Review: If is normal with mean and standard deviation , then
is the Standard Normal Distribution with mean 0 and standard deviation 1. To find the probability , you would convert to the standard normal distribution and look up the values in the standard normal table.
If is a weighted sum of normal random variables , with means , variance , and weights , then
where is the covariance between and . Note when , .
Remember: A sum of random variables is not the same as a mixture distribution! The expected value is the same, but the variance is not. A sum of normal random variables is also normal. So is normal with the above mean and variance.
Actuary Speak: This is called a stable distribution. The sum of random variables from the same distribution family produces a random variable that is also from the same distribution family.
The fun stuff:
If is normal, then is lognormal. If has mean and standard deviation , then
Recall where is the future value of an investment growing at a continuously compounded rate of for one period. If the rate of growth is a normal distributed random variable, then the future value is lognormal. The Black-Scholes model for option prices assumes stocks appreciate at a continuously compounded rate that is normally distributed.
where is the stock price at time , is the current price, and is the random variable for the rate of return from time 0 to t. Now consider the situation where is the sum of iid normal random variables each having mean and variance . Then
If represents 1 year, this says that the expected return in 10 years is 10 times the one year return and the standard deviation is times the annual standard deviation. This allows us to formulate a function for the mean and standard deviation with respect to time. Suppose we write
where is the growth factor and is the continuous rate of dividend payout. Since all normal random variables are transformations of the standard normal, we can write . The model for the stock price becomes
In this model, the expected value of the stock price at time is
Actuary Speak: The standard deviation of the return rate is called the volatility of the stock. This term comes from expressing the rate of return as an Ito process. is called the drift term and is called the volatility term.
Confidence intervals: To find the range of stock prices that corresponds to a particular confidence interval, we need only look at the confidence interval on the standard normal distribution then translate that interval into stock prices using the equation for .
Example: For example represents the 95% confidence interval in the standard normal . Suppose , , , , and . Then the 95% confidence interval for is
Which corresponds to the price interval of
Probabilities: Probability calculations on stock prices require a bit more mental gymnastics.
Conditional Expected Value: Define
This gives the expected stock price at time given that it is less than or greater than respectively.
Black-Scholes formula: A call option on stock has value at time . The option pays out if . So the value of this option at time 0 is the probability that it pays out at time , discounted by the risk free interest rate , and multiplied by the expected value of given that . In other words,
Black-Scholes makes the additional assumption that all investors are risk neutral. This means assets do not pay a risk premium for being more risky. Long story short, so . So in the Black-Scholes formula:
Continuing our derivation of but replacing with ,
For a put option with payout for and 0 otherwise,
These are the famous Black-Scholes formulas for option pricing. When derived on the back of a cocktail napkin, they are indispensable for impressing the ladies at your local bar. :p