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.

A central methodological point is the separation of the forecasting stage from the portfolio construction stage. Forecasting produces information about the future, while portfolio construction transforms that information into an implementable trading decision under explicit constraints. This separation makes the portfolio-construction theory as general as possible: the classical Markowitz mean–variance framework becomes only one special case, corresponding to a particular choice of predictive inputs, risk criterion, and admissible trading instruments.

Throughout this approach, we work under the deterministic notion of arbitrage. This is the notion that genuinely governs market prices: if a portfolio produces a nonnegative payoff in every possible market state and a strictly positive payoff in at least one state, while requiring no positive net investment, then market prices cannot remain unchanged. Deterministic arbitrage therefore gives hard, model-free restrictions on prices and leads naturally to deterministic arbitrage-free notions of fair prices.

This also clarifies a point that is often confused in the literature. A fair price is a mathematical concept: it is the price at which two quantities associated with the buyer and the seller are balanced. It is not necessarily the transaction price observed in the market. The actual selling price is formed by supply and demand, liquidity, order flow, market sentiment, inventory effects, and other economic forces.

Therefore, if one wants to forecast the future selling price of a contract, one must build a genuine forecasting mechanism that incorporates the main factors affecting that price. Once such a mechanism has been specified, one may then compute sensitivities with respect to those factors. The fair-price calculation and the selling-price forecast are thus distinct tasks: the first is a mathematical construction problem, while the second is an empirical prediction problem.

Under this perspective, any method that does not deliver a feasible hedging strategy should be interpreted only as a forecasting device. This is precisely the status of Black–Scholes-type models and their extensions, as well as binomial-type models and their extensions, whenever their outputs are not accompanied by an executable hedge under the actual market constraints. Forecasting techniques for vanilla call and put options may still be useful in some liquid markets, but forecasting prices for more specialized derivatives is far less convincing from a practical point of view, especially when the contracts are illiquid, bespoke, or weakly supported by observable market data. Consequently, papers based on PDEs or stochastic analysis that do not produce a feasible hedging strategy should be understood primarily as predictive or theoretical exercises, not as complete pricing-and-hedging methodologies; and, in many specialized option-pricing problems, the practical value of such prediction may be limited.