Table Of ContentDiscussion Paper No. B{442
Pseudo{Arbitrage
A new Approach to Pricing
and Hedging in Incomplete Markets
Daniel Sommer
November 1998
Department of Statistics
Faculty of Economics
University of Bonn
Adenauerallee 24-42
D-53113 Bonn, Germany
and
KPMG
Risk Management Advisory Group
Marie{Curie{Str. 30
D-60439 Frankfurt am Main
e-mail:
sommer@addi.(cid:12)nasto.uni-bonn.de
TheauthorisgratefulforhelpfuldiscussionswithHans{PeterBurghof,Ru(cid:127)digerFrey,DietmarLeisen,
PhilippSch(cid:127)onbucher,DieterSondermannandthemembersoftheStatisticsDepartmentattheFaculty
of Economics at the University of Bonn. The opinions expressed in this paper are the author’s
personal views and do not necessarily represent the views of KPMG. All errors are the author’s
responsibility. Financial support by the Deutsche Forschungsgemeinschaft Sonderforschungsbereich
A
303 at the University of Bonn is gratefully acknowledged. Document typeset in LTEX.
Abstract
We develop a new approach to pricing and hedging contingent claims in incomplete
markets. Mimicking as closely as possible in an incomplete markets framework the
no{arbitrage arguments that have been developed in complete markets leads us to
de(cid:12)ning the concept of pseudo{arbitrage. Building on this concept we are able to
extend the no{arbitrage idea to a world of incomplete markets in such a way that
based on a concept of risk compatible with the axioms of Artzner et al. we can
derive unique prices and corresponding optimal hedging strategies without invoking
speci(cid:12)cassumptionson preferences(otherthan monotonicityandriskaversion). Price
processes of contingent claims are martingales under a unique martingale measure. A
comparison to a version of the Hull and White stochastic volatility model shows that
in contrast to their approach explicitlytaking into account optimalhedging strategies
leads to positivemarketpricesof risk for volatilityevenif the latter is instantaneously
uncorrelatedwiththestock priceprocess. Our resultsare, however,inagreementwith
the (cid:12)ndings of Lamoureux and Lastrapes.
JEL Classi(cid:12)cation: G12 G13
Keywords: arbitrage, pseudo{arbitrage, pricing of contingent claims, hedging, incom-
plete markets, stochastic volatility
CONTENTS 1
Contents
1 Introduction 2
2 An Example 6
3 The Two{Period Model 10
3.1 Basic De(cid:12)nitions and Results : : : : : : : : : : : : : : : : : : : : : : : 10
3.2 Pseudo{arbitrage free Prices : : : : : : : : : : : : : : : : : : : : : : : : 13
4 The Multi{Period Model 20
5 An Application to Stochastic Volatility 25
6 Conclusion 28
AppendixA 30
AppendixB 31
AppendixC 32
2 1. INTRODUCTION
1 Introduction
Since the pathbreaking papers by Black/Scholes [BS73] and Merton [Mer73] pricing
derivatives in completemarkets by no{arbitrage arguments has become standard. As
is well known the normative pricing theory these authors have developed is so com-
pelling for two main reasons. First, in order to derive their prices for derivatives it is
not necessary to invoke speci(cid:12)c preference relations of individual investors. This is so
because, second, given the prices of the primitive securities there are dynamic trad-
ing strategies in the latter assets that perfectly duplicate the payo(cid:11) of any derivative.
Henceanydeviationof thepriceof a derivativefromtheinitialpriceof theduplicating
portfolio strategy gives rise to trading strategies that imply riskless pro(cid:12)t opportu-
nities. These can, however, not prevail in equilibrium if investors have monotonic
preferences, which can be taken for granted.
The preceding can be looked upon from a slightly di(cid:11)erent angle. Suppose in a
worldofcompletemarketssomebodyproposedanalternativenormativepricingtheory
suggesting pricesforderivativesthatweredi(cid:11)erentfromarbitrage freeprices. Thenan
investorfollowing the theory of pricing by arbitrage and the trading strategies that go
withitwouldhavearisklesspro(cid:12)tablestrategyagainst anysuchpricingtheory. Hence
deviatingfromthe alternativenormativepricingtheorywould be individuallyrational
for every risk averse investor with monotonic preferences. In complete markets the
pricing theory for derivativesis thus an immediateconsequence of the hedging theory.
Unfortunately actual markets are incomplete. There are many reasons for that. A
number of prominent examples are stochastic volatility, jump risk, transaction costs,
short sale restrictions or evencreditrisk. Although the present paper is not primarily
on stochastic volatility the reason for incompleteness we have in mind in writing this
paper is essentially volatility risk. Therefore, as examples of how the problem of
market incompleteness has been addressed in the (cid:12)nance literature we shall mainly
review in this introduction the literature on stochastic volatility.
The obvious di(cid:11)erence between the pricing and hedging problem in completemarkets
as compared to incomplete markets is that in incomplete markets hedging arguments
are not su(cid:14)cient to deriveunique prices for derivatives. Rather the hedging approach
is used to derive partial di(cid:11)erential equations for the prices of derivatives that un-
fortunately are speci(cid:12)ed only up to the market prices of the non{traded sources of
risk. Hence in order to arrive at fully speci(cid:12)ed pricing equations assumptions need
to be made about these market prices of risk. These assumptions are often based on
more or less carefully developed general equilibrium arguments and include the use
of speci(cid:12)c utility functions often of the logarithmic or CRRA types. Another typical
approach to solving the problem of specifying a marketprice of risk is to assume that
therespectiveriskcan be diversi(cid:12)edaway which impliesa marketpriceof zero for this
source of risk. Based on these assumptions unique price processes for derivatives are
calculated, which are subsequently used to derive fully speci(cid:12)ed hedging strategies.
Intheliteratureonstochastic volatilitythereare anumberofpapers that takethisap-
proach. Important examplesare Hull/White[HW87], Wiggins [Wig87], Scott [Sco87],
Stein/Stein [SS91], Heston [Hes93], Ball/Roma [BR94], Duan [Dua95], and Renault
3
and Touzi [RT96]. However, while these papers claim to propose normative pricing
theories for derivatives under stochatic volatility the prices they propose rely essen-
tiallyon the assumptionsthat are madeon themarketpricesof volatilityrisk. Infact,
taking as a typical example a European call option on a stock, as a consequence of
the independent work of Frey and Sin [FS97] and Cvitani(cid:19)c, Pham and Touzi [CPT97]
it turns out that in di(cid:11)usion models for stochastic volatility where the support of the
squared volatilityis (0; ) one can always specify the marketpriceof risk of volatility
1
in such a way that any price for the option between its inner value and the stock price
itself can be supported. Given that neither the speci(cid:12)c assumptions on preferences
underlying the results in these papers nor the assumption of diversi(cid:12)abilityof volatil-
ity risk (in fact it is a common experience among traders that volatilitiesfor di(cid:11)erent
stocks tend to move together) can safely be regarded as empirically warranted it is
unclear in what sense the above papers do actually provide a normative pricing the-
ory for derivatives under stochastic volatility. In any case these papers leave open the
question why even if the market followed a di(cid:11)erent pricing theory than the one they
propose, adhering to the respective pricing theories in these papers should be individ-
ually rational for a trader or investor who most likely does not have the preference
relation assumed in the respective paper and who certainly has no knowledge about
the preference relations of all the other market participants that he faces.
The essential question to ask would be whether based on these theories it would be
possible to exploit market prices for derivatives that deviate from the prices given in
these papers. This issue has been addressed in Chesney and Scott [CS89] who claim
that option{trading based on their stochastic volatility model allows small gains to
be made. However, in their study they leave out the risk dimension. Based on
their stochastic volatility model they set up delta{neutral portfolios including a sec-
ond option to hedge volatility risk. However, their hedging portfolios will only be
delta{neutral if the marketactually follows their pricing theory for the option in their
hedging portfolio. This problemis recognisedin Bakshi et al. [BCC96] who in a study
of hedging e(cid:11)ectiveness consider minimal variance hedges in the underlying and the
bond besides delta{neutral hedges that also include a second option. Conceptually
both these papers point in the direction that the present paper takes, namely devel-
oping a normativepricing theory for derivativesin incompletemarketsthat similarto
the theory in complete markets draws its justi(cid:12)cation from a theory of hedging but
due to market incompleteness necessarily takes risk explicitly into account.
In that the present paper tries to develop a normative pricing theory it also distin-
guishes itself clearly from a recent strand in the literature that focus entirely on the
question of hedging. The best example of this approach is probably the literature on
super hedging such as it has been developed in the papers by Avellaneda, Levy and
Paras [ALP95] or El Karoui, Jeanblanc{Picquet and Shreve [KJPS98]. While this is
an ingenious theory it is often impractical because the initial amount of money one
has to invest in order to set up a super hedging strategy for a certain contingent claim
is muchhigher than the price one can achieve for this contingent claimin the market.
As examples we may again point to the previously mentioned results of Frey and Sin
[FS97] and Cvitani(cid:19)c, Pham and Touzi [CPT97] on the range of option prices when
the support of the volatility is (0; ). Hence the papers dealing with super hedging
1
4 1. INTRODUCTION
typically assume that volatility is bounded and under this assumption determine su-
per hedging strategies. If the actual volatility leaves the assumed boundaries nothing
is said about the quality of the hedge that these strategies provide. This problem is
avoided in the concept of quantile hedging proposed by F(cid:127)ollmer and Leukert [FL98].
Fora givenmaximalprobabilityofmakinga loss inthefuturetheseauthors determine
the cheapest hedging strategy that will achieve the desired loss probability.
Neither super nor quantile hedging can, however, give rise to a normative pricing
theory. Rather they either take as an input the prices that the marketis ready to pay
for a contingent claim and determine the least loss probability or the widest interval
of volatilitythat can be attained with an optimalhedging strategy whose costs do not
exceedthe givenmarketpriceof thecontingentclaimconsidered. Or theystart with a
given loss probability or a given interval for the volatilityand determinethe cheapest
super or quantile hedging strategy that is compatible with these assumptions.
The situation seems to be slightly di(cid:11)erent for the concept of (local) risk minimisa-
tion suggested by F(cid:127)ollmer and Sondermann [FS86], Schweizer [Sch91], F(cid:127)ollmer and
Schweizer[FS90], and Hofmann, Platen and Schweizer [HPS92]. Given the stochastic
processes of primitivesecurities, in their incomplete market models these authors are
able to determine unique (locally) risk minimising hedging strategies for contingent
claims based on the primitive securities. One might be tempted to regard the value
processes of these strategies as suitable candidates for the price processes of the con-
tingent claims under consideration. If this could actually be done their theory would
indeed combine in incomplete markets a hedging theory with a normative pricing
theory.
However, there are a number of problems with this approach. The least seems to be
that there exists an inconsistency between discrete{timemodels and continuous{time
models. While in continuous{time models the value processes of locally risk min-
imising hedging strategies can be determined by taking conditional expectations of
the payo(cid:11)s of the contingent claims considered under the so called minimal martin-
gale measure no such martingale measure need exist in discrete{time models. This
problem has been addressed by Elliott and Madan [EM98] who in discrete time have
introduced a di(cid:11)erent concept of risk minimisation than the above authors. With
this modi(cid:12)cation even in discrete{timemodels they can represent the value processes
of their hedging strategies as martingales under a speci(cid:12)c probability measure. The
two connected problems that neither of the two approaches addresses are (cid:12)rst that
the objective function i.e. the concept of risk used in this literature is unintuitive
and economicallyinappropriate for the problem at hand and second that if the actual
price process of the contingent claim is di(cid:11)erent from the value process of the locally
risk minimising strategy then simply no locally risk minimising strategy exists and
the theory fails to give any answer as to how to hedge optimally. In particular this
literature uses quadratic risk functions. This implies that any deviation | even pos-
itive | of the value of the hedging portfolio from the payo(cid:11) of the contingent claim
is considered a manifestation of risk. Obviously this is in sharp contrast to the ideas
underlying the literature on super hedging which only regards as risk the danger of
a shortfall of the value of the hedging portfolio behind the payo(cid:11) of the contingent
claimto be hedged. Clearly only the latter makes economic sense since in hedging no
5
one is interested in hitting precisely the payo(cid:11) of a contingent claim. What counts is
not to lose money.
In view of this literature it seems fairly di(cid:14)cult to (cid:12)nd a normative pricing theory in
incompletemarketsthatdrawsitsjusti(cid:12)cationfromatheoryofhedging. Nevertheless,
this is exactly what this paper tries to achieve. Just as has been done in the literature
on super or quantile hedging we assume an asymmetric concept of risk namely the
concept of an expected decrease in portfolio value. What still allows us to derive
unique prices for contingent claims is that just as in arbitrage theory we do not only
look at short positions in contingent claims but also at long positions. The question
we ask is whether at given market prices for a contingent claim and for the primitive
assets itis possible to makea strictlypostive expectedpro(cid:12)t fromenteringthe market
for the contingent claim while running the least possible risk from doing so. Finding
an answer to this question will sometimes require going long sometimes it will imply
going short the contingent claim. With this approach we are able to modify the no{
arbitrage concept so that it becomes extendable to the case of incomplete markets.
The extended concept will be called the concept of pseudo{arbitrage.
Using this concept we obtain unique prices for contingent claims that do not give
rise to pseudo{arbitrage opportunities. These prices are justi(cid:12)ed by the fact that
whenever market prices deviate from pseudo{arbitrage free prices then similar to
what is the case in completemarketswe can o(cid:11)er trading strategies that yieldstrictly
postive expected payo(cid:11)s while being risk minimal in well de(cid:12)ned sense. In fact, our
concept of risk minimality in multi{period securities markets will be an adaptation
of the concept of local risk minimsation introduced by Schweizer. Thus similar to
what is the case in complete markets our theory o(cid:11)ers an integrated approach to
pricing, hedging and risk management. Moreover, in our frameworkpseudo{arbitrage
free prices can be represented as expected payo(cid:11)s of the contingent claims under a
unique martingale measure even in the discrete{timeframework, which we shall work
in throughout the entire paper. Finally we shall apply our theory to a version of
the Hull and White [HW87] stochastic volatility model. We shall show that unlike
what was assumed by Hull and White our theory does not lead to a zero market
price of risk for volatilityeven if the volatility process is instantaneously uncorrelated
with the stock price process. Rather we obtain a positive market price of risk that
diminishes with the size of the stock price volatility itself. This result is line with the
interpretation that Lamoureux and Lastrapes [LL93] giveto their empiricalresults on
the predictive power of implied volatilities.
The present paper is organised as follows. In section two we shall explain our ideas
in a simple two{period example. In section three we shall develop the general theory
for the two{period case. This will be transferred to the multy{period framework in
section four. Section (cid:12)ve contains our analysis of the modi(cid:12)ed Hull/White stochastic
volatility model. Section six (cid:12)nally concludes and gives some suggestions for further
work.
6 2. AN EXAMPLE
2 An Example
The purpose of this section is to provide an intuitive understanding of the economic
ideas that underly our approach to the pricing and hedging of contingent claims in
incomplete markets.
Consider the two{period model represented in Figure 1. We assume that there are
frictionless marketsfor two securities, a stock S and a riskless bond B. The prices for
the stock and the bond are assumed given as are the transition probabilities p1j from
the origin to the di(cid:11)erent states in the second period. We want to (cid:12)nd a unique price
C00 in period 0 for the contingent claimwhich has payo(cid:11)s C1; in period 1. With these
(cid:1)
payo(cid:11)s the contingent claim can be interpreted as a call{option on the stock with a
strike of 50.
C = 0 C = 0 C = 20
S 1 ,2 = 30 S 1 ,1 = 45 S 1 ,0 = 70
1,2 1,1 1,0
B = 1 B = 1 B = 1
1,1 1,0
1,2
p = 0,1 p = 0,1 p = 0,8
1,2 1,1 1,0
C = ?
0,0
S = 50
0,0
B = 1
0,0
Figure 1: Market Structure
It is easy to see that by no{arbitrage arguments C00 must be between 4 and 10. Our
aim is now to argue that there is an intuitive extension of the no{arbitrage concept
that singles out a unique price C00 inside this interval.
To thisend assumethat C00 was slightlybelow10. Therewould no longer bea riskless
hedging strategy in stocks and bonds for the contingent claim. However, we can still
ask how much risk an investor would inevitably have to accept if he went long or
short the contingent claimand hedged this position in stocks and bonds in such a way
that his risk of losing money from the entire transaction became minimal. Against
this backgrond consider taking a short position in the contingent claim at a price of
C00 = 10 (cid:15). Assumeyou wanted to investthis amount of moneyin the stock and the
(cid:0)
bond so as to minimise the overall expected loss from this portfolio. This quantity is
given by the following expression
0:1[’(30 50)+(10 (cid:15))](cid:0)+0:1[’(45 50)+(10 (cid:15))](cid:0)+0:8[ 20+’(70 50)+(10 (cid:15))](cid:0):
(cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0)
It is easy to see that its minimumis attained by choosing ’, the holdings in the stock,
equal to 0:05(10+(cid:15)). This impliesthat in states (1;0) and (1;1) in the second period
7
the value of this portfolio will be nonnegative. In state (1;2) it will have a value of
2(cid:15). The expected loss from this portfolio will hence be 0:2(cid:15). On the other hand the
(cid:0)
expected value of the portfolio is 0:025(30 13(cid:15)), which is positive for (cid:15) < 30=13 or
(cid:0)
C00 > 100=13.
Compare these results to those from going long the contingent claim and (cid:12)nancing
this with a portfolio in stocks and bonds. The expected loss is then given by
0:1[’(30 50) (10 (cid:15))](cid:0)+0:1[’(45 50) (10 (cid:15))](cid:0)+0:8[20+’(70 50) (10 (cid:15))](cid:0):
(cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0)
This expression is minimisedby choosing ’ = 0:05(10+(cid:15)) i.e. by selling 0:05(10+(cid:15))
(cid:0)
units of the stock. The remaining funds are invested in the bond. In this case the
portfolio value in the second period will be nonnegative in states (1;0) and (1;2),
whereas it will be negative in state (1;1). The minimal expected loss turns out to be
1=8(6 (cid:15)), whichexceedsthe minimalriskofthe short position inthecontingentclaim
(cid:0)
for (cid:15) < 30=13. Moreover, the expected value of this portfolio will be 13=40(cid:15) 3=4,
(cid:0)
which is negative for (cid:15) < 30=13 or C00 > 100=13.
These calculations reveal that as long as C00 > 100=13 there is actually a whole range
of short positions in the contingent claim,which at zero cost o(cid:11)er a positive expected
payo(cid:11) at a lower risk (in terms of expected loss) than the minimal risk of any long
position in the contingent claim that could be entered into at zero cost. Hence, even
without specifyingan exact tradeo(cid:11) betweenrisk and return wemayconcludethat for
anyC00 > 100=13 itispossibleto (cid:12)nd riskaverseinvestorswithmonotonicpreferences
who would prefer taking a short position in the contingent claimto both taking a long
position in this asset or staying out of this market completely.
Particular investors of this type would be arbitrageurs. In the classical sense these
would be investors seeking riskless pro(cid:12)t opportunities. Such riskless pro(cid:12)t oppor-
tunities hardly ever exist in practice since actual markets are incomplete and small
deviations of actual market prices for contingent claims from theoretical prices ob-
tained in completemarkets models need by no means indicate riskless pro(cid:12)t opportu-
nities. Nevertheless, we observe that so called \arbitrage desks" in banks are busily
trading well inside theoretical arbitrage bounds for the prices of derivatives so that
these bounds are actually never attained. Therefore, we would argue that it would be
more realistic to describe the behaviour of arbitrageurs as that of investors seeking
positive pro(cid:12)t at as low a risk as possible. We suggest that when looking at a market
for a particular contingent claim arbitrageurs would consider that zero{cost position
in this asset that would imply the lowest possible risk. If this portfolio allowed them
to achieve a payo(cid:11) that would be su(cid:14)cient to compensate them for taking the risk
they would enter the market. Otherwise they would stay out.
This vision of arbitrageurs means that we do not regard them as in(cid:12)nitely risk averse
investors. However, by the role assigned to them the set of positions or trading
strategies arbitrageurs would be allowed to choose from would be restricted. In the
market for the contingent claim they would only be allowed to take risk minimal
zero{cost positions. Such restrictions do not seem implausible since banks do assign
di(cid:11)erent trading policies to di(cid:11)erent trading books. That is they place restrictions on
the types of positions that may be taken by the traders responsible for a particular
8 2. AN EXAMPLE
book. \Arbitrage policies" comparable to the one described above are often popular
with banks since they are viewed to yield more stable if sometimes smaller pro(cid:12)ts
than taking outright positions.
If we adopted this | admittedly stylised | vision of arbitrageurs and the ensuing
extended idea of \arbitrage", in our example it would not seem reasonable to assume
that as the priceof the contingent claimdropped slightlybelow 10 all the arbitrageurs
wouldimmediatelyleavethemarket. Ratherwewouldexpectmanyofthemtoremain
activeinthemarketpreferringtheshort position to thelongposition inthecontingent
claim. However, arbitrageurs will di(cid:11)er in their perceptions of the tradeo(cid:11) between
risk and return. Hence, as the price for the contingent claim approached 100=13
arbitrage activitywould fade. AtC00 = 100=13 it would cometo a completeend since
at minimal risk no positive expected payo(cid:11) could be made and hence, whatever the
arbitrageurs’ exact risk{return tradeo(cid:11), it would be preferabel for them to stay out
of the market.
A similar argument could be made for C00 < 100=13 with arbitrageurs preferring the
long to theshort position. Wewould, therefore,argue that C00 = 100=13 is theunique
price for the contingent claim at which in the market for this asset there would be no
arbitrage activity of the type described above.
Figure 2 summarisesthe situation. The two schedulesrepresent | for di(cid:11)erentvalues
of C00 | the minimalrisk that an investor would have to face when setting up at zero
cost a short (decreasing schedule)or a long position (increasing schedule) respectively
in the contingent claim. They intersect at C00 = 100=13. This can be seen as a
point of indi(cid:11)erencebetween long and short positions in the contingent claim since at
this price risk and expected payo(cid:11) from both positions in this asset are equal. This
provides a further argument for why at this point all arbitrage activity should come
to a halt since for an investor interested only in taking positions with minimal risk
there would be no reason any more to prefer one position to the other.
MRL
MRS
6 / 13
4 100 / 13 10 C
0,0
Figure 2: Minimum Risk Schedules
Summingup we argue that it makes sense to assume that there is some sort of \arbi-
trage activity" going on even within the arbitrage bounds for the prices of contingent
Description:Pseudo Arbitrage. A new Approach to Pricing and Hedging in Incomplete Markets. Daniel Sommer. November 1998. Department of Statistics.
