Using Itô’s Lemma to Model Prices and Frame an Options Trade
Summary
The article explains Itô’s lemma as the stochastic counterpart to the ordinary chain rule and applies it to a stock price modeled by geometric Brownian motion. It derives the dynamics of log price, showing how the drift is adjusted by half the variance, and relates the adjustment to the difference between arithmetic and geometric growth. The worked example estimates a future price range from drift and volatility, then uses the range to motivate selling an out-of-the-money call and put.
The article reports an illustrative Microsoft calculation and option quotes, but treats the resulting probability and premium as simplified. It explicitly cautions that the normal-return assumption may fail, transaction costs and taxes are omitted, and the range has not been backtested or forward tested. Margin use, opportunity cost, implied volatility, and the possibility of price moves beyond the strikes also affect the trade. The derivation is useful for understanding the model, but the example does not establish a reliable trading edge or a 95% chance of profit.
Key ideas
- Itô’s lemma adds a quadratic variation term to the chain rule for stochastic processes.
- Under geometric Brownian motion, log-price drift is reduced by half the variance.
- Drift and volatility estimates can be used to construct a model-based future price range.
- The example uses that range to motivate selling an out-of-the-money call and put.
- Normality, costs, margins, implied volatility, and untested historical coverage limit the trade illustration.
Tags
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.