2.5 Regularizing the calibration problem
2.5.2 Convergence of regularized solutions
In this section we study the convergence of solutions of the regularized calibration problem (2.27) to the solutions of the minimum entropy least squares calibration problem (2.11) when the noise level in the data tends to zero.
Theorem 2.17. Let {CMδ } be a family of data sets of option prices such that kCM− CMδ k ≤ δ,
let P ∈ LN A∩ L+B and suppose that there exist a solution Q of problem (2.4) with data CM (a
least squares solution) such that I(Q|P ) < ∞. If kCQ− C
Mk = 0 (the constraints are reproduced exactly), let α(δ) be such that α(δ) → 0
and α(δ)δ2 → 0 as δ → 0. Otherwise, let α(δ) be such that α(δ) → 0 and α(δ)δ → 0 as δ → 0. Then every sequence {Qδk}, where δ
k→ 0 and Qδk is a solution of problem (2.27) with data
Cδk
M, prior P and regularization parameter α(δk), has a weakly convergent subsequence. The
limit of every convergent subsequence is a solution of problem (2.11) (MELSS) with data CM
Proof. By Lemma 2.6, there exists at least one MELSS with data CM and prior P , that has
finite relative entropy with respect to the prior. Let Q+ be any such MELSS. Since Qδk is the
solution of the regularized problem, for every k, kCQδk − Cδk
Mk2+ α(δk)I(Qδk|P ) ≤ kCQ +
− Cδk
Mk2+ α(δk)I(Q+|P ).
Using the fact that for every r > 0 and for every x, y ∈ R,
(1 − r)x2+ (1 − 1/r)y2 ≤ (x + y)2≤ (1 + r)x2+ (1 + 1/r)y2, we obtain that (1 − r)kCQδk − CMk2+ α(δk)I(Qδk|P ) ≤ (1 + r)kCQ+− CMk2+ 2δk2 r + α(δk)I(Q +|P ), (2.34)
and since Q+ is a least squares solution with data C
M, this implies for all r ∈ (0, 1) that
α(δk)I(Qδk|P ) ≤ 2rkCQ + − CMk2+ 2δ2 k r + α(δk)I(Q + |P ). (2.35)
If the constraints are reproduced exactly, then kCQ+
−CMk = 0 and with the choice r = 1/2,
the above expression yields:
I(Qδk|P ) ≤ 4δ 2 k
α(δk)
+ I(Q+|P ).
Since, by the theorem’s statement, in the case of exact constraints δ2k
α(δk) → 0, this implies that
lim sup
k {I(Q
δk|P )} ≤ I(Q+|P ). (2.36)
If kCQ+ − CMk > 0 (misspecified model) then the right-hand side of (2.35) achieves its
maximum when r = δkkCQ +
− CMk−1, in which case we obtain
I(Qδk|P ) ≤ 4δk
α(δk)kC Q+
− CMk + I(Q+|P ).
Since in the case of approximate constraints, δk
α(δk) → 0, we obtain (2.36) once again.
Inequality (2.36) implies in particular that I(Qδk|P ) is uniformly bounded, which proves,
by Lemmas 2.10 and 2.4, that {Qδk} is relatively weakly compact in M ∩ L+ B.
Choose a subsequence of {Qδk}, converging weakly to Q∗ ∈ M ∩ L+
B. To simplify notation,
this subsequence is denoted again by {Qδk}
k≥1. Substituting r = δ into Equation (2.34) and
making k tend to infinity shows that lim sup k kC Qδk − CMk2 ≤ kCQ + − CMk2.
Together with Lemma 2.2 this implies that
kCQ∗− CMk2 ≤ kCQ +
− CMk2,
hence Q∗ is a least squares solution. By weak lower semicontinuity of I (cf. Lemma 2.11) and using (2.36),
I(Q∗|P ) ≤ lim inf
k I(Q
δk|P ) ≤ lim sup k
I(Qδk|P ) ≤ I(Q+|P ),
which means that Q∗ is a MELSS. The last assertion of the theorem follows from the fact that in this case every subsequence of {Qδk} has a further subsequence converging toward Q+.
Numerical implementation of the
calibration algorithm
Before solving the calibration problem (2.27) numerically, we reformulate it as follows:
• The calibration problem (2.27) is expressed in terms of the characteristic triplets (A, νQ, γQ) and (A, νP, γP) of the prior L´evy process and the solution. This can be done using Equa-
tions (1.1), (1.25) and (1.26) (option pricing by Fourier transform) and Equation (2.19) (expression of the relative entropy in terms of characteristic triplets).
• The L´evy measure νQ is discretized (approximated by a finite-dimensional object). This discussed in detail in Section 3.1.
The prior L´evy process P is a crucial ingredient that must be specified by the user. Section 3.2 suggests different ways to do this and studies the effect of a misspecification of the prior on the solutions of the calibration problem. Section 3.3 discusses the methods to choose the regularization parameter α based on the data and Section 3.4 treats the choice of weights wi of
different options. Section 3.5 details the numerical algorithm that we use to solve the calibration problem once the prior, the regularization parameter and the weights have been fixed.
The calibration algorithm, including the automatic choice of the regularization parameter, has been implemented in a computer program levycalibration and various tests have been carried out, both on simulated option prices (computed in a known exponential L´evy model) and using real market data. Section 3.6 discusses the results of these tests.
3.1
Discretizing the calibration problem
A convenient way to discretize the calibration problem is to take a prior L´evy process P with L´evy measure supported by a finite number of points:
νP =
M −1X k=0
pkδ{xk}(dx). (3.1)
In this case, by Proposition 1.5, the L´evy measure of the solution necessarily satisfies νQ νP, therefore νQ= M −1 X k=0 qkδ{xk}(dx), (3.2)
that is, the solution belongs to a finite-dimensional space and can be computed using a numerical optimization algorithm. The advantage of this discretization approach is that we are solving the same problem (2.27), only with a different prior measure, so all results of Section 2.5 (existence of solution, continuity etc.) hold in the finite-dimensional case.
Taking L´evy measures of the form (3.1) we implicitly restrict the class of possible solutions to L´evy processes with bounded jumps and finite jump intensity. However, in this section we will see that this restriction is not as important as it seems: the solution of a calibration problem with any prior can be approximated (in the weak sense) by a sequence of solutions of calibration problems with priors having L´evy measures of the form (3.1). Moreover, in Section 3.6.1 we will observe empirically that smiles produced by infinite intensity models can be calibrated with arbitrary precision by such jump-diffusion models.
We start with a lemma showing that every L´evy process can be approximated by L´evy processes with atomic L´evy measures.
Lemma 3.1. Let P be a L´evy process with characteristic triplet (A, ν, γ) with respect to a continuous bounded truncation function h, satisfying h(x) = x in a neighborhood of 0, and for every n, let Pn be a L´evy process with characteristic triplet (A, νn, γ) (with respect to the same
truncation function) where νn:= 2n X k=1 δ{xk}(dx)µ([xk− 1/ √ n, xk+ 1/√n)) 1 ∧ x2 k ,
xk := (2(k − n) − 1)/√n and µ is a finite measure on R, defined by µ(B) :=RB(1 ∧ x2)ν(dx)
Proof. For a function f ∈ Cb(R), define fn(x) := 0, x ≥ 2√n, 0, x < −2√n, f (xi), x ∈ [xi− 1/√n, xi+ 1/√n) with 1 ≤ i ≤ 2n, Then clearly Z (1 ∧ x2)f (x)νn(dx) = Z fn(x)µ(dx).
Since f (x) is continuous, fn(x) → f(x) for all x and since f is bounded, the dominated conver-
gence theorem implies that lim n Z (1 ∧ x2)f (x)νn(dx) = lim n Z fn(x)µ(dx) = Z f (x)µ(dx) = Z (1 ∧ x2)f (x)ν(dx). (3.3)
With f (x) ≡ h1∧x2(x)2 the above yields:
lim n Z h2(x)νn(dx) = Z h2(x)ν(dx).
On the other hand, for every g ∈ Cb(R) such that g(x) ≡ 0 on a neighborhood of 0, f(x) := 1∧xg(x)2
belongs to Cb(R). Therefore, from Equation (3.3), lim n
R
g(x)νn(dx) = Rg(x)ν(dx), and by
Proposition (1.7), Pn⇒ P .
Theorem 3.2. Let P, {Pn}n≥1 ⊂ LN A∩ L+B such that Pn ⇒ P , let α > 0, let CM be a data
set of option prices and for each n let Qn be a solution of the calibration problem (2.27) with
prior Pn, regularization parameter α and data CM. Then the sequence {Qn}n≥1 has a weakly
convergent subsequence and the limit of every weakly convergent subsequence of {Qn}n≥1 is a
solution of the calibration problem (2.27) with prior P .
Proof. By Lemma 2.12, there exists C < ∞ such that for every n, one can find ˜Qn∈ M∩L with
I( ˜Qn|Pn) ≤ C. Since, by Lemma 2.2, kCQ˜n − CMk2w ≤ S02 for every n and Qn is the solution
of the calibration problem, I(Qn|Pn) ≤ S02/α + C < ∞ for every n. Therefore, by Lemma
2.10, {Qn} is tight and, by Prohorov’s theorem and Lemma 2.4, weakly relatively compact in
M ∩ L+B. Choose a subsequence of {Qn}, converging weakly to Q ∈ M ∩ L+B. To simplify
notation, this subsequence is also denoted by {Qn}n≥1. It remains to show that Q is indeed a
Lemma 2.11 entails that
I(Q, P ) ≤ lim infn I(Qn, Pn), (3.4)
and since, by Lemma 2.2, the pricing error is weakly continuous, we also have
kCQ− CMk2w+ αI(Q, P ) ≤ lim infn {kCQn− CMk2w+ αI(Qn, Pn)}. (3.5)
Let φ ∈ Cb(Ω) with φ ≥ 0 and EP[φ] = 1. Without loss of generality we can suppose that for
every n, EPn[φ] > 0 and therefore Q0
n, defined by Q0n(B) := E Pn[φ1B]
EPn[φ] , is a probability measure
on Ω. Clearly, Q0nconverges weakly to Q0 defined by Q0(B) := EP[φ1B]. Therefore, by Lemma
2.2, lim n kC Q0 n− C Mk2w = kCQ 0 − CMk2w. (3.6) Moreover, lim n I(Q 0 n|Pn) = lim n Z Ω φ EPn[φ]log φ EPn[φ]dPn = lim n 1 EPn[φ] Z Ω φ log φdPn− lim n log Z Ω φdPn= Z Ω φ log φdP. (3.7) For the rest of this proof, for every φ ∈ L1(P ) with φ ≥ 0 and EP[φ] = 1 let Q
φ denote the
probability measure on Ω, defined by Qφ(B) := EP[φ1B] for every B ∈ F. Using (3.5–3.7) and
the optimality of Qn, we obtain that for every φ ∈ Cb(Ω) with φ ≥ 0 and EP[φ] = 1,
kCQ− CMk2w+ I(Q, P ) ≤ kCQφ− CMk2w+ I(Qφ|P ) (3.8)
To complete the proof of the theorem, we must generalize this inequality to all φ ∈ L1(P ) with
φ ≥ 0 and EP[φ] = 1.
First, let φ ∈ L1(P ) ∩ L∞(P ) with φ ≥ 0 and EP[φ] = 1. Then there exists a sequence
{φn} ⊂ Cb(Ω) such that φn → φ in L1(P ), φn ≥ 0 for all n and φn are bounded in L∞ norm
uniformly on n. Moreover, φ0
n:= φn/EP[φn] also belongs to L1(P ), is positive and φ0n L1(P )
−−−−→ φ because by the triangle inequality,
kφ0n− φkL1 ≤ 1
EP[φ
n] kφn− φkL
1 + kφ − φEP[φn]kL1−−−→
n→∞ 0.
In addition, it is easy to see that Qφ0
n ⇒ Qφ. Therefore,
lim
n kC
Qφ0n
Since φ0
nare bounded in L∞ norm uniformly on n, φ0nlog φ0nis also bounded and the dominated
convergence theorem implies that limnI(Qφ0
n|P ) = I(Qφ|P ). Passing to the limit in (3.8), we
obtain that this inequality holds for every φ ∈ L1(P ) ∩ L∞(P ) with φ ≥ 0 and EP[φ] = 1.
Let us now choose a nonnegative φ ∈ L1(P ) with EP[φ] = 1. If I(Qφ|P ) = ∞ then surely
(3.8) holds, therefore we can suppose I(Qφ|P ) < ∞. Let φn= φ ∧ n. Then φn → φ in L1(P )
because kφn− φkL1 ≤ Z φ≥n φdP = Z φ≥n φ log φ log φ dP ≤ I(Qφ|P ) log n → 0. Denoting φ0n:= φn/EP[φn] as above, we obtain that
lim
n kC
Qφ0n
− CMk2w = kCQφ− CMk2w
Since, for a sufficiently large n, |φn(x) log φn(x)| ≤ |φ(x) log φ(x)|, we can once again apply the
dominated convergence theorem: lim n Z φ0nlog φ0ndP = 1 limnEP[φn] lim n Z φnlog φndP − lim n log E P[φ n] = Z φ log φdP
Therefore, by passage to the limit, (3.8) holds for all φ ∈ L1(P ) with φ ≥ 0 and EP[φ] = 1, which completes the proof of the theorem.
To approximate numerically the solution of the calibration problem (2.27) with a given prior P , we need to construct, using Lemma 3.1, an approximating sequence {Pn} of L´evy processes
with atomic measures such that Pn⇒ P . The sequence {Qn} of solutions corresponding to this
sequence of priors will converge (in the sense of Theorem 3.2) to a solution of the calibration problem with prior P .