VIII. ARBITRAGE AND HEDGING WITH OPTIONS

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.

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

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.

Valuing the One-Period Option

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

Two Time Periods The hedge ratio for time zero is -.75 and the hedge ratio in time one is either -.1875 or -1, depending on whether the share price increases or decreases in the first period

Extending the Binomial Model to n Time Periods

Illustration: Three Time Periods

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.

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

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

Greeks Calculation

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