A down-and-out call can be replicated — approximately — by a static portfolio of vanilla options, chosen so that the portfolio's value matches the barrier option's value at the barrier level , for all future times. This is the quasi-static hedge: static (no delta rebalancing needed between now and the barrier hitting time) but approximate (the hedge is exact only on the barrier, not everywhere).
The theory behind the quasi-static hedge is put-call symmetry: under log-normal dynamics, a European call struck at with underlying and a European put struck at (the "reflected" strike) with the same underlying have a precise relationship. This allows the payment made when the barrier is hit to be replicated by a vanilla option.
Learning objectives. By the end of this note you should be able to:
- state put-call symmetry under log-normal dynamics;
- construct a quasi-static hedge for a down-and-out call;
- explain when the quasi-static hedge fails and what model assumptions it requires.
In this lesson
- 3.1 Put-call symmetry
- 3.2 The quasi-static hedge for a down-and-out call
Before you start
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