2019/04/03 by Damiano Brigo, Brigo, Damiano
Economics, Econometrics and Finance · Decision Sciences · #Stochastic processes and financial applications #Forecasting Techniques and Applications #Stock Market Forecasting Methods
paper · pdf · doi:10.48550/arxiv.1904.01889
We investigate whether it is possible to formulate option pricing and hedging\nmodels without using probability. We present a model that is consistent with\ntwo notions of volatility: a historical volatility consistent with statistical\nanalysis, and an implied volatility consistent with options priced with the\nmodel. The latter will be also the quadratic variation of the model, a pathwise\nproperty. This first result, originally presented in Brigo and Mercurio (1998,\n2000), is then connected with the recent work of Armstrong et al (2018, 2021),\nwhere using rough paths theory it is shown that implied volatility is\nassociated with a purely pathwise lift of the stock dynamics involving no\nprobability and no semimartingale theory in particular, leading to option\nmodels without probability. Finally, an intermediate result by Bender et al.\n(2008) is recalled. Using semimartingale theory, Bender et al. showed that one\ncould obtain option prices based only on the semimartingale quadratic variation\nof the model, a pathwise property, and highlighted the difference between\nhistorical and implied volatility. All three works confirm the idea that while\nhistorical volatility is a statistical quantity, implied volatility is a\npathwise one. This leads to a 20 years mini-anniversary of pathwise pricing\nthrough 1998, 2008 and 2018, which is rather fitting for a talk presented at\nthe conference for the 45 years of the Black, Scholes and Merton option pricing\nparadigm.\n