Probability-free models in option pricing: statistically indistinguishable dynamics and historical vs implied volatility

We investigate whether it is possible to formulate option pricing and hedging models without using probability. We present a model that is consistent with two notions of volatility: a historical volatility consistent with statistical analysis, and an implied volatility consistent with options priced with the model. The latter will be also the quadratic variation of the model, a pathwise property. This first result, originally presented in Brigo and Mercurio (1998, 2000), is then connected with the recent work of Armstrong et al (2018, 2021), where using rough paths theory it is shown that implied volatility is associated with a purely pathwise lift of the stock dynamics involving no probability and no semimartingale theory in particular, leading to option models without probability. Finally, an intermediate result by Bender et al. (2008) is recalled. Using semimartingale theory, Bender et al. showed that one could obtain option prices based only on the semimartingale quadratic variation of the model, a pathwise property, and highlighted the difference between historical and implied volatility. All three works confirm the idea that while historical volatility is a statistical quantity, implied volatility is a pathwise one. This leads to a 20 years mini-anniversary of pathwise pricing through 1998, 2008 and 2018, which is rather fitting for a talk presented at the conference for the 45 years of the Black, Scholes and Merton option pricing paradigm.