A new representation of the risk-neutral distribution and its applications. Issue 5 (4th May 2022)

• Record Type: Journal Article
• Authors: Cui, Zhenyu; Xu, Yuewu
• Is Part Of: Quantitative finance. Volume 22:Issue 5(2022)
• Page Start: 817
• Page End: 834
• Publication Date: 2022-05-04
• Subjects: Risk-neutral distribution -- Hermite expansion -- Spanning formula -- Dirac delta function; G12 -- G13 -- G14 -- C58; Finance -- Periodicals; Business mathematics -- Periodicals; Finance -- Mathematical models -- Periodicals; Investments -- Mathematics -- Periodicals; Economics -- Periodicals; Finances -- Modèles mathématiques -- Périodiques; 332.015118
• DOI: 10.1080/14697688.2021.2013520
• Languages: English
• ISSNs: 1469-7688

Abstract

Abstract : This paper establishes a novel model-free representation of the risk-neutral density in terms of market-observed options prices by combining exact series representations of the Dirac Delta function and the Carr-Madan asset spanning formula. Compared to the widely used method for obtaining the risk-neutral densities via the Breeden–Litzenberger device, our method yields estimates of risk-neutral densities that are model-free, automatically smooth, and in closed-form. The closed-form feature of our new representation makes it ideal for many potential applications including a new model-free representation of the local volatility function in the Dupire's local volatility model. The validity of our method is demonstrated through simulation studies as well as an empirical application using S&P 500 index option data. Extension of the method to higher dimensions is also obtained by extending the spanning formula.