arXiv 1202.25852012-02-12q-fin.CPcs.GTq-fin.PR

极小极大期权定价在极限上满足Black-Scholes定理

Minimax Option Pricing Meets Black-Scholes in the Limit

Jacob Abernethy, Rafael M. Frongillo, Andre Wibisono

备注：: 19 pages

摘要：

期权合约是一种金融衍生品，允许投资者对冲风险并对资产未来市场价格的变化进行投机。简而言之，期权具有基于未来给定日期资产市场价格的特定支付。1973年，Black和Scholes提出了一个期权定价模型，该模型基本上是在假设资产价格按照几何布朗运动波动的情况下估计资产价格的尾部风险。最近，DeMarzo等人，其中，提出了更稳健的估值方案，我们甚至可以假设对手选择了价格波动。这个框架可以被认为是投资者和自然之间的一个连续的两人零和游戏。我们分析了这个游戏在极限中的价值，投资者可以在越来越小的时间间隔内进行交易。在自然界（对手）行动的弱假设下，我们证明了极大极小期权的价格渐近逼近Black-Scholes估值。我们分析的关键部分是表明自然界的最小最大最优对偶策略在极限下收敛于几何布朗运动。

英文摘要：

Option contracts are a type of financial derivative that allow investors to hedge risk and speculate on the variation of an asset's future market price. In short, an option has a particular payout that is based on the market price for an asset on a given date in the future. In 1973, Black and Scholes proposed a valuation model for options that essentially estimates the tail risk of the asset price under the assumption that the price will fluctuate according to geometric Brownian motion. More recently, DeMarzo et al., among others, have proposed more robust valuation schemes, where we can even assume an adversary chooses the price fluctuations. This framework can be considered as a sequential two-player zero-sum game between the investor and Nature. We analyze the value of this game in the limit, where the investor can trade at smaller and smaller time intervals. Under weak assumptions on the actions of Nature (an adversary), we show that the minimax option price asymptotically approaches exactly the Black-Scholes valuation. The key piece of our analysis is showing that Nature's minimax optimal dual strategy converges to geometric Brownian motion in the limit.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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