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

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