

�� E bQ

0

�C(T )

erT

�(63)

For example, if C(T ) is a European call option, the premium is equal to the price di¤erence of two

call options. If the two call options have a close form solution, the premium does too.

In the continuous-time incomplete markets framework for di¤usion processes, Theorem 2 sum-

marizes the results of the present section. American-style securities are studied next.

Theorem 2 Assume that the price process S satis�es equation (35) and that the option price C is

characterized by the PDE equation (47), or equation (62), subject to a boundary condition C(T; S(T )).

Assume a frictionless and arbitrage-free market. Then, the optimal hedging portfolio bh = CS(1:N)+ bgis given by equation (41); the market prices of risk associated with the probability measures bQy andbQ are given by equation (53) and (55), respectively; the total hedging error is given by equation (56);and the option price decomposition is given by equation (60) or (61). �

18

4.5 American�style Securities

The price of an American option can be factorized into three components, which is proved for a �nite

state space discrete-time model in Section 3.3. We give two di¤erent decompositions.

In the present continous-time setting we prove this result as follows. An American option can

be decomposed into an early-exercise premium plus an equal European option. This result was

developed for the standard American put option and then extended to more general problems. In

particular, we follow Broadie and Detemple (2004, Section 3.4, 1156-57) who review and explain this

result. We adapt this result to our incomplete markets framework.

To relate the American option to its equivalent European option, we also consider the two RNP

measures associated with the equivalent European option ( bQ and bQy). Let denote by CAt the priceof the American option, by It = I(St; E) the intrinsic value, and by b�(t) 2 [0; T ] the associatedoptimal stopping-time. First, in the exercise region (i.e., fb�(t) = tg), CAt = It and d

�e�rtCAt

�=

e�rt (dIt � rItdt). In the continuation region (i.e., fb�(t) > tg), e�rtCAt is a martingale under bQy, buthas a drift equal to �ae�rtyAt dt under bQ (similar to European options). Second, from e�rTCAT =

CA0 +R T0 d

�e�rtCAt

�, then CAT can be written as

e�rTCAT = CA0 +

Z T

01fb�(t)=tgd �e�rtCAt �+ Z T

01fb�(t)>tgd �e�rtCAt �

= CA0 +

Z T

01fb�(t)=tge�rt (dIt � rItdt) +

Z T

01fb�(t)>tgd �e�rtCAt � : (64)

Third, we take expectations under bQ and under bQy, respectively. Under bQ,CA0 = E

bQ0

�Z T

01fb�(t)=tg rItdt� dItert

�+ aE

bQ0

�Z T

01fb�(t)>tg yAtertdt

�+ E

bQ0

�C(T )

erT

�= E

bQ0

"Z T

01fb�(t)=tg rItdt�

�dIt + ay

At dt�

ert

#+ aE

bQ0

�Z T

0

yAtertdt

�+ E

bQ0

�C(T )

erT

�; (65)

where the term 1fb�(t)=tge�rtyAt dt is added and subtracted in the second equality. Under bQy,CA0 = E

bQy0

�Z T

01fb�(t)=tg rItdt� dItert

�+ E

bQy0

�C(T )

erT

�(66)

= EbQy0

�Z T

01fb�(t)=tg rItdt� dItert

�+ aE

bQ0

�Z T

0

ytertdt

�+ E

bQ0

�C(T )

erT

�; (67)

where the second equality follows from the decomposition of the European option. The last term

EbQ0

�e�rTC(T )

�is the price of the hedging portfolio of the equivalent European option, and both

factorizations are like those of the �nite state discrete-time model of Section 3.3.9

9We have taken bQ from the European option and thus we can relate the American option with the European one.

Note that this RNP measure is also associated with the American option (for yA = 0), since the hedging portfolio

and the PDE in the continuation region are of the same type than for European options (see equations (41) and (47),

respectively). See Broadie and Detemple (2004, Section 3.3) for other approaches to American options.

19

4.6 Valuation of a Portfolio of Derivative Securities

We are also interested in option-pricing from a portfolio perspective. In a complete market, the price

of a portfolio of (any kind of) n securities is equal to the sum of the n individual prices because of

linear pricing. In an incomplete market, however, this result does not necessarily hold.

We apply the decomposition to a portfolio of n European securities, CpT =�C(1)T ; C

(2)T ; :::; C

(n)T

�,

and assume that all securities have the same maturity. Then,

nXi=1

C(i)0 =

nXi=1

EbQ0

he�rTC

(i)T

i+

nXi=1

EbQ0

�Z T

0e�rty

(i)t dt

�and (68)

Cp0 = EbQ0

"e�rT

nXi=1

C(i)T

#+ E

bQ0

�Z T

0e�rtypt dt

�; (69)

where y(i)t and ypt are the risk premium of every individual security and of the portfolio p, respectively.

Both prices di¤er only in the valuation of the residual risk.

Let show a simple example with n = 2 securities. We assume that the instantaneous residual

risks associated with each security, Y (1)t and Y (2)t , respectively, are N (0; dt) with correlation �. Thefollowing table yields the variance and the standard deviation of them, separately and for a portfolio.

Risk Risk (Y pt ) = Risk�Y(1)t + Y

(1)t

�Risk

�Y(1)t

�+Risk

�Y(1)t

�Variance 2(1 + �)dt 2dt

Standard Deviationp2(1 + �)

pdt 2

pdt

Using the standard deviation as a measure of risk, which seems more appropriate than the vari-

ance, sincep2(1 + �) < 2 if � < 1. Then, ypt < y

(1)t + y

(2)t is a reasonable speci�cation for all t and

Cp0 is less expensive; i.e., Cp0 < C

(1)0 + C

(2)0 . On the other hand, if � = 1 (e.g., if the two securities

are the same), ypt = y(1)t + y

(2)t ; Hence, C

p0 = C

(1)0 + C

(2)0 .

5 Examples: Basis Risk

We price a real option or an option subject to basis risk. An European call option C depends on an

underlying asset V which is not traded or is illiquid. Yet there is a second traded asset S which is

correlated with V . For example, the option is de�ned on an illiquid commodity (the Gulf of Mexico

oil), but one could use a correlated and more liquid asset (the Texas oil future) as the hedging asset.

Then, it is possible to partially hedge the option and to derive pricing implications.10

10There are several applications of the basis risk model. V is a small stock, S is a correlated but more liquid stock.

V is a basket of assets, S is an index. V is the short-term interest rate, S are the prices of liquid bonds. In emerging

markets one �nds at most one or two liquid bonds. V is in�ation, S is a long-term bond. Executive stock options in

the company V , where the executive can trade in any stock S except V . Another related problem is that of hedging of

long-term exposures by rolling over short-term futures contracts (see Ross (1997)).

20

This problem is also studied by Cochrane and Saá-Requejo (2000), and we use a similar notation.

The dynamics of both assets under the objective lognormal probability measure, P, are given by

dS = �sSdt+ �sSdz1;t and (70)

dV = �vV dt+ �vV (�dz1;t +p1� �2dz2;t); (71)

where dz1;t and dz2;t are two standard orthogonal Brownian motions, the parameter � measures the

correlation between the returns of V and S, and there exists a risk-free asset with return equal to r.

The normality of returns allows us to obtain close form solutions and thus provide further intuition.

Let T be the option maturity and E the strike price, C(V (T )) = fV (T ) � Eg+. Because S is atradable asset, the no-arbitrage bounds of a call option C(S) are

�S � Ee�r(T�t)

+< C(S) < S,

Merton (1973). However, since V is nontradable, if j�j < 1, one can show that the no-arbitrage

bounds of C(V ) are much more unconstrained, i.e., 0 < C(V ) < 1. Therefore, any non-negativeprice is feasible as it does not allow arbitrage opportunities and the arbitrage bounds are unpractical.

The Hedging Strategy. By applying Itô�s Lemma we can decompose the return of dC into

dC = (Ct + �vV CV +1

2�2vV

2CV V )dt+ �vV CV (�dz1;t +p1� �2dz2;t): (72)

Consider the minimum variance hedging portfolio, since dz2 is orthogonal to dS,

bh1 = �vV CV�sS

�: (73)

Then, the return of the hedging portfolio, a�bh1S � C�, is equal to

a�bh1dS � dC� = �a�Ct + ��v � �s ��v�s

�V CV +

1

2�2vV

2CV V

�dt� a�vV CV

p1� �2dz2;t: (74)

The PDE equation. If we forget for a moment the residual risk, dz2;t, then the return of this

portfolio, a�bh1S � C�, is risk free. The law of one price implies that�a�Ct +

��v � �s

��v�s

�V CV +

1

2�2vV

2CV V

�= �a

�C � ��vV CV

�s

�r: (75)

If j�j = 1; this is a standard complete markets problem and we obtain the same no-arbitrage conditionon the drift process.

However, we still have the residual risk, dY bht = �vV CVp1� �2dz2;t;. Let ytdt (where yt = 0 ifj�j = 1) be this risk premium. Then,

�a�Ct +

��v � �s

��v�s

�V CV +

1

2�2vV

2CV V

�= �a

�C � ��vV CV

�s

�r + yt: (76)

The investor in C obtains an extra premium ytdt for carrying extra risk on dz2;t: Since a2 = 1, the

latter equation can be rewritten as

Ct + �vV CV +1

2�2vV

2CV V ��s � r�s

��vV CV = rC � ayt: (77)

21

Note the hedging portfolio in equation (73) depends on the risk premium from the option Delta, CV .

Examples. If the risk premium is proportional to the option price, i.e., yt = ��vp1� �2C, then

Ct + �vV CV +1

2�2vV

2CV V ��s � r�s

��vV CV = (r � a��vp1� �2)C; (78)

i.e., the risk-neutral return of the option, under bQ, is equal to �r � a��vp1� �2.�That is, underbQ, the discount rate is not r, but a lower (larger) rate for the upper (lower) bound since � > 0. Arisk-premium proportional to the option price is similar to the recovery market value assumption in

defaultable term structure models (Du¢ e and Singleton (1999)).

If the risk premium is proportional to the option Gamma, i.e., yt = 12�p1� �2�2vV 2CV V , and

� > 0. We have

Ct +

��v �

�s � r�s

��v

�V CV +

1

2�2v

�1 + a�

p1� �2

�V 2CV V = rC: (79)

Interestingly, the risk-neutral volatility under bQy,11 �vq1 + a�p1� �2; is di¤erent from, �v, thevolatility under the actual probability measure if j�j 6= 1. Whereas both volatilities must be equal ina complete market model, this constraint does not necessarily apply in an incomplete market. This

premium, which depends on the convexity of the call option, is intuitive if there are transaction costs

associated with trading and delta-hedging.

Assume now that yt = eA�vV CVp1� �2, i.e., yt is proportional to the hedging error standarddeviation, and note that yt > 0 if eA > 0, since CV > 0 for call payo¤s. Then,

Ct + �vV CV +1

2�2vV

2CV V = rC +

��s � r�s

�� a eAp1� �2��vV CV : (80)

This risk-premium guarantees that the option price is arbitrage-free if these call options start to be

traded and are liquid. The parameter eA can be related to the relative-risk aversion of a representativeinvestor (Heston (1993)).

The Associated RNP Measure. Let us extract the RNP measure bQy and call ��bh;ys ; �bh;ys

�and

��bh;yv ; �

bh;yv

�the risk neutral parameters under bQy for S and V , respectively. Clearly,

��bh;ys ; �

bh;ys

�=

��s �

�s � r�s

�s; �s

�= (r; �s) and

��bh;yv ; �

bh;yv

�=

��v �

�s � r�s

��v + aytV CV

; �v

�from equation (77). In other words, �1 =

�s�r�s

and �2 = �ayt 1

�vV CVp1��2

are the market prices of

risk associated with dz1 and dz2, respectively. These two prices of risk and therefore bQy are unique.11Let � � 0. We assume that 1+a�

p1� �2 � 0, and equivalently, �a� �

�p1� �2

��1. Thus, for the upper bound

(a = +1), this inequality holds for � � 0. For the lower bound (a = �1), � is constrained, 0 � � ��p

1� �2��1

,

which makes sense to avoid negative option prices as the lower arbitrage bound is zero.

22

For example, if yt = 0, �2 = 0; if yt = ��vp1� �2C, �2 = �a �C

V CV(which is related to the inverse

of the option price elasticity since �2 = �a��@C@V

VC

��1); and if yt = eA�vV CVp1� �2, �2 = �a eA.

If yt = 12�p1� �2�2vV 2CV V , the risk-neutral drift and volatility parameters can be interpreted

di¤erently, and �2 = �a12��vV CV VCV

, which is related to the curvature of the option price. The price

of risk �2 is well de�ned if the lower option price bound is non-negative.

The Price Decomposition. Let us assume a risk premium yt = eA�vV CVp1� �2. First,the total hedging error is simply the sum of the one-period hedging errors plus the one-period risk

premiums, �nanced or invested at the riskless rate r; i.e.,

YbhT = a

�Xbh(T )� C(T; S(T ))� = �vp1� �2 Z T

0er(T�t)V CV

� eAdt� adz2;t� : (81)

That is, C(T; S(T )) = Xbh(T )�aY bhT , where Xbh(T ) = Xbh(0)+�v� R T0 er(T�t)V CV dz1;t is the risk thatcan be hedged and �aY bhT = �v

p1� �2

R T0 e

r(T�t)V CV dz2;t, besides the risk premium, is the risk

that cannot be hedged. Second, the associated risk premium is given by

aEbQ0

��vp1� �2 eAZ T

0e�rtV CV dt

�: (82)

The PDE equation (80) (like equations (78) and (79)) have a close form solution of the Black-

Scholes type. The risk premium in equation (82) also has a close form solution given by the di¤erence

of two call options, which both satisfy equation (80) (with eA > 0 and eA = 0). See Ibáñez (2005) forthese solutions and additional numerical examples for call options.

5.1 The Decomposition of the American Put Option with Basis Risk

We now consider the valuation of one American put option on the same non-traded security V .

Let P and PA denote the price of the European and the American put, respectively. We �rst price

the European put, which is equivalent to provide a risk-neutral dynamic for V . We assume a risk

premium yt = � eA�vV PVp1� �2. The sign in front of eA is changed since PV < 0 for put payo¤s.For the American put, the risk premium is similar but depends on PAV , y

At = � eA�vV PAV p1� �2.

From equations (75) and (80), �V ���S�� a�2

p1� �2

��V is the risk-neutral drift of V , with

�S = �s�r�s, and where �2 = 0 for bQ and �2 = � eA for bQy, respectively. We de�ne q = r ��

�V ���S�� a�2

p1� �2

��V

�. We also de�ne � = r �

��V �

��S�+ a eAp1� �2��V �, which is

similar to a dividend.

Now, we apply and extend the results of Kim (1990) and Carr et al. (1992). The price of the

American put can be decomposed into an early-exercise premium plus the price on an equivalent

European put. Kim (1990, 560) provides analytical results if the security pays a dividend, � 6= 0.12

12These results are used, e.g., to develop e¢ cient algorithms to price American options (see Ibáñez (2003)).

23

We provide two factorizations from the results of Section 4.5. Under bQ, we have q = � �a eAp1� �2�V and E bQ

t [dIt] = (r � q)Vtdt. Then, if fb�(t) = tg, It = E � Vt, PAV = �1, andrItdt�

�EbQt [dIt] + ay

At dt�

= r (E � Vt) dt���(r � q)Vt + a eA�vVtp1� �2� dt

= rEdt��q + a eA�Vp1� �2�Vtdt = rEdt� �Vtdt:

Consequently,

PA(0; V0) =

Z T

0

h��V0e�qtN

bQ(�d1 (V0; Bt; t)) + rEe�rtN bQ(�d2 (V0; Bt; t))i dt�aE bQ

0

��vp1� �2 eAZ T

0e�rtV PAV dt

�+ E

bQ0

�P (T )

erT

�; (83)

and where Bt, 0 � t � T , is the optimal exercise frontier at time t.Under bQy, q = �, E bQy

t [dIt] = (r � q)Vtdt, and rItdt� Et [dIt] = rEdt� �Vtdt as above. Then,

PA(0; V0) =

Z T

0

h��V0e��tN

bQy(�d1 (V0; Bt; t)) + rEe�rtN bQy(�d2 (V0; Bt; t))i dt+ P (0; V0); (84)Note that P (0; V0) = E

bQy0

�e�rTP (T )

�. The European put price P (0; V0) can be divided into the

price of a hedging portfolio plus a risk-premium by applying Theorem 2. Consequently,

PA(0; V0) =

Z T

0

h��V0e��tN

bQy(�d1 (V0; Bt; t)) + rEe�rtN bQy(�d2 (V0; Bt; t))i dt�aE bQ

0

��vp1� �2 eAZ T

0e�rtV PV dt

�+ E

bQ0

�P (T )

erT

�; (85)

where, d1 (V0; Bt; t) =ln(V0=Bt)+(r�q+0:5�2v)t

�vpt

and d2 (V0; Bt; t) = d1 (V0; Bt; t)��vpt, in equations (83)

and (85). And note q takes di¤erent values under bQ and bQy.The price of the American put is equal to the price of a hedging portfolio, a risk premium

associated with the residual risk, and an early-exercise premium. The risk premium is positive

(negative) for the upper (lower) bound, as a = +1 (a = �1) and �PV > 0 and �PAV > 0. Moreover,both P and PA depend on the same risk-preference parameter, eA. Note how the risk premium termsin equations (83) and (85) depend on PAV and PV , respectively. If it is not optimal to early exercise

(e.g., if r = 0 and � � 0, see Kim (1990, 560)), then Bt = 0, 0 � t � T , and PA(0; V0) = P (0; V0).Intuitively, for small �t, from t to t + �t, the American put provides two premiums to her/his

owner if the option is alive (i.e., Vt > Bt): a risk premium and an early-exercise premium, respectively,

�a�vp1� �2 eAVtPAV ��t

+h��Vte�q�tN

bQ(�d1 (Vt; Bt+�t; t+�t)) + rEe�r�tN bQ(�d2 (Vt; Bt+�t; t+�t))i��t:(86)

24

6 Concluding Remarks

This paper shows that the price of one European option can be divided into two orthogonal com-

ponents. This decomposition is interesting and has several applications for option�pricing under

complete and incomplete markets. First, we can see the price of an option as the sum of two separate

parts. One part is robust and is priced by arbitrage. A second part depends on a risk orthogonal to

the traded securities. Therefore, certain misspricings in option markets can be directly related to the

second part. For example, Ibáñez (2006) quanti�es the part of the negative option premium, which

is associated with stochastic volatility.

Second, it is easy to de�ne upper and lower bounds, or bid/ask prices, in an incomplete market,

since the price of the hedging portfolio is the same for both bounds. For instance, one can consider

positive (negative) risk-premiums for the upper (lower) bound. Therefore, it is also easy to constrain

the arbitrage bounds by constraining the risk premiums. Third, as another application, we consider

the valuation of a portfolio of derivative securities. Whereas this problem is trivial for a complete

market, our results show that, for an incomplete market, the di¤erence between the price of this

portfolio and the sum of the individual prices is due to the valuation of the residual risk. Since there

can be some diversi�cation in the portfolio of derivatives, this portfolio can be cheaper.

Moreover, these results extend to American options, which have a third component �an additional

early-exercise premium. We provide explicit results for the American put under basis risk. Jumps

(see Björk and Slinko (2006)) and market frictions (see Ibáñez (2005)), such as portfolio constraints

or transaction costs, prevent perfect hedging and are other good examples of market incomplete-

ness which deserve future research. Figlewski and Green (1999) show that even the most liquid

and developed option markets bear many residual risks. It would be interesting to obtain similar

decompositions and to empirically study these problems.

REFERENCES

Bernardo, A. and O. Ledoit, 2000, �Gain, Loss, and Asset Pricing,�Journal of Political Economy,

108, 144-172.

Bertsimas, D., L. Kogan and A. Lo, 2001, �Hedging Derivative Securities and Incomplete Markets:

An �-Arbitrage Approach,�Operations Research, 49, 372-397.

Björk, T. and I. Slinko, 2006, �Towards a Theory of Good-Deal Bounds,�Review of Finance,

forthcoming.

Black, F. and M. Scholes, 1973, �The Pricing of Options and Corporate Liabilities,�Journal of

Political Economy, 81, 637-654.

Broadie M. and J.B. Detemple, 2004, �Option Pricing: Valuation Models and Applications,�

Management Science, 50, 1145-1177.

25

Carr P., H. Geman and D.B. Madan, 2001, �Pricing and Hedging in Incomplete Markets,�Journal

of Financial Economics, 62, 131-167.

Carr P., R. Jarrow and R. Myneni, 1992, �Alternative Characterization of American Put Op-

tions,�Mathematical Finance, 2, 87,106.

Cerný, A., 2003, �Generalized Sharpe Ratios and Asset Pricing in Incomplete Markets,�European

Finance Review, 7 (2), 191-233.

Cochrane, J.H. and J. Saá-Requejo, 2000, �Beyond Arbitrage: Good-Deal Asset Price Bounds in

Incomplete Markets,�Journal of Political Economy, 108, 79-119.

Detemple, J. and S. Sundaresan, 1999, �Nontraded Asset Valuation with Portfolio Constraints:

A Binomial Approach,�Review of Financial Studies, 12, 835-872.

Duarte, J., 2004, �Evaluating an Alternative Risk Preference in A¢ ne Term Structure Models,�

Review of Financial Studies, 17, 379-404.

Du¤ee, G., 2002, �Term Premia and Interest Rate Forecasts in A¢ ne Models,� Journal of Fi-

nance, 57, 405-443.

Du¢ e, D., 2001, �Dynamic Asset Pricing Theory,�Third Edition, Princeton University Press,

Princeton, New Jersey.

Du¢ e, D. and H. Richardson, 1991, �Mean-Variance Hedging in Continuous Time,�Annals of

Applied Probability, 1, 1-15.

Du¢ e, D. and K.J. Singleton, 1999, �Modeling Term Structures of Defaultable Bonds,�Review

of Financial Studies, 12, 687-720.

Figlewski S. and T.C. Green, 1999, �Market Risk and Model Risk for a Financial Institution

writing Options,�Journal of Finance, 54, 1465-1499.

Harrison M. and D. Kreps, 1979, �Martingales and Arbitrage in Multiperiod Securities Markets,�

Journal of Economic Theory, 20, 381-408.

Heath, D., E. Platen and M. Schweizer, 2001, �A Comparison of Two Quadratic Approaches to

Hedging in Incomplete Markets,�Mathematical Finance, 11, 385-413.

Heston, S., 1993, �A Closed-Form Solution for Options with Stochastic Volatility with Applica-

tions to Bond and Currency Options,�Review of Financial Studies, 6, 327-344.

Ibáñez, A., 2003, �Robust Pricing of the American Put Option: A Note on Richardson Extrapo-

lation and the Early Exercise Premium,�Management Science, 49, 1210-1228.

Ibáñez, A., 2005, �Option-Pricing in Incomplete Markets: The Hedging Portfolio plus a Risk

Premium-Based Recursive Approach,�Working Paper, Universidad Carlos III de Madrid.

Ibáñez, A., 2006, �On the Negative Market Volatility Risk Premium: Bridging the Gap Between

Option Returns and the Pricing of Options,�Working Paper, Universidad Carlos III de Madrid.

Ingersoll, J. E., 1987, �Theory of Financial Decision Making,�Rowman & Little�eld.

26

Kim, I. J., 1990, �The Analytical Valuation of American Options,�Review of Financial Studies,

3, 547-572.

Merton, R., 1973, �The Theory of Rational Option Pricing,� Bell Journal of Economics and

Management Science, 4, 141-183.

Merton, R., 1976, �Option Pricing when the Underlying Stock Returns are Discontinuous,�Jour-

nal of Financial Economics, 5, 125-144.

Merton, R., 1998, �Applications of Option-Pricing Theory: Twenty-Five Years Later,�American

Economic Review, 88, 323-349.

Naik V. and M. Lee, 1990, �General Equilibrium Pricing of Options on the Market Portfolio with

Discontinuous Returns,�Review of Financial Studies, 3, 493-521.

Pliska, S., 1997, �Introduction to Mathematical Finance: Discrete time Models,�Blackwell Pub-

lishers.

Rubinstein, M., 1976, �The Valuation of Uncertain Income Streams and the Pricing of Options,�

Bell Journal of Economics and Management Science, 7, 407-425.

Ross, S., 1978, �A Simple Approach to the Valuation of Risky Streams,�Journal of Business, 51,

453-475.

Ross, S., 1997, �Hedging Long Run Commitments: Exercises in Incomplete Market Pricing,�

Economic Notes by Banca Monte, 26, 99-132.

Schweizer, M., 1992, �Mean-variance Hedging for General Claims,�Annals of Applied Probability,

2, 171-179.

27