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Advances in Nonlinear Analysis

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A constructive method for convex solutions of a class of nonlinear Black-Scholes equations

Mostafa Abounouh
  • Department of Mathematics, Cadi Ayyad university, Faculty of science and technology, Av. Abdelkarim El Khattabi, Gueliz, 40000 Marrakech, Marocco, Monaco
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/ Hassan Al Moatassime
  • Department of Mathematics, Cadi Ayyad university, Faculty of science and technology, Av. Abdelkarim El Khattabi, Gueliz, 40000 Marrakech, Marocco, Monaco
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/ Aicha Driouch
  • Department of Mathematics, Cadi Ayyad university, Faculty of science and technology, Av. Abdelkarim El Khattabi, Gueliz, 40000 Marrakech, Marocco, Monaco
  • LAMFA CNRS UMR 7352, Université de Picardie Jules Verne, 33 rue Saint-Leu 80039, Amiens Cedex, France
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/ Olivier Goubet
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  • LAMFA CNRS UMR 7352, Université de Picardie Jules Verne, 33 rue Saint-Leu 80039, Amiens Cedex, France
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Published Online: 2019-07-20 | DOI: https://doi.org/10.1515/anona-2020-0019

Abstract

In this work, we are concerned with the theoretical study of a nonlinear Black-Scholes equation resulting from market frictions. We will focus our attention on Barles and Soner’s model where the volatility is enlarged due to the presence of transaction costs. The aim of this paper is to give a constructive mathematical approach for proving the existence of convex solutions to a non degenerate fully nonlinear deterministic problem with nonlinear dependence upon the highest derivative. The existence of a strong solution to the original equation is shown by considering a monotone sequence satisfying an abstract Barenblatt equation and converging toward the solution of a limit problem.

Keywords: Fully nonlinear PDE; Black-Scholes equations; Barenblatt equation

PACS: Primary 35K65; Secondary 35K55

1 Introduction

In financial modeling, option pricing has gained popularity since the establishment of the Black-Scholes theory [1] in 1973. The authors assumed that the price S(t) satisfies the stochastic differential equation

dS(t)=μS(t)+σS(t)dBt,(1)

where μ is the expected rate of return, σ is the constant volatility and Bt is the standard Brownian motion on the underlying asset S(t). This model gives us an estimate of the European call option price [1]. The aim of pricing derivative products, such as options is particularly interesting since they are used in order to minimize the damage caused by variations in the stock price. Using Itô’s lemma, the option price becomes the solution to the deterministic linear equation

Vt+12σ2S2VSS+rSVSrV=0,S>0,t(0,T).(2)

The unknown V is the European Call option, S is the underlying asset, T is the maturity of the option and r is the risk-free interest rate constant. Despite the success of F. Black and M. Scholes model, it is based on unrealistic assumptions. In fact, they assumed that the market is not subject to transaction costs; this is not true in the real world, as pointed by H. Leland [2]. Actually in this case the hedging of the portfolio becomes tremendously expensive since it requires an infinite number of adjustments. H. Leland had suggested in his work a discrete frequent revision of the portfolio. This idea was later developed by [3, 4, 5, 6] and was studied numerically in [7, 8]. Another model was proposed by G. Barles and H. Soner in [9], they suggested a Black-Scholes extension with an enlarged volatility σ͂ where

σ~(S,t,VSS)2=σ21+Ψ(er(Tt)a2S2VSS).(3)

The option price satisfies then the nonlinear PDE

Vt+12σ~(S,t,VSS)2S2VSS+rSVSrV=0,S>0,t(0,T).(4)

The constant a is a nonnegative parameter that measures transaction costs (see for an overview [2, 7, 10]) and Ψ denotes the solution to the nonlinear ordinary differential equation

Ψ(x)=Ψ(x)+12xΨ(x)x,x0,Ψ(0)=0.(5)

The analysis of (5) performed in [9] yields

Graph of the solution Ψ to the nonlinear ordinary differential equation (5).
Fig. 1

Graph of the solution Ψ to the nonlinear ordinary differential equation (5).

limx+Ψ(x)x=1,limxΨ(x)=1.(6)

Equation (4) has been introduced in [9] by using the theory of stochastic optimal control; then the authors have proved that there exists a unique viscosity solution. Another theoretical study for (4) was given recently by D. Ševčovič in [11] by transforming (4) into a quasilinear parabolic equation. Since there is no exact solution to (4) many authors studied (4) numerically (for instance, see [7, 8, 12, 13, 14]). There are also some new studies in the literature in the case of pricing options under variable transaction costs (see [15], for more details). In this paper, we give an alternate and constructive theoretical approach of Barles and Soner’s model (4). By constructive method, we mean methods that lead naturally to numerical schemes to approximate the solution; this will appear in a forthcoming work. Besides, the variation of the hedging strategy is an important matter for option traders. This variation is measured by the Gamma parameter which is the second derivative of the option price (Γ = VSS). In the sequel, we consider the case where we have a non-negativity constraint on the Γ (i.e Γ ≥ 0). This is a natural assumption since the delta hedge of a European Call or Put is increasing when the underlying is non decreasing. For the sake of simplicity, wee restrict our attention to the case of Ψ(x) = x, that is a suitable approximation of Ψ for large x; the general case can be handled with similar but more lengthy methods (see Section 4.1 in the sequel). With this assumption, equation (4) becomes

Vt+σ22(1+er(Tt)a2S2VSS)S2VSS+rSVSrV=0.(7)

Equation (7) is supplemented with the boundary conditions

V(t,0)=0t(0,T),V(t,S)+S(S+),(8)

and a final condition (Pay-off of the option V), given by V(T, S) = (SK)+ where S ∈ [0, +∞[ and K is the strike price.

The terminal value of the option V.
Fig. 2

The terminal value of the option V.

In this article, we restrict our study to Call options, since in our opinion from the mathematical point of view they are more difficult to handle than Put options (we shall address this issue in a forthcoming work). This article is organized as follows. In Section 2, we set the mathematical framework, transforming the problem (7) into an abstract Barenblatt equation. We then introduce a suitable approximation of this equation by a time discretization. We prove existence and uniqueness of a solution to this approximation in Section 3. Then, we state our main convergence result in Section 4. Eventually we provide a conclusion and perspectives in a last section.

2 Mathematical framework

2.1 Preparing the equation

We first introduce the new function

v(t,S)=a2exp(r(Tt))V(t,Sert).

This function is solution to

vt+σ22(1+S2vSS)S2vSS=0,S>0,t(0,T)v(T,S)=a2erT(SKerT)+,S>0v(t,0)=0,t(0,T)v(t,S)+a2erTSt(0,T).(9)

Since we prefer to deal with a initial value problem, we perform the shift (t, S) = v(t, s) – a2SerT, the change of time tTt, and a time dilation to get rid of σ22 (say V(t, S) = v~(2(Tt)σ2,S)) to simplify (9) into

Vt(1+S2VSS)S2VSS=0,S>0,t(0,T)V(t,0)=0,V(0,S)=a2erT((SKerT)+S),limS+VS(t,S)=0.(10)

Let us emphasize that we supplement (10) with the constraint VSS ≥ 0, that is we seek convex solutions.

Remark 1

  • At this stage this is not clear why the far field boundary condition V(t, S) + S transforms to VS → 0. We will clarify this point in the sequel.

  • Seeking a solution to equation (10) is equivalent to seek a solution to the original equation (7) since we can perform backward the change of variable. A variable of interest for (10) is the function V(2Tσ2) that corresponds to the Call option at t = 0 for the original equation. We will prove below the convergence of this quantity to the corresponding initial Call option for the linear Black-Scholes equation when the transaction costs goes to 0.

2.2 Functional analysis

Throughout this paper, we use the following notations. For Ω = ℝ+, the set D(Ω) is the space of smooth compactly supported functions in Ω and D′(Ω) its dual space. We set C0(Ω) for the set of continuous functions V such that V(0) = 0. We denote by Llocp(Ω) the space of locally integrable functions in Ω with 1 ≤ p < ∞. Finally we denote by LSp(Ω) the weighted Lebesgue space with a weight decaying at infinity

LSp(Ω)=VLlocp(Ω);ΩV(S)pS2dS<,(1p<).(11)

We also define the Hilbert space

V=VLloc1(Ω);ΩVS2(S)dS<,(12)

as the closure of D(Ω) for the norm ∥V𝕍 = (ΩVS2(S)dS)12.

Lemma 2.1

The set 𝕍 is a subset of LS2(Ω) ∩ C0(Ω). Moreover if V belongs to 𝕍 then for any S > 0 the inequality V(S)28S||V||V2 is valid.

Proof

The proof of 𝕍 ⊂ C(Ω) is standard and then omitted. For a test function V in D(Ω) we have that, appealing Cauchy-Schwarz inequality

ΩV2(S)S2ds=2ΩV(S)VS(S)dSS2||V||V(ΩV2(S)S2ds)12.(13)

We then have the Hardy inequality,

ΩV2(S)S2ds4||V||V2,(14)

and eventually the embedding 𝕍 ⊂ LS2(Ω) is valid thanks to a density argument. Moreover the identity

V2(S)S=S(V(s)2s22V(s)VS(s)s)ds8||V||V2

completes the proof of Lemma 2.1.□

Let us now define the unbounded operator A in LS2(Ω) defined as AV = –S2VSS. In other words A is defined by

(AV,W)LS2(Ω)=ΩVS(s)WS(s)ds.(15)

For later use we state a maximum principle for the operator A.

Lemma 2.2

Consider V in 𝕍 such that AV ≤ 0 in 𝕍′. Then V(S) ≤ 0 for any S ≥ 0.

Proof

We would like to point out that the boundary condition respectively at S = 0 and at S ∼ +∞ are hidden in the assumption V ∈ 𝕍. We prove the result for smooth V and we appeal a density argument to conclude. Assume then that VSS ≥ 0. Then V is a convex function that satisfies for s < S the inequality

V(S)V(s)+VS(s)(Ss).

We infer from this inequality, dividing by S and letting S diverge to +∞, appealing Lemma 2.1, that VS ≤ 0. Then the result.□

Remark 2

Let us assume that V satisfies the assumptions of Lemma 2.2. Since V(S)2CS one can prove that actually limS→+∞ VS(S) = 0. Then the condition at S ∼ +∞ in (10) makes sense.

In the following we will use an approach inspired by [16]. Let us define β : ℝ+ ↦ ℝ+ as the inverse function of yy2 + y, that is

β2(y)+β(y)=yi.eβ(y)=y+1412.(16)

Equation (10) becomes

Vtβ1(AV)=0.(17)

Rather than solving (17) with respect to the boundary conditions in (10) we solve

β(Vt)+AV=0,S>0,t(0,T)V(t,0)=0,V(0,S)=a2erT(SKerT)+S,limS+VS(t,S)=0.(18)

The equation above belongs to the class of Barenblatt’s equations. To handle β(Vt) we need another functional space. Hence we introduce a suitable Orlicz space (see [17, 18]). Let us consider the function J defined by

J:R+R+yβ(y)y.

Proposition 1

The function J is convex and nondecreasing on+. Moreover for y ≥ 0 we have 2J(y) ≤ J(2y) ≤ 4J(y).

Proof

deriving J implies

J(y)=y+1412+y2y+140,

and

J(y)=12y+14+y+124(y+14)320.

Hence J is convex and monotone increasing on ℝ+. Since J is convex and J(0) = 0 then 2J(y) ≤ J(2y). Since β is concave β(2y) ≤ 2β(y) and then J(2y) ≤ 4J(y).□

Definition 1

The weighted Orlicz space associated to J is the vector space generated by vLloc1(Ω) that satisfies

ΩJ(|v(S)|)S2dS=Ωβ(|v(S)|)|v(S)|S2dS<+,(19)

this space is equipped with the Luxemburg norm

LJ(v)=infλ>0/Ωβ|v(S)|λ|v(S)|λdSS21.(20)

Actually, since 2J(y) ≤ J(2y) ≤ 4J(y) it is standard to prove that there exists c > 0 such that

cmin(||v||LJ,||v||LJ2)ΩJ(|v(S)|)dSS21cmax(||v||LJ,||v||LJ2).

Proposition 2

Actually this weighted Orlicz space LJ is LS2(Ω)+LS32(Ω) that is the dual space of LS2(Ω)LS3(Ω). Moreover V belongs to LJ if and only if min(V2,|V|32) belongs to LS1(Ω).

Proof

We know that β(y)=y+1412=yy+14+12. Hence β(y) ≤ min(y, y). Conversely we have y = β(y) + β2(y), thus y ≤ 2 max(β(y), β2(y)). Either β(y) ≥ 1 and thus y ≤ 2β2(y) or β(y) ≤ 1 and thus y ≤ 2β(y). Then, there exists c > 0 such that c min(y, y) ≤ β(y). Therefore V belongs to LJ if and only if min(V2,|V|32) belongs to LS1(Ω). To prove that LJ is LS2(Ω)+LS32(Ω) is then standard.□

2.3 Main results

Theorem 2.3

There exists a unique convex solution to (10) that belongs, for any 𝔗 > 0 to C(0, 𝔗; 𝕍) and such that Vt belongs to the dual space of L2([0, 𝔗] × Ω) ∩ L3(0, 𝔗 × Ω).

In the next section we handle the proof of the theorem.

3 Proof of the main theorem

3.1 Approximation scheme

Now we approximate (18) by a discrete-time scheme, and then we use the monotone iterative technique [19]. Given Vn in a set that will be defined below and τ > 0, we solve recursively the problem, with V0(S) = a2erT((SKerT)+S).

βVn+1VnτS2VSSn+1=0S>0,Vn+1(0)=0,limS+VSn+1(S)=0.(21)

Proposition 3

Let us consider the convex set Kn that reads

Kn=VV/VnV0,VVnLJ.(22)

For all n ≥ 0, there is a unique Vn+1Kn solution of (21).

Proof

We assume that Vn is known and we prove the existence of Vn+1. Let us observe that the Lipschitz constant of β is 1. We solve recursively in 𝕍 the equation

τS2WSSm+1+Wm+1=Wmτβ(WmVnτ),(23)

with initial data W0 = Vn. Assume that WmVn belongs to LJ. Then the right hand side of (23) belongs to 𝕍′ and (23) can be solved by Lax-Milgram Theorem. Since the map G(v)=vτβ(vVnτ) is increasing, we can prove recursively by the Maximum Principle (see Proposition 2.2) that WmWm+1. Moreover Wm ≤ 0 since 0 is a supersolution above V0. Hence the Perron sub and supersolution method converges and Vn+1 = supm Wm is a weak solution to (21). Besides, we know that WmVn = (WmV0) – (VnV0). The function VnV0 is in LJ and the non negative function WmV0 that is bounded by above by –V0 is also in LJ. Then Vn+1Vn is in LJ and the righ hand side of (21) is in 𝕍′. Then Vn+1 is in 𝕍.

We prove the uniqueness as follows. Consider n+1 another solution. Then

τΩ(β(Vn+1Vnτ)β(V~n+1Vnτ))(Vn+1VnτV~n+1Vnτ)dSS2+(A(Vn+1V~n+1),(Vn+1V~n+1))=0.(24)

Since the two terms in the left hand side of (24) are non negative, the conclusion follows promptly.□

3.2 Passing to the limit

Let us fix a 𝔗 > 0. For τ > 0 let N be such as = 𝔗 and let us consider the function

Vτ(t,S)=n=0+σ(tτn)Vn(S),(25)

where σ is defined by σ(t) = max(0, 1 – |t|). We plan to let N → +∞ (and so τ → 0). Note that in the time interval In = ], (n + 1)τ[ we have

Vτ(t,S)=(n+1tτ)Vn(S)+(tτn)Vn+1(S),(26)

dVτdt(t,S)=Vn+1(S)Vn(S)τ.(27)

Lemma 3.1

The following assertions hold true

  1. The sequence dVτdt is bounded in LS2( Ω × [0, 𝔗]) + LS32(Ω × [0, 𝔗]).

  2. The sequence VSτ is bounded in L([0, 𝔗]; L2(Ω)).

  3. The sequence β(dVτdt) is bounded in LS2(Ω × [0, 𝔗]) ∩ LS3(Ω × [0, 𝔗]).

  4. The sequence Vτ is bounded in L(Ω × [0, 𝔗]).

Proof

1. Summing (24) for n+1 = Vn with respect to n yields

n=0+τΩβVn+1VnτVn+1VnτdSS2+12Ω|VSn+1|2dS12Ω|VS0|2dS.

Hence 1. and 2. of the Lemma are proved. The point 3. is straightforward due to the properties of β that maps L2 + L32 into L2L3. Besides, since Vτ is a convex combination of Vn and Vn+1 in In then V0Vτ ≤ 0 everywhere. This completes the proof of the Lemma.□

We know pass to the limit when τ → 0 using a monotonicity argument [19].

For (t, S) in In × Ω, we have

βdVτdtS2VSSτ=S2(n+1tτ)(VSSn+1VSSn).(28)

We first consider the scalar product in LS2(Ω) by a test function ϕD(Ω); then we have

ΩβdVτdtS2VSSτϕ(S)S2dS=ϵn.(29)

where

ϵn=(n+1tτ)ΩVn+1VnττϕSS.

Let us observe that

ϵnCτ||dVτdt||LS2(Ω)+LS32(Ω)||ϕSS||LS2(Ω)LS3(Ω),

and then this term converges towards 0 when τ → 0. Considering now a subsequence Vτ that converges towards V in L(0, 𝔗; 𝕍) and such that (dVτdt,β(dVτdt)) converges weakly towards (dVdt,χ)inLJ×LJ, we have that χ = S2VSS.

We now prove that χ=βdVdt. We then compute

0Qτ=0T<βdVτdtβ(v),dVτdtv>LS2,(30)

where v is a test function. By weak convergence we have

0T(<βdVτdt,v>LS2+<β(v),dVτdt>LS2)0T(<χ,v>LS2+<β(v),dVdt>LS2).(31)

Moreover, observing that

0T<βdVτdt,dVτdt>LS2=nΩVSSn+1(Vn+1Vn)dS12Ω|VSτ(T)|2dS+12Ω|VS0(0)|2dS,

we have, by weak convergence

lim¯0T<βdVτdt,dVτdt>LS212Ω|VS(T)|2dS+12Ω|VS(0)|2dS=120Tddt||V|||V2dt.

Therefore

lim¯Ω0TβdVτdtdVτdt120Tddt||V|||V2=0T<χ,dVdt>LS2.(32)

Thus we have

0lim¯Qτ0T<β(v)χ,vdVdt>LS2.(33)

We choose v=dVdt+ϵϕ where ϕ is a given test function; then we have

0ϵ0T<βdVdtχ,ϕ>LS2+o(ϵ).(34)

By changing ϕ to –ϕ and tending ϵ to 0+ we obtain

βdVdt=χa.e.(35)

This completes the proof of the existence of a solution V that belongs to L(0, 𝔗, 𝕍) such that dVdt belongs to L2(Ω × (0, 𝔗)) + L32(Ω × (0, 𝔗)). To prove that V is continuous in t with values in 𝕍 is a consequence of

||V(t)V(s)||V||dVdt||L2(Ω×(0,T))+L32(Ω×(0,T))max(|ts|12,|ts|13).

3.3 Uniqueness of the solution

Let V and be two solutions of (18), and W = V. We have

<βdVdtβdV~dt,dWdt>LS2+12ddt||W|||V2=0.(36)

Thus V = .

4 Miscellaneous results

4.1 Handling the original Barles-Soner equation

For the sake of simplicity of exposure, we have chosen to set Ψ(x) = x in this article. We know indicate how to handle the original Barles-Soner equation. The main difference is the very definition of the Orlicz space that is defined through a function (y) = β͂(y) y where β͂(y) is solution to (1 + Ψ(β͂(y))β͂(y) = y. This Orlicz space has similar properties than LS2(Ω)+LS32(Ω) because β͂ and β have the same behavior at y ∼ 0 and y ∼ ∞, but computations are more lengthy.

4.2 Comparison with the solution of the linear equation

Rescaling the solution V of Theorem 2.3 to Va = a–2 V we then have a solution to the equation

Vta=(1+a2S2VSSa)S2VSSa,

with initial condition V0(S) = erT((SKerT)+S). We will compare this solution to the solution of the linear Barles-Soner equation Vl

Vtl=S2VSSl,

supplemented with the same initial data. Actually, if the transaction costs are small, then a → 0. To begin with we prove a result that implies that the so-called Gamma parameter VSSa remains bounded in some space when a converges to 0.

Lemma 4.1

There exists a constant C that depends ont the data r, K, T but that is independent of a such that

0tΩ(S2VSSa)2dSS2dτC.(37)

Proof

We know that

βa(dVadt)=ΔSVa,

where βa(y) is the positive solution to a2βa + βay = 0. Multiplying this equation by S2dVadt and integrating in Ω × [0, t] leads to

0tΩβa(dVadt)dVadtdSS2dτ12ΩVS0dS=C.(38)

On the other hand

ΔSVaΔSVa+a2(ΔSVa)2=dVadt,

and then (ΔSVa)2βa(dVadt)dVadt that completes the proof of the Lemma.□

We know state

Proposition 4

We have that VaVl everywhere, and that there exists a constant C that depends on K, r, T such that

||VaVl||LS1(Ω)Ca2.

Proof

Set W = VaVl. Since WtS2WSS ≥ 0 then by the maximum principle W ≥ 0. On the other hand

WtS2WSSa2(S2VSSa)2.

Multiplying by S–2 and integrating in Ω yields

ddt||VaVl||LS1(Ω)=WS(0)+a2ΩS2(VSS(a))2(S)dS,

since WS(S) → 0 when S diverges towards +∞ due to Lemma 2.1. On the other hand, since W(0) = 0 and W(S) ≥ 0 for S > 0 we then have –WS(0) ≤ 0. Then

ddt||VaVl||LS1(Ω)a2ΩS2(VSS(a))2(S)dS,

Then integrating in time concludes, thanks to Lemma 4.1, concludes the proof of the Proposition.□

The result of Proposition 4 means that the option price when transaction costs occur is more expensive than the option price in the linear Black-Scholes framework. This is an expected result since pricing in this case implies additional costs, in fact the higher the parameter a is, the greater option price becomes. Proposition 4 can also be seen formally as a deterministic way of proving that hedging errors vanish when transaction costs converge toward 0 where the contingent claim can be replicated with the delta-hedge strategy; see for instance [3, 20].

4.3 Conclusion

We have provided here a deterministic approach to construct a solution of a nonlinear Black-Scholes equation with an enlarged volatility. We have transformed the problem into a Barenblatt equation in order to overcome the difficulty coming from the nonlinearity. Our method gives a constructive approximation for convex solutions in a suitable Orlicz space, in contrast to the general method of viscosity solutions [9]. We then have compared the solution to Barles and Soner’s equation to solution of the linear Black-Scholes model. For future work we will be interested in performing the numerics to approximate the solution of the Barles-Soner equation using our approximation procedure. We are also interested in extending our theory for the two dimensional problem of (4) where the option depends on two underlying assets.

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About the article

Received: 2018-12-08

Accepted: 2019-02-01

Published Online: 2019-07-20

Published in Print: 2019-03-01


Citation Information: Advances in Nonlinear Analysis, Volume 9, Issue 1, Pages 654–664, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2020-0019.

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© 2020 M. Abounouh et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

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