Second-order integro-differential parabolic variational inequalities arising from the valuation of American option
© Sun et al.; licensee Springer. 2014
Received: 14 November 2012
Accepted: 3 December 2013
Published: 3 January 2014
This article deals with the existence and uniqueness of solution to the -valued parabolic variational inequalities with integro-differential terms which arise from the valuation of American option. The authors use the penalty method to construct a sequence of approximation parabolic problem and hence obtain the existence and uniqueness of solution to the approximation problem by using fixed point theory. Then the solution of parabolic variational inequalities is obtained by showing that the solution of this penalty problem converges to the variational inequalities. The uniqueness of the solution is also proven.
which is a continuous integral operator as defined in .
(1) becomes the linear variational inequality based on the famous Black-Scholes equation (see ). Such a variational inequality arises in many applications of American option pricing (see [3, 4]). To deal with this problem, some scholars often introduce a free and moving boundary problem. By adding a certain penalty term to the Black-Scholes equation, the solution of this variational inequality is extended to a fixed domain. Furthermore, this penalty term forces the solution to stay above the payoff function at expiry. Throughout the last decade, a number of papers addressing penalty schemes for American options have been published (see, for instance, [5–9] and references therein).
where is a kind of levy process ( for details, see [10, 11]). The authors in  generalize (2) and prove the existence and uniqueness of a classical solution to a more general problem in the parabolic domain , where Ω is an open, unbounded subset of , with a smooth boundary ∂ Ω. A related work in the context of quantum mechanics has been studied in [12, 13].
Therefore, the authors of this paper intend to study a more complex variational inequality involved in American option based on the more complicated PIDE type Black-Scholes equation than (2). And we will consider -valued parabolic variational inequality (1) using the penalty method. The rest of this article is as follows. In Section 2, we state the main result. Section 3 discusses the penalty problem which will be used to prove our main result. In Section 4, we show the proof of our main result.
2 Main result
Moreover, we use the following lemmas to show the existence and uniqueness of the solution for the penalty approximation of our main problem.
Then is a Banach space endowed with the norm .
Definition 2.2 A mapping is called to be compact if and only if the sequence is precompact for each bounded sequence , that is, there exists a subsequence such that converges in X.
Lemma 2.3 (Schaefer’s fixed point theorem)
is bounded. Then A has a fixed point.
The following two lemmas from the linear theory of parabolic partial differential equations can be found in .
Lemma 2.4 (Energy estimates)
where C is a positive constant depending only on Ω, T and the operator L.
Lemma 2.5 (Improved regularity)
where C is a positive constant depending only on Ω, T and the operator L.
Throughout this section, we impose the following assumptions:
(A1) The coefficients , , belong to the Holder space .
(A4) and belong to the Holder spaces and , respectively. Moreover, for all , we have that , .
hold for all , .
for all , where is a positive constant independent of .
(A8) If , then . If in , then in .
Using these assumptions, we will elaborate our main result.
The proof of Theorem 2.6 will be given in Section 4.
3 The penalty problem
For further details, see Theorem 10.4.1 in .
for all .
for any . The proof is quite standard and can be found in .
where C is independent of v.
where is a positive constant independent of v.
Thus, the proof is ended by letting . □
This is obviously contradictory. Therefore, we conclude that for any . □
where N is a positive constant independent of v.
where v is derived from w via (21).
Here we plan to prove the existence and uniqueness by Schaefer’s fixed point theorem. So that we need to present the continuity and compactness of the mapping M. In this proof we only prove the continuity of the mapping M. The compactness can be obtained by following similar arguments, so we omit it here.
Further, by Lemma 3.3, we have that is bounded. Hence the existence and uniqueness of this theorem are proven by using Lemma 2.3 with .
Therefore, the proof is complete. □
4 The proof of the main result
This is obviously contradictory. Therefore, we conclude that problem (1) has a unique solution. Moreover the estimate can be easily obtained by Lemma 3.5 and (28) with .
This work was supported by the National Nature Science Foundation of China (Grant No. 71171164) and the Doctorate Foundation of Northwestern Polytechnical University (Grant No. CX201235). The authors are sincerely grateful to the referees and the associate editor handling the paper for their valuable comments.
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