# A Study on the Impact of Nonlinear Source Term in Black-Scholes Option Pricing Model

Authors: P. Sozhaeswari; R. Sowrirajan; K. Loganathan; Sonam Gyeltshen
Journal: Abstract and Applied Analysis
 (2022)

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## 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.


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## Body

## 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.(1)V~t+12σ2S2V~SS+rSV~S−rV~=φ~∫0TfS,t,V~SdtV~,where S>0, t∈0,T, and Ω∗⊂Rn.Here,V~ is the option price; σ is the volatility constant; S is the stock price; r is the risk-free interest rate; φ~ is the potential function; time T>0 and the nonlinear term fS,t,V~Sdt 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 f:Ω∗×0,T×Ω∗⟶Ω∗ is assumed to satisfy the Caratheodory conditions that, f.,.,V~ is measurable over all V~∈Ω∗. In addition, the function f.,t,. and its derivatives are bounded and continuous throughout its entire region for all time t. More details on the function spaces are explained in [6].The nonlinear termΦ~·V~ 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 “V~” 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
(2)S=ex,t=T−2τσ2V~S,t=v~x,τ=v~lnS,σ22T−t.The derivatives of transformation are(3)∂V~∂t=−σ22v~τ,∂V~∂S=1Sv~x,∂2V~∂S2=1S2v~xx−v~x.Substituting the derivatives of Black-Scholes equation, we get(4)V~t+12σ2S2V~SS+rSV~S−rV~=−v~τ+v~xx−v~x+2rσ2v~x−2rσ2v~.Settingk=2r/σ2, the equation becomes
(5)V~t+12σ2S2V~SS+rSV~S−rV~=−v~τ+v~xx+k−1v~x−kv~,for −∞<x<∞,0≤τ≤σ2/2T. As more modifications on variables are required to make the unnecessary terms be removed, let
(6)v~x,τ=eαx+βτu~x,τ=ϕu~.Computing the partial derivatives ofv~ and using this in (5), we get
(7)V~t+12σ2S2V~SS+rSV~S−rV~=−u~τ+u~xx+2α+k−1u~x+α2+k−1α−k−βu~,since α and β are arbitrary constants, introduced to eliminate the u~x and u~ terms
(8)α=−k−12,β=α2+k−1α−k=−k+124.Then,α and β form the coefficients of 0, which eliminate u~x and u~ terms, respectively. The equation is then reduced to a one-dimensional heat equation, we can attain that
(9)V~t+12σ2S2V~SS+rSV~S−rV~=−u~τ+u~xx#.Substituting (9) in (1) and replacing the variables on the R.H.S of (1) S=x, t=τ, and V~=u~. Thence, we have
(10)u~τ−u~xx+φ~∫0Tfx,τ,u~xdτu~=0,with conditions
(11)u~τ=0=u~0,u~∂Ω∗=0,τ,x∈0,T×Ω∗,where Ω∗ lies in a multidimensional bounded domain C2. Obviously, the significance of the nonlinear term (9) is its dependence on
(12)u~T≔∫0Tfx,τ,u~xdτ.That is, while the existence on0,T is unknown at whole interval, T>0 is given a precursor.From [23, 24], the existence of the solution of (1) should be nonnegative, nondecreasing potential φ~ for which x⟼φ~xx is derivable. We present a brief demonstration on its solvability based on generic assumptions about potential φ~.The proof depends on fixed-point theorem due to Schauder and the fact that under acceptable presumptions,Bu~T≔−∆D+φ~u~T from a semigroup e−τBu~T;τ≥0 on LpΩ∗, where −∆D stands for Laplacian, which is subject to Dirichlet boundary conditions. The expected form of solution (1)
(13)u~τ=e−τBu~Tu~0,τ∈0,T,and, as a result, u~T is described by the relation
(14)u~T=∫0Te−τBu~Tu~0dτ.Theorem 1.
Letφ~∈CR,R be nonnegative and n<2p. If one of the relations
(15)u~0∈L∞Ω∗,or
(i)
φ~x≤a1+x,x∈R holds, then for some a>0 and u~0∈LpΩ∗.There exists at least one possible solution(16)u~∈C0,T,LpΩ∗∩C10,T,LpΩ∗∩C0,T,Wp2Ω∗,to (1). Moreover, u~τp≤u~0p for τ∈0,T. If 0≤u~0, then 0≤u~τ for τ∈0,T.Proof.
Part (i).
Take a nonnegative functionφ~∈CR,R. Based on the boundedness of φ~ and by defining u~0∈L∞Ω∗, we arrive at
(17)φ~∈CL∞Ω∗,L∞Ω∗,
is bounded.
SetR0≔Tu~0∞ and let
(18)XT≔B¯L∞Ω∗0,R0,indicate a closed ball whose cent is at origin and radius R0. Take 2p∈n,∞ with the fact that, for any u~T∈XT, the mapping φ~u~T≔Ω∗↦φ~u~TΩ∗ satisfies without a doubt,
(19)φ~u~TLLpΩ∗≤φ~u~T∞,≤max−R0,R0φ~,u~T∈XT.
Combining the results and the fact−∆D∈HWp,D2Ω∗,LpΩ∗(20)Bu~T≔−∆D+φ~u~T∈HWp,D2Ω∗,LpΩ∗;κ,ωR0,for some positive ωR0 and 1≤κ. Moreover, for a nonnegative φ~, the operator −Bu~T generates a positive contraction semigroup e−τBu~Tτ≥0 on each LjΩ∗ for j∈1,∞. Hence,
(21)e−τBu~TLLpΩ∗≤1,τ≥0,j∈1,∞.
Let us delineate,(22)Φ~u~T≔∫0Te−τBu~Tu~0dτ,u~T∈XT.
Then, (21) signifies that
(23)Φ~u~T∞≤∫0Te−τBu~TLLpΩ∗u~0∞dτ≤Tu~0∞≤R0,with Φ~:XT⟶XT. Using the condition 2γ∈n/p,2 and (20) together with 1≤KR0 and positive ζR0, such that
(24)e−τBu~TLLpΩ∗,Wp,D2γΩ∗≤KR0eζR0ττ−γ,τ>0.
Therefore,(25)Φ~u~TWp,D2γΩ∗≤∫0Te−τBu~TLLpΩ∗,Wp,D2γΩ∗u~0pdτ,≤KR01−γeζR0TT1−γu~0p≤cR0,
Φ~XT is bounded in Wp,D2γΩ∗, the latter is being compact ingrained in CΩ¯∗ since 2γ∈n/p. To validate the property of continuity of Φ~,(26)e−τBu~T−e−τBv~T=−∫0τddxe−τ−xBu~Te−xBv~Tdx,=−∫0τe−τ−xBu~Te−xBv~T−Bv~T+Bu~Tdx,=−∫0τe−τ−xBu~Te−xBv~Tφ~u~T−φ~v~Tdx,thereby using (19), (21), and (24), we get
(27)e−τBu~T−e−τBv~TLLpΩ∗,Wp,D2γΩ∗≤∫0τe−τ−xBu~TLLpΩ∗,Wp,D2γΩ∗φ~u~T−φ~v~TLLpΩ∗e−xBv~TLLpΩ∗dx,≤cR0ecR0Tτ1−γφ~u~T−φ~v~T∞.
The continuous embedding ofWp,D2γΩ∗ in CΩ¯∗(28)Φ~u~T−Φ~v~T∞≤cΦ~u~T−Φ~v~TWp,D2γΩ∗≤c∫0Te−τBu~T−e−τBv~TLLpΩ∗,Wp,D2γΩ∗u~0pdτ,using (25) and (27), we get
(29)Φ~u~T−Φ~v~T∞≤c∫0TcR0ecR0Tτ1−αφ~u~T−φ~v~T∞u~0pdτ,≤c1R0φ~u~T−φ~v~T∞,for u~T,v~T∈XT. Hence, the required results on continuity Φ~:XT⟶XT are established through (17). Using the precompact image of Φ~∈EXT,XT, the theorem due to Schauder shows the presence of point u~T∈XTwith the property u~T=Φ~u~T. Now, define u~τ as
(30)u~τ≔e−τBu~Tu~0,τ∈0,T.in order to obtain the solution for (1). If 0≤u~0, then 0≤u~τ for τ∈0,T because the semigroup is positive. This completes proof of part (i).
Part (ii). Now, assume a nonnegativeφ~∈CR,R and that a1+x≥φ~x,x∈R, for positive a. Let us presume u~0∈LpΩ∗ with n<2p. We modify the set
(31)R0≔Tu~0∞,XT≔B¯LpΩ∗0,R0.One of the presumption onφ~ includes
(32)φ~∈BCXT,LpΩ∗.Further,Wp,D2−2εΩ∗ maintains its continuity for small n<2p in CΩ¯∗ over ε>0. Thus, for ω∈Wp,D2Ω∗, we have
(33)Φ~u~Tωp≤cφ~u~TpωWp,D2−2εΩ∗,≤cφ~u~TpωpεωWp,D2Ω∗1−ε,≤cδφ~u~Tp1/εωp+δωWp,D2Ω∗,with meager positive δ. Hence, the relation (20) holds good if the equality (33) and the corresponding results are combined. Moreover, (21) holds for j=p. Defining Φ~ as in (22), by similar arguments as in previous part, a precompact image of Φ~∈CXT,XT can be arrived (26) along with continuity that follows from the relations (21), (24), and (33) as given in the following:
(34)e−τBu~T−e−τBv~TLLpΩ∗≤∫0τe−τ−xBu~TLLpΩ∗φ~u~T−φ~v~TLLpΩ∗,Wp,D2−2εΩ∗e−xBv~TLLpΩ∗,Wp,D2−2εΩ∗dx,≤cR0φ~u~T−φ~v~Tp.The required result is achieved by applying Schauder’s fixed-point theorem.Corollary 2.
Letu~ be the solution to (1) provided by Theorem 1. If p≥2, then
(35)∇u~T22+∫Ω∗φ~u~Tu~T2dx=∫Ω∗u~0−u~Tu~Tdx≤2Tu~022.
Ifu~0∈WpγΩ∗ for some positive γ, then u~T∈Wp2Ω∗(36)−∆u~T+φ~u~Tu~T=u~0−u~TinΩ∗,u~T=0on∂Ω∗.Proof.
The proof is the direct consequence of Theorem1.


## 3. Uniqueness Result

Following the methodology adopted in [24], we demonstrate the property of uniqueness for small data in (1) if R0≔Tu~0∞ is small along with locally Lipschitz continuous
(37)φ~:R⟶R,and
(38)x⟼φ~xx,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 u~ and v~ with u~0=u~0=v~0∈L∞Ω∗. Fix ωT≔u~T−v~T.
(39)⟹φ~u~T−φ~v~T2≤LR0u~T−v~T2,=LR0ωT2,for fixed value LR0, obtained by the definition on φ~, which behaves as uniformly Lipschitz over its region of existence, (i.e., on the set −R0,R0). This leads to the fact (35) which entails that
(40)∇ωT22=−∫Ω∗u~T−v~Tφ~u~Tu~T−φ~v~Tv~Tdx+∫Ω∗ωTv~T−u~Tdx,≤v~T−u~T2ωT2.where (38) allows inequality to end. Now, owing to (21), (26), and (39), we get
(41)v~T−u~T2=e−τBu~T−e−τBv~Tu~02,≤∫0Te−τ−xBu~TLL2Ω∗φ~u~T−φ~v~T2e−xBv~TLL∞Ω∗u~0∞dx,≤R0LR0ωT2.So, (40) and (41) cover it
(42)Ω∗T22≤R0LR0Ω∗T22≤cΩ∗R0LR0∇ωT22,where cΩ∗ is borrowed from Poincare’sine quality. Now, if R0LR0cΩ∗<1, it refers to ωT≡0 in ω⟹u~T=v~T.In the following proposition, we study the monotonicity condition and its effect on uniqueness theory.Proposition 3.
Letφ~ be unknown that satisfies (37), 2p>n, and consider u~0∈LpΩ∗. There is M>0 such that (1) has a unique value
(43)u~∈C0,T,LpΩ∗∩C10,T,LpΩ∗∩C0,T,Wp2Ω∗,based on the inequality Tu~0p≤M.Proof.
The proof is obtained through Banach’s fixed-point theorem. Fix2γ∈n/p,2 and take R0>0 along with
(44)XT≔B¯Wp,D2γΩ∗0,R0.
Notice thatWp,D2γΩ∗ embeds continuously into CΩ¯∗. Thus, as φ~ is uniformly Lipschitz continuous on compact sets, there exists a constant LR0>0 with φ~(45)φ~u~T−φ~v~T∞≤LR0u~T−v~TWp,D2γΩ∗,u~T,v~T∈XT.
Especially from the proof of Theorem1, we can write
(46)Φ~u~TWp,D2γΩ∗≤KR01−γeζR0TT1−γu~0p,
Moreover, (27) and (45) combines to give
(47)e−τBu~T−e−τBv~TLLpΩ∗,Wp,D2γΩ∗≤cR0ecR0TT1−γu~T−v~TWp,D2γΩ∗,for some u~T,v~T∈XT and positive cR0. Therefore,
(48)Φ~u~T−Φ~v~TWp,D2γΩ∗≤∫0Tu~0pe−τBu~T−e−τBv~TLLpΩ∗,Wp,D2γΩ∗dτ,≤c1R0u~0pecR0TT2−γu~T−v~TWp,D2γΩ∗.
The inequality together with (46) shows that Φ~:XT⟶XT behaves like a contraction mapping with Tu~0p being very small. Consequently, if Tu~0p is very small, then there exists a unique u~T∈XT with Φ~u~T=u~T.


## 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.



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