CSI 779 / STAT 789

Nonconstant Volatility Models

Volatility of rates of return is a rather complicated concept. Empirical evidence strongly suggests that it is not constant.
We can speak of
  • historical volatility or
  • implied volatility
    but what we are really interested in is instantaneous volatility or future volatility. We sometimes use the phrase "local volatility" to refer to a current prediction of future volatility during a specific time period.

    There are various approaches for accounting for nonconstant volatility in models of asset prices:

  • expressing the volatility as a deterministic function of time
  • use of jumps in the parameters or random component
  • expressing the volatility as a correlated stochastic process

    If we could express volatility as a deterministic function of time, we can easily modify the Black-Scholes formulae to account for the nonconstant volatility.

    A useful approach is to model volatility as a mean-reverting Ornstein-Uhlenbeck process. This model has parameters for the strength of the reversion to the mean, the mean volatility, and the volatility of the volatility.

    The strength of the reversion is the key to how far the volatility can wander from its mean.

    In the bivariate process for stock prices changes and for volatility, the correlation of the two processes is important.

    Correlation = .1

    Correlation = .9
    The correlation also affects the shape of the distribution of the rates of return. For negatively correlated processes, the rates of return tend to be negatively skewed, and for positively correlated processes, the rates of return tend to be positively skewed.

    Assignment

    1. Write a program to generate a mean-reverting volatility process. Let the mean volatility be 15% and the volatility of the volatility be 15%. Consider reversion strength of .10, .20, and 1.00.
      1. For each, estimate the min and max (symmetric about the mean) within which the vollatility remains 90% of the time.
      2. For each, estimate the mean number of returns to the mean in a year.
    2. Compute histograms of stock price returns under the stochastic volatility model with correlations of -.8, -.3, 0, +.3, and +.8.
    3. For the S&P 500 index (SPX on E*Trade ), compute and plot the volatility surface for year and for enough prices around the strike price to get a reasonable picture.
    4. Have the general levels of volatility changed recently?
      Choose 5 stocks and compare the historical volatility of rates of return for the period 1/1/00 to 3/31/00 with those for the period 1/1/01 to 3/31/01. Now, do the same thing with the square root of the semivariance.