1. VIII. ARBITRAGE AND HEDGING WITH OPTIONS

2. A. Derivative Securities Markets and Hedging
As we discussed earlier, a derivative security is simply a financial instrument whose value is derived from that of another security, financial index or rate.

3. cT - pT = MAX[0, ST - X] - MAX[0, X – ST] = ST - X
B. Put-Call Parity
pT = MAX[0, X – ST]
cT - pT = MAX[0, ST - X] - MAX[0, X – ST] = ST - X
pT = cT + X – ST

4. C. Options and Hedging in a Binomial Environment
The Binomial Option Pricing Model is based on the assumption that the underlying stock follows a binomial return generating process.
This means that for any period during the life of the option, the stock's value will be only one of two potential constant values.

5. Valuing the One-Period Option

6. Extending the Binomial Model to Two Periods
First, we substitute for the hedge ratio:
Some algebra then substitute hedging probabilities:

7. Two Time Periods
The hedge ratio for time zero is -.75 and the hedge ratio in time one is either or -1, depending on whether the share price increases or decreases in the first period

8. Extending the Binomial Model to n Time Periods

9. Illustration: Three Time Periods

10. Obtaining Multiplicative Upward and Downward Movement Values
One difficulty in applying the binomial model is obtaining estimates for u and d that are required for p; all other inputs are normally quite easily obtained.

11. D. The Greeks and Hedging in a Black-Scholes Environment

12. Black Scholes Illustration
T = .5 rf = .10 X = 80 2 = .16  = .4 S0 = 75

13. Greeks Calculation

14. Delta and Gamma Neutrality
Same example as above, but add a call with X = 75