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The below formula is used to convert the implied vol into the local volatility, my question is, once I have converted it into the LV ( and have built the full surface), what models do I use to calculate the option price? Do I simulate the underlying price using Monte Carlo and just calculate the average PV of the payoff, using:

$$ dS_{t} = (r_{t}-d_{t})S_{t}\,dt + \sigma (S_{t},t)S_{t}\,dW_{t} $$

Using the LV in the BS formula would give me a different result (ie. ATM IV = 20%, transformed ATM LV = 17%, so I can't use the LV in the BS model)?

$\sigma^2 \left(T,y\right)=\frac{\frac{\partial w}{\partial T}}{1 -\frac{ y}{w} \frac{\partial w}{\partial y}+\frac{1}{2}\frac{\partial^2 w}{\partial y^2}+\frac{1}{4}\left(\frac{ y^2}{w^2}-\frac{1}{w}-\frac{1}{4}\right)\left( \frac{\partial w}{\partial y}\right)^2}$

Where y is the money-ness, defined as $y=\ln \left(\frac{ K}{F} \right)$, and w is the transformation of Black Scholes implied vol $w=\sigma_{BS}^2\,T$

I will add a few quotes from the book (https://bookdown.org/maxime_debellefroid/MyBook/all-about-volatility.html#review-of-volatility-models)

"Once the local volatilities are obtained, one can price exotic instruments with this calibrated local volatility model. Properly accounting for the market skew can have a massive impact on the price of exotics --> example: call up-and-out."

"In the Monte Carlo simulation approach, we simulate many paths and keep only the ones that finishes around the strike. We obtain a stream of trajectories that start at the initial spot and finish around the strike. We average on each date all these paths and obtain the most likely path. We can also extract the variance around this path. We obtain the implied volatility estimation from it (thanks to the most likely path and the width around it). This is well explained in Adil Reghai's book 'Quantitative Finance: Back to Basic principles'."

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Surfaces are used to compute prices of exotics. For vanilla pricing, there is no need to fit a surface.

LV is applicable for non-path-dependent exotics. LV essentially boils down to a static probability distribution on the price of the UL at expiration. It misses several real market features that severely impact path dependency -- namely stochastic volatility and jumps. So ideally you would want stochastic volatility and jumps in your surface model. I am unfamiliar with SLV, but it seems reasonable.

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    $\begingroup$ Why is LV not applicable for path dependent exotics, and what is used SLV perhaps? Is my assumption of how Is MC with LV used correct? $\endgroup$
    – Skittles
    Commented Jan 5 at 7:10
  • $\begingroup$ I edited to include more information. The MC approach seems fine; there are other references on SE about it as well. To me it makes most sense to simulate and adjust the volatility based on where the underlying ends up. Re: LV vs BS -- the BS numbers are just a reference. Once we've decided to use LV, we've declared that BS itself is insufficient. The best way to try to understand is see if you can construct an arbitrage. So if LV says the BS number is too high, sell the option and continuously delta hedge it. $\endgroup$
    – Yike Lu
    Commented Jan 5 at 18:53

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