CSI 779 / STAT 789

Models of Financial Asset Prices

In an open market the price of goods and services is determined by supply and demand.

A "supply curve" is a schedule of quantities producers or owners are willing to sell at various prices, and a "demand curve" is a schedule of quantities desired by consumers or investors willing to purchase at various prices. The "equilibrium price" is the price in the price-quantity point where these two curves intersect (if they intersect).

The prices and quantities bid or asked financial assets depend on a complicated interaction of prices, preferences, and probabilities -- the "three P's (Lo, 1999, Financial Analysts Journal 55, 13--26). Models of financial asset prices usually relate the price to the present value of all future cashflows to which the owner of the asset is entitled. Both future cashflows and discount rates (to determine present value) are uncertain, so probability distributions and expected values must be incorporated into the model. The uncertainty is accommodated differently by different investors, so individual preferences must be allowed for.

A less ambitious goal than development of a comprehensive model of prices of financial assets is to analyze price changes over time. Two common approaches to this are the binomial model and the geometric Brownian motion model.

These models are not used for predicting asset prices. Their chief use is in relating the prices of derivative assets to underlying assets.

Characteristics of Probability Densities

For most continuous random variables we can write a probability density function (PDF), which yields the relative probability that the random variable takes values in given intervals.

Three of the most important characteristics of random variables are

  • the range
  • the moments (mean, variance, etc.) and
  • the shape of the density (assuming it exists).

    The normal distribution is the standard for comparison. The shapes of distributions may be

  • normal
  • skewed
  • heavy-tailed
  • light-tailed
  • bimodal (or multimodal)
  • or not

    The relative frequencies and the order statistics of a random sample from a given distribution will be "similar to" those of the parent distribution.

    A histogram and a q-q plot of the sample are useful in assessing the shape.

    Normal: xx<-rnorm(500)

    Skewed; Lognormal: xx<-rlnorm(500)

    Heavy-Tailed; t(5): xx<-rt(500,5)

    Light-Tailed

    Bimodal: xx[1:250]<-rnorm(250,-2) xx[251:500]<-rnorm(250,2)

    Mixture that is not bimodal: xx[1:250]<-rnorm(250,-1) xx[251:500]<-rnorm(250,1)

    Optimal Portfolio Construction

    In the allocation of financial assets to form a portfolio, it is desirable to maximize expected return (growth) of the portfolio and to minimize overall risk. As we have seen, a commonly accepted measure of risk is the variability in the rate of return. The "expected return" is generally measured by the sample mean of past rates of return, and the variances and covariances of the rates of return are generally measured by the sample variance-covariance matrix of the past rates of return.

    Standard mean-variance portfolio optimization seeks to determine an "efficient frontier" on which for any given level of risk, the expected return of the portfolio is maximized.
    Reference: Chapters 1, 2, and 3 of Michaud (1998).

    Assignment

    1. Continue developing the program to identify patterns.
    2. The geometric Brownian motion model leads to a distribution of asset prices (lognormal) with an expected value that decreases proportionally to the variance of the rate of return. Study this decrease by Monte Carlo, and develop an intuitive explanation why it should decrease.
    3. Define a project in financial data analysis. This can be a study of real data or it can be a simulation study of models/methods.