Lookback Options

1. Lookback Options Presented by P. V. Krishnarao Enron Research Group Houston, Texas 9 October, 2000

2. Lookback Options Definition of standard lookbacks  Uses of lookbacks  Comparison of premium to European  Pricing and hedging  Other types of lookbacks Lookback Options

3. Standard Lookback Options  A standard lookback option grants the right to – Buy at the lowest price (call) or – Sell at the highest price (put) reached during the life of the option Lookback Options

4. Lookback Call Price (F) Payoff = F(T) - min(F) Time T Lookback Options

5. Lookback Put Price (F) Payoff = max (F) - F(T) Time T Lookback Options

6. Uses of Lookback Options Useful for those wishing to buy volatility Lookback call + Lookback put pays max(F) - min(F) Example: Oil put warrant offered by Bankers Trust called U-strike gave the buyers the right to lock in a strike price at the most favorable level reached over an initial period Useful building block for understanding other path-dependent options Lookback Options

7. Lookback vs. European Call Premium Strike = Price = $50 Volatility = 30% Risk-free rate = 7% Call Premium ($) Time to Expiration (months) Lookback Options

8. Lookback Pricing and Hedging Idea is seductive, but customers shy from the expensive premium Lookback premium is about two times the premium of an at-the-money European  Lookbacks are hedgeable (no discontinuous deltas) Pricing needs to account for the sampling frequency Lookback Options

9. Other Types of Lookbacks  Options on Extrema: - Put payoff = max {K-min(F), 0} - Call payoff = max {max(F) - K, 0} where K is the strike price, and min(F) and max(F) are the minimum and maximum prices reached during the option period  Perpetual Lookback, also called Russian option, is an American-style lookback that lives until it is exercised Lookback Options