Calibrating Local Volatility to American Option Prices

Summary

The document asks how to infer a local volatility function from a surface of American option prices. It considers a stock model with diffusion driven by a state- and time-dependent volatility, with the aim of reproducing the observed prices across strikes and maturities. The author notes that the implied volatility surface is derived from American option prices, but the standard Dupire formula relies on European option prices and their terminal-payoff structure.

The author reports that applying Dupire directly yields model prices materially different from the American price surface. Direct least-squares calibration to that surface is identified as a possible brute-force alternative, though it is described as time-consuming. Discrete dividends add another modeling requirement. The post asks for practical or industry methods and cites no solution, numerical experiment, or evidence that a particular calibration approach works. It therefore frames the modeling challenge rather than resolving it.

Key ideas

• The question concerns fitting local volatility to a surface of American option prices.
• The author notes that the usual Dupire approach is based on European option prices.
• Direct least-squares calibration is raised as an alternative but described as computationally slow.
• Discrete dividends are an additional feature the model would need to handle.

Tags

# How to price the american options using local volatility


# How to price the american options using local volatility












I have given with a surface of american option prices $C_{am}(T, K)$. From these american option prices the implied volatility surface is deduced.

Now I want to find the local volatility $\sigma(s,t)$ for my model, such that the price with respect to the price process $$dS_t=rS_tdt+\sigma(S_t,t)S_tdW_t$$ replicate all prices on the price surface.

There is the Dupire Formula for vanilla European option. Clearly, I can't use the Dupire formula to convert my implied volatility surface to the local volatility surface, as Dupire Formula bases heavily on the assumption of European Option(especially, the price should only depend on the stock price at final time). As it turns out, if we use the Dupire Formula, the price obtained by my local volatility model is significant different than the one on the price surface $C_{am}(T, K)$.

Is there another way to do that? I know there exists some brute force method using a least square optimization to calibrate the local volatility directly to the price surface $C_{am}(T, K)$. However, this method is very time consuming. here for the paper Also I have to adapt the model to discrete dividend. Can anybody provide any helpful materials or method used in the industry?

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.