On the nature of ill-posedness of an inverse problem arising in option

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1328 THeinandBHofmann Theorem 3.5. Let {un = N(Sn)} ∞ n=1 with N from formula (11) be a sequence of arbitragefree noisy option price functions satisfying the assumptions 3.1 and 3.3 that converges in the Banach space B2 = C[0, T ] to the fair option price function u = N(S). Thenthe associated sequence of functions {Sn} ∞ n=1 also converges to S in the Banach space B3 = C[0, T ]. Proof. In view of the positivity and continuity of the partial derivative ∂UBS(X, K, r, t, s) on the domain (t, s) ∈ [0, T ] × (0, ∞) (see lemma 2.1) we have, ∂s for fixed t ∈ (0, T ], � �−1 ∂UBS(X, K, r, t, Sim(t)) |Sn(t) − S(t)| � |un(t) − u(t)| ∂s with intermediate values Sim(t) between the positive values Sn(t) and S(t). Now, for given sufficiently small ε>0wechoose tε ∈ (0, T ]suchthat S(tε) = ε 4 .Sincethe function UBS is increasing with respect to s > 0, the functions Sn and S are increasing for t ∈ [tε, T ]andthere holds limn→∞ un(tε) = u(tε) >max(X − K ertε , 0) as well as limn→∞ un(T ) = u(T )0 (0 < t � T ).

On the nature of ill-posedness of an inverse problem arising in option pricing 1329 4. Solving the outer equation of the IP in L p -spaces for noisy option data In this section we measure deviations of the functions u δ and S δ from u and S on the interval [0, T ]bymeans of L p -norms. We consider the Banach spaces B2 = L q (0, T ) and B3 = L p (0, T ) with 1 � p, q < ∞ for the outer equation (16) of the IP. The positive data function u δ (t)(0 � t � T ) of observed maturity-dependent option prices is not necessarily smooth and arbitrage free in the sense of assumptions 3.1 and 3.3, but it satisfies assumption 4.1. Assumption 4.1. The non-negative data function uδ ∈ Lq approximated by the estimate (0, T ) (1 � q < ∞) is �u δ − u�L q (0,T ) � δ (27) the fair option price function u = F(a) = N(S) for a given noise level δ>0. Moreover, let a ∈ L∞ (0, T ) hold for the volatility function, where we assume an upper bound ¯c � �a�L ∞ (0,T ) implying 0 � S(t) � κ(0� t � T ) with κ := ¯cT. We apply a variant of the method of quasisolutions exploiting the fact that D κ + :={˜S ∈ D+ :0� ˜S(t) � κ(0� t � T ), ˜S(t1) � ˜S(t2) (0 � t1 < t2 � T )} is a compactum in the Banach space L p (0, T )(1 � p < ∞) (see, e.g., [4, example 3, p 26]). As an approximate solution of the outer equation (16) we use a quasisolution associated with the data uδ ,whichisaminimizer Sδ ∈ Dκ + of the extremal problem �N( ˜S) − u δ �L q (0,T ) −→ min, subject to ˜S ∈ D κ + . Then we can prove the following convergence assertion. Theorem 4.2. Let {Sδn ∞ } n=1 be a sequence of quasisolutions associated with a sequence of data {uδn ∞ } n=1 satisfying the inequality (27), where δn → 0 as n →∞.Thenthe convergence properties lim n→∞ �Sδn − S�L p (0,T ) = 0 (1 � p < ∞) (28) and lim n→∞ �Sδn − S�L ∞ (0,γ ) = 0 for all 0

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