In hedging barrier and Asian options, we do not need an extra asset in addition to the underlying stock. That is to say, although the payoff and value of the barrier (Asian) option does depend on the running extremum (average) process, such a process does not introduce an extra source of uncertainty (however they do introduce an extra dimension in numerical pricing).
- For barrier option, the running extremum can be handled using reflection principle.
- For Asian option, the running average can be handled by having an auxiliary process. Once again, no additional source of uncertainty is introduced by doing this.
See Shreve.
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