The Black–Scholes model of option prices if individual's utilities are admitted

The principle of no arbitrage in Black and Scholes's framework is manifested in their assumption of risk neutrality. It turned out that while the martingale probability measure which arises from having no arbitrage is sufficient for their unique pricing of options, the criterion of utility maximisation was left out in their discussion. Thus, its inclusion into the Black–Scholes model is the theme of this paper. To this end the uncertainty which prevails in financial environments is reconciled here with the principle of risk-neutral portfolios by designing a typical investor's portfolio which maintains extraneous full certainty. In particular, the full certainty is achieved by using external (i.e., peculiar) portfolios comprising shares and options on them such that any source of uncertainty in the portfolio will be neutralised by the investor via a martingale probability measure in a Markov process. The resulting equilibrium will be a Black–Scholes fair option price.