From actuarial to financial valuation principles
Martin Schweizer
Technische Universität Berlin, Fachbereich Mathematik, MA 7-4, Straße des 17. Juni 136, D-10623 Berlin, Germany
Received 1 July 1999; received in revised form 1 July 2000; accepted 19 September 2000
Abstract
A valuation principle is a mapping that assigns a number (value) to a random variable (payoff). This paper constructs a transformation on valuation principles by embedding them in a financial environment. Given an a priori valuation ruleu, we define the associated a posteriori valuation rulehby an indifference argument: Theu-value of optimally investing in the financial market alone should equal theu-value of first selling the payoff at itsh-value and then choosing an optimal investment strategy inclusive of the payoff. In anL2-framework, we explicitly construct in this way the financial transforms of the variance principle and the standard deviation principle. © 2001 Elsevier Science B.V. All rights reserved.
MSC:62P05; 91B28 Subj. Class:G10
Keywords:Finance; Insurance; Variance principle; Standard deviation principle; Variance-optimal martingale measure
1. Introduction
In recent years, there has been an ever increasing trend to bring finance and insurance closer together. The motivation for the present paper could almost be quoted from Bühlmann (1997) who in this context also coined the term “actuary of the third kind” in Bühlmann (1987): “. . . finance and insurance mathematics should be presented to today’s students as one discipline”. In accordance with this, our starting point is the observation that the valuation of random amounts is an important topic in both actuarial and financial mathematics and has been studied extensively in both fields. Almost any textbook includes a treatment under headings like premium principles or derivative pricing. We attempt to bring these approaches together by embedding an actuarial valuation principle in a financial environment.
The basic idea is simple. We begin with an a priori valuation rule which assigns a number (premium,price) to any random payoff from a suitable class. Typically, this rule is given or motivated by an actuarial premium principle. But the payoffs we consider do not exist in a vacuum; they are surrounded by a financial environment described by the outcomes of trades available to participants in a financial market. Such trades can be used to reduce the risk one has contracted by the sale or purchase of some random amount like an insurance claim or a financial obligation. To value a given payoff in this environment, we compare two procedures. One is to ignore the payoff completely and simply trade on the financial market in a subjectively optimal way. More precisely, one tries to obtain via trading from a given initial capital a final outcome with maximal value, where the value is computed according to the given
E-mail address:mschweiz@math.tu-berlin.de (M. Schweizer).
a priori rule. Alternatively, one can first sell the payoff under consideration to increase one’s initial capital and then look for a trade whose resulting net final outcome (trading outcome minus payoff) has again maximal value. The selling price for the payoff is then defined implicitly by equating these two maximal values; it thus compensates for the payoff since one becomes indifferent between optimal trading alone and the combination of selling and optimal trading inclusive of the payoff. The resulting a posteriori valuation is called the financial transform of the a priori valuation rule.
Of course, this abstract program is too general to be feasible. We therefore specialize the financial environment to a frictionless market modelled by a linear subspaceG ofL2with a riskless assetB. We consider two specific examples of actuarial valuation principles and explicitly determine their financial transforms. As a whole, the paper is a joint venture between finance and insurance. Finance makes explicit the transformation mechanism and insurance provides the input on which the mechanism can operate. In particular, an actuarial justification for the choice of one particular a priori valuation could via this approach lead to a foundation for pricing in an incomplete financial market. An alternative economic approach to price insurance contracts has recently been suggested by Kliger and Levikson (1998). Their idea is to determine the premium for an insurance product by maximizing some form of expected profit that involves, in particular, a specific type of insolvency costs. However, their method does not include any genuine financial component and thus is rather different from what we propose here.
In some form or other, most of the ideas in our program have appeared before. Using an indifference argument to implicitly determine a valuation is a standard approach from economics; it has recently been taken up again in a number of papers on option pricing under transaction costs. Pricing and hedging with a mean–variance utility has been studied by Mercurio (1996) in a special case of our framework forG. His paper in fact provided the main single source of motivation for our work. Replacing the variance by the standard deviation has been suggested by Aurell and ˙Zyczkowski (1996) in the very special case where the financial environment is generated by a trinomial model for stock prices. The main contribution of the present paper is that it combines all these isolated ideas into a unified framework and explicitly points out that this induces a transformation on the set of valuation rules. It would be very interesting to see other examples of financial transforms of particular valuation principles.
The paper is organized as follows. Section 2 explains the financial background(G, B), introduces a no-arbitrage type condition and presents some of its implications. In particular, we obtain a decomposition of any random payoff into an attainable part and a non-hedgeable part. The former is riskless and the latter contains all the financial riskiness of the payoff. Section 3 describes a change of measure required for working with discounted payoffs under general interest rates and defines the important concept of theB-variance-optimal signed(G, B)-martingale measureP˜. Section 4 gives a more formal description of the basic idea and determines the financial transform of the actuarial variance principle. This turns out to have a very intuitive structure. It is again a variance principle, but the expectation is taken underP˜and the variance component (risk-loading) is based only on the non-hedgeable part of the payoff. A similar result is obtained in Section 5 where we find the financial transform of the actuarial standard deviation principle. Section 6 contains two basic examples and an extension.
2. Approximate profits and projections inLLL222
This section presents the abstract financial background and some of its basic properties. Let (Ω,F, P )be a probability space andL2 = L2(Ω,F, P )the space of all square-integrable real random variables with scalar product(Y, Z)=E[YZ] and normkYk =pE[Y2]. For any subsetYofL2, we denote byY⊥:= {Z∈L2
|(Z, Y )= 0 for allY ∈Y}the orthogonal complement and byY¯the closure ofYinL2. FixB ∈L2withB >0P-a.s., letGbe a fixed subset ofL2and setA:=RB+G. We denote byπthe projection inL2on the closed linear subspaceG⊥.
expressing everything in discounted quantities. The setAis the space of those random payoffs which are strictly attainable in the sense that they can be obtained as the final wealth of some trading strategy with some initial capital. Later on, we shall assume thatGis a linear subspace ofL2; this corresponds to a financial market without frictions like transaction costs, constraints or other restrictions on strategies. Square-integrability is imposed to obtain a nice Hilbert space structure and because we want means and variances to exist.
Example. LetT ⊆ R+ be a time index set andX = (Xt)t∈T anRd-valued semimartingale with respect toP and a filtrationF=(Ft)t∈T on(Ω,F). LetΘbe the space of allRd-valuedF-predictableX-integrable processes ϑ=(ϑt)t∈T such that the stochastic integral processG(ϑ ):=RϑdXis in the spaceS2of semimartingales. Then we could takeB ≡1 andG:=GT(Θ), whereT :=supT is the time horizon of our economy. In this example,
Xmodels the discounted price evolution ofd risky assets and eachϑ ∈Θ can be interpreted as a self-financing dynamic portfolio strategy so thatGT(ϑ )describes the total gains that result from trading according toϑ. For a
continuous-time model whereT =[0, T] for someT ∈(0,∞], the spaceGT(Θ)has been studied by Delbaen
et al. (1997) and Choulli et al. (1998, 1999), among others. Ford = 1 andT = {0,1, . . . , T}withT ∈ N, Schweizer (1995) has studied the projection inL2onGT(Θ), and Mercurio (1996) has introduced and computed
mean–variance utility prices under an additional (restrictive) condition on X. Also in discrete time, Aurell and ˙Zyczkowski (1996) have examined mean-standard-deviation utility prices for a very special processX. The results of Mercurio (1996), and Aurell and˙Zyczkowski (1996) have been generalized by Gharagozlou (1997) and Schweizer (1999).
Definition. We say thatGadmits no approximate profits inL2ifG¯does not containB.
With the preceding interpretations, this notion is very intuitive. It says that one cannot approximate (in the L2-sense) the riskless payoffB by a self-financing strategy with initial wealth 0. This is one way to impose a no-arbitrage condition on the financial environment; loosely speaking, it should be impossible to turn nothing into something without incurring costs. For the case whereB≡1 andGconsists of elementary stochastic integrals with respect to a given square-integrable stochastic process, a very similar condition has been studied by Stricker (1990).
Lemma 1. IfGis a linear subspace ofL2,the following assertions are equivalent: 1. Gadmits no approximate profits inL2.
2. π(B)is not P-a.s. identically0. 3. E[Bπ(B)]>0.
4. G⊥∩(RB+ ¯G)6= {0}.
5. There is some Z inG⊥∩(RB+ ¯G)with(B, Z) >0.
Proof. SinceG is a linear subspace ofL2,G⊥⊥ = ¯G. ThusL2 = G⊥⊕ ¯G and so the riskless payoffB has the orthogonal decomposition
B=π(B)+(B−π(B)).
In particular,π(B)∈G⊥∩(RB+ ¯G)and
E[Bπ(B)]=E[(π(B))2]≥0.
This shows that (1)–(4) are all equivalent and imply (5) withZ =π(B). As (5) clearly implies (4), the proof is
complete.
Actually, it is easy to show that linearity ofGimplies thatG⊥∩(RB+ ¯G)=π(RB), but we shall not use this
Lemma 2. LetGbe a linear subspace ofL2admitting no approximate profits inL2.Then: 1. A⊥⊥= ¯A=RB+ ¯G=RB+G⊥⊥.
2. Ais closed inL2if and only ifGis closed inL2.
Proof.
1. SinceAandGare both linear subspaces, the first and third equalities are clear without further assumptions. Any g ∈ ¯Gis the limit inL2 of a sequence(gn)inG; hencecB+gn = anis a Cauchy sequence inAand thus
converges inL2to a limita∈ ¯Aso thatcB+g=a∈ ¯A. This gives the inclusion “⊇” for the middle equality. For the converse, we use the assumption thatG admits no approximate profits inL2to obtain from Lemma 1 an elementZ ∈G⊥∩(RB+ ¯G)with(B, Z) >0. For anya ∈ ¯A, there is a sequencean =cnB+gninA
converging toainL2. SincecnB+gn∈RB+Gfor alln, we conclude that
(an, Z)=(cnB+gn, Z)=(cnB, Z)=cn(B, Z)
converges inRto(a, Z)and since(B, Z) >0,(cn)converges toc=(a, Z)/(B, Z). Therefore,gn=an−cnB
converges inL2tog :=a−cBand since this limit is inG¯, we havea =cB+g∈ RB+ ¯Gwhich proves the inclusion “⊆”.
2. The “if” part is immediate from (1). Conversely, letgnbe any sequence inGconverging inL2to someg∞which
is inA¯sinceG ⊆A. IfAis closed inL2, we obtaing∞∈A, henceg∞=cB+gfor somec∈Randg∈G. BecauseGadmits no approximate profits inL2, we can chooseZas in Lemma 1 to obtain
c(B, Z)=(cB+g, Z)=(g∞, Z)= lim
n→∞(gn, Z)=0,
sincegn∈GandZ∈G⊥. But since(B, Z) >0, we conclude thatc=0 so thatg∞=g∈Gand this proves the
“only if” part.
The next result provides an elementary but important decomposition for arbitrary payoffs inL2. Its interpretation is deferred until the end of Section 3.
Corollary 3. LetGbe a linear subspace ofL2admitting no approximate profits inL2.Then everyH ∈L2has a unique decomposition as
H =cHB+gH +NH withcH ∈R, gH ∈ ¯GandNH ∈ ¯A⊥. (2.1)
In particular, we haveE[BNH]=0andE[NHg]=0for allg∈ ¯G.
Proof. WritingL2= ¯A⊕ ¯A⊥yieldsH=aH+NHwith unique elementsaH =cHB+gH ∈ ¯AandNH ∈ ¯A⊥. IfaH =cB+gfor somec∈Randg∈ ¯Gand ifZis as in Lemma 1, we obtainc(B, Z)=(aH, Z)=cH(B, Z) which proves the uniqueness of the decomposition (2.1).
Consider now the optimization problem
MinimizeE[(H −cB−g)2] over allc∈Rand allg∈G (2.2) for a fixedH ∈L2. SinceA=RB+G, this can equivalently be rewritten as
Lemma 4. LetGbe a linear subspace ofL2admitting no approximate profits inL2.For every fixedH ∈L2,we then have in terms of the decomposition(2.1)that
J0:= inf
(c,g)∈R×GE[(H −cB−g) 2]
= inf
a∈AkH−ak 2
=E[(NH)2].
IfgH is inG,then the solution of(2.2)and(2.3)is given by(cH, gH)∈ R×G.In particular, this is always the case ifGis closed inL2.
Proof. Clearly, we haveJ0=infa∈AkH−ak2=infa∈ ¯AkH−ak2. By Corollary 3, the projection ofHinL2on ¯
AiscHB+gH and so
J0= kH−cHB−gHk2= kNHk2=E[(NH)2].
The remaining assertions are clear.
3. Choosing the measure
In this section, we introduce some additional measures on(Ω,F)for two reasons. Firstly, we can then give alternative expressions for the quantityJ0in Lemma 4 and the constantcH in the decomposition (2.1) ofH. More importantly, however, one of these measures is also needed in the formulation of our a priori valuation rules in the next section.
To motivate the next definition, we first reconsider the problem (2.2) of minimizingE[(H −cB−g)2] over all c∈Randg∈G. Think ofHas some random payoff to be made at the final date in our environment. If we choose some initial capitalcand some self-financing strategy represented by someg∈G, our final wealth will becB+g and thus leave us with a net loss ofH−cB−gat the end. Hence (2.2) is just the problem of minimizing the expected squared net loss by choice of an appropriate strategy(c, g).
In this formulation, all payoffs are measured in the originally given monetary units. If we want to re-express this in units of our riskless assetB, we have to divide all final date payments byB; this is also called discounting with respect to B. In these new units, we can then write (2.2) as minimizing
E
"
B2
H
B −c− g B
2#
over (c, g)∈R×G,
or equivalently as
MinimizeEB
"
H B −c−
g B
2#
over allc∈Rand allg∈G,
where the new probability measurePB≈P is defined by
dPB
dP := B2
E[B2]. (3.1)
Here and in the sequel, all expectations and variances without indices refer toP. It is clear that a random variable H is in L2 = L2(P )if and only if H /B is inL2(PB)and thusPB is the natural measure for working with B-discounted quantities.
Lemma 5. If H ∈ L2 has a decomposition H = cHB +gH +NH as in(2.1), thenEB[NH/B] = 0 and
CovB(g/B, NH/B)=0for allg∈ ¯G.This means that underPB,NH/Bhas zero mean and is uncorrelated with the space(1/B)G¯of B-discounted gains.
Proof. From the definition ofPB, we obtainEB[NH/B]=(1/E[B2])E[BNH]=0 by (2.1). Hence, we have for
The next concept has been introduced by Schweizer (1996) forB ≡ 1; the extension to positiveB follows Gouriéroux et al. (1998).
A signed(G, B)-martingale measureP˜is calledB-variance-optimal if
for all signed(G, B)-martingale measuresQ.
(To indicate the dependence onB, we could writeP˜B instead ofP˜, but asB is fixed throughout, this would only clutter up the notation.)
ThusP˜isB-variance-optimal if and only if itsB-discounted density with respect toPBhas minimalPB-variance among all signed(G, B)-martingale measuresQ; hence the terminology.
2. As already pointed out in Delbaen and Schachermayer (1996a,b), one has to consider signed measures because without additional assumptions,P˜ does turn out to become negative on some sets. This reflects the fact that a quadratic “utility function” (which underlies the idea of a mean–variance criterion) is not increasing.
Lemma 7. LetGbe a linear subspace ofL2.The B-variance-optimal signed(G, B)-martingale measureP˜ exists if and only ifGadmits no approximate profits inL2.In that case,P˜ is unique and given by
measure, then(1/B)dQ/dP ∈G⊥by definition and so
This shows that the set of signed(G, B)-martingale measures is nonempty if and only ifπ(B)6≡0, hence by Lemma 1 if and only ifG admits no approximate profits inL2. Since the set of allB-discounted densities(1/B)dQ/dP of signed(G, B)-martingale measuresQis convex and closed inL2, it is clear that theB-variance-optimal signed (G, B)-martingale measures then exists and is unique. To prove (3.2), we first note thatBπ(B)/E[Bπ(B)] is well defined by Lemma 1 sinceGadmits no approximate profits inL2and soP˜ defined by the first equation in (3.2) is a signed(G, B)-martingale measure. Moreover,π(B)is inRB+ ¯Gaccording to the proof of Lemma 1, hence of the formcB+gfor somec∈Randg∈ ¯G, and so
is constant over all signed(G, B)-martingale measuresQ. IfQis now an arbitrary signed(G, B)-martingale measure, then so isR:=2Q− ˜P, andQ= ˜P +12(R− ˜P ). Hence, we obtain
because the mixed term disappears thanks to (3.3). This proves thatP˜defined by the first expression in (3.2) is indeed B-variance-optimal. The alternative second expression follows if we use the first half of (3.2) and the definition of PBto obtain
For future reference, we compute the minimal variance VarB[dP /dP˜ B] explicitly in terms ofP , Bandπ(B).
We first observe that
Corollary 8. LetGbe a linear subspace ofL2admitting no approximate profits inL2.ForH ∈L2,consider the representationH=cHB+gH +NH from Corollary3.Then
whereE˜denotes expectation with respect to the B-variance-optimal signed(G, B)-martingale measureP˜.Moreover,
Proof. By (3.2) and the proof of Lemma 1,(1/B)dP /dP˜ is inG⊥∩(RB+ ¯G)=G⊥∩ ¯Aby Lemma 2. Since gH ∈ ¯GandNH ∈ ¯A⊥, we thus obtain
˜
E
H
B
= 1
B dP˜ dP, c
HB
+gH+NH
!
=cHP˜[Ω]=cH.
Moreover, Lemmas 4 and 5 imply that
J0=E[(NH)2]=E[B2]EB
"
NH B
2#
=E[B2] VarB
NH
B
.
Corollary 8 allows us to give a very intuitive financial interpretation for the decomposition
H =cHB+gH +NH (3.6)
in Corollary 3. It tells us thatH splits naturally into an attainable partcHB+gH ∈ ¯Aand a non-hedgeable part NHwhich is orthogonal to the spaceA¯of attainable payoffs. The constantcH is the initial capital of the attainable part; it can be computed in a simple way as theP˜-expectation of theB-discounted payoffH /B. Moreover, (3.5) tells us that thePB-variance of theB-discounted non-hedgeable part ofH is a measure for the intrinsic financial risk ofH in terms of approximation inL2by attainable, hence perfectly replicable payoffs.
Remark. In the present generality, one has to be a bit careful about the above interpretation. In fact, it may happen that a nonnegative payoffH ≥0 hascH <0 becauseP˜is in general a signed measure. The intuition behind such a situation is that a quadratic “utility” has also a part where one is actually risk-seeking instead of risk averse and that the values ofHlie to a large extent in that part.
4. The financial variance principle
This section explains the basic idea behind our approach and shows in a first example how it works. We consider a fixed random variableH ∈L2and think of this as a random payoff as in Section 3. In a financial context,Hcould represent the net payoff of some derivative product, e.g., a European call option. In insurance terms,−H should be thought of as a claim amount to be paid by the insurer. Since financial derivatives are often termed contingent claims, it seems appropriate to callHa general claim. How much should we charge or pay if we sell or buyγunits of this claim?
This question has a standard answer from classical option pricing theory in the special case whereH is strictly attainable in the sense thatH ∈A. Then we can writeH =cHB+gHwithcH ∈RandgH ∈G, and the price per unit ofHmust becHto avoid arbitrage. To understand the idea behind this argument, suppose for instance thatHis nonnegative and offered at a pricex < cH. Then one can buycH/xunits ofHand finance this initial outlay ofcH with(−cH,−gH), i.e., by borrowing the amountcHand trading according to the strategy associated to−gH. The terminal payoff from this costless deal is then(cH/x)H−cHB−gH =((cH/x)−1)H ≥0; thus one has turned nothing into something, but absence of arbitrage postulates that this is impossible. A similar argument applies if one can sellH for more thancH. However, this entire reasoning depends crucially on the attainability ofH and therefore breaks down in a general incomplete financial market. A typical contingent claim there is not attainable and its price will depend on subjective preferences.
Just applying someutoHis of course one possible way to arrive at a valuation. But from a financial perspective, this is too simplistic because it ignores the trading opportunities represented by G. We therefore use a utility indifference argument to obtain our financial valuation rule as a transform ofu. This is a well-known general approach in economics and we follow here Mercurio (1996), and Aurell and˙Zyczkowski (1996) who combined this with anL2-framework to determine option prices in some special cases. For a general formulation of the basic idea, we start with an initial capitalc∈Rand an a priori valuation principleu. In order to decide on the price ofγunits ofH, we compare the following two alternatives:
1. Invest optimally into a trading strategy with initial capitalcand ignore the possibility of sellingH. This amounts to maximizingu(cB+g)over allg∈Gand we denote the value of this control problem by
v(c,0):=sup
g∈G
u(cB+g).
2. Sellγunits ofHfor the amounth(c, γ )∈Rto increase the initial wealth toc+h(c, γ ). This can then be invested into a trading strategy and leads to a total final wealth of(c+h(c, γ ))B+g−γ Hsince we have to pay out the claims fromH at the end. For an optimal investment, we thus have to maximizeu((c+h(c, γ ))B+g−γ H ) over allg∈Gand the value of this second control problem is
v(c, γ ):=sup
g∈G
u((c+h(c, γ ))B+g−γ H ).
Indifference with respect touprevails ifh(c, γ )is chosen in such a way that neither of these alternatives is preferred to the other. More formally, we have the following definition.
Definition. We callh(c, γ )anu-indifference price forγunits ofH ifv(c, γ )=v(c,0), i.e., ifh(c, γ )satisfies sup
g∈G
u((c+h(c, γ ))B+g−γ H )=sup
g∈G
u(cB+g). (4.1)
Due to the generality of the preceding approach, not much can be said about the properties ofh(c, γ )at this stage. Of course,h(c,0)=0 is always anu-indifference price for 0 units ofH, althoughh(c, γ )need not be unique. If H =cHB+gH ∈Ais strictly attainable, it is also easily checked thath(c, γ )=γ cH is anu-indifference price whateveruis. Thus our approach is consistent with absence of arbitrage.
Recall now the probability measurePB ≈P introduced in Section 3. As explained there, this is the measure one should use to work withB-discounted quantities in anL2-context. Our goal is to determine theu-indifference prices for the two valuation principles
u1(Y ):=EB
Y
B
−AVarB
Y
B
, u2(Y ):=EB
Y
B
−A
s
VarB
Y
B
, (4.2)
whereA >0 is a risk aversion parameter. The correspondingu-indifference prices will be denoted byh1,2(c, γ ), respectively.
Lemma 9. Foru∈ {u1, u2},the u-indifference priceh(c, γ )for anyH ∈L2does not depend on c and is for all γ , c∈Rgiven by
h(c, γ )=h(γ )=sup
g∈G
u(g)−sup
g∈G
u(g−γ H )=sup
g∈ ¯G
u(g)−sup
g∈ ¯G
u(g−γ H ).
Proof. Since the variance of a random variable does not change if we add a constant,
u1,2((c+x)B+g−γ H )=EB
c+x+g−γ H B
−A
VarB
c+x+g−γ H B
β1,2
for anyc, x∈R, whereβ1=1 andβ2=12. This implies that For the third equality, it is obviously enough to show that for anyY ∈L2, we have
L:=sup
fornsufficiently large, hence
L≥EB
Theorem 10. LetGbe a linear subspace ofL2admitting no approximate profits inL2.For anyH ∈L2and any
γ , c∈R,theu1-indifference price forγ units of H is then
Proof. By Corollary 3,Hcan be decomposed asH=cHB+gH +NH as in (2.1) so that
u1(g−γ H )=u1(−γ cHB+g−γ gH−γ NH)= −γ cH+u1(g−γ gH−γ NH).
SinceGis a linear subspace ofL2, the mappingg7→g′ :=g−γ gH is a bijection ofG¯into itself for every fixed H ∈L2andγ ∈Rand so Lemma 9 implies that
h1(γ )=sup
g∈ ¯G
u1(g)+γ cH −sup
g′∈ ¯G
u1(g′−γ NH). (4.5)
ButEB[NH/B]
=0 and CovB(g′/B, NH/B)
=0 for allg′∈ ¯Gby Lemma 5 and so we obtain
u1(g′−γ NH)=EB
g′
−γ NH B
−AVarB
g′
−γ NH B
=EB
g′
B
−AVarB
g′
B
−Aγ2VarB
NH
B
=u1(g′)−Aγ2VarB
NH
B
.
Combining this with (4.5) yields
h1(γ )=sup
g∈ ¯G
u1(g)+γ cH −sup
g′∈ ¯G
u1(g′)+Aγ2VarB
NH
B
and this together with Corollary 8 implies the assertion.
The valuation formula (4.4) in Theorem 10 has a number of very attractive features. To see the most striking of these, let us interpret±h1(±1)as the buying(γ = −1)and selling(γ = +1)prices for one unit ofH, respectively. Then we see from (4.4) that
±h1(±1)= ˜E
H
B
±AVarB
NH
B
. (4.6)
This looks very similar to (4.2), but differs by two important points. First of all, the expectation of theB-discounted claimH /Bis not taken under the original measurePB, but under theB-variance-optimal signed(G, B)-martingale measure P˜. Secondly, the variance component in (4.6) is not based on the entire claim H, but only on its non-hedgeable partNH. Thus we have to pass from the real-world measurePB to the (appropriate) risk-neutral measureP˜ for computing expectations, and we do not add or subtract a risk-loading (underPB) for that part ofH which can be hedged away by judicious trading. In view of the perfect analogy to (4.2), the prescription (4.6) could be called the financial variance principle.
IfHis attainable in the sense thatH =cHB+gH is inA¯, then (4.6) reduces to±h1(±1)=cH. This is exactly what we expect from the classical arbitrage arguments from option pricing. For a general claimH, we obtain a bid-ask spread
h1(+1)+h1(−1)=2AVarB
NH
B
=2A J0 E[B2]
proportional to the risk aversionAand the intrinsic financial riskJ0ofH. Finally, note that the valuation in (4.4) does not depend on the initial capitalcand is not linear in the numberγof claims. It might be interesting to compare this to empirically observed prices.
can also think ofh1(γ )as a benchmark value against which one can compare one’s own subjective assessments. In particular, the fact thath1(zγ )6=zh1(γ )does not entail arbitrage opportunities because it may not be possible to actually trade at the “prices” suggested byh1.
5. The financial standard deviation principle
In this section, we determine theu-indifference pricesh2(c, γ )for the valuation principle
u2(Y ):=EB
By repeating the arguments in the proof of Theorem 10, we get
h2(γ )=γ cH +sup
We start by analyzing the last term.
Lemma 11. LetGbe a linear subspace ofL2 admitting no approximate profits inL2.For anyy ∈ R,we then
two steps by first restricting attention to thosegsatisfying the constraintEB[g/B]=m. In analogy to Corollary 16 of Schweizer (1996), we therefore define
gm:=cm(B−π(B)):=
and so we deduce that
This implies that
Together with (3.4), this proves the assertion.
The next result is elementary analysis; its proof is only included for completeness.
Lemma 12. For anyy∈R,let
undefined forC <1.
Proof. Fixy ∈Rand letf (x):=x−pCx2+y2forx ∈R. Then
this shows thatf ≤ 0. If C = 1, the numerator does not depend onx and the denominator goes to +∞for x → +∞. Hence we conclude thats≡0 in this case. IfC >1 andy =0, the maximum off ≤0 is attained in x =0 so thats(0)=0 forC >1. Finally, ifC >1 andy2 >0,f is continuously differentiable with derivative f′(x)=1−(Cx/pCx2+y2). Sincef′ >0 forx ≤ 0, it is easily checked thatf′vanishes at the unique point x∗=py2/C(C−1)andf (x∗)= −py2√1−(1/C)by computation. By (5.4),f must have its maximum atx∗ and so the assertion follows.
Theorem 13. LetGbe a linear subspace ofL2admitting no approximate profits inL2.For anyH ∈L2and any
whereE˜ denotes expectation with respect to the B-variance optimal signed(G, B)-martingale measureP˜.
Proof. We first observe that ifB ∈G⊥, thenπ(B)=Band therefore VarB[dP /dP˜ B]=0 by (3.2). This shows
From Lemma 12 and Corollary 8, we thus obtain
h2(γ )=γE˜ Like (4.4), the valuation formula (5.5) has a very appealing interpretation. If we write
±h2(±1)=
given by
h2(+1)+h2(−1)=
2A
s
1−Var
B[dP /dP˜ B]
A2
s
VarB
NH
B
forA2≥VarB
"
dP˜ dPB
#
,
undefined forA2<VarB
"
dP˜
dPB
#
.
In contrast to (4.4), the valuation (5.5) is piecewise linear in the numberγ of claims; hence the resulting selling and buying “prices” for an arbitrary amount ofH are proportional to the selling and buying “price” of 1 unit of H, respectively. A second major difference to the last section is that all these results require a sufficiently high risk aversion forh2(c, γ )to be well defined. The lower bound onAdepends on thePB-variance of the density ofP˜with respect toPB. It is thus determined by the global properties of the financial environment(G, B)and in particular independent of the individual claim under consideration. In a very special case, a result like Theorem 13 has also been obtained by Aurell and ˙Zyczkowski (1996) by means of rather laborious calculations.
6. Two basic examples and an extension
In this section, we briefly discuss the two extreme cases of an inexistent and a complete financial market and mention an extension of our approach to take into account already existing positions. For reasons of space, explicit examples for the environment(G, B)that allow for more concrete computations with the resulting financial valuation rule will be discussed elsewhere. We refer to Mercurio (1996), Aurell and ˙Zyczkowski (1996), and Gharagozlou (1997) for some preliminary studies and to Møller (1999, 2000) for applications in an actuarial context.
6.1. The case of no financial market
We first consider the caseG= {0}; this describes a situation without a financial market because every self-financing trading strategy has outcome 0. Since there is nothing to optimize, the general indifference condition (4.1) reduces to
u(cB)=u(cB+h(c, γ )B−γ H );
this is very similar to the principle of zero utility as in Gerber (1979). Foru∈ {u1, u2}, we obtain as in Lemma 9
h1,2(c, γ )=h1,2(γ )= −u1,2(−γ H )=γ EB
H
B
+A
γ2VarB
H
B
β1,2
,
so that the actuarial and financial valuation principles coincide in this case. Moreover,G⊥=L2implies thatπis the identity mapping, henceπ(B) =B, and so comparing (3.1) and (3.2) shows thatP˜ =PB, i.e., the original measurePBis already variance-optimal.
6.2. The case of a complete financial market
The other extreme case occurs if(G, B)describes acomplete financial marketwhere every claimH ∈ L2is attainable. Note that this does not mean thatG=L2because then we would haveB ∈Gand therefore arbitrage. The correct description is thatBis not inG¯(so thatGadmits no approximate profits inL2) and thatA¯=RB+ ¯G=L2. By Corollary 3, the non-hedgeable partNHof every claimH∈L2is then 0, and so (4.1) takes the form
sup
g∈G
u(cB+g)=sup
g∈G
Of course, the mappingg7→g′:=g−γ gH is a bijection ofG¯into itself. If we have sup
g∈G
u(g+Y )=sup
g∈ ¯G
u(g+Y ) (6.1)
for everyY ∈L2, we can therefore conclude that h(c, γ )=h(γ )=γ cH =γE˜
H
B
is au-indifference price; this slightly generalizes the remark in Section 4. Condition (6.1) is trivially satisfied for any choice ofuifGis closed inL2. Without this assumption, (6.1) still holds foru ∈ {u1, u2}; see the proof of Lemma 9.
6.3. Extension to an initial position
Up to now, we have assumed that we start at time 0 with just an initial capital ofcand then value a claimH on that basis. But in more realistic situations, one has to examine cases where one already has some position (a book of options or an insurance portfolio) and is adding a new claimHon top of that. If we describe the existing position by its final time payoffH0 ∈L2, we have to replace the value functions of the optimization problems (1) and (2) in Section 4 by
v(c,0, H,0;H0):=sup
g∈G
u(cB−H0+g), v(c, γ , H, x;H0):=sup
g∈G
u((c+x)B−H0+g−γ H ). In words,v(c, γ , H, x;H0)is the maximal subjective value achievable by trading inG if one has initial capitalc and positionH0and sellsγ units ofH for an amount ofx. The condition (4.1) for the utility indifference price h(c, γ , H;H0)then becomes
v(c, γ , H, h(c, γ , H;H0);H0)=v(c,0, H,0;H0). (6.2) Essentially the same arguments as in Section 4 lead to
h1(c, γ , H;H0)=h1(γ , H;H0)=γE˜
H
B
+Aγ2VarB
NH
B
+2Aγ CovB
NH
B , H0
B
=γE˜
H
B
+Aγ2VarB
NH
B
+2AγCovB
NH
B , NH0
B
, (6.3)
and so the valuation contains an additional term to account for the correlation between the existing positionH0and the non-hedgeable partNH of the new claimH. Note that as for the variance component, the attainable partgH does not enter; the same is true forgH0 since CovB(NH/B, gH0/B)
=0 by Lemma 5. If we repeat the arguments of Section 5, we also find
h2(c, γ , H;H0)=h2(γ , H;H0)
=γE˜
H
B
+A
s
1−Var
B[dP /dP˜ B]
A2
s
VarB
NH0+γ NH
B
−
s
VarB
NH0
B
(6.4)
forA2≥VarB[dP /dP˜ B], whereash
2(c, γ , H;H0)is again undefined forA2<VarB[dP /dP˜ B].
Hand splitHintoH1=HIDandH2=HIDc, whereDis the event that a specified person dies before a specified
date. BothH1andH2 are then the so-called unit-linked life insurance products; the above person may want to purchaseH2as part of a pension plan while his/her partner or heirs may be more interested inH1. (I am grateful to D. Lando for this colorful example.) Then both Theorems 10 and 13, applied separately toH1andH2, would charge in total more than the arbitrage-free price ofHwhich clearly should not happen.
The flaw in the above reasoning is of course that one does not take into account the changed position onceH1 orH2has been sold. Once this is remedied by appropriately using (6.2) instead of (4.1), we obtain in fact the very natural result that
h(c, γ , H1+H2;H0)=h(c, γ , H1;H0)+h(c+h(c, γ , H1;H0), γ , H2;H0+γ H1). (6.5) Put into words, this says that the price for selling directlyγunits ofH1+H2equals the sum of the prices for selling firstγ units ofH1and thenγ units ofH2if the second sale takes into account that the first sale has changed both the initial capital (fromctoc+h(c, γ , H1;H0)) and the position (fromH0toH0+γ H1).
To see why (6.5) is true, we setx1:=h(c, γ , H1;H0), x2:=h(c+x1, γ , H2;H0+γ H1)andx:=h(c, γ , H1+ H2;H0), so that we have to show thatx =x1+x2. Since clearly
v(y,0, H,0;J )=v(y,0, H′,0;J )
for any choice ofy, H, H′, J, using repeatedly (6.2) and the definition ofvgives
v(c, γ , H1+H2, x;H0)=v(c,0, H1+H2,0;H0)=v(c,0, H1,0;H0)=v(c, γ , H1, x1;H0)
=sup
g∈G
u((c+x1)B−(H0+γ H1)+g)=v(c+x1,0, H2,0;H0+γ H1)
=v(c+x1, γ , H2, x2;H0+γ H1)=sup
g∈G
u((c+x1+x2)B−H0+g−γ (H1+H2))
=v(c, γ , H1+H2, x1+x2;H0).
Under the mild assumption that (6.2) uniquely determinesh(c, γ , H;H0), we deduce (6.5). For the special cases ofu1, u2, this could also be obtained by direct computations from (6.3) or (6.4).
References
Aurell, E.,˙Zyczkowski, K., 1996. Option pricing and partial hedging: theory of Polish options. Preprint. Royal Institute of Technology, Stockholm. Bühlmann, H., 1987. Actuaries of the third kind?. ASTIN Bulletin 17, 137-138.
Bühlmann, H., 1997. The actuary: the role and limitations of the profession since the mid-19th century. ASTIN Bulletin 27, 165-171. Choulli, T., Krawczyk, L., Stricker, C., 1998.E-martingales and their applications in mathematical finance. Annals of Probability 26, 853-876. Choulli, T., Krawczyk, L., Stricker, C., 1999. On Fefferman and Burkholder–Davis–Gundy inequalities forE-martingales. Probability Theory
and Related Fields 113, 571-597.
Delbaen, F., Schachermayer, W., 1996a. The variance-optimal martingale measure for continuous processes. Bernoulli 2, 81-105. Delbaen, F., Schachermayer, W., 1996b. Amendments and corrections. Bernoulli 2, 379-380.
Delbaen, F., Monat, P., Schachermayer, W., Schweizer, M., Stricker, C., 1997. Weighted norm inequalities and hedging in incomplete markets. Finance and Stochastics 1, 181-227.
Gerber, H.U., 1979. An Introduction to Mathematical Risk Theory. Huebner Foundation Monograph No. 8, Irwin, Illinois.
Gharagozlou, A., 1997. Zur Bewertung von Optionen mit einer mean-variance Nutzenfunktion. Diploma Thesis. Technical University of Berlin, Berlin.
Gouriéroux, C., Laurent, J.P., Pham, H., 1998. Mean-variance hedging and numéraire. Mathematical Finance 8, 179–200.
Kliger, D., Levikson, B., 1998. Pricing insurance contracts — an economic viewpoint. Insurance: Mathematics and Economics 22, 243-249. Mercurio, F., 1996. Variance-optimal hedging and risk preferences. Preprint. Erasmus University, Rotterdam.
Møller, T., 1999. On indifference pricing for insurance contracts. Preprint. Laboratory of Actuarial Mathematics, University of Copenhagen. Møller, T., 2000. On transformations of actuarial valuation principles. Proceedings of the 10th AFIR Colloquium, Tromsø, pp. 517–545. Schweizer, M., 1995. Variance-optimal hedging in discrete time. Mathematics of Operations Research 20, 1-32.
Schweizer, M., 1996. Approximation pricing and the variance-optimal martingale measure. Annals of Probability 24, 206-236. Schweizer, M., 1999. Risky options simplified. International Journal of Theoretical and Applied Finance 2, 59-82.
