Chapter 11 Stochastic Models For Security Prices Flashcards
Describe the continuous time lognormal model.
The conventional continuous time lognormal model assumes that log prices form a random walk. If St denotes the market price of an investment, then the model states that, for u>t, log returns are given by: log(Su) - log(St) ~N(nu(u-t), sigma-squared (u-t)) where nu is the drift in the log price and sigma is the volatility or diffusion coefficient.
SDE for the log price:
d(logSt) =nu.dt +sigma.dWt
SDE for the price:
dSt=(nu+0.5(sigma-squared))Stdt + (sigma-squared)StdWt
Where (nu+0.5(sigma-squared)) is the average rate of drift of the price itself.
(Su/St)~logN((nu)(u-t), (sigma-squared)(u-t))
Notice the %return us not dependent on St.
Discuss the 5 properties of the lognormal model.
1) Mean and Variance of the log returns are proportional to the length of the interval considered. (u-t).
2) Mean, variance and standard deviation tend to infinity as the length of the interval increases since there is a dependence on the the length of the interval.
If nu=0 and sigma=0 this is not the case.
If sigma=0 there is no randomness in the movement of the security price process. Therefore the model is deterministic(no risk). Therefore return on the security should equal the risk free rate of interest.
If nu>0 represents upward drift of log prices due to growth in company profits(linked to other economic factors.
Mean and standard deviation changes in the log of the share price are constant.
3) Returns over non-overlapping intervals are independent of each other. Since normal variables generating the random variation in the log of the share price are assumed to be independent.
4) Value of the investment at time u can be written as: Su=St.exp(Xu-t) where Xu-t~N(nu(u-t), sigma-squared(u-t))
5)Su is ~logN
E[Su] =St.exp(nu(u-t) +0.5(sigma-squared)(u-t))
Var[Su] =St-squared.exp(2.nu(u-t) +0.5(sigma-squared)(u-t)).exp(sigma-squared(u-t) - 1)
Give 5 arguments for the assumptions underlying the continuous time lognormal model and for the model itself.
1) Model is mathematically more tractable than other more complex models.
If everyone uses the same model, prices are more comparable.
2) Share prices cannot be negative in this model, since exponential functions can’t take on negative values.
3) Variance of returns in a particular period is proportional to the length of that period, seems intuitively reasonable.
4) Lognormal model for security prices with independent non-overlapping periods of time is consistent with the assumption that markets are efficient (at least in their weak form).
5) Return and risk characteristics of the underlying share (given by nu and sigma) are expressed as proportions of the current share price rather than in absolute terms.
Give 6 arguments against the assumptions underlying the continuous time lognormal model as well as against the model itself.
1) In reality share price movements may not be consistent with such a process which generates continuous sample paths. Since prices may ‘jump’ ie: change by a significant amount suddenly.
2) In reality nu and sigma are not constant.
Nu should vary over time with varying interest rates. Since investors require a risk premium on equities relative to risk free bonds.
Sigma varies depending on the time period considered and how frequently samples are taken. Sigma is greater in times of recession and financial crisis.
3) Historical evidence suggests that large movements in prices are more common than the lognormal model would suggest. There are fatter tails(more extreme events) exists more volatility.
4) Historical evidence suggests that days of little or no movement in prices is more common than lognormal model would suggest. There are higher peaks.
5) The assumption of efficient markets may be invalid.
6) The evidence that the share price exhibits momentum effects(rise one day is more likely to be followed by another rise the next day) in the short term and mean reversion in the long term. This contradicts the property of independent increments.
Demonstrate a basic understanding of mean reversion.
A mean reverting market is one where rises are more likely following and market fall, and falls are more likely following a market rise.
Average return revert back toward their long run trend level.
Evidence of this rests heavily on the aftermath of a small number of dramatic crashes.
