A Guaranteed Deterministic Approach to Superhedging: No Arbitrage Properties of the Market

Guaranteed Deterministic Approach to Superhedging: Case of Binary European Option

“…☐ Proposition 2. Let assumptions (2) and (4) be satisfied, trading constraints be absent (in this case, condition ( 5) is obviously fulfilled), i.e., D t ð•Þ ≡ ℝ n , and the condition NDAO of no arbitrage opportunities be satisfied (in this case, the NDAO condition is equivalent to a geometric one: 0 lies in the relative interior of convex hull of K t ð•ÞÞ, t = 1, ⋯, N; see [10]). Then, for European options, the solutions of the Bellman-Isaacs equation (1) are monotonically decreasing in time, i.e.,…”

“…Proof. When there are no trading constraints and the condition NDAO of no arbitrage opportunities is fulfilled, we can assume that this is a special case of American options with payout functions (in principle, a weaker condition NDSA of no guaranteed arbitrage is sufficient for this; see [10])…”

“…Problem Statement. The present paper joins a series of publications (in particular, [9] describes the market model in detail and provides a literature review) [9][10][11][12][13][14][15] that develop a financial market model consistent with an uncertain deterministic price evolution with discrete time: asset prices evolve deterministically under uncertainty described using a priori information about possible price increments. Namely, they are assumed to lie in the given compacts that depend on the prehistory of the prices (such a model is an alternative to the traditional probabilistic market model (in our proposed deterministic approach, the reference probability measure is not initially set, as it is supposed in the probabilistic approach, see, e.g., [16])).…”