Is Dupire's local volatility model path independent to recover historical option price?

Question

Generally when we implement Dupire's local volatility model, we follow the steps below:

1. Calculate implied volatility from given historical data
2. Fit the implied volatility skew. So we also know the corresponding call option prices.
3. Use the Dupire's Formula to calculate the local volatility $$\sigma_{loc}^2(K,T) = \frac{\partial_TC(K,T) + rK\partial_KC(K,T)}{\frac{1}{2}K^2\partial_{KK}C(K,T)}$$.
4. Use the SDE $$dS_t = r_tS_tdt + \sigma_{loc}(S_t, t) S_t dW_t$$ to do Monte Carlo simulation to get any call option price we want.

I have a few questions:

1. What's the differences between the call option prices from step 2 and step 4?
2. Using historical data, we can generate discrete points in the implied volatility surface. If we fix those points and regardless of the shape of the volatility skew, could we perfectly recover the historical call option prices using Dupire's Formula regardless of simulation path?

Is there a formal proof or a detailed explanation of the above two?

Answer 1

A few thoughts on the steps please:

1):You would normally calibrate Dupire based on current option prices; so you won't calculate implied volatility from historical prices of the underlying (assuming this is meant), but the current option prices. The prices are usually quoted in terms of implied volatilities, so in most cases you wont need to calculate the implied volatility.

2): This is one of the many possible ways to approach the calibration problem, and is needed to generate more granular input data - market price quotes (which could be in the form of IV) can be very sparse, and we need more granular data along both the strike and the maturity dimension.

3): You will need some finite approximation formulae to estimate the derivatives that appear in this formula, but I assume this is implicit in the statement.

Now regrading the two questions:

1): If the model is correctly implemented, the price produced by the MC should match the input prices. The only conceptual difference between the two prices could be down to the fact that the market prices are quoted in terms of Black Scholes' IV, and the Dupire in terms of market prices directly, but that is only a presentation matter.

2): The short answer is no, as Dupire's model uses and tries to explain the current option prices (not the historical or future).

Hope this helps!