Financial Markets for Commodities

Appendix The Cox, Ross and Rubinstein model

One of the major advantages of the Cox, Ross and Rubinstein (CRR) model is its relative mathematical simplicity. This makes it easier to approach than the Black and Scholes (BS) model. However, if, in CRR, we divide time into a number of periods that tend to infinity, then both models converge. Another major advantage that both these models offer is that the common method underlying both can be extended to any asset that results in random financial flows. The CRR and BS models were constructed to evaluate European buying options (calls), which can only be exercised at the time of maturity. The underlying support or asset for these options is a share that does not yield any dividends between the time the option is created and the time it matures. We also assume that the interest rates are constant over this period. In the CRR model, the period between the date the option was created, at t = 0, and its date of maturity, at t = T , can be divided into n number of periods that are arbitrarily chosen. The CRR model (constructed using the discrete time hypothesis) then becomes equivalent to the BS model (constructed on a continuous time hypothesis) when n tends toward infinity. We propose the hypothesis that at each instant t ∈ [0, T [, the support can change in only two ways. It may increase, being multiplied by a factor u > 1, or it may diminish, being multiplied by a factor d ∈ ]0, 1[. This hypothesis always appears surprising at first. ...