Brownian Motion, Quadratic Variation, and Asset Price Modeling
Summary
The document builds standard Brownian motion from a scaled sequence of random coin tosses. Scaling each step by the square root of its time interval keeps the walk’s quadratic variation finite as the number of steps grows. In the continuous-time limit, the resulting Wiener process has independent, normally distributed increments whose variance equals the elapsed time. The article also describes its continuity, lack of differentiability, unbounded variation, and Markov and martingale properties.
These features make Brownian motion a foundation for stochastic differential equations and later option-pricing derivations. The article notes that standard Brownian motion itself is insufficient as a direct asset-price model and points toward geometric Brownian motion as a more suitable construction. It gives conceptual explanations rather than empirical tests or calibration evidence; the limiting argument is asserted without technical proof. Its discussion is therefore an introduction to the mathematical model and its implications, not a validation that real market prices follow Brownian paths.
Key ideas
- Scaling random-walk steps by the square root of elapsed time gives a finite, nonzero quadratic variation in the continuous-time limit.
- A standard Brownian motion starts at zero and has independent normal increments with variance equal to the interval length.
- Brownian paths are continuous but nowhere differentiable and have unbounded variation.
- Brownian motion has Markov and martingale properties and underpins stochastic differential equations.
- The document argues that geometric Brownian motion is more appropriate than standard Brownian motion for asset-price modeling.
Tags
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.