Three new weak formulations of the problem of American call options valuation

Three new weak formulations of the problem of American call options valuation are given. The first of these is a parabolic obstacle problem in a finite domain. The second is a parabolic variational inequality with a convex and Lipschitz-continuous functional and the last one is a semilinear parabolic equation with a discontinuous spatial operator. All these problems are equivalent – they have the same unique solution. Different formulations can be used both for theoretical research and for constructing numerical methods.


Introduction
The option is one of the most important financial derivatives, and a wide variety of options are traded in exchanges. Its gives to its owner the right to buy (call option) or to sell (put option) a fixed quantity of assets of a specified stock at a fixed price (exercise or strike price). There are two major types of traded options. One is the American option that can be exercised at any time prior to its expiry date, and the other option, which can only be exercised on the expiry date, is called the European option. It was shown by Black and Scholes that the value of an European option is governed by a second-order differential equation of degenerate parabolic type [1]. This is now referred to as the Black-Scholes equation. The value of an American option is determined by a linear complementarity problem involving the Black-Scholes differential operator and a constraint on the value of the option (cf. e.g. [2]). In the case of put options this complementarity problem can also be formulated in a weak form as a variational inequality with an obstacle inside a domain (cf. e.g. [3]).
In this paper we consider call options problem and propose three new formulations of it: a parabolic variational inequality in a finite domain with an obstacle inside a domain, a parabolic variational inequality with a convex and Lipschitz-continuous functional and a semilinear parabolic equation with a discontinuous spatial operator. The existence of a unique solution is proved, the same for all the constructed problems. Different formulations of the problem are useful from the point of view of investigating the properties of its solution, and also from the point of view of constructing various approximations [4], [5], [6].

Formulation of the problem
Let u denote the value of an American call option with strike price K , expiry date T , and let x be the price of the underlying asset of the option. It is known that u satisfies the following linear complementarity problem almost everywhere (a.e.) in Here t denote the time until expiry, At is the Black-Scholes operator defined by the equality where  is a local volatility, d is a variable dividend yield and r is an interest rate. We focus on the case of a vanilla call, when the payoff function is : is called the region of exercise and its boundary is called the exercise boundary or free boundary. Let it admits a parameterization The function  (early exercise curve) is a hidden unknown of the considered problem. Further we assume that the region of exercise has nonzero Lebesgue measure, since otherwise due to (3 the complementarity problem reduces to Black-Scholes We make some assumptions on ,, dr  , which ensures the existence of a weak solution of the problem (1)-(4), its regularity and also the non-emptiness of the region of exercise: || xt Further we shall prove that this remains true in the general case under the assumption   1 H (see the theorem 1 below).

Function spaces
with the norm and the inner product We note also that the embedding VH  is not compact.
We introduce the subspace are continuous and dense, so, we can extend the inner product of H to the duality pairing between * V and We call We also use well-known Sobolev spaces . We also define the space

Perpetual call options
Let us define the operator The solution of this problem is well known (cf. e.g. \cite [7, p. 259). It has the form We fix an arbitrary 0 LL  and we will assume that the function

is a bounded function and
It is easy to verify that () ux  is a solution of the following complementarity problem on (0, ) L :

Variational inequality with an obstacle inside a domain
Further the Black-Scholes operator will be considered as a linear operator from () (5). We define operator * 22 a.e. on (0, ) T . We note some important structural properties of these operators.

A t w t w t A t w t w t
Proof. The existence of a unique solution of the problem (P) can be proved by a penalty method in the same way as in [8] for the case of put options. Let us prove that It is the solution of the inequality ( , ) ( , ) .