In financial modeling, option pricing has gained popularity since the establishment of the Black-Scholes theory  in 1973. The authors assumed that the price S(t) satisfies the stochastic differential equation
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 . 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
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 . 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 , they suggested a Black-Scholes extension with an enlarged volatility σ͂ where
The option price satisfies then the nonlinear PDE
Equation (4) has been introduced in  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  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 , 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
Equation (7) is supplemented with the boundary conditions
and a final condition (Pay-off of the option V), given by V(T, S) = (S – K)+ where S ∈ [0, +∞[ and K is the strike price.
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
This function is solution to
Since we prefer to deal with a initial value problem, we perform the shift v͂(t, S) = v(t, s) – a2SerT, the change of time t ↦ T – t, and a time dilation to get rid of (say V(t, S) = ) to simplify (9) into
Let us emphasize that we supplement (10) with the constraint VSS ≥ 0, that is we seek convex solutions.
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 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 (Ω) the space of locally integrable functions in Ω with 1 ≤ p < ∞. Finally we denote by (Ω) the weighted Lebesgue space with a weight decaying at infinity
We also define the Hilbert space
as the closure of D(Ω) for the norm ∥V∥𝕍 =
The set 𝕍 is a subset of (Ω) ∩ C0(Ω). Moreover if V belongs to 𝕍 then for any S > 0 the inequality V(S)2 ≤ is valid.
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
We then have the Hardy inequality,
and eventually the embedding 𝕍 ⊂ (Ω) is valid thanks to a density argument. Moreover the identity
completes the proof of Lemma 2.1.□
Let us now define the unbounded operator A in (Ω) defined as AV = –S2VSS. In other words A is defined by
For later use we state a maximum principle for the operator A.
Consider V in 𝕍 such that AV ≤ 0 in 𝕍′. Then V(S) ≤ 0 for any S ≥ 0.
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
We infer from this inequality, dividing by S and letting S diverge to +∞, appealing Lemma 2.1, that VS ≤ 0. Then the result.□
In the following we will use an approach inspired by . Let us define β : ℝ+ ↦ ℝ+ as the inverse function of y ↦ y2 + y, that is
Equation (10) becomes
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
The function J is convex and nondecreasing on ℝ+. Moreover for y ≥ 0 we have 2J(y) ≤ J(2y) ≤ 4J(y).
deriving J implies
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).□
The weighted Orlicz space associated to J is the vector space generated by v ∈ (Ω) that satisfies
this space is equipped with the Luxemburg norm
Actually, since 2J(y) ≤ J(2y) ≤ 4J(y) it is standard to prove that there exists c > 0 such that
Actually this weighted Orlicz space LJ is that is the dual space of . Moreover V belongs to LJ if and only if min belongs to (Ω).
We know that Hence β(y) ≤ min(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, ) ≤ β(y). Therefore V belongs to LJ if and only if min belongs to (Ω). To prove that LJ is is then standard.□
2.3 Main results
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 . Given Vn in a set that will be defined below and τ > 0, we solve recursively the problem, with V0(S) = a2erT((S – Ke–rT)+ – S).
Let us consider the convex set Kn that reads
For all n ≥ 0, there is a unique Vn+1 ∈ Kn solution of (21).
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
with initial data W0 = Vn. Assume that Wm – Vn 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 is increasing, we can prove recursively by the Maximum Principle (see Proposition 2.2) that Wm ≤ Wm+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 Wm – Vn = (Wm – V0) – (Vn – V0). The function Vn – V0 is in LJ and the non negative function Wm – V0 that is bounded by above by –V0 is also in LJ. Then Vn+1 – Vn 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 V͂n+1 another solution. Then
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 Nτ = 𝔗 and let us consider the function
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 = ]nτ, (n + 1)τ[ we have
The following assertions hold true
The sequence is bounded in ( Ω × [0, 𝔗]) + (Ω × [0, 𝔗]).
The sequence is bounded in L∞([0, 𝔗]; L2(Ω)).
The sequence β() is bounded in (Ω × [0, 𝔗]) ∩ (Ω × [0, 𝔗]).
The sequence Vτ is bounded in L∞(Ω × [0, 𝔗]).
1. Summing (24) for V͂n+1 = Vn with respect to n yields
Hence 1. and 2. of the Lemma are proved. The point 3. is straightforward due to the properties of β that maps L2 + into L2 ∩ L3. Besides, since Vτ is a convex combination of Vn and Vn+1 in In then V0 ≤ Vτ ≤ 0 everywhere. This completes the proof of the Lemma.□
We know pass to the limit when τ → 0 using a monotonicity argument .
For (t, S) in In × Ω, we have
We first consider the scalar product in (Ω) by a test function ϕ ∈ D(Ω); then we have
Let us observe that
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 converges weakly towards we have that χ = S2VSS.
We now prove that We then compute
where v is a test function. By weak convergence we have
Moreover, observing that
we have, by weak convergence
Thus we have
We choose where ϕ is a given test function; then we have
By changing ϕ to –ϕ and tending ϵ to 0+ we obtain
This completes the proof of the existence of a solution V that belongs to L∞(0, 𝔗, 𝕍) such that belongs to L2(Ω × (0, 𝔗)) + (Ω × (0, 𝔗)). To prove that V is continuous in t with values in 𝕍 is a consequence of
3.3 Uniqueness of the solution
Let V and V͂ be two solutions of (18), and W = V – V͂. We have
Thus V = V͂.
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 J͂(y) = β͂(y) y where β͂(y) is solution to (1 + Ψ(β͂(y))β͂(y) = y. This Orlicz space has similar properties than 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
with initial condition V0(S) = erT((S – Ke–rT)+ – S). We will compare this solution to the solution of the linear Barles-Soner equation Vl
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 remains bounded in some space when a converges to 0.
There exists a constant C that depends ont the data r, K, T but that is independent of a such that
We know that
where βa(y) is the positive solution to a2βa + βa – y = 0. Multiplying this equation by and integrating in Ω × [0, t] leads to
On the other hand
and then that completes the proof of the Lemma.□
We know state
We have that Va ≥ Vl everywhere, and that there exists a constant C that depends on K, r, T such that
Set W = Va – Vl. Since Wt – S2WSS ≥ 0 then by the maximum principle W ≥ 0. On the other hand
Multiplying by S–2 and integrating in Ω yields
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
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].
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 . 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
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.
© 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