My question might be very straight forward but I have seen both approaches being followed in practice so I am curious to see if there are arguments in favor or against each one. I am explaining my question via the following example.
Let's say we want to value a twin-win double knock-out option on a single stock X (see payoff in the picture). Essentially at maturity if the underlying has moved up or down more than 10% we gain a return but if the movement is above 25% we gain nothing.
This payoff can be easily decomposed into the sum of two options, an up-out call option with strike at 1.1/knock-out at 1.25 and a down-out put option with strike at 0.9/knock-out at 0.75.
Approach 1: I value the options separately so i might use a Black-Scholes for each and add the results. That would mean that probably I would use a different implied vola for the two separate calculations.
Approach 2: I simulate the underlying (via MC for example) and for each path I apply the payoff function below. For that I would need to make sure that I match the relevant parts of the vola surface so I would use maybe a local vol model.
Question: People from big banks told me that that they always decompose such cases and value each component completely separately and sometimes using different models. I who do not work in a big bank :p , always used the second approach. As per my understanding the second approach is more appropriate because in such a way you can calibrate the model to the relevant products quoted in the market while with the second you have a quite different representation of the underlying (probably not matching the market quotes simultaneously)
Am I missing something very basic here?