I suppose the usual BS assumptions (and no repo, no dividend) and would like to understand the following hedging strategy, even if it is a naive and therorical one. I am long a call K=100 : when the stock goes to 100.01, I sell 1 stock and when it goes to 99.99, I buy it back. Of course, if I am short gamma, that will cost me one tick each side so this shoud provide a B/O for my option. I am quite aware that there are transaction costs and that there could be jumps on the strike and that in reality, there is a lag between the moment I see the price and I execute the order. But this is not really important here since I suppose I am in a BS world.If the interest rate is not zero, this strategy is not self-financing I think. If I am short gamma, each time the stock goes above 100, I have to buy it and borrow money. The thing is that I have to borrow money for the time the stock is above 100. So I could try to determine the time the stock is above 100 and come to the price of my option. So all the problem of this pricing and hedging strategy is to estimate the time the stock is above 100. I suppose that the variance of my PNL will be huge since it really depends of the path the stock takes (and of how much time it will stay above 100), no ? And the price at which I should buy the option is one tick * each time I cross the strike + risk-free rate over the time I am above the strike (all this using Monte-Carlo of course!) ? So basically, if I consider that I can estimate the time I am above the strike, what is wrong with this strategy ?An even more interesting case is the one where interest rate = 0 (Japan ?). Here there is no question of self-financing strategy since there is no interest rate. So I suppose there is another reason why this hedging strategy does not work. And my delta and price are clearly not the same as in BS.Finally, to reduce the variance of my PNL, I could perhaps pass a delta to account for the time I stay above the strike ?Thanks in advance !

how do you estimate the time that the stock price is above the strike?

I don't really know. I suppose that with some stochastic calculus, you may be able to do so but I have not done it yet. And, you can also do it numerically. But this is not that important when interest rates are zero which is the interesting case.

For a Brownian motion(less than 3d), if it crosses a spot once, it will cross it infinitely many times, that means your naive hedging will kill you in transaction cost.

QuoteOriginally posted by: alexandreChow do you estimate the time that the stock price is above the strike?Same way you would price a Parisian.QuoteOriginally posted by: MartingaleFor a Brownian motion(less than 3d), if it crosses a spot once, it will cross it infinitely many times, that means your naive hedging will kill you in transaction cost. But brownian motion means your moves, trades, and costs are all infinitesimal, and as an integral, should converge in expectation. In reality, both are finite and efficiently charged for. In fact, I have seen some hedges done this way as a bet deferring frequent delta hedging in exchange for having to do a lot of trading in round lots if the strike is hit. In individual cases I have seen, the strikes seemed to act as support and resistance levels, so the amount of trading you would have to do is less than you might think, but the error of gapping through is greater than the model.

QuoteBut brownian motion means your moves, trades, and costs are all infinitesimal, and as an integral, should converge in expectation. In reality, both are finite and efficiently charged for. In fact, I have seen some hedges done this way as a bet deferring frequent delta hedging in exchange for having to do a lot of trading in round lots if the strike is hit. In individual cases I have seen, the strikes seemed to act as support and resistance levels, so the amount of trading you would have to do is less than you might think, but the error of gapping through is greater than the modelTrue, I was talking about the theoretic aspect of this....

This precise issue was addressed in a 1990 article by Robert Jarrow and Peter Carr. There is a reference here. The one sentence answer is that the loss one experiences by implementing that strategy is related to the local time of geometric Brownian motion, and that loss is exactly equal to the Black-Scholes price of the option.

Thank you very much sehrlich !

See also p198 of "Dynamic hedging" by N. Taleb.

QuoteOriginally posted by: sehrlichThis precise issue was addressed in a 1990 article by Robert Jarrow and Peter Carr. There is a reference here. The one sentence answer is that the loss one experiences by implementing that strategy is related to the local time of geometric Brownian motion, and that loss is exactly equal to the Black-Scholes price of the option.Also, in one of the more recent editions of Hull, the question is adressed. He mentions some MC experiments that show that the cost does not converge to zero. Still, Carr's and Jarrow's paper give the full insight. You might need to consult Karatzas and Shreve for background.