When the option pricing constraints do not determine the exponential L´evy model completely (this is for example the case if the number of constraints is finite), additional information may
be introduced into the problem by specifying a prior model : we suppose given a L´evy process P and look for the solution of the problem (2.4) that has the smallest relative entropy with respect to P . For two probabilities P and Q on the same measurable space (Ω, F), the relative entropy of Q with respect to P is defined by
I(Q|P ) = EPhdQdP logdQdPi, if Q P, ∞, otherwise, (2.10)
where by convention x log x = 0 when x = 0. For a time horizon T ≤ T∞ we define IT(Q|P ) :=
I(Q|FT|P |FT).
Minimum entropy least squares calibration problem Given prices CM of call options
and a prior L´evy process P , find a least squares solution Q∗ ∈ QLS, such that
I(Q∗|P ) = inf
Q∈QLSI(Q|P ). (2.11)
In the sequel, any such Q∗will be called a minimum entropy least squares solution (MELSS) and the set of all such solutions will be denoted by QM ELS.
The prior probability P must reflect our a priori knowledge about the risk-neutral distribu- tion of the underlying. A natural choice of prior, ensuring absence of arbitrage in the calibrated model, is an exponential L´evy model, estimated from the time series of returns. The effect of the choice of prior on the solution of the calibration problem and the possible ways to choose it in practice are discussed in Section 3.2.
Using relative entropy for selection of solutions removes, to some extent, the identification problem of least-squares calibration. Whereas in the least squares case, this was an important nuisance, now, if two measures reproduce market option prices with the same precision and have the same entropic distance to the prior, this means that both measures are compatible with all the available information. Knowledge of many such probability measures instead of one may be seen as an advantage, because it allows to estimate model risk and provide confidence intervals for the prices of exotic options. However, the calibration problem (2.11) remains ill-posed: since the minimization of entropy is done over the results of least squares calibration, problem (2.11) may only admit a solution if problem (2.4) does. Also, QLS is not necessarily a compact set,
so even if it is nonempty, the least squares solution minimizing the relative entropy may not exist. Other undesirable properties like absence of continuity and numerical instability are also inherited from the least squares approach. In Section 2.5 we will propose a regularized version of problem (2.11) that does not suffer from these difficulties.
The choice of relative entropy as a method for selection of solutions of the calibration problem is driven by the following considerations:
• The relative entropy is a convenient notion of distance for probability measures. Indeed, it is convex, nonnegative functional of Q for fixed P , equal to zero if and only if dQdP = 1 P -a.s. To see this, observe that
EP dQ dP log dQ dP = EP dQ dP log dQ dP − dQ dP + 1 ,
and that z log z − z + 1 is a convex nonnegative function of z, equal to zero if and only if z = 1.
• The relative entropy of two L´evy processes is easily expressed in terms of their character- istic triplets (see Theorem 2.9).
• Relative entropy, also called Kullback-Leibler distance, is a well-studied object. It appears in many domains including the theory of large deviations and the information theory. It has also already been used in finance for pricing and calibration (see Section 2.3). We can therefore use the known properties of this functional (see e.g. [93]) as a starting point of our study.
The minimum entropy least squares solution need not always exist, but if the prior is chosen correctly, that is, if there exists at least one solution of problem (2.4) with finite relative entropy with respect to the prior, then the MELSS will also exist, as shown by the following lemma. We recall that LN A stands for the set of L´evy processes, satisfying the no arbitrage conditions
of Proposition 1.8.
Lemma 2.6. Let P ∈ LN A∩ L+B for some B > 0 and suppose that the problem (2.4) admits a
solution Q+ with I(Q+|P ) = C < ∞. Then the problem (2.11) admits a solution.
Proof. The proof of this lemma uses the properties of the relative entropy functional (2.10) that will be proven in Section 2.4 below. Under the condition of the lemma, it is clear that the
solution Q∗ of problem (2.11), if it exists, satisfies I(Q∗|P ) ≤ C. This entails that Q∗ P ,
which means by Proposition 1.5 that Q∗ ∈ L+B. Therefore, Q∗ belongs to the set L+B∩ {Q ∈ M ∩ L : kCQ− CMk = kCQ
+
− CMk} ∩ {Q ∈ L : I(Q|P ) ≤ C}. (2.12)
Lemma 2.10 and the Prohorov’s theorem entail that the level set {Q ∈ L : I(Q|P ) ≤ C} is relatively weakly compact. On the other hand, by Corollary 2.1, I(Q|P ) is weakly lower semicontinuous with respect to Q for fixed P . Therefore, the set {Q ∈ P(Ω) : I(Q|P ) ≤ C} is weakly closed and since by Lemma 2.4, M ∩ L+B is also weakly closed, the set M ∩ L+B∩ {Q ∈
L : I(Q|P ) ≤ C} is weakly compact. Lemma 2.2 then implies that the set (2.12) is also weakly compact. Since I(Q|P ) is weakly lower semicontinuous, it reaches its minimum on this set.