"Given some reasonable assumptions about the random behaviour of stock returns, a lognormal distribution is implied."Ã¯Â¿Â½ Discuss the relevance of those assumptions and their implications.
A popular model which examines the evolution of stock prices in continuous time and one that has received wide coverage in the finance and statistics literature is the lognormal distribution. It largely results from the effects of a large number of independent but multiplicative sources of variation. It is upward skewed, with a mean larger than its mode [Black, 1997, p.277]. Although it is possible to establish upper and lower bounds for option prices using general arbitrage arguments, "precise"Ã¯Â¿Â½ option pricing requires some additional assumptions about the probability of possible price changes in the underlying asset. These assumptions characterise the lognormal distribution in a very intuitive manner.
ÃÂÃÂ· A1. Stock returns are independently distributed.
ÃÂÃÂ· A2. Stock returns are identically distributed.
ÃÂÃÂ· A3. The expected return of the continuously compounded returns is constant.
ÃÂÃÂ· A4. The variance of the continuously compounded returns is constant.
Assumptions A1 and A2 together imply a random walk, which is one form of the Markov Process. The hypothesis states that share prices move without any memory of price movements, and therefore follow no pattern i.e. only the stock's current price is useful in forecasting future prices. This links in with the martingale hypothesis that tomorrow's price is expected to equal today's price, irrespective of the asset's entire price history [Merton, 1996, p.30].
These ideas are consistent with the notion of a weak-form efficient market i.e. a market in which the information contained in past prices is instantly, fully and perpetually reflected in the assets current price. Weak-form efficiency implies that the market is extremely greedy for information, and will use all available information because someone or the other will try...