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When Local Volatility and Gaussian Copula Prices Match

Article Quant Q&A · Author: NullSpace
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Summary

The document compares pricing an at-the-money European call on a two-stock basket with a multivariate local volatility model using constant correlation and a Gaussian copula using the same correlation. It reports a result for models calibrated to each asset’s smile at a single maturity: a suitable local volatility construction can reproduce the Gaussian copula price for any European payoff when the copula uses those calibrated marginals and the same correlation matrix.

The matching local volatility is described as one derived from a Markov-functional model built on the maturity-specific smiles. The equivalence is limited to that maturity and calibration setup. Other local volatility models can fit the same maturity smile while generating different shorter-maturity smiles, so using the same correlation alone does not guarantee equal prices. The document points readers to a book section for a fuller explanation but gives no numerical comparison or procedure for measuring price differences.

Key ideas

  • A local volatility model and Gaussian copula can produce equal European payoff prices under a specific single-maturity calibration.
  • The local volatility construction that matches the copula price is associated with a Markov-functional model.
  • A single maturity smile does not uniquely determine local volatility behavior at shorter maturities.
  • Using the same correlation coefficient alone does not establish price equivalence.

Tags

Multi-assetDerivatives pricingVolatilityOptions
Full text
# Difference between Local Vol and Copula


# Difference between Local Vol and Copula












Let's assume we have ATM European call on a basket of two stocks and price it with:

1) Multivariate Local Vol with constant correlation

2) Gaussian copula

Assuming we use the same correlation coefficient, will we always get the same (or almost the same) result? How we can quantify the difference between both approaches?

## Answer by TheHare (score 1)

https://quant.stackexchange.com/a/36878

He says the following:

Let's use a multi-asset local volatility model calibrated for each stock on its market smile of maturity $T$ (a one-maturity smile), and with the Brownian motions correlated through a correlation matrix $\rho$

Then there exists a local volatility for each asset such that: (1) the smile of maturity $T$ for each asset is recovered, (2) the resulting multi-asset local volatility price is equal to the Gaussian copula price with a correlation martrix equal to $\rho$ and the marginals calibrated on the $T$-maturity smiles. This is true for any European payoff.

Given a $T$-maturity smile, there exist many different local volatilities calibrated to this single maturity smile. They generate different smiles for shorter maturities. The local volatility that recovers the copula price is the one generated by a Markov-functional model built on the $T$-maturity smile.

He explains it better than me in section 2.10 of his book. Chapter 2 of his book is posted for free on his website: www.lorenzobergomi.com.

## Answer by TheHare (score 0)

https://quant.stackexchange.com/a/36829

The answer to your question is in Lorenzo Bergomi's book "Stochastic Volatility Modeling", section 2.10.

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.