Using Local Volatility to Price Vanilla and Exotic Options
Summary
The discussion asks how to use a local volatility surface derived from implied volatility data to price options. It proposes simulating the underlying under a risk-neutral process whose volatility depends on both spot and time, then discounting simulated payoffs. It also distinguishes this from inserting a local volatility value into the Black-Scholes formula: the local volatility surface is calibrated to represent the market’s strike and maturity structure, while vanilla prices can be read directly from the implied volatility surface without fitting a local volatility model.
The answer says local volatility is useful for pricing exotic options, particularly those sensitive to the distribution of the underlying at expiry. It cautions that local volatility does not capture important path-dependent market behavior such as stochastic volatility and jumps, so it may be inadequate for path-dependent products; stochastic or combined models may be more suitable. The discussion gives no numerical valuation or simulation results, and its comments are conceptual rather than a full implementation guide.
Key ideas
- A local volatility model uses volatility that varies with the underlying price and time in risk-neutral pricing.
- Monte Carlo simulation can estimate option values by averaging discounted payoffs across simulated paths.
- A calibrated implied volatility surface can price vanilla options without first fitting a local volatility model.
- Local volatility can support exotic pricing by representing the market’s strike and maturity skew.
- Local volatility alone omits stochastic volatility and jumps that can materially affect path-dependent options.
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# Option price calculation using Local Volatility and Monte Carlo
# Option price calculation using Local Volatility and Monte Carlo
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'."
## Answer by Yike Lu (score 2)
https://quant.stackexchange.com/a/77909
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.