Tikhonov Regularization

An important issue in quantitative nance is model calibration. The calibration problem is the inverse of the pricing problem. Instead of computing prices in a model with given values for its parameters, one wishes to compute the values of the model parameters that are consistent with observed prices. Now, it is well-known by physicists that such inverse problems are typically ill-posed. So, if one perturbs the data (e.g., if the observed prices move from some small amount between today and tomorrow), it is quite typical that a numerically determined best t solution of the calibration problem switches from one `basin of attraction' to the other, thus the numerically determined solution isunstable. To achieve robustness of model (re)calibration, we need to introduce some regularization. The most widely known and applicable regularization method is Tikhonov( Phillips) regularization method. In this paper we provide a survey about Tikhonov regularization and we illustrate it by application to the problem of calibrating a local volatility model. 1 Financial Motivation An important issue in quantitative nance is model calibration. The calibration problem is the inverse of the pricing problem. Instead of computing prices in a model with given values for its parameters, one wishes to compute the values of the model parameters that are consistent with observed prices (up to the bid ask spread). Now, it is well-known by physicists that such inverse problems are typically ill-posed. Recall that a problem is well-posed (as de ned by Hadamard) if its solution exists, is unique, and depends continuously on its input data. Thus there are three reasons for which a problem might be illposed: • it admits no solution, or/and • it admits more than one solution, or/and • the solution(s) of the inverse problem do(es) not depend on the input data in a continuous way. In the case of calibration problems in nance, except for trivial situations, there exists typically no instance of a given class of models which is exactly consistent with a full calibration data set, including a number of option prices, a zero-coupons curve, an expected dividend yield curve on the underlying, etc. But there are often various instances of a given class of models that t the data within the bid ask spread. In this case, if one perturbs the data (e.g., if the observed prices move from some small amount between today and tomorrow), it is quite typical that a numerically determined best t solution of the calibration problem switches from one `basin of attraction' to the other, thus the numerically determined solution is not stable either. In order to get a well-posed problem, we need to introduce some regularization. The most widely 2 Tikhonov Regularization known and applicable regularization method is Tikhonov( Phillips) regularization method [17, 15, 10]. 2 Tikhonov regularization of non-linear inverse problems We consider a Hilbert space H, a closed convex non-void subset A of H, a direct operator (`pricing functional') H ⊇ A 3 a Π −→ Π(a) ∈ R , (so a corresponds to the set of model parameters), noisy data (`observed prices') π, and a prior a0 ∈ H (a priori guess for a). The Tikhonov regularization method for inverting Π at π, or estimating the model parameter a given the observation π, consists in: • reformulating the inverse problem as the following nonlinear least squares problem: mina∈A ∥∥Π(a)− πδ∥∥2 (1) to ensure existence of a solution, • selecting the solutions of the previous nonlinear least squares problem that minimize ‖a− a0‖ over the set of all solutions, and • introducing a trade-o between accuracy and regularity, parameterized by a level of regularization α > 0, to ensure stability. More precisely, we introduce the following cost criterion: J α (a) ≡ ∥∥Π(a)− πδ∥∥2 + α ‖a− a0‖ . (2) Given α, δ and a further parameter η, where η represents an error tolerance on the minimization, we de ne a regularized solution to the inverse problem for Π at π, as any model parameter a α ∈ A such that J α ( a α ) ≤ J α (a) + η , a ∈ A . Under suitable assumptions, one can show that the regularized inverse problem is well-posed, as follows. We rst postulate that the direct operator Π satis es the following regularity assumption. Assumption 2.1 (Compactness) Π(an) converges to Π(a) in R if an weakly-converges to a in H. We then have the following stability result. Theorem 2.1 (Stability) Let πn → π, ηn → 0 when n → ∞. Then any sequence of regularized solutions ann α admits a subsequence which converges towards a regularized solution a δ,η=0 α . Assuming further that the data lie in the range of the model leads to convergence properties of regularized solutions to (unregularized) solutions of the inverse problem as α→ 0. Let us then make the following additional assumption on Π. Assumption 2.2 (Range property) π ∈ Π(A). By an a0 solution to the inverse problem for Π at π, we mean any a ∈ Argmin {Π(a)=π} ‖a − a0‖. Note that the set of a0-solutions is non-empty, by Assumption 2.2. Theorem 2.2 (Convergence; see, for instance, Theorem 2.3 of Engl et al [11]) Let the perturbed parameters αn, δn, ηn and the perturbed data πn ∈ R satisfy (n ∈ N) ‖π − πn‖ ≤ δn,