Actuarial-Financial Mathematics Lab

Historical Background and Motivation

The modern mathematical study of option pricing is usually traced back to Louis Bachelier (1900), who made the first systematic attempt to formulate the pricing problem in mathematical terms. Much later, in 1973, the Black–Scholes–Merton theory provided a decisive breakthrough: for the first time, option valuation was placed in direct connection with dynamic hedging and no-arbitrage arguments in continuous time.

One of the most striking implications of that theory was the idealized possibility of perfect replication. At the same time, however, this viewpoint relied on strong assumptions, such as continuous rebalancing, frictionless trading, and a highly stylized market environment. In practice, these assumptions are not literally satisfied.

In 1979, the Cox–Ross–Rubinstein binomial model reformulated the pricing and hedging problem in discrete time. This made the logic of replication particularly transparent and brought the theory closer to feasible trading decisions. Its limitation, however, was that the underlying asset price was described through a finite-state tree.

The material presented on this page is motivated by a more modern viewpoint: we preserve the central role of hedging and discrete-time trading, but we do not restrict the terminal stock price to finitely many values. Instead, we work under the minimal condition that the terminal asset price satisfies S_T ≥ 0.

Within this framework, one can study both static hedging portfolios and dynamic hedging strategies, while also developing corresponding notions of fair option pricing. The slides, notes, books, and research material collected below are intended to guide the reader from the classical foundations to these more recent data-driven and discrete-time perspectives.