1. The Pricing of Stock Options Using Black-Scholes Chapter 12

2. Assumptions Underlying Black-Scholes
We assume that stock prices follow a random walk
Over a small time period dt,the change in the stock price is dS. The return over time dt is dS/S
This return is assumed to be normally distributed with mean dt and standard deviation

3. The Lognormal Property
These assumptions imply ln ST is normally distributed with mean:
and standard deviation:
Since the logarithm of ST is normal, ST is lognormally distributed  T

4. The Lognormal Property continued
where m,s] is a normal distribution with mean m and standard deviation s
If T=1 then ln(ST/S) is the continuously compounded annual stock return.

5. The Lognormal Distribution

6. The Expected Return Two possible definitions:
 is the arithmetic average of the returns realized in may short intervals of time
 – 2/2 is the expected continuously compounded return realized over a longer period of time
 is an arithmetic average
  – 2/2 is a geometric average
Notice the geometric (compound) return is less than the average with the difference positively related to .

7. Expected Return
Suppose  = 10% and  =0. Then annual compound return = 10%
Suppose  = 10% and  =5. Then annual compound return   – 2/2 = .1 – (.05)(.05)/ = 9.875%
Suppose  = 10% and  =20%. Then annual compound return   – 2/2 = .1 – (.2)(.2)/ = 8.0%

8. The Volatility  T The standard deviation of the return in time T is
The volatility is the standard deviation of the continuously compounded rate of return in 1 year
The standard deviation of the return in time T is
If a stock price is $50 and its volatility is 25% per year what is the standard deviation of the price change in one day?  T

9. Estimating Volatility from Historical Data
1. Take observations S 0, S 1, , Sn at intervals of  years
2. Define the continuously compounded return as:
3. Calculate the standard deviation of the ui ´s (=s)
4. The volatility estimate is

10. The Concepts Underlying Black-Scholes
The option price & the stock price depend on the same underlying source of uncertainty
We can form a portfolio consisting of the stock & the option which eliminates this source of uncertainty
The portfolio is instantaneously riskless & must instantaneously earn the risk-free rate

11. Assumptions
Stock price follows lognormal model with constant parameters
No transactions costs
No dividends
Trading is continuous
Investors can borrow or lend at a constant risk-free rate

12. The Black-Scholes Formulas

13. Properties of Black-Scholes Formula
As S becomes very large c tends to S-Xe-rT and p tends to zero
As S becomes very small c tends to zero and p tends to Xe-rT-S

14. The N(x) Function
N(x) is the probability that a normally distributed variable with a mean of zero and a standard deviation of 1 is less than x
See tables at the end of the book

15. Applying Risk-Neutral Valuation
1. Assume that the expected return from the stock price is the risk-free rate
2. Calculate the expected payoff from the option
3. Discount at the risk-free rate

16. Implied Volatility
The volatility implied by a European option price is the volatility which, when substituted in the Black-Scholes, gives the option price
In practice it must be found by a “trial and error” iterative procedure

17. Implied Volatility
The implied volatility of an option is the volatility for which the Black-Scholes price equals the market price
The is a one-to-one correspondence between prices and implied volatilities
Traders and brokers often quote implied volatilities rather than dollar prices

18. Causes of Volatility
To a large extent, volatility appears to be caused by trading rather than by the arrival of new information to the market place
For this reason days when the exchnge are closed are usually ignored when volatility is estimated and when it is used to calculate option prices