Static Stock and Option Hedges for a Perpetual Up-and-Out Call
Summary
The document discusses hedging a perpetual up-and-out call with a strike of 110 and barrier of 120. One proposed static hedge buys stock in an amount intended to cover the option’s maximum intrinsic payoff at the barrier. This is a buy-and-hold superhedge for a short option position, but the answer notes that it can cost substantially more than the option itself. Another suggestion uses layers of barrier options with different strikes and barriers to manage the chance that the position leaves value unprotected.
A separate response assumes no dividends and argues that the perpetual structure has constant delta, making a static stock hedge possible under those assumptions. The discussion is brief and offers no pricing derivation or market validation. Its conclusions depend on the payoff interpretation, exercise mechanics, and assumptions about dividends and price jumps; the layered option approach is also only sketched, without specifying hedge ratios or costs.
Key ideas
- A stock position sized against the option’s maximum payoff can provide a static hedge for a short call.
- The proposed stock hedge may be expensive relative to the option’s value.
- Layers of barrier options with different strikes and barriers are suggested as an alternative protection approach.
- The constant-delta argument for a static hedge assumes no dividends and a perpetual option.
- The responses provide limited pricing detail and do not establish hedge performance in market conditions.
Tags
Full text
# How to hedge a perpetual barrier option? # How to hedge a perpetual barrier option? 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? MY ## Answer by Ivan (score 6) https://quant.stackexchange.com/a/44520 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. ## Answer by vrume21 (score 1) https://quant.stackexchange.com/a/44488 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. ## Answer by Chris (score 1) https://quant.stackexchange.com/a/44523 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.
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