Definition
Derivatives pricing is the discipline of assigning a present value to a contract whose payoff depends on the future state of some underlying asset — a stock, an index, an interest rate, a commodity, another derivative. Because the payoff is contingent, the price cannot be quoted directly from supply and demand alone; it must be reasoned out from a model of how the underlying might evolve and what it would cost to hedge the resulting exposure.
The canonical methods — Black-Scholes closed-form solutions, binomial and trinomial trees, Monte Carlo simulation, and partial differential equation (PDE) solvers — all share a common backbone: under no-arbitrage assumptions, the fair price of a derivative is the expected present value of its payoff, computed under a risk-neutral measure rather than the real-world probability measure. The methods differ in how they represent the underlying's dynamics and the contract's features, not in their economic logic.
Why it matters
How it works
The risk-neutral pricing argument runs as follows. If the underlying follows a known stochastic process and an investor can continuously rebalance a hedging portfolio of the underlying and a risk-free bond to replicate the derivative's payoff, then by no-arbitrage the derivative must trade at the cost of that replicating portfolio. The math falls out: the price is the discounted expected payoff under a measure in which the underlying drifts at the risk-free rate. Black-Scholes solves this analytically for a European call or put on a non-dividend-paying stock under geometric Brownian motion. Binomial trees discretize the time grid and the underlying's possible moves, then roll the contract value back from expiry to today, optionally checking at each node whether early exercise is optimal — which is what makes them the natural method for American-style options.
Monte Carlo simulation goes the other direction: generate many sample paths of the underlying, compute the payoff along each, average, and discount. It excels at path-dependent payoffs (Asian options, barrier options, exotics) and high-dimensional underlyings (basket options on many stocks). PDE solvers — finite-difference grids over the Black-Scholes equation — handle complex boundary conditions and are the workhorse for many fixed-income derivatives. Every method's accuracy is bounded by the model's grip on reality: assuming constant volatility, no jumps, and continuous trading is convenient mathematics but a known idealization. Practitioners patch this by calibrating richer models (stochastic volatility, jump-diffusion, local volatility) to observed option-market prices — making the implied dynamics match the market by construction, then using the calibrated model to price the contract at hand.