二元期权超级对冲问题中的近似和渐近问题

IF 0.8 Q4 BUSINESS, FINANCE
Sergey Smirnov, Dimitri Sotnikov, Andrey Zanochkin
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引用次数: 0

摘要

本文研究了科洛科尔佐夫(Kolokoltsov)的无交易约束市场价格动态乘法模型。在一般假设和单调报酬函数条件下,我们证明了从博弈论角度解释的保证确定性方法在超级套期保值问题上与概率方法产生了相同的结果。我们在保证确定性方法(GDA)中详细分析了特殊单调报酬函数(即欧式二元期权)的超级套期保值问题。与概率方法不同,GDA 可以直接描述最不利的混合市场策略。我们得到了相应的贝尔曼-伊萨克方程的解的一些有趣的分析性质,提供了在到期时支付期权报酬所需的最小储备金(也称为超级套期保值价格)。与最不利的市场情况相对应的条件分布的价格过程可以在对数尺度上用带有两个吸收障碍的随机漫步来近似。我们还证明,在适当的归一化条件下,当离散时间模型的步数趋于无穷大时,价格过程会弱收敛于行权价处有一个吸收障碍的几何布朗运动。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

Approximation and asymptotics in the superhedging problem for binary options

Approximation and asymptotics in the superhedging problem for binary options

This paper considers Kolokoltsov’s multiplicative model of market price dynamics witout trading constraints. Under general assumptions and monotonic payoff functions, we show that the guaranteed deterministic approach, having a game-theoretic interpretation, yields the same result in the superhedging problem as in the probabilistic approach. We analyze in detail the superhedging problem for a special monotonic payoff function, i.e., a European-style binary option, within the guaranteed deterministic approach (GDA). Unlike the probabilistic counterpart, GDA allows a direct description of the most unfavorable mixed market strategy. We obtain some interesting analytical properties of the solutions of the corresponding Bellman–Isaacs equations, providing the minimal required reserves (also called the superhedging price) to cover the option payoff at the expiration time. The price process with the conditional distributions corresponding to the most unfavorable market scenarios can be approximated on a logarithmic scale by a random walk with two absorbing barriers. We also prove that, under an appropriate normalization, the price process weakly converges to the geometric Brownian motion with one absorbing barrier at the strike price when the discrete-time model number of steps tends to infinity.

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来源期刊
Annals of Finance
Annals of Finance BUSINESS, FINANCE-
CiteScore
2.00
自引率
10.00%
发文量
15
期刊介绍: Annals of Finance provides an outlet for original research in all areas of finance and its applications to other disciplines having a clear and substantive link to the general theme of finance. In particular, innovative research papers of moderate length of the highest quality in all scientific areas that are motivated by the analysis of financial problems will be considered. Annals of Finance''s scope encompasses - but is not limited to - the following areas: accounting and finance, asset pricing, banking and finance, capital markets and finance, computational finance, corporate finance, derivatives, dynamical and chaotic systems in finance, economics and finance, empirical finance, experimental finance, finance and the theory of the firm, financial econometrics, financial institutions, mathematical finance, money and finance, portfolio analysis, regulation, stochastic analysis and finance, stock market analysis, systemic risk and financial stability. Annals of Finance also publishes special issues on any topic in finance and its applications of current interest. A small section, entitled finance notes, will be devoted solely to publishing short articles – up to ten pages in length, of substantial interest in finance. Officially cited as: Ann Finance
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