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Ricci流與Black-Scholes-Merton方程的關系

Grisha Perelman曾寫過這篇文章

Ricci-flow方程,一種熱方程,是一個   債券交易者的Black-Scholes方程的遠親   世界各地都在使用股票和債券期權定價。

Wilmot已從BS方程推導出熱方程,但想知道是否有任何證據表明您可以從Ricci流中獲得BS方程。

最佳答案

嗯,這似乎是這些概念的流行帳戶,但在很高的層次上,連接如下:

The Ricci flow "is a process that deforms the metric of a Riemannian manifold in a way formally analogous to the diffusion of heat, smoothing out irregularities in the metric." [Wikipedia]

enter image description here

現在Black-Scholes方程在數學上基於幾何布朗運動,它描述了概率的擴散底層價格路徑的分布。

enter image description here [Source]

The connection between Black-Scholes and diffusion becomes especially clear when you have a look at how Black-Scholes' differential equation is solved by transforming it to the diffusion equation, see also this question and answers therein:
Transformation from the Black-Scholes differential equation to the diffusion equation - and back

因此,Ricci flow和Black-Scholes都是(基於)擴散模型的數學描述。我認為除此之外還沒有更多的東西。

轉載註明原文: Ricci流與Black-Scholes-Merton方程的關系