K.K. Aase Insurance: Mathematics and Economics 27 2000 345–363 351
Here N t ; dy is a random Poisson measure, meaning that N
A t
= R
A
N t ; dy is a Poisson process with parameter
νA =
R
A
νdy, and N
A t
and N
B t
are independent if A and B are disjoint sets of reals. Thus, ˜ R is a process with
paths of bounded variation. Any P , F
˜ R
t
-martingale M can be represented as Mt
= M0 + Z
t
Z
−∞,∞
θ s, y {Nds; dy − νdy ds}.
16 This representation property is obviously different from a representation in the form of linear spanning. Thus, our
model is incomplete. The price process of a risky asset having return process ˜
R
t
we assume to be given by the following: St
= S e
˜ R
t
, ≤ t T .
17 In other words, we interpret ˜
R
t
as the cumulative logarithmic returns by time t . In Section 8, we alternatively model S by the semimartingale exponential, implying that we view the term ˜
R
t
as the cumulative simple returns for the period [0, t ].
3. Preference-free equilibrium pricing of derivative securities
We now turn to the pricing of derivative securities in the present model, where we first consider the case with one risky and one riskfree asset. Using the pricing rule 4, the equilibrium market price π
φ
of a contingent claim having payoff φST at the expiration time T is given by
π
φ
S, t =
1 u
′
c
t
, t E
t
{u
′
c
T
, T φ S
T
}, 18
where c is the aggregate consumption of all the consumers, accumulated from time 0 to time t , having probability density given in 5. Here the price of the underlying asset equals ST at time T and there are no intermediate
dividends paid out to the holder of the contingent claim.
5
Below we shall consider the European call option as an illustration, where φ x = x − k
+
. In the case where the contingent claim also pays dividends, where D
φ
= D
φ t
≤t≤T
equals the accumulated dividend process, then the market price in 18 changes to
π
φ,D
S, t =
1 u
′
c
t
, t E
u
′
c
T
, T φ S
T
+ Z
T t
u
′
c
s
, s dD
φ
s +
Z
T t
d[D
φ
, u
′
c]s |F
t
, 19
We start by considering the real price St at time. It satisfies St
= 1
u
′
ct , t E
t
{u
′
cT , T ST } =
St u
′
ct , t E
t
{u
′
cT , T exp { ˜
R
T
− ˜ R
t
}}, ≤ t ≤ T ,
20 where we have used 17 and the fact that S is an F
t
-adapted process. Conditional on c
T −t
, the random variable R
T
− R
t
is normally distributed having mean T − tµ + βc
T −t
and variance c
T −t
. In Eq. 20, we now condition on c
T
− c
t
, which gives u
′
c
t
, t = E
t
{u
′
c
T
, T exp[β +
1 2
c
T
− c
t
] } e
µT −t
. 21
Let us now make use of our assumption that the intertemporal marginal utility function of the market is given by u
′
c, t = e
−ρt
e
−ηc
, where ρ is the market’s impatience rate and η the intertemporal coefficient of absolute risk
5
The case with intermediate dividends of the L´evy type can be handled by formula 3.
352 K.K. Aase Insurance: Mathematics and Economics 27 2000 345–363
aversion in the market. We want to demonstrate that the market values of contingent claims in the above model will not directly depend on the parameters ρ, η. Thus, we obtain a similar kind of preference-free evaluations in the
present model as in the equilibrium version of the model of Black and Scholes, a result we may have expected to be true only in a complete market see, e.g., Aase, 1998.
6
To this end, observe that under the present conditions, expression 21 reduces to E
t
{exp{β − η +
1 2
c
T
− c
t
}} = e
T −tρ−µ
. 22
Using Eq. 7, we get e
T −tρ−µ
= E
t
{exp{β − η +
1 2
c
T
− c
t
}} = e
T −tδ
√ ψ
− √
ψ +2η−2β−1
. Thus, we obtain the following relationship between the parameters:
ρ − µ = δ
p ψ
− p
ψ + 2η − 2β − 1.
23 The second equation between the parameters of the model is obtained as follows. The equilibrium interest rate rt
satisfies the following equation: rt
= − µ
p
t pt
, pt
= u
′
c
t
, t , where µ
p
t is the conditional expected rate of change in the process p see, e.g., Duffie, 1988; Aase, 1993b, which can be written as follows:
e
−ρt−ηc
t
= E Z
s t
e
−ρu−ηc
u
ru du + E
Z
T s
e
−ρu−ηc
u
ru du |F
s
F
t
. Since E
{ R
T s
e
−ρu−ηc
u
ru du |F
s
} = e
−ρs−ηc
s
, we obtain that 1
= e
ρt +ηc
t
E Z
s t
e
−ρu−ηc
u
ru du + e
−ρs−ηc
s
|F
t
. We now use that the local characteristics of a Lévy process are deterministic and Eqs. 3.3–4 in Aase 1993b to
conclude that r is constant. Using relation 7, we get 1
= r e
ρt
Z
s t
e
−ρu
e
−u−tδ √
ψ −
√ ψ
+2η
du + e
−ρs
e
−s−tδ √
ψ −
√ ψ
+2η
, which directly leads to the equilibrium interest rate r being given by
r = ρ + δ
p ψ
− p
ψ + 2η,
24 where ψ
= α
2
− β
2
, and δ 0 is the scale parameter. Eq. 24 is the second equilibrium relation between the parameters.
7
Note that since η ≥ 0, r ≤ ρ, which we
normally would expect to be the case. Also notice that if η = 0, so the market is risk neutral, then r = ρ. Clearly
Eqs. 23 and 24 can be solved and η, ρ found as functions of the parameters δ, µ, α, β and r. We therefore propose the following concept.
Definition 1. An equilibrium model is preference-free if the expression for the market price of any risky security does not contain the preference parameters.
6
We may remark here that if all agents are identical with the above utility function, then the competitive equilibrium is fully Pareto optimal whatever the market structure, since there are no gains from trade Hart, 1975.
7
Observe how probability distributions of asset prices are here being determined in equilibrium.
K.K. Aase Insurance: Mathematics and Economics 27 2000 345–363 353
Because of jumps of random sizes at unpredictable time points in the pricing process of the primitive assets, our model is not complete according to the standard definition, but as already noted, the general pricing rule 2 still
applies to contingent claims. It may be useful to resort a notion of complete markets due to Borch 1962, who defined a market model to be
complete if it assigns a unique value, its market price, to an arbitrary random process.
8
This notion turns out to be useful in our setting.
Clearly a preference-free model is also complete in the meaning of Borch’s definition, but a Borch-complete model need not be preference-free. The word preference-free in the above definition should not be taken literally,
since equilibrium prices will normally depend upon preferences, e.g., through the equilibrium riskfree interest rate which normally depends on the preference parameters. However, the prices on risky securities are not allowed to
depend explicitly on these parameters according to the above definition. The importance of this concept seems rather obvious from a practical point of view, since the statistical estimation of the process parameters is a straightforward
task compared to the statistical estimation of the preference parameters. We have essentially shown the following theorem.
Theorem 1. The model of this section, with constant, intertemporal absolute risk aversion η and constant impatience rate ρ, is preference-free.
Proof. Any computation of the price of a contingent claim using formula 18 will depend upon the parameters ρ, η as well as on the rest of the parameters of the model. However, ρ and η can be eliminated using Eqs. 23
and 24, and these equations are derived independent of any contingent claim possibly existing in the model. We obtain that the unique solution is
η =
1 8c
2
{ψ
2
+ b
2
+ a
4
− 2ψb − 2ψa
2
− 2ba
2
}, 25
ρ = r + δ
p ψ
+ 2η − p
ψ, 26
where b
= ψ − 2β − 1, a
= 2 p
ψ +
1 δ
µ − r.
27 It is worth emphasizing again that all the return and consumption-related parameters δ, µ, α, and β can readily
be jointly estimated from return and consumption data, and r is presumably observed in the market.
9
Thus, the estimated values can be substituted for the unknown preference parameters η and ρ, leading to numerical values
for market price of risky assets even if the preference parameters are not known. Alternatively, one can view the above equations as relations from which the preference parameters can indirectly be estimated from market
data.
4. An option pricing formula
