On some nonlinear boundary value problems related to a Black-Scholes model with transaction costs

Figueroa and Grossinho Boundary Value Problems (2015) 2015:145 DOI 10.1186/s13661-015-0410-9 RESEARCH Open Access On some nonlinear boundary value problems related to a Black-Scholes model with transaction costs Rubén Figueroa1* and Maria do Rosário Grossinho2 *Correspondence: ruben.fi[email protected] 1Department of Mathematical Analysis, University of Santiago de Compostela, Campus Vida, Santiago de Compostela, 15782, Spain Full list of author information is available at the end of the article Abstract We deal with some generalizations on a Black-Scholes model arising in financial mathematics. As a novelty in this paper, we consider a variable volatility and abstract functional boundary conditions, which allow us to treat a very large class of problems involving Black-Scholes equation. Our main results involve the existence of extremal solutions in presence of lower and upper solutions. Some examples of applications are provided too. Keywords: Black-Scholes equation; functional conditions; discontinuous ODE’s 1 Introduction In this paper we are concerned with the following nonlinear boundary value problem: x(V)(x)+p(x)xV(x)+q(x)(xV(x)–V(x))=, B(V(c),V)=, B(V(d),V)=, () where p,qare nonnegative bounded functions which could be discontinuous in [c,d], c,d>,andBi:R×C([c,d]) −→R,i= ,, are functions which satisfy some conditions that we will state later. We observe that under this framework, a large class of boundary conditions is included, namely: () Dirichlet conditions: B(V(c),V)=V(c)–Vc,B(V(d),V)=V(d)–Vd; () Initial-integral conditions: B(V(c),V)=V(c)–d ck(x)V(x)dx; () Multipoint conditions: B(V(d),V)=V(d)–n j= V(xj), with Vc,Vd∈R. This study follows and generalizes the results contained in [] with respect to the problem x(V)(x)+pxV(x)+q(xV(x)–V(x)) =, V(c)=Vc,V(d)=Vd.() In [], it is assumed that p,qare positive constants and Vc<Vd.Thecontributionsof the present paper are the following. First we address this problem but drop the condition Vc<Vd, and replace the constants p,qby two functions p(x), q(x); second, we replace ©2015 Figueroa and Grossinho. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Figueroa and Grossinho Boundary Value Problems (2015) 2015:145 Page 2 of 14 theDirichletconditionsbyfunctional boundaryconditions,whichallowsustoconsidera very large class of problems for the equation of (). These problems are related to financial option pricing since they address the existence of stationary solutions of a class of generalizations of the classical Black-Scholes model (BS),introducedin[], with equation ∂V ∂t+ σS∂V ∂S+rS∂V ∂S–V=, () where Vrepresents the value of a call or put option, depending on an underlying asset S and on time t,ris the interest short rate and σis the volatility of the asset price. In the (BS) model, Sis modeled as a geometric Brownian motion, and no costs are considered when financial transactions hold. Supposethattransactioncostsareincludedinthemodelundertheassumptionthatthey are a percentage of the transaction, given as in [] by a linear function hof the number of shares traded, i.e.,h(η)=a–bη,whereηisthenumberofsharestradedanda,b>. The following nonlinear BS type equation is obtained, where tis the interval between transactions, Vt+ σSVSS –aσS πt|VSS|+bSσV SS +r(SVS–V)=. Then, if ais small enough and VSS >(see[, , ]), the following nonlinear version of ()isobtained: ∂V ∂t+ ˜σS∂V ∂S+bσS∂V ∂S+rS∂V ∂S–V=, () where ˜σ=σ(–a σ πt)>andiscommonlydenotedbyadjustedvolatility.Now,ifwe consider the stationary version of (), we obtain the above ordinary differential equation ()wherep=˜σ bσand q=r bσare constants. This paper is organized as follows. In Section , we introduce an auxiliary nonlinear boundary value problem and the notions of upper and lower solutions used later. In Section , we consider problem () with Dirichlet boundary conditions. So, we start from paper[]andwegeneralizeitbyconsideringp(x),q(x)insteadofconstantsp,q,whichcorresponds to variable volatility in the (BS) model, and by dropping the condition Vc<Vd. InSection,wedealwithproblem()onitsfullversion,thatis,withfunctionalboundary conditions. Namely, we provide a result on the existence of extremal solutions between lowerandupper solutionsbyusing a generalized iterationwith Dirichletproblems. Some examples of application are provided, too. 2 Auxiliary problem and upper and lower solutions If we look to the equation of () xV(x)+p(x)xV(x)+q(x)xV(x)–V(x)=, Figueroa and Grossinho Boundary Value Problems (2015) 2015:145 Page 3 of 14 and, as in [, ]and[], solve it algebraically in order to V,weobtain V(x)=–p(x)x±p(x)x–xq(x)(xV(x)–V(x)) x.() Since we are interested in convex solutions of (), they can exist only if V(x)=–p(x)x+p(x)x–xq(x)(xV(x)–V(x)) x,() and –xV(x)–V(x)≥. () These facts lead us to consider first the following related problem: V(x)+H(x,V(x),V(x))=, B(V(c),V)=, B(V(d),V)=, () where H(x,y,z)=p(x)x–p(x)x+xq(x)|xz–y| x.() Then we will see how solutions of this problem are solutions of our original problem. Wewillusethemethodof upperandlower solutions forthisproblemandwe will begin byconsideringtheclassicalnotionsofC-loweranduppersolutions.However,afterwards, wewillusesomeweakernotions,namelywewillneedanotionthatallowslowersolutions to have ‘angles’. In fact, we will consider a maximum of two classical C-lower solutions which is not necessarily differentiable and can exhibit an ‘angle’. So, denoting by D–f(x) and D+f(x), respectively,thelowerleft-hand and the upperright-hand Dini-derivatives of the function fat x, we introduce the following definitions (see []). Definition. We say that α∈AC([c,d]) is a lower solution for problem ()if Bα(c),α≤, Bα(d),α≤, and for each x∈(c,d) one of the following conditions holds: () D–α(x)<D+α(x); () There exists an open interval Jsuch that x∈J⊂(c,d),α|J∈W,(J),andfor almost all x∈Jwe have α(x)+Hx,α(x),α(x)≥. We say that β∈AC([c,d]) is an upper solution for problem ()if Bβ(c),β≥, Bβ(d),β≥, and for each x∈(c,d) one of the following conditions holds: () D–β(x)>D+β(x); Figueroa and Grossinho Boundary Value Problems (2015) 2015:145 Page 4 of 14 () There exists an open interval Jsuch that x∈J⊂(c,d),β|J∈W,(J),andfor almost all x∈Jwe have β(x)+Hx,β(x),β(x)≤. Notice that if α,βare classical C-lower and upper solutions for problem (), then they are also lower and upper solutions in the sense referred above. 3 Nonlinear problem with Dirichlet conditions Consider problem ()with BV(c),V=V(c)–Vc,BV(d),V=V(d)–Vd, that is, with standard Dirichlet conditions. So, we have in this case x(V)(x)+p(x)xV(x)+q(x)(xV(x)–V(x))= , V(c)=Vc,V(d)=Vd.() The auxiliary problem referred in the previous sections is now V(x)+H(x,V(x),V(x))=, V(c)=Vc,V(d)=Vd,() where the function His given by (). Fromthestudyofproblem(),wewilldeducelaterexistenceandlocalizationresultsfor problem (). Next proposition establishes adequately the existence of classical C-upper and lower solutions for (). Proposition. The following assertions hold: () If Vd d≤Vc c,then the function α(x)=Vd dx() is a C-lower solution for problem (). The converse is also valid. () Take k>such that k≥Q cmax x∈[c,d]k x–cd+cVd–dVc d–c, () where Q=max x∈[c,d]q(x). Then the function αk(x)=k x+Vd–Vc d–c–k (d+c)x+k cd–cVd–dVc d–c() is a C-lower solution for problem (). Figueroa and Grossinho Boundary Value Problems (2015) 2015:145 Page 5 of 14 () The function β(x)=Vd–Vc d–cx+dVc–cVd d–c,x∈[c,d] () is a C-upper solution for problem (). Proof () If Vd d≤Vc c, the thesis follows since α(c)=Vd dc≤Vc,α(d)=Vd, and, in (c,d), α (x)+Hx,α(x),α (x)=. Conversely, if αis a lower solution for problem (), then α(c)=Vd dc≤Vc, which implies Vd d≤Vc c. () Observe that αk(c)=Vc,αk(d)=Vd. On the other hand, as Hx,αk(x),α k(x)≥p(x)x–p(x)x–xq(x)|xα k(x)–αk(x)| x ≥–q(x) xxα (x)–αk(x), ≥–Q cmax x∈[c,d]k x–cd+Vdc–Vcd d–c, then α k(x)+Hx,αk(x),α k(x)≥k–Q cmax x∈[c,d]k x–cd–Vc+Vd–Vc d–cc. Since k> satisfies hypothesis (), we derive α k(x)+Hx,αk(x),α k(x)≥. Then the assertion holds. () The thesis follows easily from the fact that β(x)+Hx,β(x),β(x)=Hx,β(x),β(x)≤ and β(c)=Vc,β(d)=Vd. Remark . () Notice that there is no ambiguity in considering kbig enough such that condition () holds. In fact, it is easy to see that the maximum in ()dependsonkand Figueroa and Grossinho Boundary Value Problems (2015) 2015:145 Page 6 of 14 is attained in the following way: max x∈[c,d]k x–cd+Vdc–Vcd d–c =maxk d–cd+Vdc–Vcd d–c,k c–cd+Vdc–Vcd d–c. So, it is clear that in the proof of the previous proposition, we could choose ksatisfying ()since lim k→∞ k √k=+∞. () Observe also that the lower solution αkcan be written as αk(x)=β(x)+θ(x), where θ(x)=k x–(d+c)x+cd. In fact, αk(x)=k x+Vd–Vc d–c–k (d+c)x+k cd–Vdc–Vcd d–c =Vd–Vc d–cx+dVc–cVd d–c+k x–(d+c)x+cd. The function θis quadratic, vanishes at x=cand x=d, and is negative in ]c,d[. Notation . Given two functions φ≤ψin [c,d], let us denote by [φ,ψ] the functional interval [φ,ψ]=V∈W,[c,d]:φ(x)≤V(x)≤ψ(x) for all x∈[c,d]. Next, we state an existence and localization result for problem (). Theorem. Let α,αkand βbe the functions defined in the previous proposition. (a) If Vd d≤Vc c,then problem ()has extremal W,-solutions,that is,the least and the greatest one,in the functional interval [α,β]. (b) If k>satisfies (), then problem ()has extremal W,-solutions in the functional interval [αk,β]. Proof Consider in case (a) E=(x,y,z)∈[c,d]×R:α(x)≤y≤β(x), and in case (b) E=(x,y,z)∈[c,d]×R:αk(x)≤y≤β(x). Figueroa and Grossinho Boundary Value Problems (2015) 2015:145 Page 7 of 14 It is clear that His continuous in Eand E. Moreover, we have that H(x,y,z)≤p(x) x+q(x) xβ(x)+q(x) x|z| for all (x,y,z)∈Eor (x,y,z)∈E.Then,putting ˆ A=max x∈[c,d]p(x) x+q(x) xβ(x),ˆ B=max x∈[c,d]q(x) x,() we derive that H(x,y,z)≤ˆ A+ˆ B|z|() for (x,y,z)∈Eor (x,y,z)∈E, respectively. This inequality guarantees that the function Hsatisfies the (classical) Nagumo condition both in Eand E.Usingthefactthatα,αk are C-lower solutions in cases (a) and (b), respectively, and βis a C-upper solution for problem ()suchthat α≤β,αk≤β, the conclusion holds by application of a well-known result contained in [].  Corollary . Let Vd d≤Vc cand k >satisfy () and consider the functions α,αkand β defined,respectively,by (), (), (). Define α(x)=maxα(x),αk(x). Then problem ()has the extremal W,-solutions in the interval [α,β]=V∈W,[c,d]:α(x)≤V(x)≤β(x)for all x ∈[c,d]. Proof We observe that α(x)=maxα(x),αk(x) is a lower solution for problem ()inthesensedefinedintheprevioussection(notnecessarily C)and,asbefore, βisaC-uppersolutionforproblem().Similarlytotheproof of Theorem ., the Nagumo condition holds in E=(x,y,z)∈[c,d]×R:α(x)≤y≤β(x). Then the result follows from an existence theorem contained in [].  Proposition. The following assertions hold: (a) Every solution Vof problem ()is convex. Figueroa and Grossinho Boundary Value Problems (2015) 2015:145 Page 8 of 14 (b) Consider αdefined by (). Then every solution Vof problem ()such that V≥α satisfies,for all in x∈[c,d], xV(x)–V(x)≤. Proof (a) Clearly, the convexity of solutions of () derives from the fact that V(x)=–Hx,V(x),V(x)≥ for all x∈]c,d[ and from the continuity in cand d. (b) Let Vbe a solution of problem ()suchthatV≥αin [c,d]. We claim that Vd d≥ V(d). In fact, α=Vd dx≤V(x)impliesthatVd d≥V(x)–Vd x–din [c,d], and then, letting x→d, weobtain Vd d≥V(d).Thisinequalitytogetherwiththefactthatthefunctionx∈[c,d]−→ xV(x)–V(x) is nondecreasing implies that xV(x)–V(x)≤ for all x∈[c,d].  Next theorem establishes the relations between the convex solutions of ()andof(). Theorem. Consider problems ()and (). Then (a) If Vd d≤Vc c,every solution Vof ()provided by Theorem . is a convex solution of (). (b) Every convex solution Vof problem ()is a convex solution of problem (). Proof As for (a), let Vd d≤Vc c.Thenαis a lower solution of ()andeverysolutionVof ()providedbyTheorem.satisfiesV(x)≥α(x).Hence,byProposition.,Visconvex and xV(x)–V(x)=–xV(x)–V(x). So V(x)=–Hx,V(x),V(x)=–p(x)x+p(x)x–q(x)x(xV(x)–V(x)) x, which shows clearly that Visaconvexfunctionthatsolves(). As for (b),let Vbe a convex solution of (). Then xV(x)+p(x)xV(x)+q(x)xV(x)–V(x)=, and as V ≥, p(x)≥and<c≤x≤d,then q(x)xV(x)–V(x)≤. Then xV(x)+p(x)xV(x)=xq(x)xV(x)–V(x), that is, by adding  p(x)xto both members, xV(x)+ p(x)x= p(x)x+xq(x)xV(x)–V(x). Figueroa and Grossinho Boundary Value Problems (2015) 2015:145 Page 9 of 14 Applyingthesquareroot,weobtain xV(x)+ p(x)x= p(x)x+xq(x)xV(x)–V(x), and then V(x)=–p(x)x+p(x)x+xq(x)|xV(x)–V(x)| x=–Hx,V(x),V(x). So, Vsolves ().  FromTheorem.,Theorem.andCorollary.,itisclearthatthefollowingexistence and localization result holds. Theorem. Consider problem () with standard Dirichlet conditions,that is, x(V)(x)+p(x)xV(x)+q(x)(xV(x)–V(x))= , V(c)=Vc,V(d)=Vd. () If Vd d≤Vc c,then this problem has the extremal convex W,-solutions in the functional interval [α,β],where αand βare provided,respectively,by ()and (); () If Vd d≤Vc cand k>satisfies (), then this problem has the extremal convex W,-solutions in the functional interval [α,β],where α(x)=maxα(x),αk(x) and α,αk,βare provided,respectively,by (), (), (). Remark. Under the hypotheses of the above theorem, observe that if Vd d=Vc cthen α isasolutionofproblem().Ontheotherhand,intheperiodiccase,Vc=Vd,theconstant function V≡Vcis a solution of (). Example. Consider problem ()intheinterval[c,d]=[,],with p(x)=+x,q(x)=[x], where[·] denotesintegerpart,andboundaryconditionsV()=,V()=.Noticethatin this case it is Vc>Vd. From condition (), k> must satisfy k≥ max|–k–|,|k–|, so simple computations show that k=  satisfies (). Then the function α(x)=x–x+ is a lower solution for this problem. At the same time, β(x)=–x is an upper solution.