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Professional · Lesson 3 of 6

Binomial trees and American options

8 minute read · By Krishna Avadhani, Vestora Academy · Updated 18 October 2026

The Cox-Ross-Rubinstein (1979) binomial model splits time into steps. At each step the price moves up by a factor u or down by d. Working backwards from expiry using risk-neutral probabilities gives the option value today.

Because you can check at every node whether early exercise beats holding, binomial trees price American options, which Black-Scholes cannot do directly. As the number of steps increases, the binomial price for a European option converges to Black-Scholes.

Monte Carlo simulation is a third approach: simulate thousands of price paths and average the discounted payoffs. It handles complex, path-dependent options (such as Asian options) well. Vestora's retirement planner uses the same Monte Carlo idea.

Formula

u = e^(σ√Δt), d = 1/u
p = (e^(rΔt) - d) / (u - d)
V = e^(-rΔt) [p V_up + (1 - p) V_down]

Key takeaways

  • Binomial trees value options step by step backwards.
  • They handle early exercise (American options).
  • Monte Carlo suits path-dependent payoffs.

Check your understanding

1. Main advantage of binomial trees over Black-Scholes?

  • Faster for all options
  • They can value early exercise
  • No volatility input needed
  • They predict prices
Show the answer

They can value early exercise. Each node checks whether early exercise is optimal.

2. As binomial steps increase, European option prices converge to...

  • Zero
  • The Black-Scholes price
  • The strike
  • Intrinsic value
Show the answer

The Black-Scholes price. The binomial model converges to Black-Scholes.

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General education only, not financial advice. Examples are illustrations. Rules, tax settings and products change; check official sources.