I have been looking at pricing a barrier option that has payoff of your usual European Call option, $\max(S_T - K, 0)$ if the stock price exceeds a horizon $A$ and then afterwards drop under some level $B$. We have the constraints $B < S_0 < A$ and $B < K$ and are also assuming zero interest rate. I am told it can be priced by the pricing formula for European call options alone. I just don't see how the indicator functions on the stopping times can be set up to reduce it to the regular European calls. Any nudge in the right direction would be highly appreciated.

  • $\begingroup$ From your question it is not immediately clear to me what exactly the payout is. Are you talking about a double knockin, or a double knockout? i.e. if and only if both barriers are hit then you receive the call (KI) or is it you have a call but if it hits both barriers then it knocks out? $\endgroup$ – mbison Nov 21 '18 at 19:50
  • $\begingroup$ Is it necessary that level $A$ is reached by $S_t$ before going below $B$? $\endgroup$ – Daneel Olivaw Nov 21 '18 at 21:57
  • $\begingroup$ If that's the case, I don't think you can construct a portfolio with simple options, the case where $S_t$ goes below $B$ first, then reaches $A$ only to go below $B$ again afterwards cannot be hedged with vanilla options, you need something like a knock-in+knock-out combined with a knock-in, or a double knock-in. $\endgroup$ – Daneel Olivaw Nov 21 '18 at 22:03
  • $\begingroup$ In "regular European calls" do you include European (single) knock-ins and knock-outs? $\endgroup$ – Daneel Olivaw Nov 21 '18 at 22:08
  • $\begingroup$ Perhaps I wasn't as clear in my explanation. We want the stock to cross the level A, and then afterwards fall below B, to get the payoff. To add to the context, right before this I proved for a question that a European up and in put option with strike K and barrier B is given by K/B(C(B^2/K)), where C is just the payoff of the (European) call option. That I did by some manipulation of the indicators and call put symmetry. I assumed this one would also be something similar. Maybe devolve the payoff into a bunch of barrier options and then use the above equivalence to get it in C. $\endgroup$ – Anonymous Nov 23 '18 at 10:24

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