There is a clear trajectory in mathematical finance starting from Black, Scholes, and Merton’s Nobel Economic Prize winning work demonstrating how to replicate an option payoff by delta hedging. The value of the option is the cost of setting up the initial hedge. Their Platonic Ideal of a bond, stock, and option with perfect liquidity and trading in continuous time is evolved to any collection of instruments and accurately model features relevant business needs.

Ross showed how to extend this to any portfolio of instruments and should have won a Nobel Economic Prize for figuring out option valuation does not require the complicated machinery of Ito processes and partial differential equations.

Since then, more general option payoffs and multi-period models were developed to extend theory to more closely align with trading reality.

There are still some basic questions that existing theory cannot answer. Perhaps these were obscured by mathematical artifacts due to using a continuous time stochastic process to model stock prices. Scholes and Merton won a Nobel prize for showing the initial hedge does not involve the return on the underlying, only the rate used for funding the dynamic hedge and the volatility of the underlying instrument.

The first question a trader has after putting on an initial hedge is when, and how much, to adjust that as market prices move. Current theory has no answer for that. A trader aphorism is “hedge when you can, not when you have to.” They also know