08. Introduction to Brownian motion

Understanding Brownian Motion in Financial Mathematics

• Origin: Named after Scottish Botanist Robert Brown, who observed pollen movement in water in 1827.

• Concept:

  • Represents a stochastic or random process, denoted as W_t.
  • Typically starts at zero: When t = 0, W_t = 0.
  • At any later time, W_t has a normal distribution with mean 0 and variance equal to t.

• Increments:

  • Follow a normal distribution with mean 0 and variance equal to the time lapse's length.
  • For times t_1 and t_2, the increment W_t_1 - W_t_1 has variance t_2 - t_1.

• Financial Application:

  • Used to model the relative returns of stock investments.
  • Relative return example: Buying a stock at $100 and selling at $110 gives a 10% return.
  • Modeled as at + (SigmaW_t), where:
  • a is a constant (discussed later).
  • Sigma represents stock volatility.

• Volatility and Time:

  • Longer holding times and higher volatility increase return uncertainty.