In the preceding section, we have seen the importance of martingale measures for option pricing. In some situations, there is no doubt what the risk-neutral measure is: It is already determined by the condition that the discounted price processes of a set of basic securities are martingales. A famous example is the Samuelson (1965) model. Here the price of a single stock is described by a geometric Brownian motion.
St=S0exp (à−σ2/2)t+σWt
(t≥0), (1.17)
whereS0is the stock price at time t= 0,à∈IRand σ ≥ 0are constants, and whereW is a standard Brownian motion. This model is of the form (1.2), with a Lộvy processLt= (à−r−σ2/2)t+σWt. If the filtration in the Samuelson (1965) model is assumed to be the one generated by the stock price process,9 there is only one locally equivalent measure under which e−rtSt is a martingale. (See e. g.
Harrison and Pliska (1981).) This measure is given by the following density processZ with respect to P.
Zt= e (r−à)/σ
Wt
E
e (r−à)/σ
Wt
. (1.18)
The fact that there is only one such measure implies that derivative prices are uniquely determined in this model: If the condition thate−rtSt is a martingale is already sufficient to determine the measure Q, then necessarilyQmust be the risk-neutral measure for any market that consists of the stockS and derivative securities that depend only on the underlyingS. The prices of these derivative securities are then determined by equations of the form (1.10). For European call options, these expressions can be evaluated analytically, which leads to the famous Black and Scholes (1973) formula.
9This additional assumption is necessary since obviously the condition thatexp(Lt)be a martingale can only determine the measure of sets in the filtration generated byL.
Note that the uniquely determined density process (1.18) is of the form (1.6) withθ= (r−à)/σ2. That is, it is an Esscher transform.
If one introduces jumps into the stock price model (e. g. by choosing a more general Lévy process instead of the Brownian motionW in (1.17)), this in general results in losing the property that the risk-neutral measure is uniquely determined by just one martingale condition. Then the determination of prices for derivative securities becomes difficult, because one has to figure out how the market chooses prices.
Usually, one assumes that the measure Q—that is, prices of derivative securities—is chosen in a way that is optimal with respect to some criterion. For example, this may involve minimizing theL2-norm of the density as in Schweizer (1996). (Unfortunately, in general the optimal densitydQ/dP may become negative; so, strictly speaking, this is no solution to the problem of choosing a martingale measure.) Other approaches start with the problem of optimal hedging, where the aim is maximizing the expectation of a utility function. This often leads to derivative prices that can be interpreted as expectations under some martingale measure. In this sense, the martingale measure can also be chosen by maximizing expected utility. (Cf. Kallsen (1998), Goll and Kallsen (2000), Kallsen (2000), and references therein.)
For a large class of exponential Lévy processes other than geometric Brownian motion, Esscher trans- forms are still a way to satisfy the martingale condition for the stock price, albeit not the only one. They turn out to be optimal with respect to some utility maximization criterion. (See Keller (1997), Section 1.4.3, and the reference Naik and Lee (1990) cited therein.) However, the utility function used there depends on the Esscher parameter. Chan (1999) proves that the Esscher transform minimizes the relative entropy of the measures P and Qunder all equivalent martingale transformations. But since his stock price is the stochastic exponential (rather than the ordinary exponential) of the Lévy process employed in the Esscher transform, this result does not hold in the context treated here.
Below we present another justification of the Esscher transform: If Conjecture 1.16 holds, then the Esscher transform is the only transformation for which the density process does only depend on the current stock price (as opposed to the entire stock price history.)
Martingale Conditions
The following proposition shows that the parameterθof the Esscher transform leading to an equivalent martingale measure satisfies a certain integral equation. Later we show that an integro-differential equa- tion of similar form holds for any functionf(x, t)for whichf(Lt, t)is another density process leading to an equivalent martingale measure. Comparison of the two equations will then lead to the characterization result for Esscher transforms.
Proposition 1.13. LetLbe a Lévy process for whichL1 possesses a finite moment generating function on some interval(a, b)containing0. Denote byκ(v)the cumulant generating function, i. e. the logarithm of the moment generating function. Then there is at most oneθ∈IRsuch thateLtis a martingale under the measuredPθ= (eθLt/E[eθL1]t)dP. This valueθsatisfies the following equation.
b+θc+ c 2 +
Z
eθx(ex−1)−x
F(dx) = 0, (1.19)
where(b, c, F)is the Lévy-Khintchine triplet of the infinitely divisible distributionPL1.
Remark: Here we do not need to introduce a truncation function h(x), since the existence of the mo- ment generating function implies thatR
{|x|>1}|x|F(dx) <∞, and henceLis a special semimartingale according to Jacod and Shiryaev (1987), Proposition II.2.29 a.
Proof of the proposition. It is well known that the moment generating function is of the Lévy-Khintchine form (1.2), withu replaced by−iv. (See Lukacs (1970), Theorem 8.4.2.) Since Lhas stationary and independent increments, the condition thateLt be a martingale underPθis equivalent to the following.
E h
eL1 eθL1 E[eθL1]
i
= 1.
In terms of the cumulant generating functionκ(v) = lnE[exp(vL1)], this condition may be stated as follows.
κ(θ+ 1)−κ(θ) = 0.
Equation (1.19) follows by inserting the Lévy-Khintchine representation of the cumulant generating function, that is,
κ(u) =ub+ c 2u2+
Z
eux−1−ux
F(dx).
(1.20)
The density process of the Esscher transform is given byZwith Zt= exp(θLt)
exp(κ(θ)t).
Hence it has a special structure: It depends onω only via the current valueLt(ω)of the Lévy process itself. By contrast, even on a filtration generated byL, the value of a general density process at timet may depend on the whole history of the process, that is, on the pathLs(ω),s∈[0, t].
Definition 1.14. Letτ(dx)be a measure on(IR,B1)withτ(IR)∈(0,∞). ThenGτ denotes the class of continuously differentiable functionsg: IR→IRthat have the following two properties
For allx∈IR, Z
{|h|>1}
g(x+h)
|h| τ(dh)<∞. (1.21)
If
Z g(x+h)−g(x)
h τ(dh) = 0for allx∈IR, thengis constant.
(1.22)
In (1.22), we define the quotient g(x+h)h−g(x)to beg0(x)forh= 0.
Lemma 1.15. a) Assume that the measureτ(dx)has a support with closureIR. For monotone continu- ously differentiable functionsg(x), property (1.21) implies property (1.22).
b) If the measureτ(dx)is a multiple of the Dirac measureδ0, thenGτ contains all continuously differ- entiable functions.
Proof. a) Without loss of generality, we can assume that g is monotonically increasing. Then
g(x+h)−g(x)
h ≥ 0 for all x, h ∈ IR. (Keep in mind that we have set g(x+0)0−g(x) = g0(x).) Hence R g(x+h)−g(x)
h τ(dh) = 0implies thatg(x+h) =g(x) forτ(dh)-almost everyh. Since the closure of the support ofτ(dx)was assumed to beIR, continuity ofg(x)yields the desired result.
b) Now assume thatτ(dx) =αδ0for someα >0. Condition (1.21) is trivially satisfied. For the proof of condition (1.22), we note thatR g(x+h)−g(x)
h αδ0(dh) =αg0(x).But if the derivative of the continuously differentiable functiong(x)vanishes for almost allx∈IR, then obviously this function is constant.
Conjecture 1.16. In the definition above, if the support ofτ has closureIR, then property (1.21) implies property (1.22).
Remark: Unfortunately, we are not able to prove this conjecture. It bears some resemblance with the integrated Cauchy functional equation (see the monograph by Rao and Shanbhag (1994).) This is the integral equation
H(x) = Z
IR
H(x+y)τ(dy) (almost allx∈IR), (1.23)
whereτ is a σ-finite, positive measure on(IR,B1). Ifτ is a probability measure, then the functionH satisfies (1.23) iff
Z
IR
H(x+y)−H(x)
τ(dy) = 0 (almost allx∈IR).
According to Ramachandran and Lau (1991), Corollary 8.1.8, this implies thatH has every element of the support ofτ as a period. In particular, the cited Corollary concludes thatHis constant if the support ofτ contains two incommensurable numbers. This is of course the case if the support ofτ has closure IR, which we have assumed above. However, we cannot apply this theorem since we have the additional factor1/hhere.
As above, denote by∂if the derivative of the functionf with respect to itsi-th argument. Furthermore, the notation XãY means the stochastic integral ofX with respect to Y. Note thatY can also be the deterministic processt. HenceXãtdenotes the Lebesgue integral ofX, seen as a function of the upper boundary of the interval of integration.
We are now ready to show the following theorem, which yields the announced uniqueness result for Esscher transforms.
Theorem 1.17. Let Lbe a Lévy process with a Lévy-Khintchine triplet (b, c, F(dx))satisfying one of the following conditions
1. F(dx)vanishes andc >0.
2. The closure of the support ofF(dx) is IR, andR
{|x|≥1}euxF(dx) < ∞ for u ∈ (a, b), where a <0< b.
Assume thatθ∈(a, b−1)is a solution ofκ(θ+ 1) =κ(θ), whereκis the cumulant generating function of the distribution ofL1. SetG(dx) :=cδ0(dx) +x(ex−1)eθxF(dx). Define a density processZ by
Zt:= exp(θLt)
exp(tκ(θ)) (t∈IR+).
Then under the measure Q loc∼ P defined by the density process Z with respect to P, exp(Lt) is a martingale.10 Z is the only density process with this property that has the formZt = f(Lt, t) with a functionf ∈C(2,1)(IR×IR+)satisfying the following: For everyt >0,g(x, t) :=f(x, t)e−θxdefines a functiong(ã, t)∈ GG.
10See Assumption 1.1 for a remark why a change of measure can be specified by a density process.
Proof. First, note that the condition onF(dx) implies that the distribution of everyLt has supportIR and possesses a moment generating function on (a, b). (The latter is a consequence of Wolfe (1971), Theorem 2.)
We have already shown thateLt indeed is aQ-martingale.
Letf ∈C(2,1)(IR×IR+)be such thatf(Lt, t)is a density process. Assume that under the transformed measure,eLt is a martingale. Thenf(Lt, t)as well asf(Lt, t)eLt are strictly positive martingales under P. By Ito' s formula for semimartingales, we have
f(Lt, t) =f(L0,0) +∂2f(Lt−, t)ãt +∂1f(Lt−, t)ãLt
+ (1/2)∂11f(Lt−, t)ã hLc, Lcit +
f(Lt−+x, t)−f(Lt−, t)−∂1f(Lt−, t)x ∗àLt
and
f(Lt, t)eLt =f(L0,0)eL0 +∂2f(Lt−, t) exp(Lt−)ãt + (∂1f(Lt−, t) +f(Lt−, t)) exp(Lt−)ãLt
+ (1/2) ∂11f(Lt−, t) + 2∂1f(Lt−, t) +f(Lt−, t)
ã hLc, Lcit +
f(Lt−+x, t)ex−f(Lt−, t)−(∂1f(Lt−, t) +f(Lt−, t))x
eLt−
∗àLt.
Since both processes are martingales, the sum of the predictable components of finite variation has to be zero for each process. So we have
0 =∂2f(Lt−, t)ãt+∂1f(Lt−, t)bãt+ (1/2)∂11f(Lt−, t)cãt +
Z Z
f(Lt−+x, t)−f(Lt−, t)−∂1f(Lt−, t)x
F(dx)dt and
0 =∂2f(Lt−, t) exp(Lt−)ãt+ ∂1f(Lt−, t) +f(Lt−, t)
bexp(Lt−)ãt + (1/2) ∂11f(Lt−, t) + 2∂1f(Lt−, t) +f(Lt−, t)
cãt +
Z Z
f(Lt−+x, t)ex−f(Lt−, t)−(∂1f(Lt−, t) +f(Lt−, t))x
eLt−
F(dx)dt.
By continuity, we have for anyt > 0andyin the support ofL(Lt−)(which is equal to the support of L(Lt), which in turn is equal toIR)
0 =∂2f(y, t) +∂1f(y, t)b+ (1/2)∂11f(y, t)c+ Z
f(y+x, t)−f(y, t)−∂1f(y, t)x
F(dx) and
0 =∂2f(y, t) + (∂1f(y, t) +f(y, t))b+ (1/2)(∂11f(y, t) + 2f1(y, t) +f(y, t))c +
Z
f(y+x, t)ex−f(y, t)−(∂1f(y, t) +f(y, t))x
F(dx).
Subtraction of these integro-differential equations yields 0 =f(y, t)b+f1(y, t)c+f(y, t)c/2 +
Z
f(y+x, t)(ex−1)−f(y, t)x
F(dx) (y∈IR).
Division byf(y, t)results in the equation 0 =b+ f1(y, t)
f(y, t)c+c 2+
Z f(y+x, t)
f(y, t) (ex−1)−x
F(dx). (y∈IR) (1.24)
By Proposition 1.13, the Esscher parameterθsatisfies a similar equation, namely 0 =b+θc+ c
2 + Z
eθx(ex−1)−x
F(dx).
(1.25)
Subtracting (1.25) from (1.24) yields 0 =
f1(y, t) f(y, t) −θ
c+
Z f(y+x, t) f(y, t) −eθx
(ex−1)F(dx) (y∈IR).
For the ratiog(y, t) :=f(y, t)/eθy, this implies11 0 = g1(y, t)
g(y, t)c+
Z g(y+x, t) g(y, t) −1
(ex−1)eθxF(dx) (y∈IR).
Multiplication byg(y, t)finally yields 0 =g1(y, t)c+
Z
g(y+x, t)−g(y, t)
(ex−1)eθxF(dx) (1.26)
=g1(y, t)c+
Z g(y+x, t)−g(y, t)
x x(ex−1)eθxF(dx)
=
Z g(y+x, t)−g(y, t)
x G(dx) (y∈IR),
where we set again g(y+0,t)0−g(y,t) :=g0(y). The measureG(dx) =cδ0+x(ex−1)eθxF(dx)is finite on every finite neighborhood ofx= 0. Furthermore,G(dx)is non-negative and has supportIR\{0}. Since we have assumed thatθlies in the interval(a, b−1)(where(a, b)is an interval on which the moment generating function ofL1is finite), we can find >0such that the interval(θ−, θ+ 1 +)is a subset of (a, b). Using the estimation|x| ≤(e−x+ex)/, which is valid for allx∈IR, it is easy to see that one can form a linear combination of the functionse(θ−)x,e(θ+)x,e(θ+1−)x, ande(θ+1+)xthat is an upper bound for the functionx(ex−1)eθx. Choosing >0small enough, all the coefficients in the exponents lie in(a, b). Therefore the measureG(dx)is finite. Since by assumptiong∈ GG, equation (1.26) implies that g(ã, t) is a constant for every fixedt, sayg(x, t) = c(t)for all x ∈ IR, t > 0. By definition ofg, this impliesf(x, t) = c(t)eθx for all x ∈ IR, t > 0. It follows from the relation E[f(Lt, t)] = 1that c(t) = 1/E
eθLt
= exp(−tκ(θ)).
11Note that
g1(y, t)
g(y, t) =(∂1f(y, t)−θf(y, t))e−θy
g(y, t) =∂1f(y, t) f(y, t) −θ and
g(y+x, t)
g(y, t) exp(θx) =f(y+x, t) f(y, t) .
Chapter 2
On the Lévy Measure
of Generalized Hyperbolic Distributions