Normal Price Limits from the Infinite-Step Binomial Tree

Summary

The document derives a limiting asset-price distribution from a multi-step binomial model under simplifying assumptions: zero interest rates, equal up and down probabilities, and an expected expiry price equal to today’s spot. The step changes are constructed to have zero mean and a variance that preserves total expiry variance as the number of steps grows. Using independent coin-toss increments and the Central Limit Theorem, the article shows how the normalized sum converges to a standard normal distribution, then applies risk-neutral expectation to value a derivative payoff.

This is an introductory stage in a progression toward Black–Scholes, not a realistic finished pricing model. The article flags that its model permits negative stock prices, represents absolute rather than relative price moves, omits interest rates, and excludes a real-world risk premium. It provides a theoretical derivation, but no empirical pricing comparison or calibration evidence; its assumptions limit direct application to traded options.

Key ideas

• Equal-probability independent increments with preserved total variance lead to a normal limit as tree steps increase.
• Under the stated assumptions, risk-neutral derivative value is the expected expiry payoff under the limiting asset distribution.
• The assumptions include zero interest rates and an expected expiry asset price equal to spot.
• The model allows negative prices and absolute moves, making it unsuitable as a realistic final stock-price model.

Tags

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.