以類神經網路預測 S&P 500 的風險中立機率分布

Abstract

根據論文調查，找出並過濾許多實作出風險中立機率分佈 (Q) 的方法，最後採用其中 Jackwerth 的 Fast and Stable Method;接著同樣透過論文調查，找出實作出真實機率分佈 (P) 的方法，最後是採用 Jackwerth 的研究報告中的其中一篇延伸論文，作為我們實作的方法。 接著，實作出了風險中立機率分佈 (Q) 與真實機率分佈 (P) 以後，下一步就是希望透 過把歷史的真實機率分佈 (P) 當作輸入，另外加上許多當期所觀測得到的總體經濟變數也當作輸入，丟進深度學習的類神經網路模型中，希望能對現今之後，也就是未來的風險中 立機率 (Q)，能夠有好的預測表現。 背後的原理是這樣子:我們希望透過歷史的真實機率分佈 (P) 來類比出現今的真實機率分佈 (P)(其原因是我們觀測不到現今的真實機率分佈 (P))，並再透過現今的真實機率 分佈 (P)，經過定價核的轉換方式，得到現今的風險中立機率分佈 (Q)。 但本篇論文不處理這類比與定價核和的轉換方式，而是希望嘗試透過深度學習的模型，來學習結合這兩個步驟，並預測出現今的風險中立機率分佈 (Q)，這是本篇論文期望做到的事。

This paper presents a novel approach for predicting the risk-neutral probability distribution (Q) for the future by leveraging historical real probability distribution (P) and current macroeconomic variables. The method involves identifying and filtering numerous methods for estimating risk-neutral probability distributions, ultimately selecting Jackwerth's Fast and Stable Method. Similarly, a comprehensive survey is conducted to identify techniques for estimating real probability distributions, with the chosen approach stemming from an extended research paper within Jackwerth's portfolio. Once the risk-neutral probability distribution (Q) and real probability distribution (P) are established, the subsequent step entails integrating historical real probability distribution (P) and contemporaneous macroeconomic variables as inputs into a deep learning neural network model. The primary objective is to forecast the future risk-neutral probability distribution (Q) with high predictive accuracy. The underlying principle hinges on analogizing the present-day real probability distribution (P) based on historical real probability distribution (P) due to the unobservable nature of the current real probability distribution (P). This approach then combines the present-day real probability distribution (P) with a pricing kernel transformation to derive the current risk-neutral probability distribution (Q). Notably, this paper does not delve into the specifics of the analogizing process and pricing kernel transformation, instead opting to explore the integration of these two steps using a deep learning model. The ultimate goal of this paper is to achieve accurate predictions of the current risk-neutral probability distribution (Q) by melding these methodologies.