Abstract
In this work, we study the effect of nonlinear source term in Black-Scholes model by finding the solution of it. We use the mathematical concepts of existence and uniqueness to arrive the conclusion. The transformation of the nonlinear equation into heat equation leads to the existence of solution through fixed-point theorems, semigroup theory, and certain regularity conditions imposed on variables.
1. Introduction
The Black-Scholes equation is a mathematical model that plays an important role in dynamics of financial and economic identity. In order to increase the efficiency of Black-Scholes model, the condition on the parameters is made uncertain, and their behaviours are studied in [1–4]. Agyeman and Oduro in [5]studied the existence with nonlinear force term depending on this deviation of the option price.
Based on this theory, let us study the solvability of the following nonlinear Black-Scholes type equation. where , , and .
Here, is the option price; is the volatility constant; is the stock price; is the risk-free interest rate; is the potential function; time and the nonlinear term represent the various price effects that happens during the period. Assume that the force effect happen as a multiple of the option price. In [1], the function is assumed to satisfy the Caratheodory conditions that, is measurable over all . In addition, the function and its derivatives are bounded and continuous throughout its entire region for all time . More details on the function spaces are explained in [6].
The nonlinear term on the right hand side represents the monetary assistance provided to the investor. Pilant and Rundell in [7] studied the existence and uniqueness of the partial differential equations (PDEs) with nonlinear term using fixed-point technique. Shu and Shi in [8] studied the mild solution of impulsive fractional evolution equations through fixed-point technique. Following this, Nanda and Das [9] extended the aforementioned method for solving various forms heat conduction problems involving nonlinear source terms. The similar problem is studied by one of the authors in [10], where fixed-point technique is applied but for its inverse theory. In [11, 12], the authors adopted fixed-point technique in the discussion on the existence of various forms of solution of an almost periodic stochastic differential equations. The works in [13–15] studied the existence of solutions for PDE under nonlocal conditions. Moreover, the works in [16–20] also deals with the different forms of Black-Scholes equation and studied the existence theory. In [21], the author used the semigroup theory and operators to study the Black-Scholes model. Hence, considering the mentioned points, it is clear that the discussion on Black-Scholes model using fixed-point theory is of great interest. The problem under consideration is a special one as it depends on the forcing term that reflects today’s economic outlook.
The novelty of this problem lies in the form under investigation. This is one of the most difficult forms of the Black-Scholes equation. Physically, this problem is interpreted as follows: find the value for option “” that make up for any losses incurred while exercising the option. The integral term is defined to check the cash flow required for offsetting. The loss must be within the specified range. This paper is organized as follows: the existence part is discussed is Section 2, the uniqueness property is established in Section 3, and the conclusion in Section 4. This discussion is based on the technique adapted in [22].
2. Existence Result
In this segment, transform the left hand side of (1) in the form of heat equation and thereby the direct application of the notations and procedure of [22], we obtain the necessary result. Also, the integrodifferential forcing term can be neutralized through the method discussed in [14]. Let
The derivatives of transformation are
Substituting the derivatives of Black-Scholes equation, we get
Setting , the equation becomes for . As more modifications on variables are required to make the unnecessary terms be removed, let
Computing the partial derivatives of and using this in (5), we get since and are arbitrary constants, introduced to eliminate the and terms
Then, and form the coefficients of 0, which eliminate and terms, respectively. The equation is then reduced to a one-dimensional heat equation, we can attain that
Substituting (9) in (1) and replacing the variables on the R.H.S of (1) , , and . Thence, we have with conditions where lies in a multidimensional bounded domain Obviously, the significance of the nonlinear term (9) is its dependence on
That is, while the existence on is unknown at whole interval, is given a precursor.
From [23, 24], the existence of the solution of (1) should be nonnegative, nondecreasing potential for which is derivable. We present a brief demonstration on its solvability based on generic assumptions about potential .
The proof depends on fixed-point theorem due to Schauder and the fact that under acceptable presumptions, from a semigroup on , where stands for Laplacian, which is subject to Dirichlet boundary conditions. The expected form of solution (1) and, as a result, is described by the relation
Theorem 1. Let be nonnegative and If one of the relations or (i) holds, then for some and
There exists at least one possible solution to (1). Moreover, for If then for
Proof. Part (i).
Take a nonnegative function . Based on the boundedness of and by defining we arrive at
is bounded.
Set and let
indicate a closed ball whose cent is at origin and radius . Take with the fact that, for any the mapping satisfies without a doubt,
Combining the results and the fact for some positive and . Moreover, for a nonnegative , the operator generates a positive contraction semigroup on each for . Hence,
Let us delineate,
Then, (21) signifies that
with Using the condition and (20) together with and positive , such that
Therefore,
is bounded in , the latter is being compact ingrained in since . To validate the property of continuity of thereby using (19), (21), and (24), we get
The continuous embedding of in using (25) and (27), we get
for Hence, the required results on continuity are established through (17). Using the precompact image of , the theorem due to Schauder shows the presence of point with the property . Now, define as
in order to obtain the solution for (1). If then for because the semigroup is positive. This completes proof of part (i).
Part (ii). Now, assume a nonnegative and that , for positive . Let us presume with . We modify the set
One of the presumption on includes
Further, maintains its continuity for small in over . Thus, for , we have with meager positive . Hence, the relation (20) holds good if the equality (33) and the corresponding results are combined. Moreover, (21) holds for . Defining as in (22), by similar arguments as in previous part, a precompact image of can be arrived (26) along with continuity that follows from the relations (21), (24), and (33) as given in the following:
The required result is achieved by applying Schauder’s fixed-point theorem.
Corollary 2. Let be the solution to (1) provided by Theorem 1. If then If for some positive , then
Proof. The proof is the direct consequence of Theorem 1.
3. Uniqueness Result
Following the methodology adopted in [24], we demonstrate the property of uniqueness for small data in (1) if T is small along with locally Lipschitz continuous and is a function which is nondecreasing.
Assume that the existence of solutions is true and the relations (37) and (38) hold good, let the solutions of (1) be and with . Fix . for fixed value , obtained by the definition on , which behaves as uniformly Lipschitz over its region of existence, (i.e., on the set ). This leads to the fact (35) which entails that where (38) allows inequality to end. Now, owing to (21), (26), and (39), we get
So, (40) and (41) cover it where is borrowed from Poincare’sine quality. Now, if , it refers to in .
In the following proposition, we study the monotonicity condition and its effect on uniqueness theory.
Proposition 3. Let be unknown that satisfies (37), , and consider . There is such that (1) has a unique value based on the inequality
Proof. The proof is obtained through Banach’s fixed-point theorem. Fix and take along with Notice that embeds continuously into . Thus, as is uniformly Lipschitz continuous on compact sets, there exists a constant with Especially from the proof of Theorem 1, we can write Moreover, (27) and (45) combines to give for some and positive . Therefore, The inequality together with (46) shows that behaves like a contraction mapping with being very small. Consequently, if is very small, then there exists a unique with .
4. Conclusion
As discussed, the nonlinear force term on the right hand side plays an important role in establishing the existence of solution. It is clear that the cash flows to the market needs to be examined and must always be within limits. This fixed value must be compared with the actual price of the option and the investment decision that must be made.
Data Availability
No data were used to support the finding of this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors contributed equally to this work and read and approved the final version of the manuscript.