CoxRossRubinsteinModell

The Cox-Ross-Rubinstein model, also known as the Cox-Ross-Rubinstein Modell in German literature, is a discrete-time, recombining binomial model for pricing options. Introduced by John Cox, Martin Rubinstein, and Stephen Ross in 1979, it provides a simple framework to replicate stock price dynamics and to value European and American options by risk-neutral valuation.

In each time step the stock price moves up by a factor u or down by a factor d. The parameters are chosen to prevent arbitrage and to approximate continuous price behavior: typically u = e^{σ√Δt} and d = e^{-σ√Δt}, where σ is the annualized volatility and Δt is the time step. The risk-free rate r is used to determine the risk-neutral up-move probability p = (e^{rΔt} − d) / (u − d).

The model uses a recombining binomial tree, so after n steps there are only n+1 possible stock prices. Option values are computed by backward induction, starting from the payoff at maturity. European options are priced exactly in this framework, while American options can be exercised early if such an action is optimal at a given node.

The Cox-Ross-Rubinstein model is closely linked to the Black-Scholes framework: as Δt → 0 with the CRR parameterization, the binomial prices converge to Black-Scholes prices. The model is widely taught and used because of its intuition and straightforward implementation; however, it assumes constant volatility and interest rates, and minimal treatment of dividends unless extended.

Extensions include dividends, multiple assets, and calibration to observed market prices. The CRR model remains a foundational tool in financial engineering and derivatives valuation.