Chapter 11: Stochastic models of security prices

Question 1

What does the lognormal model assume about the process that generates security prices

Answer 1

Share prices follow a geometric brownian motion, inplying that their log follows a brownian motion

Question 2

Advantage of using GBM

Answer 2

• ALways positive
• Result in closed form asset prices

Question 3

Why is Brownian motion not good for modelling indicies in the short run

Answer 3

• Brownian motion without a positive drift is certain to hit negative values eventually
• Even with a positive drift, there is a possibilty of negative security prices in the future
• Brownian motion predicts that daily movements of size 10 would occur as frequently when the process is at level 10 as when it is at level 10000

Question 4

Name emperical reasons against the use of the lognormal model to model share prices

Answer 4

• Volatility
• The drift parameter
• Mean reversion
• Momentum effects
• Normality assumptions

Question 5

Explain the emperical reason why the following is used against modelling share prices using the lognormal model
* Volatility
* The drift parameter
* Mean reversion
* Momentum effects
* Normality assumptions

Answer 5

The volatility depends on when the information was taken, even within the same time interval, as well as how frequently the samples are taken.

Question 6

Explain the emperical reason why the following is used against modelling share prices using the lognormal model
* Volatility
* The drift parameter
* Mean reversion
* Momentum effects
* Normality assumptions

Answer 6

• The drift parameter is not constant with time.
• There is reason to assume that the risk premium of equities will be relative to the price of bonds
• For example, when bond expected returns are higher, a higher risk premium would be required to incentivise investors to invest in equities

Question 7

Explain the emperical reason why the following is used against modelling share prices using the lognormal model
* Volatility
* The drift parameter
* Mean reversion
* Momentum effects
* Normality assumptions

Answer 7

• A mean revertin market is one where highs are likely following lows and lows are likely following highs
• There is some evidence of this, but concentrated in a small number of crashes
• However, this would violate the lognormal model as it assumes that prices over non-overlapping time periods are independent

Question 7

Explain the emperical reason why the following is used against modelling share prices using the lognormal model
* Volatility
* The drift parameter
* Mean reversion
* Momentum effects
* Normality assumptions

Answer 7

• A rise in one day is likely to be followed by a rise the next day
• Which is inconsistent with the independence of returns in the lognormal model

Question 8

Explain the emperical reason why the following is used against modelling share prices using the lognormal model
* Volatility
* The drift parameter
* Mean reversion
* Momentum effects
* Normality assumptions

Answer 8

• Emperiical evidence suggests that extreme events occur more frequently than suggested by the normal distribution
• It also suggests the presence of discontinuities in share prices
• Thus the distribution of returns appears to have fatter tailes, higher peaks and some discontinuities, which is inconsistent with the normal model