Approximating Functional of Local Martingale Under the Lack of Uniqueness of Black-Scholes PDE

arXiv:1102.2285v3 [q-fin.PR] 21 Sep 2012 Approximating Functional of Local Martingale Under the Lack of Uniqueness of Black-Scholes PDE Qingshuo Song ∗ November 20, 2018 Abstract When the underlying stock price is a strict local martingale process under an equivalent local martingale measure, Black-Scholes PDE associated with an European option may have multiple solutions. In this paper, we study an approximation for the smallest hedging price of such an European option. Our results show that a class of rebate barrier options can be used for this approximation. Among of them, a specific rebate option is also provided with a continuous rebate function, which corresponds to the unique classical solution of the associated parabolic PDE. Such a construction makes existing numerical PDE techniques applicable for its computation. An asymptotic convergence rate is also studied when the knocked-out barrier moves to infinity under suitable conditions. Keywords. Black-Scholes PDE, Non-Uniqueness, Financial bubbles; Local martingales; Convergence rate; 1 Introduction In a financial market equipped with the unique equivalent local martingale measure (ELMM) P, the smallest hedging price of an European option is the conditional expectation of the payoffwith respect to the probability P, see [5, 7]. In contrast to the probabilistic repre- sentation, option price can be also characterized as the unique solution of its associated Black-Scholes PDE, provided that PDE has a unique classical solution. The necessary and sufficient condition for the unique solvability of the parabolic PDE is that, the underlying stock price is Martingale process, see [1]. In other words, if the stock price is a strict local Martingale, then there exists multiple solutions for Black-Scholes PDE. Moreover, the option price may be one of the many solutions, see [4]. The difference of the multiple solutions of PDE is termed as financial bubbles, see [2, 4, 6] and the references ∗Department of Mathematics, City University of Hong Kong, song.qingshuo@cityu.edu.hk, The re- search of Q.S. is supported in part by the Research Grants Council of Hong Kong No. CityU 103310. 1 therein. In this work, we will consider the following problem proposed by Fernholz and Karatzas [5]: (Q) How can one find a feasible numerical solution convergent to the option price under the lack of uniqueness of Black-Scholes PDE? We first examine the existing numerical schemes on CEV model of Example 1, where the option price can be explicitly identified. There are typically two kinds of numerical schemes in this vein [11]. One is Monte Carlo method by discretizing the probability representation, the other is PDE numerical method by discretizing the truncated version of PDE. Unfortunately, Example 2 and Example 3 shows that classical Euler-Maruyama ap- proximation (for Monte Carlo method) and finite difference method (for PDE numerical method) leads to a strictly larger value than the desired option price. Motivated from these two examples, question (Q) boils down into the following two problems: (Q1) Find a feasible approximation for a Monte Carlo method, and its convergence rate; (Q2) Find a feasible approximation for PDE numerical method, and its convergence rate. In short, this work intends to find a feasible approximation to the smallest superhedging price V (x, t). It turns out that the value function can be obtained by a limit of a series of appropriate rebate option prices, which can be estimated by usual Monte Carlo method, see the details in Corollary 2. However, Corollary 2 may not be utilized for the approximation by PDE numerical method, since it may cause a discontinuity at the corner of the terminial- boundary datum. Therfore, a specific rebate option is proposed with its price continuous up to the boundary, so that its price corresponds to the unique classical solution of its associated parabolic PDE. Such a construction makes existing numerical PDE techniques applicable for computations, see Theorem 3. The rest of the paper is outlined as follows. In the next section, we give precise formu- lation of the problem. Section 3 presents main results, and related proofs is relegated to Section 4 for the reader’s convenience. The last section summarizes the work. 2 Problem formulation Throughout this paper, we use K as a generic constant, and R+ = (0, ∞), ¯R+ = R+ ∪{0}. If A is a subset of R × [0, T], then C(A) denotes the set of all continuous real functions on A, C2,1(A) denotes a collection of all functions ϕ : A 7→R such that ϕxx and ϕt belong to C(A). Dγ(A) denotes the set of all measurable functions ϕ : A →¯R+ satisfying growth condition ϕ(x, t) ≤K(1 + |x|γ), ∀(x, t) ∈A. (2.1) Cγ(A) = C(A)∩Dγ(A) denotes the set of all continuous functions satisfying γ-growth. We also denote the parabolic domain Q := R+ ×(0, T), truncated domain Qβ := (0, β)×(0, T), and Qα β := (α, β) × (0, T) for 0 < α < β. 2 We consider a single stock in the presence of the unique equivalent local martingale me

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