Chapter 10 - Stochastic Models Of Security Prices

Question 1

Evidence against the continuous-time lognormal model (7)

Answer 1

1. The volatility parameter σ may not be constant over time. Estimates of volatility from past data are critically dependent on the time period chosen for the data and how often the estimate is re-parameterised.
2. The drift parameter μ may not be constant over time. In particular, bond yields will influence the drift. (if interest rates are high, we might expect the equity drift to be high as well)
3. There is evidence in real markets of mean-reverting behaviour, which is inconsistent with the independent increments assumption. (rises are more likely following a market fall, and falls are more likely following a rise)
4. There is evidence in real markets of momentum effects, which is inconsistent with the independent increments assumption. (a rise in one day is more likely to be followed by another rise the next day)

Normality assumptions:

5. The distribution of security returns log(Su/St) has a taller peak in reality than that implied by the normal distribution. This is because there are more days of little or no movement in financial markets than the normal distribution suggests.
6. The distribution of security returns log(Su/St) has fatter tails in reality that that implied by the normal distribution. This is because there are more extreme movements in security prices, both on the upside and on the downside, than is consistent with such a model. In particular, market crashes appear more often than one would expect from a normal (or lognormal) distribution.
7. The sample paths of security prices are not continuous, but instead appear to jump occasionally.
   While the random walk produces continuous price paths, jumps or discontinuities seem to be an important feature of real markets.

Question 2

Cross-sectional property

Answer 2

A cross-sectional property fixes a time horizon and looks at the distribution over all the simulations.
Cross-sectional properties are difficult to validate from past data, since year of past history typically started from a different set of conditions. However, the prices of deivatives today should reflect market views of a cross-sectional distribution. Cross-sectional information can therefore sometimes be deduced from the market prices of options and other derivatives.

Question 3

Longitudinal property

Answer 3

A longitudinal property picks one simulation and looks at a statistic sampled repeatedly from that simulation over a long period of time.
For some models, this longitudinal distribution will converge to some limiting distribution as the time horizon lengthens. Furthermore, the limiting distribution is common to all simulations. Such convergence results are sometimes called ergodic theorems. The resulting distribution is an ergodic distribution.
Unlike cross-sectional properties, longitudinal properties do not reflect market conditions at a particular date but, rather, an average overall likely future economic conditions. Most statistical properties computed from historical data are effectively longitudinal properties.

Question 4

Advantages of models based on economic theory (2)

Answer 4

1. Capable of an economic interpretation

2. Consistent with economic principles, such as arbitrage-free markets

Question 5

State the key properties of the lognormal model for security prices

Answer 5

1. The proportional change in the price is lognormally distributed, so the returns over any interval do not depend on the initial value of the investment St.
2. The mean and variance of the log returns are proportional to the length of the interval considered (u-t), and so the standard deviation of the log returns (often taken to be a measure of volatility) increases with the square root of the interval.
3. The dependence of the length of the interval means that the mean, variance and standard deviation tend to infinity as the length of the interval increases. The exceptions are the simplified cases where there is no drift in the log price (μ=0) or where there is no volatility (σ=0).
4. It is assumed that returns over non-overlapping intervals are independent of each other.

Question 6

One advantage and one disadvantage of using a non-normal distribution to model share prices

Answer 6

Advantage:
A non-normal distribution can provide an improved description of the actual returns observed (in particular, the greater frequency of more extreme events than would be the case under the lognormal model).
Disadvantage:
The improved fit to empirical data comes at the cost of losing the tractability of working with normal (and lognormal) distributions.

Question 7

Discuss the implications of the empirical deviations from the random walk for market efficiency

Answer 7

It is important to appreciate that many of the empirical deviations from the random walk do not imply market inneficiency.

For example, periods of high and low volatility could easily arisenif new information sometimes arrived in large measure and sometimes in small.
Market jumps are consistent with the arrival of information in packets rather than continuously.
Even mean reversion can be consistent with efficient markets. After a crash, many investors may have lost a significant proportion of their total wealth. It is not irrational for them to be more averse to the risk of losing what remains. As a result, the prospective equity risk premium could be expected to rise.