Minimizing Expected Loss of Hedging in Incomplete and Constrained Markets.
We study the problem of minimizing the expected discounted loss  $$ E\left[e^{-\int_0^Tr(u)du}( C- X^{x,\pi}(T))^+\right] $$  when hedging a liability C at time t=T, using an admissible portfolio strategy $\pi(\cdot)$ and starting with initial wealth x. The existence of an optimal solution is established in the context of continuous-time Ito process  incomplete market models, by studying an appropriate dual problem. It is shown that the optimal strategy is of the form of a knock-out option with payoff C, where the "domain of the knock-out" depends on the value of the optimal dual variable. We also discuss a dynamic measure for the risk associated with the liability C, defined as the supremum over different scenarios of the minimal expected loss of hedging C.
Introduction
In a complete financial market which is free of arbitrage opportunities, any sufficiently integrable
random payoff (contingent claim) C, whose value has to be delivered and is known
at time can be hedged perfectly: starting with a large enough initial capital x, an
agent can find a trading strategy - that will allow his wealth X x;- (\Delta) to hedge the liability
C without risk at time that is
while maintaining "solvency" throughout [0; T ]. (For an overview of standard results in
complete and some incomplete markets in continuous-time, Ito processes models, see, for
example, Cvitani'c 1997). This is either no longer possible or too expensive to accomplish in
a market which is incomplete due to various "market frictions", such as: insufficient number
of assets available for investment, transaction costs, portfolio constraints, problems with
liquidity, presence of a "large investor", and so on. In this paper we concentrate on the
case in which incompleteness arises due to some assets not being available for investment,
and the more general case of portfolio constraints. Popular approaches to the problem of
hedging a claim C in such contexts have been to either maximize the expected utility of the
difference \GammaD := X x;- (T minimize the risk of D. In particular, one of the most
studied approaches is to minimize E[D 2 ], so-called quadratic hedging of F-ollmer-Schweizer-
(for recent results and references see Pham, Rheinlaender and Schweizer 1996,
for example). An obvious disadvantage of this approach is that one is penalized for high
profits, and not just high losses. On the other hand, Artzner, Delbaen, Eber and Heath
have shown in a static hedging setting that the only measure of risk that satisfies
certain natural "coherence" properties is of the type E[ -
(or a supremum of these over
a set of probability measures), where -
is the discounted value of the positive part of
D. Motivated by this work, Cvitani'c and Karatzas (1998) solve the problem of minimizing
in a context of a complete continuous-time Ito process model for the financial market.
We solve in this paper the same problem in a more difficult context of incomplete or constrained
markets. Recently, Pham (1998) has solved the problem of minimizing E[(D
discrete-time models, and under cone constraints. Moreover, independenly
from Pham and the present paper, F-ollmer and Leukert (1998b) analyze the problem of
minimizing loss function l and in general incomplete semimartingale
models, emphasizing the Neyman-Pearson lemma approach, as opposed to the duality ap-
proach. The former approach was used by the same authors in F-ollmer and Leukert (1998a)
to solve the problem of maximizing the probability of perfect hedge P [D - 0]. Some early
work on problems like these is presented in Dembo (1997), in a one-period setting. A very
general study of the the duality approach and its use in the utility meximization context
can be found in Kramkov and Schachermayer (1997).
Suppose now that, in addition to the genuine risk that the liability C represents, the
agent also faces some uncertainty regarding the model for the financial market itself. Following
Cvitani'c and Karatzas (1998), we capture such uncertainty by allowing a family
P of possible "real world probability measures", instead of just one measure. Thus, the
"max-min" quantity
represents the maximal risk that the agent can encounter, when faced with the "worst
possible scenario" P 2 P: In the special case of incomplete markets and under the condition
that all equivalent martingale measures are included in the set of possible real-world measures
P, we show that
sup
In other words, the corresponding fictitious "stochastic game" between the market and the
agent has a value. The trading strategy attaining this value is shown to be the one that
corresponds to borrowing just enough money from the bank at time as to be able to
have at least the amount C at time
We describe the market model in Section 2, and introduce the optimization problem
in Section 3. As is by now standard in financial mathematics, we define a dual problem,
whose optimal solution determines the optimal terminal wealth X x;- (T ). It turns out that
this terminal wealth is of the "knock-out" option type - namely, it is either equal to C or
to 0 or to a certain (random) value depending on whether the optimal dual
variable is less than, larger than, or equal to one, respectively. What makes the dual problem
more difficult than in the usual utility optimization problems (as in Cvitani'c and Karatzas
1992) is that the objective function fails to be everywhere differentiable, and the optimal
dual variable (related to the Radon-Nikodym derivative of an "optimal change of measure")
can be zero with positive probability. Nevertheless, we are able to solve the problem using
nonsmooth optimization techniques for infinite dimensional problems, which can be found
in Aubin and Ekeland (1984). We discuss in Section 4 the stochastic game associated with
(1.2) and (1.3).
2 The Market Model
We recall here the standard, Ito processes model for a financial market M. It consists of one
bank account and d stocks. Price processes S 0 (\Delta) and S 1 of these instruments
are modeled by the equations
d
Here is a standard d\Gammadimensional Brownian motion on a complete
probability
endowed with a filtration
augmentation of F W (t) := oe(W the filtration generated
by the Brownian motion W (\Delta). The coefficients r(\Delta) (interest rate),
(vector of stock return rates) and 1-i;j-d (matrix of stock-volatilities) of the
model M, are all assumed to be progressively measurable with respect to F. Furthermore,
the matrix oe(\Delta) is assumed to be invertible, and all processes r(\Delta), b(\Delta), oe(\Delta), oe \Gamma1 (\Delta) are
assumed to be bounded, uniformly in (t; !) 2 [0; T
The "risk premium" process
bounded and F\Gammaprogressively measurable. Therefore,
the process
ds
is a P \Gammamartingale, and
is a probability measure equivalent to P on F(T ). Under this risk-neutral equivalent martingale
measure P 0 , the discounted stock prices S 1 (\Delta)
become martingales, and the
process
becomes Brownian motion, by the Girsanov theorem.
Consider now an agent who starts out with initial capital x and can decide, at each time
proportion - i (t) of his (nonnegative) wealth to invest in each of the stocks
d. However, the portfolio process (- 1 has to take values in a given
closed convex set K ae R d of constraints, for a.e. t 2 [0; T ], almost surely. We will also
assume that K contains the origin. For example, if the agent cannot hold neither short
nor long positions in the last stocks we get a typical example of
an incomplete market, in the sense that not all square-integrable payoffs can be exactly
replicated. (One of the best known examples of incomplete markets, the case of stochastic
volatility, is included in this framework). Another typical example is the case of an agent
who has limits on how much he can borrow from the bank, or how much he can go short or
long in a particular stock.
chosen, the agent invests the amount
in the bank account, at time t, where we have denoted X(\Delta) j X x;- (\Delta) his
wealth process. Moreover, for reasons of mathematical convenience, we allow the agent to
spend money outside of the market, and -(\Delta) - 0 denotes the corresponding cumulative
consumption process. The resulting wealth process satisfies the equation
d
d
d
Denoting
R tr(u)du X(t); (2.6)
the discounted version of a process X(\Delta), we get the equivalent equation
It follows that -
X(\Delta) is a nonnegative local P 0 \Gammasupermartingale, hence also a P 0 \Gammasupermartingale,
by Fatou's lemma. Therefore, if - 0 is defined to be the first time it hits zero, we have
so that the portfolio values -(t) are irrelevant after that happens. Accordingly,
we can and do set
More formally, we have
Definition 2.1 (i) A portfolio process
F\Gammaprogressively measurable
and satisfies
as well as
almost surely. A consumption process -(\Delta) is a nonnegative, nondecreasing, progressively
measurable process with RCLL paths, with
(ii) For a given portfolio and consumption processes -(\Delta), -(\Delta), the process X(\Delta) j
defined by (2.7) is called the wealth process corresponding to strategy (-) and
initial capital x.
(iii) A portfolio-consumption process pair (-(\Delta); -(\Delta)) is called admissible for the initial
capital x, and we write (-) 2 A(x), if
holds almost surely.
We refer to the lower bound of (2.9) as a margin requirement. The no-arbitrage price
of a contingent claim C in a complete market is unique, and is obtained by multiplying
("discounting") the claim by H taking expectation. Since the
market here is incomplete, there are more relevant stochastic discount factors other than
along the lines of Cvitani'c and Karatzas (1993), hereafter [CK93],
and Karatzas and Kou (1996), hereafter [KK96], as follows: Introduce the support function
of the set \GammaK , as well as its barrier cone
~
For the rest of the paper we assume the following mild conditions.
Assumption 2.1 The closed convex set K ae R d contains the origin; in other words, the
agent is allowed not to invest in stocks at all. In particular, ffi(\Delta) - 0 on ~
K. Moreover, the
set K is such that ffi(\Delta) is continuous on the barrier cone ~
K of (2.11).
Denote by D the set of all bounded progressively measurable process -(\Delta) taking values in ~
a.e.
on\Omega \Theta [0; T ]. In analogy with (2.2)-(2.5), introduce
ds
a P - \GammaBrownian motion. Also denote
Note that
From this and (2.7) we get, by Ito's rule,
for all - 2 D. Therefore, H - (\Delta) -
X(\Delta) is a P \Gammalocal supermartingale (note that ffi(- 0 - 0
K), and from (2.9) thus also a P \Gammasupermartingale, by Fatou's lemma.
Consequently,
3 The minimization problem and its dual
Suppose now that, at time the agent has to deliver a payoff given by a contingent
claim C, a random variable in L
Introduce a (possibly infinite) process
ess sup
almost surely, the discounted version of the process
C(\Delta)
We have denoted
the discounted value of the F(T )\Gamma measurable random variable C. We impose the following
assumption, throughout the rest of the paper (see Remark 3.3 for a discussion on the
relevance of this assumption).
Assumption 3.1 We assume
The following theorem is taken from the literature on constrained financial markets (see, for
example, [CK93], [KK96], or Cvitani'c (1997)).
Theorem 3.1 (Cvitani'c and Karatzas 1993). Let C - 0 be a given contingent claim. Under
Assumption 3.1, the process C(\Delta) of (3.3) is finite, and it is equal to the minimal admissible
wealth process hedging the claim C. More precisely, there exists a pair (- C
such that
and, if for some x - 0 and some pair (-) 2 A(x) we have
then
Consequently, if x - C(0) there exists then an admissible pair (-) 2 A(x) such that
Achieving a "hedge without risk" is not possible for x ! C(0). Motivated
by results of Artzner et al. (1996) (and similarly as in a complete market setting of Cvitani'c
and Karatzas 1998) we choose the following risk function to be minimized:
In other words, we are minimizing the expected discounted net loss, over all admissible
trading strategies.
above, we can find a wealth
process that hedges C. Moreover, the margin requirement (2.9) implies that x - 0, so we
assume from now on that
Note that we can (and do) assume X x;- (T our optimization problem
(3.8), since the agent can always consume down to the value of C, in case he has more than
C at time T . In particular, if 0, we can (and do) assume X x;- (T;
means that the set not relevant for the problem (3.8), which motivates
us to define a new probability measure
(see also Remark 3.3 (ii)). Denote by E C the associated expectation operator.
The problem (3.8) has then an equivalent formulation
We approach the problem (3.11) by recalling familiar tools of convex duality: starting
with the convex loss function its Legendre-Fenchel transform
~
(where z The minimum in (3.12) is attained by any number I(z; b) of the
Consequently, denoting
we conclude from (3.12) that for any initial capital x 2 (0; C(0)) and any (-) 2 A(x),
Thus, multiplying by E[ -
C], taking expectations and in conjunction with (2.20), we obtain
This is a type of a duality relationship that has proved to be very useful in constrained
portfolio optimization studied in Cvitani'c and Karatzas (1993). The difference here is that
we have to extend it to the random variables in the set
It is clear that H is a convex set. It is also closed in L
in L 1
exists a (relabeled) subsequence fH n g n2N converging to H
C]E C [HY x;- x, for all
By Theorem 3.1 we have Consequently,
we have Y C(0);- C ;-
where we extend a random variable H to the probability
on 0g. Similarly, since 0 2 K, taking -
in the definition (3.17) of H, we
see that
Moreover, since E[ -
D, and by (2.20), we get
Remark 3.1 The idea of introducing the set H is similar to and inspired by the approach
of Kramkov and Schachermayer (1998), who work with the set of all nonnegative processes
G(\Delta) such that G(\Delta) -
X(\Delta) is a P \Gammasupermartingale for all admissible wealth processes X(\Delta).
Next, arguing as above (when deducing (3.16)), we obtain
~
where we have denoted
~
It is easily seen that \Gamma ~
R is a convex, lower-semicontinuous
and proper functional, in the terminology of convex analysis; see, for example, Aubin and
Ekeland (1984), henceforth [AE84].
Remark 3.2 It is straightforward to see that the inequality of (3.21) holds as equality for
some (-
z - 0, -
only if we have
and
for some F(T )\Gammameasurable random variable -
B that satisfies 0 -
a.s. We also
set
If (3.23) and (3.24) are satisfied, then (-
-) is optimal for the problem (3.11), under the
"change of variables" (3.14), since the lower bound of (3.21) is attained. Moreover, -
is optimal for the auxiliary dual problem
~
~
If we let
the conditions (3.23) and (3.24) become
and
H=1g
for some F(T )\Gammameasurable random variable -
B that satisfies 0 -
is the terminal wealth of the strategy (-) which is optimal for the problem (3.8).In light of the preceding remark, our approach will be the following: we will try to find
a solution -
H to the auxiliary dual problem (3.25), a number -
z ? 0, a random variable -
as above, and a pair (-) 2 A(x) such that (3.23) and (3.24) (or, equivalently, (3.27) and
are satisfied.
Theorem 3.2 For any given z ? 0, there exists an optimal solution -
for the
auxiliary dual problem (3.25).
Proof: Let H n 2 H be a sequence that attains the supremum in (3.25), so that
~
Note that, by (3.18), H is a bounded set in L
so that by Koml'os theorem
(see Schwartz 1986, for example) there exists a random variable -
a (relabeled) subsequence fH i g i2N such that
Fatou's lemma then implies -
by the Dominated Convergence
Theorem and concavity of ~
J(\Delta; z) we get
~
~
J(n
~
Thus, -
Lemma 3.1 The function ~
V (z) is continuous on [0; 1).
Proof: Let H 2 H and assume first z 1 ; z 2 ? 0. We have
~
Taking the supremum over H 2 H we get ~
do the same while interchanging the roles of z 1 and z 2 , we have shown continuity on (0; 1).
To prove continuity at z note that, by duality and (3.19), we have
~
for all z 1 ? 0; y ? 0. Choosing first y large enough and then z 1 small enough, we can make
the two terms on the right-hand side arbitrarily close to zero, uniformly in H 2 H.Proposition 3.1 For every
that attains the
supremum sup z-0 [ ~
Proof: Denote
Note that first show that
lim sup
so that the supremum of ff(z) over [0; 1) cannot be attained at z = 1. Suppose, on the
contrary, that there exists a sequence z n !1 such that lim n ff(z n
the optimal dual variable of Theorem 3.2 corresponding to z = z n . We have then
z n
by Dominated Convergence Theorem, a contradiction.
Consequently, being continuous by Lemma 3.1, function ff(z) either attains its supremum
at some -
z ? 0, or else ff(z) - Suppose that the latter is true. We
have then
~
z
z
for all z ? 0 and H 2 H. In particular, we can use the Dominated Convergence Theorem
while letting z ! 0 to get
for all H 2 HD . Taking the supremum over H 2 HD we obtain x - C(0), a contradiction
again.Denote -
z the optimal dual variable for problem (3.25), corresponding to
z of
Proposition 3.1. We want to show that there exists an F(T )\Gammameasurable random variable
such that the optimal wealth for the primal problem is given by CI(-z -
B),
where I(z; b) is given in (3.13). In order to do that, we recall some notions and results from
convex analysis, as presented, for example, in [AE84].
First, introduce the space
with the norm
and its subset
It is easily seen that G is convex, by the convexity of H. It is also closed in L. Indeed, if
we are given subsequences z n - 0 and H n 2 H such that (z n H in L, then
also have, from (3.18),
so that zH n ! Z in L
and we are done. If
z ? 0, we get H n ! Z=z in L
closed in
and we are done again. The closedness of G has been confirmed.
We now define a functional ~
~
It is easy to check that ~
U is convex, lower-semicontinuous and proper on L. Moreover, since
we have
~
from Proposition 3.1, and in the notation of Theorem 3.2, it follows that the pair -
G :=
optimal for the dual problem
~
Let L   := L
R be the dual space to L and let N(-z -
z) be the normal cone
to the set G at the point (-z -
z), given by
by Proposition 4.1.4 in [AE84]. Let @ ~
z) denote the subdifferential of ~
U at (-z -
z),
which, by Proposition 4.3.3 in [AE84], is given by
@ ~
Then, by Corollary 4.6.3 in [AE84], since (-z -
z) is optimal for the problem (3.35), we obtain
Proposition 3.2 The pair (-z -
G is a solution to
In other words, there exists a pair ( -
which belongs to the normal cone N(-z -
and such that \Gamma( -
belongs to the subdifferential @ ~
z).
From (3.36) and (3.37), this is equivalent to
and
It is clear from (3.40) (by letting z ! \Sigma1 while keeping Z fixed) that necessarily
On the other hand, if we let - z = z in (3.39), we get
Moreover, letting
H in (3.39), and recalling -
we obtain
H]:
Similarly, we get the reverse inequality by letting -
H in (3.39) (recall
that -
z ? 0 by Proposition 3.1), to obtain finally
This last equality will correspond to (3.23) with -
if we can show the following
result and recall (3.14).
Proposition 3.3 There exists an admissible pair (-
and such that (3.27) is satisfied.
(Here we set -
Proof: This follows immediately from (3.41) and (3.42), which can be written as
Y H]
(with 0g). Indeed, Theorem 3.1 tells us that the right-hand side is no
smaller than the minimal amount of initial capital needed to hedge C -
there exists a
that does the hedge.In order to "close the loop", it only remains to show (3.24).
Proposition 3.4 Let \Gamma (Y; y) 2 @ ~
Y is of the form
for some F(T )\Gammameasurable random variable B that satisfies a.s.
Proof: We have already seen that y = \Gammax. Define a random variable A by
From (3.40) with -
be such that
Then,
by (3.45). This implies
A - 0 on f-z -
for otherwise we could make Z arbitrarily small (respectively, large) on f-z -
(respectively, on f-z -
to get a contradiction in (3.46).
Suppose now that P C [A ! 0; - z -
There exists then
beacuse of (3.47). For a given " ? 0, let
on f-z -
in (3.46). This gives
The left-hand side is greater than
H!1g
contradiction to (3.48). Thus, we have shown
Going back to (3.46), this implies
for all Z 2 L
If we set now
we get from (3.50) and (3.47)
Using (3.49) and (3.51) in (3.45), we obtain
Suppose now that P C [A ?
z -
There exists then
(for a given " ? 0), (3.52) implies
The left-hand side is greater than ffi +P C [-z -
so that from (3.53) we conclude
contradiction. Therefore,
Together with (3.44), (3.47), (3.49) and (3.51), this completes the proof.We now state the main result of the paper.
Theorem 3.3 For any initial wealth x with there exists an optimal pair
for the problem (3.8) of minimizing the expected loss of hedging the claim C.
It can be taken as that strategy for which the terminal wealth X x;- (T ) is given by (3.28),
i.e.,
H=1g
Here (-z; -
H) is an optimal solution for the dual problem (3.35), and -
B can be taken as the
random variable B in Proposition 3.4, with (Y; y) replaced by some ( -
z)g, which exists by Proposition 3.2.
Proof: It follows from Remark 3.2. Indeed, it was observed in that remark that a pair
is optimal for the problem (3.8) if it satisfies (3.27) and (3.55) for some
F(T )\Gammameasurable random variable -
z - 0, -
The existence of
such a pair (-
established in Proposition 3.3 in conjunction with Proposition
3.4, with -
B, -
z and -
H as in the statement of the theorem. 2
The following simple example is mathematically interesting from several points of view.
It shows that the optimal dual variable -
H can be equal to zero with positive probabil-
ity, unlike the case of classical utility maximization under constraints (as in Cvitani'c and
Karatzas 1992). Moreover, - z -
H can be equal to one with positive probability, so that the
use of nonsmooth optimization techniques and subdifferentials for the dual problem is really
necessary. It also shows why it can be mathematically convenient to allow nonzero consump-
tion. Finally, it confirms that condition (3.5) is not always necessary for the dual approach
to work.
Example 3.1 Suppose r(\Delta) j 0 for simplicity, and let C - 0 be any contingent claim such
that P [C - x] ? 0. We consider the trivial primal problem for which so that
there is only one possible admissible portfolio strategy -(\Delta) j 0 (in other words, the agent
can invest only in the riskless asset). We do not assume condition (3.5), which, for these
constraints, is equivalent to C being bounded. It is clear that the value V (x) of the primal
problem is duality implies
for all z - 0, H 2 H (see (3.21)). Here we can take H to be the set of all nonnegative
random variables such that E[H] - 1. Let - z := P [C - x] ? 0 and -
z -
. It is then
easily checked that -
and that the pair ( -
z) attains equality in (3.56), so that the
optimal for the dual problem (3.35). One possible choice for the optimal
terminal wealth is
According to (3.55), this corresponds to -
while -(T
Remark 3.3 (i) Assumption 3.1 is satisfied, for example, if C is bounded. We need it
in order to get existence for the dual problem (3.35), due to our use of Koml'os theorem.
Example 3.1 shows that this assumption is not always necessary: in this example the dual
problem has a solution and there is no gap between the primal and the dual problem, even
when (3.5) is not satisfied.
(ii) If we, in fact, assumed that C is bounded, the switch to the equivalent formulation
(3.11) from (3.8) would not be necessary. (The reason for this is, the dual spaces of
are then the same, up to the equivalence class determined
by the set
Remark 3.4 Numerical approximations. Suppose that we have a Markovian model in
which r(t; S(t)), b(t; S(t)) and oe(t; S(t)) are deterministic and "nice" functions of time and
current stock prices, and so is the claim could then imagine doing
the following three-step approximation procedure to solve first the dual and then the primal
problem: First, in order to have differentiability rather than having to deal with subdifferen-
tials, one could replace the loss function with the function R p
for some p ? 1, as in Pham (1998). Second, in order to be able to use standard dynamic
programming and Hamilton-Jacobi-Bellman partial differential equations (HJB PDEs), one
could replace the auxiliary dual problem (3.25) by the approximating problem
~
~
for some large n 0 , where D n consists of those elements of D which are bounded by n, almost
surely, and where ~
J p corresponds to the dual problem associated with the loss function
R p (y). After the approximate optimal dual variable -
corresponding
z
(the one maximizing ~
are found, one has to hedge, under portfolio constraints
given by set K, the claim
where I p (\Delta) corresponds to the function I(\Delta; b) of (3:13) in the case of the loss function being
R p (\Delta). If we are in a Black-Scholes model with r, b and oe constant, the (constrained) strategy
for hedging X n 0
(T ) can be found quite easily, using the results of Broadie, Cvitani'c and Soner
(1998). Otherwise, one again has to use approximating HJB PDEs to calculate the values
of the aproximate discounted wealth process -
defined analogously to (3.2), with C
replaced by X n0 (T ) and D replaced by Dm , for some large m (see section 8 of [CK93]).
We plan to investigate properties of above described numerical approximations elsewhere.
4 Dynamic measures of risk
Suppose now that we are not quite sure whether our subjective probability measure P is
equal to the real world measure. We would like to measure the risk of hedging the claim
C under constraints given by set K, and under uncertainty about the real world measure.
According to Artzner et al. (1996), and Cvitani'c and Karatzas (1998), it makes sense to
consider the following quantities as the lower and upper bounds for the measure of such a
risk, where we denote by P a set of possible real world measures:
the maximal risk that can be incurred, over all possible real world measures, dominated by
its "min-max" counterpart
sup
the upper-value of a fictitious stochastic game between an agent (who tries to choose (-) 2
A(x) so as to minimize his risk) and "the market" (whose "goal" is to choose the real world
measure that is least favorable for the agent). Here, E Q is expectation under measure Q. A
question is whether the "upper-value" (4.2) and the "lower-value" (4.1) of this game coincide
and, if they do, to compute this common value. We shall answer this question only in a very
specific setting as follows. Let P be the "reference" probability measure, as in the previous
sections. We first change the margin requirement (2.9) to a more flexible requirement
where k is a constant such that 1 ? k -
we look at the special case of the constraints given by
In other words, we only consider the case of a market which is incomplete
due to the insufficient number of assets available for investment. In this case
~
and
fbounded progress. meas. processes
We define the set P of possible real world probability measures as follows. Let E be a set of
progressively measurable and bounded processes -(\Delta) and such that
We set
in the notation of (2.14) (note that the reference measure P is not necessarily in P). In other
words, our set of all possible real world probability measures includes all the "equivalent
martingale measures" for our market, corresponding to bounded "kernels" -(\Delta). This way,
under a possible real world probability measure P - 2 P, the model M of (2.1) becomes
d
in the notation of (2.15). The resulting modified model M - is similar to that of (2.1); now
the role of the driving Brownian motion (under P - ), but the stock return rates
are different for different "model measures" P - .
The following theorem shows that, if the uncertainty about the real world probability
measure is large enough (in the sense that all equivalent martingale measures corresponding
to bounded kernels are possible candidates for the real world measure), then the optimal
thing to do in order to minimize the expected risk of hedging a claim C in the market, is
the following: borrow exactly as much money from the bank as is needed to hedge C.
Theorem 4.1 Under the above assumptions we have
In other words, the stochastic game defined by (4.1) and (4.2) has a value that is equal to
the expected loss of the strategy which borrows C(0) \Gamma x from the bank, and then invests
according to the least expensive strategy for hedging the claim C.
Proof: Let (-   ; -   ) be the strategy from the statement of the theorem, namely the one for
which we have
in the notation of (3.2). Such a strategy exists by Theorem 3.1. It is clear that (4.3) is then
satisfied, so that (-
for all Q 2 P, it also follows that
On the other hand, we have here
K, so that H -
Ito's rule gives, in analogy to (2.7) and in the notation of (2.15),
R tr(u)du d-   (t)
for all - 2 D, since - 0 (\Delta)-   (\Delta) j 0. Therefore, -
X   (\Delta) is a P - \Gammalocal supermartingale bounded
from below, thus also a P - \Gammasupermartingale, by Fatou's lemma. Consequently,
is the expectation under P - measure. Since P - 2 P for all - 2 D, (4.11) and
Jensen's inequality imply
is a consequence of (4.9) and (4.12). 2

Acknowledgements

I wish to thank Ioannis Karatzas for suggesting the use of Koml'os
theorem and for providing me with the reference Schwartz (1986), as well for thorough
readings and helpful comments on the paper.



--R

A characterization of measures of risk.
Applied Nonlinear Analysis.






Optimal portfolio replication.


On the Pricing of Contingent Claims under Constraints.
Annals of Applied Probability
The asymptotic elasticity of utility functions and optimal investment in incomplete markets.
Dynamic L p

New proofs of a theorem of Koml'os.
--TR
hedging;incomplete markets;portfolio constraints;expected loss;dynamic measures of risk