From the Cox-Ross-Rubinstein Binomial Model to Black-Scholes: A Discrete to Continuous Time Limit

Sammanfattning

This thesis explains how the discrete-time Cox–Ross–Rubinstein (CRR) binomial model converges to the continuous-time Black–Scholes model as the time step δt = T /n tends to zero. We begin by deriving the CRR one-step call price using replication and no-arbitrage, which leads to the risk-neutral probability and a backward-induction algorithm for n-step European call pricing. We then present the trading interpretation: when the market call price differs from the model value, a hedged position (hedge/reverse hedge) combined with periodic rebalancing can lock in the mispricing. Next, we derive the Black–Scholes partial differential equation as the continuous-time limit of the binomial model by choosing diffusive scaling for the up/down factors and applying Taylor expansions to the one-step risk-neutral pricing relation. Solving the resulting PDE with the European call terminal condition yields the Black–Scholes closed-form formula. Finally, MATLAB experiments compare CRR prices to the Black–Scholes benchmark for increasing n across different moneyness and maturities. The results confirm convergence C(n) → CBS , with near first-order error decay in at-the-money cases and visible lattice oscillations for out-of-the-money strikes due to discretizations around the payoff kink.