# 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?

• Can you write down how you generate scenarios in both of these techniques? – will Jun 5 '17 at 20:22
• I guess that: (1) One local volatility surface per underlying (inferred from the respective vanilla market - e.g. Dupire's formula), individual driving Brownian motions are then tied using a constant instantaneous correlation coefficient $\rho$. (2) The risk-neutral distributions at $T$ for each underlying (inferred from vanilla market - e.g. Breeden Litzenberger idendity), these marginals are then tied using a Gaussian copula with the same correlation coefficient $\rho$. The second method could be seen as a "large time step" approximation of the first (/!\ decorrelation effect). – Quantuple Jun 6 '17 at 7:39
• @Quantuple I exactly meant that. Could you elaborate on a "large time step" approximation though? – NullSpace Jun 6 '17 at 16:21
• I simply meant that you can run the copula method using 1 time step of you like. Whereas for the first you need to simulate a LV dynamics hence the usual time discretisation. Also, if the LV is flat then it is easy to show that both methods coincide (no decorrelation i.e. terminal correlation = instantaneous correlation) – Quantuple Jun 7 '17 at 7:10
• No. The correlation of the gaussian copula should be the terminal correlation. The correlation in the local vol model is instantaneous. The concept is similar to local and implied volatility – FKaria Jun 27 '17 at 12:13

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.

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

• This answer would be much more helpful if you could give a brief summary instead of just referring to the book. – LocalVolatility Nov 10 '17 at 22:39