I have encountered the following question during my interview: How to have a static hedging of a perpetual barrier up-and-out call option in practice? Strike K = 110, barrier B = 120 for example?
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Presumably the option can be exercised for intrinsic at any point. Note the interviewer asked for a static hedge using the stock, not a dynamic hedge. Hence you must find a buy and hold portfolio that will always give you at least the value of the option (if you’re short it which I suppose is the question) until it is exercised.
Note that the maximum option payoff is 10, and is attained at $S=120$. If you buy $10/120 = 0.0833...$ worth of stock for each option sold, then at any point in time, your portfolio will be worth at least the payoff. That is your static hedge. It’s very expensive vs the “true” price of the option, but that’s what your static hedge is.
There are lots of ways to do this. One simple hedge would be to just buy down-and-out options. This is a kind of volatility trading where you make money when the asset price moves up or down and the moves are large enough to compensate for the prices of the options. Up-and-out options can leave a lot of money on the table though and you might want to protect yourself from this possibility. One way to do this is to buy up-and-outs with higher strikes and higher barriers. In this case, you can construct layers of up-and-outs with different risk levels. This is nice because the riskier up-and-outs are cheaper and provide protection against leaving too much money on the table.
This assumes no dividends. By virtue of being perpetual, theta is zero hence vega is and gamma as well. So delta is constant and the hedge has to be static. The premium fully finances the hedge so in fact there is zero rho and no rates dependency. In practice things can be cheaper only because the stock can in principle gap upwards hence the derivs knocks out and leaves you with a gain on the hedge.