Binomial Option Pricing (CRR)
Price options on a tree of future prices by working backwards under a risk-neutral world.
The binomial model assumes that at each small time step the underlying price can move up or down by fixed factors, forming a recombining price tree.
Starting from maturity, the option value at each node is the discounted risk-neutral expectation of its values in the next step, with American options allowing early exercise at each node.
The tree is equivalent to constructing a self-financing hedging portfolio (replicating strategy) at each step, so pricing and hedging are two views of the same object.
As the number of steps increases and the time step shrinks, the CRR binomial model converges to Black–Scholes under standard assumptions.
Cox–Ross–Rubinstein (CRR) binomial option pricing. The chart shows how the option price converges as the number of time steps N increases. The orange line is a high-step reference; the green point is the current N. Below we also decompose the price into intrinsic and time value.
The CRR lattice approximates a continuous-time model by slicing the maturity into N steps. For small N the price can sit away from the high-step reference; as N grows, the blue curve stabilises around the orange line and the vertical Δ shrinks.
The decomposition “intrinsic + time value” is how traders read this price: intrinsic reads off the payoff if you exercised now, time value is what you pay for optionality and volatility until expiry. In-the-money options with very low time value are close to behaving like forwards.
The risk-neutral probability p should lie in [0,1]; if it needs clamping, your u/d choice is outside the textbook no-arbitrage region. In practice, a binomial tree is a numerical control grid: N is like your re-hedging frequency. More steps push you towards the continuous model but increase computation; American features justify the lattice when early exercise or path structure matters.