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On modifications of the Bachelier model
Annals of Finance Pub Date : 2021-01-18 , DOI: 10.1007/s10436-020-00381-1
Alexander Melnikov , Hongxi Wan

Mathematically, stock prices described by a classical Bachelier model are sums of a Brownian motion and an absolute continuous drift. Hence, stock prices can take negative values, and financially, it is not appropriate. This drawback is overcome by Samuelson who has proposed the exponential transformation and provided the so-called Geometrical Brownian motion. In this paper, we introduce two additional modifications which are based on SDEs with absorption and reflection. We show that the model with reflection may admit arbitrage, but the model with an appropriate absorption leads to a better model. Comparisons regarding option pricing among the standard Bachelier model, the Black–Scholes model and the modified Bachelier model with absorption at zero are executed. Moreover, our main findings are also devoted to the Conditional Value-at-Risk based partial hedging in the framework of these models. Illustrative numerical examples are provided.



中文翻译:

关于Bachelier模型的修改

在数学上,经典巴切里尔模型描述的股票价格是布朗运动和绝对连续漂移的总和。因此,股票价格可以取负值,从财务上讲,这是不合适的。Samuelson克服了这一缺点,他提出了指数变换并提供了所谓的几何布朗运动。在本文中,我们介绍了基于带吸收和反射的SDE的两个附加修改。我们表明,具有反射的模型可以允许套利,但是具有适当吸收的模型会导致更好的模型。在标准Bachelier模型,Black-Scholes模型和吸收率为零的改良Bachelier模型之间进行了关于期权定价的比较。而且,我们的主要发现还致力于这些模型框架下基于条件风险值的部分对冲。提供了说明性的数值示例。

更新日期:2021-03-13
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