I'm trying to understand why we would need the Heston, SABR and other volatility models. So far, my understanding is that, we can use the Black-Scholes to back-out market implied volatility. Those implied volatility data points are discrete and therefore need to be interpolated. Heston and SABR can give us a model to interpolate between those data-points while making sure the surface fits to those data points. If this is the case, I don't understand why can't we just use a simper piecewise linear interpolation?

you need a model for the dynamics of the spot and vol for a lot of derivatives

I'm sure you're correct but I don't think I understand. Can you please give a simple example?

lets say you want to price a simple knock out barrier. you will need to model the fact that as equities rise, the volatility falls and as they fall volatility rises. difficult to do with a 1 factor BS model.

1. one should clear understand what does BS pricing state and missing the fact might lead us to other subproblems. BS concept states that if we assign to call option the price in the form of BSE solution and sell short delta underlying stocks at t ( ie we get correspondent amount of money) then your balance at t+dt represent rate of return equal to risk free interest rate. If we replace short selling we replaced by selling delta shares of stocks then looking at derivation of the BSE we could not use the value S ( t + dt ) of the short stocks at t + dt and derivation of the BSE would be incorrect. Assume that market agrees with BS and buyers of the call always sell short underlying stocks otherwise we can not claim that market uses something similar to BS price buying or selling options. Also one should be sure that buyer of the option should borrow portfolio value at t but not the value of the option itself. 2. Volatility in original BS pricing is a constant though the derivation is correct when sigma is a random function on time t, ie for each t there exists in the theory a unique sigma ( t ). When one uses historical data over say 1 month period we implicitly replace sigma (t ) by one period average which can be close or say insufficiently close to sigma (t) over the one month period. If BS theoretical price does not correspond to market it looks natural to add more parameters which can help us better reconciliation practice and theory. Exotics options have more complex payoff and deviations between theory and practice probably higher then for plain vanilla options.3. Changes in value of an equity can be interpreted in different ways as the equity value is a sum of 2 components drift and fluctuations. Hence based on historical data one can explain risen value as changes of drift when sigma does not changes or when changes in value of stock are follow from changes of sigma when drift does not change. Note that changes can be defined as changes in drift and volatility simultaneously. Because the real drift does not involved in BS formula (ie in theory if mu changed from 2% to 0.5% BS does not change the practice suggests that changes in mu it is reasonable assign to sigma. Then we can apply Heston or other innovations of the classical BS pricing.