Model Definition

  • Consider time interval
    • total time
    • intervals
    • : duration of one interval ()
  • : price of security at time .
    • is given as a constant
  • Gross return
    • It’s called gross because its in the form of or , not
    • Is defined as a Bernouilli Distribution:
  • : number of upticks. Is a random variable with Binomial Distribution
  • Sample Space
    • is for -period binomial tree
  • Final Price
    • is a random variable.
    • i.e. probability that price will be equal to there being upticks is the probability that there will be upticks. (no shit)
  • Expectation of final price:
  • To find number of upticks from final price we use
  • Dividends:
    • See Dividend Discount Model for dividends in non-binomial tree model
    • Total Return (incl. dividends):
    • Capital Gains Return (excl. dividends):
    • (Dividends Return )

Assumptions & Definitions

  • Log returns: Is also a random variable
  • i.e. the stock ticking up then down is same as no movement at all
  • Instantaneous Rate of Return
  • Drift: Instantaneous Expected Log-Return
  • Log Variance: Instantaneous Variance of Log-Return = Volatility ()
  • Dividend Rate


  • (Proofs in notes)

thm. Parameter Triple. Given a security with we determine that:

Continuous Time Model

Model Definition

Instead of assuming an uptick-downtick gross return is a Bernouilli Distribution, we instead think of the log returns. (Use Lindenberg CLT)

⇒ Thus we have


  • is a lognormal random variable
    • i.e. where is the standard normal random variable
  • are i.i.d.