There are many ways to derive the Black-Scholes option price. One of them is to show that that in the Black-Scholes world, the BS price is the arbitrage-free price (see e.g. here). The price being arbitrage-free depends on the ability to constantly be updating a hedge and that ability depends on prices being continuous.
If you change the Black-Scholes world by dropping the requirement for continuous asset prices, the whole construction falls apart. Essentially, the Black-Scholes formula is a solution to a particular stochastic differential equation and if the underlying process is not continuous, the math breaks down.
The real world, however, is not the Black-Scholes world and there ain’t no such thing as a “Black-Scholes based strategy which was making you steady money”.
Well, that’s not quite what I mean.
There are many ways to derive the Black-Scholes option price. One of them is to show that that in the Black-Scholes world, the BS price is the arbitrage-free price (see e.g. here). The price being arbitrage-free depends on the ability to constantly be updating a hedge and that ability depends on prices being continuous.
If you change the Black-Scholes world by dropping the requirement for continuous asset prices, the whole construction falls apart. Essentially, the Black-Scholes formula is a solution to a particular stochastic differential equation and if the underlying process is not continuous, the math breaks down.
The real world, however, is not the Black-Scholes world and there ain’t no such thing as a “Black-Scholes based strategy which was making you steady money”.