- EdisonCruise
**Posts:**117**Joined:**

Suppose the underlying now is 70, once itreaches 100, the option?s pay off is 1. This option never expires and assumesinterest rate is 0.I think one can hedge this option byholding 0.01 share of this underlying, once it reaches 100, then earn (100-70)*0.01=0.3.So the price of this option should be 1-0.3=0.7.On the other hand, the underlying hits 100almost surely, so it will pay off with probability 1. And it payoff is 1, so itseems its price should be 1 too. Then what?s wrong with this argument?

It never expires, does that mean I can never exercise???? Or do you mean I can exercise as soon as it hits the barrier 100? To me, this looks like a barrier option with zero interest rate and infinite time-to-maturity. Can anyone here confirm it?In the question, I think you have two different kinds of payoff. You claimed a payoff of 1 when the underlying hits the barrier. But then, you claimed a European payoff (100-strike) where the strike is 70 when it hits the barrier.

- EdisonCruise
**Posts:**117**Joined:**

I mean you can exercise as soon as it hits the barrier 100.If I receive option fee 0.7, and buy 0.01 shares of the underlying at 70, when it researches 100, I can make a profit of 0.3. This profit plus the option fee is equal to the option payoff 1. The payoff of this option is 1 once it hits 100. No hitting, no payoff and no expiration.

Last edited by EdisonCruise on June 17th, 2015, 10:00 pm, edited 1 time in total.

QuoteOriginally posted by: EdisonCruiseSuppose the underlying now is 70, once itreaches 100, the option?s pay off is 1. This option never expires and assumesinterest rate is 0.I think one can hedge this option byholding 0.01 share of this underlying, once it reaches 100, then earn (100-70)*0.01=0.3.So the price of this option should be 1-0.3=0.7.On the other hand, the underlying hits 100almost surely, so it will pay off with probability 1. And it payoff is 1, so itseems its price should be 1 too. Then what?s wrong with this argument?Hint: examine the self-consistency of your assumptions under GBM.

Is Black-Scholes equation still valid? If so, can you solve it?

The problem, as stated, is under specified. First, if upward jumps are allowed, pretty much all bets are off. Second, even with continuous sample paths, there are delicate issues with the infinite time horizon. This is also true in a Black-Scholes economy, depending in part on the magnitude of the (P measure) expected rate of return on the underlying. Finally, I can think of no circumstance where the (Q measure) hitting time probability equals one; it certainly won't under GBM. For whatever it is worth, the problem is more interesting with r>0.

Question: Can we price it with a barrier option with r==0 and time-to-maturity be infinite?

this question gets asked every few yearssee past discussionsIs this a simple question?Perpetual barrier option

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