I see things about if a position is losing, you buy something else such that deltas are even. Like if your delta is -400, you go over one strike and sell enough that the delta of your position is around 400.
What does that do?
In the Black-Scholes model with GBM, an option can be replicated exactly by delta-hedging the option. In fact the Black-Scholes PDE we derived earlier was obtained by a delta-hedging / replication argument.
I'm gonna try explain how it works. Think of classic arbitrage: you buy a stock with $10 on some exchange and at the same time sell it with $15 on another exchange. You make $5 risk-free. Conversely you may short a stock for $15 and buy it simultanously from elsewhere at a price of $10.
This is exactly how delta-hedging works.
We're talking two instances of the same product here:
1) The real option that you buy or sell on the options market.
2) A "virtual", exactly-the-same price and characteristics option that "replicates", "clones", however you wanna call it, "emulates" #1.
So two cases:
a) The bid price for the option on the real exchange is $15, hence you can sell it for $15 but the "theoretical fair value" (average price) you would get through delta-hedging for the same options is $10.
In this case you sell the real option for $15 and start constructing a virtual portfolio which by all practical purposes behaves as if you would have bought the very same option. You "delta-hedge", meaning you sell and buy stock in various quantity throughout the life of the option and magically, at the end the virtual portfolio will have cost you on average only $10. So you capitalized on an arbitrage: sold the real option for $15 and "bought it" for $10, leaving you with $5 - ON AVERAGE!
b) The ask price for the option on the real exchange is $10, hence you can buy it for $10 but the "theoretical fair value" (average price) you would get through delta-hedging for the same options is $15.
In this case you buy the real option for $10 and start constructing a virtual portfolio which by all practical purposes behaves as if you would have sold the very same option. You "delta-hedge", meaning you buy and sell stock in various quantity throughout the life of the option and magically, at the end the virtual portfolio will have made you an average $15. So you capitalized on an arbitrage: bought the real option for $10 and "sold it" for $15, leaving you with $5 - ON AVERAGE!