Covered call writers often plan to buy back the written call if the stock price drops sufficiently. The logic is that the written call at that point has little €œupside,€ and, if the stock recovers, the position could sustain a loss from the written call.
a. Explain in general how this buy-back strategy could be implemented using barrier options.
b. Suppose S = $50, σ = 0.3, r = 0.08, t = 1, and δ = 0. The premium of a written call with a $50 strike is $7.856. We intend to buy the option back if the stock hits $45. What is the net premium of this strategy?
A European lookback call at maturity pays STˆ’ ST . A European lookback put at maturity pays STˆ’ ST . (Recall that ST and ST are the maximum and minimum prices over the life of the option.) Here is a formula that can be used to value both options:

where

The value of a lookback call is obtained by setting ËœSt = ST andω = 1. The value of a lookback put is obtained by setting ËœSt = ST and ω=ˆ’1.

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