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
In this section, we determine the u-indifference prices h
2
c, γ for the valuation principle
u
2
Y := E
B
Y B
− A s
Var
B
Y B
. 5.1
By repeating the arguments in the proof of Theorem 10, we get h
2
γ = γ c
H
+ sup
g ∈ ¯
G
E
B
h g
B i
− A r
Var
B
h g
B i
− sup
g
′
∈ ¯ G
E
B
g
′
B − A
s Var
B
g
′
B + γ
2
Var
B
N
H
B
.
5.2 We start by analyzing the last term.
Lemma 11. Let G be a linear subspace of L
2
admitting no approximate profits in L
2
. For any y ∈ R, we then have
sup
g ∈ ¯
G
E
B
h g
B i
− A r
Var
B
h g
B i
+ y
2
=
−A p
y
2
for B ∈ G
⊥
, sup
m ∈R
m − A
s m
2
Var
B
[d ˜ P
dP
B
] + y
2
for B ∈ G
⊥
. 5.3
Proof. If B ∈ G
⊥
, then E
B
[gB] = 1E[B
2
]E[Bg] = 0 for all g ∈ ¯ G
and so we can simply minimize the P
B
-variance of gB by choosing g = 0. For the case where B ∈ G
⊥
, the idea is to perform the maximization in two steps by first restricting attention to those g satisfying the constraint E
B
[gB] = m. In analogy to Corollary 16 of Schweizer 1996, we therefore define
g
m
:= c
m
B − πB :=
mE [B
2
] E
[B − πB
2
] B
− πB. Since B
∈ G
⊥
, we have E[BB − πB] = E[B − πB
2
] 0 and so g
m
is well defined and in G
⊥⊥
= ¯ G
since G
is linear. Moreover, E
B
h g
m
B i
= 1
E [B
2
] E
[Bg
m
] = m. If we take any g ∈ ¯
G with E
B
[gB] = m, then kg − c
m
B k
2
= kg − g
m
+ c
m
πB k
2
= kg − g
m
k
2
+ c
2 m
kπBk
2
≥ c
2 m
kπBk
2
= kg
m
− c
m
B k
2
, and so we deduce that
Var
B
h g
B i
= Var
B
h g
B − c
m
i =
1 E
[B
2
] kg − c
m
B k
2
− m − c
m 2
≥ Var
B
h g
m
B i
.
M. Schweizer Insurance: Mathematics and Economics 28 2001 31–47 43
This implies that sup
E
B
h g
B i
− A r
Var
B
h g
B i
+ y
2
g ∈ ¯
G with E
B
h g
B i
= m = E
B
h g
m
B i
− A r
Var
B
h g
m
B i
+ y
2
= m − A s
c
2 m
Var
B
1 − πB
B + y
2
. But
Var
B
1 − πB
B =
1 E
[B
2
] kB − πBk
2
− 1
E [B
2
] E
[BB − πB]
2
= 1
E [B
2
] kB − πBk
2
− 1
E [B
2
] E
[B − πB
2
]
2
, and so we get
c
2 m
Var
B
1 − πB
B = m
2
E [B
2
] E
[B − πB
2
] − 1
= m
2
kπBk
2
kB − πBk
2
. Together with 3.4, this proves the assertion.
The next result is elementary analysis; its proof is only included for completeness.
Lemma 12. For any y
∈ R, let sy
:= sup
x ∈R
x −
q Cx
2
+ y
2
for a fixed C ≥ 0. Then
s 0 − sy =
p
y
2
r 1 −
1 C
for C ≥ 1,
undefined for C
1.
Proof.
Fix y ∈ R and let f x := x − p
Cx
2
+ y
2
for x ∈ R. Then lim
x →±∞
f x |x|
= lim
x →±∞
±1 − s
C +
y
2
x
2
= ±1 − √
C. 5.4
For C 1, this implies that lim
x →±∞
f x = ±∞ and since f is continuous, we conclude that f has no finite
maximum so that s ≡ +∞ in this case. For C ≥ 1, we write f x as f x
= x
2
− Cx
2
+ y
2
x +
p Cx
2
+ y
2
= 1 − Cx
2
− y
2
x +
p Cx
2
+ y
2
; this shows that f ≤ 0. If C = 1, the numerator does not depend on x and the denominator goes to +∞ for
x → +∞. Hence we conclude that s ≡ 0 in this case. If C 1 and y = 0, the maximum of f ≤ 0 is attained in
x = 0 so that s0 = 0 for C 1. Finally, if C 1 and y
2
0, f is continuously differentiable with derivative f
′
x = 1 − Cx
p Cx
2
+ y
2
. Since f
′
0 for x ≤ 0, it is easily checked that f
′
vanishes at the unique point x
∗
= p
y
2
CC − 1 and f x
∗
= − p
y
2
√ 1 − 1C by computation. By 5.4, f must have its maximum at x
∗
and so the assertion follows. Combining the two previous results, we now obtain the following theorem.
44 M. Schweizer Insurance: Mathematics and Economics 28 2001 31–47
Theorem 13. 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
2
-indifference price for γ units of H is then h
2
c, γ = h
2
γ
=
γ ˜ E
H B
+ A|γ | s
1 − Var
B
[d ˜ P
dP
B
] A
2
s Var
B
N
H
B for A
2
≥ Var
B
d ˜ P
dP
B
, undefined
for A
2
Var
B
d ˜ P
dP
B
, 5.5
where ˜ E
denotes expectation with respect to the B-variance optimal signed G, B -martingale measure ˜
P .
Proof. We first observe that if B ∈ G
⊥
, then πB = B and therefore Var
B
[d ˜ P
dP
B
] = 0 by 3.2. This shows that for B ∈ G
⊥
, the second case in 5.5 will never occur. If B ∈ G
⊥
, then 5.2 and 5.3 imply that h
2
γ = γ c
H
+ A s
γ
2
Var
B
N
H
B = γ ˜
E H
B + A|γ |
s Var
B
N
H
B by Corollary 8. If B
∈ G
⊥
, then 5.2 and 5.3 yield h
2
γ = γ c
H
+ sup
m ∈R
m −
p Cm
2
− sup
m ∈R
m −
s Cm
2
+ A
2
γ
2
Var
B
N
H
B
,
where we have set C
:= A
2
Var
B
[d ˜ P
dP
B
] ≥ 0.
From Lemma 12 and Corollary 8, we thus obtain h
2
γ = γ ˜
E H
B + A|γ |
s 1 −
Var
B
[d ˜ P
dP
B
] A
2
s Var
B
N
H
B for C ≥ 1, while h
2
γ is undefined for C 1. This proves the assertion.
Like 4.4, the valuation formula 5.5 has a very appealing interpretation. If we write
±h
2
±1 =
˜ E
H B
± A s
1 − Var
B
[d ˜ P
dP
B
] A
2
s Var
B
N
H
B for A
2
≥ Var
B
d ˜ P
dP
B
, undefined
for A
2
Var
B
d ˜ P
dP
B
, 5.6
we see that our approach transforms the actuarial standard deviation principle 5.1 into the financial standard de- viation principle 5.6. Like 4.4, the valuation in 5.5 is based on the expectation under the B-variance-optimal
signed G, B-martingale measure ˜ P
and the intrinsic financial risk of H . The corresponding bid-ask spread is
M. Schweizer Insurance: Mathematics and Economics 28 2001 31–47 45
given by
h
2
+1 + h
2
−1 =
2A s
1 − Var
B
[d ˜ P
dP
B
] A
2
s Var
B
N
H
B for A
2
≥ Var
B
d ˜ P
dP
B
, undefined
for A
2
Var
B
d ˜ P
dP
B
. In contrast to 4.4, the valuation 5.5 is piecewise linear in the number γ of claims; hence the resulting selling
and buying “prices” for an arbitrary amount of H 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 for h
2
c, γ to be well defined. The lower bound on A depends on the P
B
-variance of the density of ˜ P
with respect to P
B
. 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
˙Z
yczkowski 1996 by means of rather laborious calculations.
6. Two basic examples and an extension
