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We investigate whether it is possible to formulate option pricing and hedgingmodels without using probability. We present a model that is consistent withtwo notions of volatility: a historical volatility consistent with statisticalanalysis, and an implied volatility consistent with options priced with themodel. The latter will be also the quadratic variation of the model, a pathwiseproperty. 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 isassociated with a purely pathwise lift of the stock dynamics involving noprobability and no semimartingale theory in particular, leading to optionmodels without probability. Finally, an intermediate result by Bender et al.(2008) is recalled. Using semimartingale theory, Bender et al. showed that onecould obtain option prices based only on the semimartingale quadratic variationof the model, a pathwise property, and highlighted the difference betweenhistorical and implied volatility. All three works confirm the idea that whilehistorical volatility is a statistical quantity, implied volatility is apathwise one. This leads to a 20 years mini-anniversary of pathwise pricingthrough 1998, 2008 and 2018, which is rather fitting for a talk presented atthe conference for the 45 years of the Black, Scholes and Merton option pricingparadigm.

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