38 M. Schweizer Insurance: Mathematics and Economics 28 2001 31–47
Proof.
By 3.2 and the proof of Lemma 1, 1B d ˜ P
dP is in G
⊥
∩ RB + ¯ G = G
⊥
∩ ¯ A
by Lemma 2. Since g
H
∈ ¯ G
and N
H
∈ ¯ A
⊥
, we thus obtain ˜
E H
B =
1 B
d ˜ P
dP , c
H
B + g
H
+ N
H
= c
H
˜ P
[Ω] = c
H
. Moreover, Lemmas 4 and 5 imply that
J = E[N
H 2
] = E[B
2
]E
B
N
H
B
2
= E[B
2
] Var
B
N
H
B .
Corollary 8 allows us to give a very intuitive financial interpretation for the decomposition H
= c
H
B + g
H
+ N
H
3.6 in Corollary 3. It tells us that H splits naturally into an attainable part c
H
B + g
H
∈ ¯ A
and a non-hedgeable part N
H
which is orthogonal to the space ¯ A
of attainable payoffs. The constant c
H
is the initial capital of the attainable part; it can be computed in a simple way as the ˜
P -expectation of the B-discounted payoff H B. Moreover, 3.5
tells us that the P
B
-variance of the B-discounted non-hedgeable part of H is a measure for the intrinsic financial risk of H in terms of approximation in L
2
by 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 payoff H ≥ 0 has c
H
0 because ˜ P
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 of H lie 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 variable H ∈ L
2
and think of this as a random payoff as in Section 3. In a financial context, H could 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 call H a general claim. How much should we charge or pay if we sell or buy γ units
of this claim?
This question has a standard answer from classical option pricing theory in the special case where H is strictly attainable in the sense that H ∈ A. Then we can write H = c
H
B + g
H
with c
H
∈ R and g
H
∈ G, and the price per unit of H must be c
H
to avoid arbitrage. To understand the idea behind this argument, suppose for instance that H is nonnegative and offered at a price x c
H
. Then one can buy c
H
x units of H and finance this initial outlay of c
H
with −c
H
, −g
H
, i.e., by borrowing the amount c
H
and trading according to the strategy associated to −g
H
. The terminal payoff from this costless deal is then c
H
xH − c
H
B − g
H
= c
H
x − 1H ≥ 0; thus one has turned
nothing into something, but absence of arbitrage postulates that this is impossible. A similar argument applies if one can sell H for more than c
H
. However, this entire reasoning depends crucially on the attainability of H 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. If we think of −H as an insurance risk, there is also a standard approach to its valuation from actuarial mathematics;
we simply apply a valuation principle appropriate for our needs. Thus we choose a mapping u from random variables Y
into R and think of uY as value or utility associated to the random amount Y . For instance, the classical actuarial variance principle would correspond to uY := expected value of Y − A times the variance of Y ; the slightly
unfamiliar choice of signs is due to our convention that −H and not H corresponds to an insurance claim.
M. Schweizer Insurance: Mathematics and Economics 28 2001 31–47 39
Just applying some u to H is 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 of u. This is a well-known general approach in economics and we follow here Mercurio 1996, and Aurell and
˙Z
yczkowski 1996 who combined this with an L
2
-framework to determine option prices in some special cases. For a general formulation of the basic idea, we start with an initial capital c ∈ R and an a priori valuation principle u. In order to decide on the price of γ units
of H , we compare the following two alternatives: 1. Invest optimally into a trading strategy with initial capital c and ignore the possibility of selling H . This amounts
to maximizing ucB + g over all g ∈ G and we denote the value of this control problem by vc,
0 := sup
g ∈G
u cB
+ g. 2. Sell γ units of H for the amount hc, γ ∈ R to increase the initial wealth to c+hc, γ . This can then be invested
into a trading strategy and leads to a total final wealth of c + hc, γ B + g − γ H since we have to pay out the claims from H at the end. For an optimal investment, we thus have to maximize uc + hc, γ B + g − γ H
over all g ∈ G and the value of this second control problem is
vc, γ := sup
g ∈G
uc + hc, γ B + g − γ H .
Indifference with respect to u prevails if hc, 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 call hc, γ an u-indifference price for γ units of H if vc, γ = vc, 0, i.e., if hc, γ satisfies
sup
g ∈G
uc + hc, 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 of hc, γ at this stage. Of course, hc, 0 = 0 is always an u-indifference price for 0 units of H , although hc, γ need not be unique. If
H = c
H
B + g
H
∈ A is strictly attainable, it is also easily checked that hc, γ = γ c
H
is an u-indifference price whatever u is. Thus our approach is consistent with absence of arbitrage.
Recall now the probability measure P
B
≈ P introduced in Section 3. As explained there, this is the measure one should use to work with B-discounted quantities in an L
2
-context. Our goal is to determine the u-indifference prices for the two valuation principles
u
1
Y := E
B
Y B
− A Var
B
Y B
, u
2
Y := E
B
Y B
− A s
Var
B
Y B
, 4.2
where A 0 is a risk aversion parameter. The corresponding u-indifference prices will be denoted by h
1,2
c, γ ,
respectively.
Lemma 9. For u
∈ {u
1
, u
2
}, the u-indifference price hc, γ for any H ∈ L
2
does not depend on c and is for all γ , c
∈ R given by hc, γ
= hγ = sup
g ∈G
ug − sup
g ∈G
ug − γ H = sup
g ∈ ¯
G
ug − sup
g ∈ ¯
G
ug − γ H .
Proof. Since the variance of a random variable does not change if we add a constant,
u
1,2
c + xB + g − γ H = E
B
c + x +
g − γ H
B − A
Var
B
c + x +
g − γ H
B
β
1,2
= c + x + u
1,2
g − γ H
40 M. Schweizer Insurance: Mathematics and Economics 28 2001 31–47
for any c, x ∈ R, where β
1
= 1 and β
2
=
1 2
. This implies that vc, γ
= sup
g ∈G
uc + hc, γ B + g − γ H = c + hc, γ + sup
g ∈G
ug − γ H ,
vc, 0 = sup
g ∈G
uc + g = c + sup
g ∈G
ug for u ∈ {u
1
, u
2
}. Since hc, γ is defined by 4.1, we obtain the first and second equalities. For the third equality, it is obviously enough to show that for any Y ∈ L
2
, we have L
:= sup
g ∈G
ug + Y = sup
g ∈ ¯
G
ug + Y =: R.
4.3 Clearly L ≤ R and so it only remains to show that L ≥ R. For any g ∈ ¯
G , there is a sequence g
n n
∈N
in G converging to g in L
2
. This implies that E
B
[g
n
+ Y B] = 1E[B
2
]E[Bg
n
+ Y ] converges to E
B
[g + Y B] and that kg
n
+ Y k
n ∈N
is bounded in n. Since |kg
n
+ Y k
2
− kg + Y k
2
| ≤ kg
n
− gk sup
n ∈N
kg
n
+ Y k + kg + Y k ,
|E[Bg
n
+ Y ]
2
− E[Bg + Y ]
2
≤ |E[Bg
n
− g]| sup
n ∈N
|E[Bg
n
+ Y ]| + |E[Bg + Y ]| ≤ kBk
2
kg
n
− gk sup
n ∈N
kg
n
+ Y k + kg + Y k ,
we conclude that Var
B
g
n
+ Y B
− Var
B
g + Y B
≤ 1
E [B
2
] |E[g
n
+ Y
2
] − E[g + Y
2
]| +
1 E
[B
2
]
2
|E[Bg
n
+ Y ]
2
− E[Bg + Y ]
2
| converges to 0 as n → ∞. Given ε 0, we thus have E
B
[g
n
+ Y B] ≥ E
B
[g + Y B] − ε and Var
B
[g
n
+ Y B] ≤ Var
B
[g + Y B] + ε for n sufficiently large. This implies that L
≥ ug
n
+ Y = E
B
g
n
+ Y B
− A Var
B
g
n
+ Y B
β
≥ E
B
g + Y B
− ε − A Var
B
g + Y B
+ ε
β
for n sufficiently large, hence L
≥ E
B
g + Y B
− A Var
B
g + Y B
β
= ug + Y by letting ε tend to 0 and since g ∈ ¯
G was arbitrary, we obtain 4.3.
Theorem 10. Let G be a linear subspace of L
2
admitting no approximate profits in L
2
. For any H ∈ L
2
and any γ , c
∈ R, the u
1
-indifference price for γ units of H is then h
1
c, γ = h
1
γ = γ c
H
+ Aγ
2
J E
[B
2
] = γ ˜
E H
B + Aγ
2
Var
B
N
H
B ,
4.4 where ˜
E denotes expectation with respect to the B-variance-optimal signed G, B
-martingale measure ˜ P
.
M. Schweizer Insurance: Mathematics and Economics 28 2001 31–47 41
Proof.
By Corollary 3, H can be decomposed as H = c
H
B + g
H
+ N
H
as in 2.1 so that u
1
g − γ H = u
1
−γ c
H
B + g − γ g
H
− γ N
H
= −γ c
H
+ u
1
g − γ g
H
− γ N
H
. Since G is a linear subspace of L
2
, the mapping g 7→ g
′
:= g − γ g
H
is a bijection of ¯ G
into itself for every fixed H
∈ L
2
and γ ∈ R and so Lemma 9 implies that h
1
γ = sup
g ∈ ¯
G
u
1
g + γ c
H
− sup
g
′
∈ ¯ G
u
1
g
′
− γ N
H
. 4.5
But E
B
[N
H
B ] = 0 and Cov
B
g
′
B, N
H
B = 0 for all g
′
∈ ¯ G
by Lemma 5 and so we obtain u
1
g
′
− γ N
H
= E
B
g
′
− γ N
H
B − A Var
B
g
′
− γ N
H
B = E
B
g
′
B − A Var
B
g
′
B − Aγ
2
Var
B
N
H
B = u
1
g
′
− Aγ
2
Var
B
N
H
B .
Combining this with 4.5 yields h
1
γ = sup
g ∈ ¯
G
u
1
g + γ c
H
− sup
g
′
∈ ¯ G
u
1
g
′
+ Aγ
2
Var
B
N
H
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 ±h
1
±1 as the buying γ = −1 and selling γ = +1 prices for one unit of H , respectively. Then we see from 4.4 that
±h
1
±1 = ˜ E
H B
± A Var
B
N
H
B .
4.6 This looks very similar to 4.2, but differs by two important points. First of all, the expectation of the B-discounted
claim H B is not taken under the original measure P
B
, but under the B-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 part N
H
. Thus we have to pass from the real-world measure P
B
to the appropriate risk-neutral measure ˜
P for computing expectations, and we do not add or subtract a risk-loading under P
B
for that part of H 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. If H is attainable in the sense that H = c
H
B + g
H
is in ¯ A
, then 4.6 reduces to ±h
1
±1 = c
H
. This is exactly what we expect from the classical arbitrage arguments from option pricing. For a general claim H , we obtain a
bid-ask spread h
1
+1 + h
1
−1 = 2A Var
B
N
H
B = 2A
J E
[B
2
] proportional to the risk aversion A and the intrinsic financial risk J
of H . Finally, note that the valuation in 4.4 does not depend on the initial capital c and is not linear in the number γ of claims. It might be interesting to compare
this to empirically observed prices.
Remark.
Let us pause here for a moment to comment on the interpretation of h
1
. Despite our terminology, we do not claim that h
1
γ is necessarily “the price for γ units of H ”; it is rather a possible value assigned to γ units of H
by someone who is a priori willing to use u
1
as valuation and then to accept the above indifference argument. One
42 M. Schweizer Insurance: Mathematics and Economics 28 2001 31–47
can also think of h
1
γ as a benchmark value against which one can compare one’s own subjective assessments. In
particular, the fact that h
1
zγ 6= zh
1
γ does not entail arbitrage opportunities because it may not be possible to
actually trade at the “prices” suggested by h
1
.
5. The financial standard deviation principle
