# Gaussian copula calibration to option price

I have an "exotic" option that is a function of two interest rates (say 3m Libor at 1y maturity and 2y maturity). I assume both the rates follow sabr model (already calibrated to vanillas), so I have both the marginals fully defined. Price of this option is observable in the market. I assume Gaussian copula to model the dependence of two rates. So only parameter left to estimate is the correlation.

How do I calibrate this copula to market prices i.e. how do I estimate the correlation? I vaguely know it will involve iteratively solving the double integral of gaussian copula to match market price but I don't know how to get the limits of the integral and also how to go from copula to option price.

Please feel free to assume a reasonable payoff of option in case it helps explain. Thanks for any pointers for this copula beginner.

• You could guess a starting value and then run an minimization routine on the squared pricing errors (market - model)**2 . Jun 8 at 22:56
• @Kermittfrog Exactly. In that process I want to understand how to go from correlation to model price. Jun 9 at 12:43

You did not mention it, but I think you also need to include the discount factor $$D$$ at the time $$T$$ of maturity of your option as a third variable. Denote the two interest rates as $$r$$ and $$s$$ and the pay-out function of your option as $$f=f(r,s).$$

The price of your option is then the expectation of the discounted cash flow: $$\text{price }=\mathbb{E}[f(r,s)D]$$ under your pricing measure. Denote the density function of your pricing measure by $$\phi$$. Since it is the joint density of the three risk factors $$\phi=\phi(r,s,D)$$. I assume that the measure, hence also the density, is supported on $$\mathbb{R}^3.$$

Then $$\mathbb{E}[f(r,s)D] = \int_{-\infty}^\infty\int_{-\infty}^\infty\int_{-\infty}^\infty f(r,s)D\, \phi(r,s,D) \,d r\,d s\,d D.$$

Up to now this is almost completely generic. To introduce your copula assumption you need to write the joint density in terms of the marginal densities and the copula density. It is true in general that a density can be written as: $$\phi(r,s,D)=m_1(r)m_2(s)m_3(D)*c(r,s,D)$$ where $$m_i$$ are the marginal and $$c$$ is the copula density (see for example Proposition 4.2.14 in Actuarial Theory for Dependent Risks). Pulling it all together your price is the triple integral $$\text{price }= \int_{-\infty}^\infty\int_{-\infty}^\infty\int_{-\infty}^\infty f(r,s)D\, m_1(r)m_2(s)m_3(D)*c(r,s,D) \,d r\,d s\,d D.$$

Note that this Gaussian copula $$c$$ requires three correlation parameters, not just one!

• Thank you, very helpful of you! Jun 21 at 13:09

If I remember e.g. McNeil et. al (2005: Proposition 5.29) correctly and if I understand you correctly, it should be possible to estimate the correlation parameter $$\varrho$$ of the Gaussian copula from two time series of the desired interest rates via calculating Kendall's $$\tau$$ or Spearman's rank correlation $$\rho$$ using the relationships:

$$\varrho = \sin(\frac{\pi}{2}\tau)$$

or

$$\varrho = 2\sin(\frac{\pi}{6}\rho)$$

• Thanks but time series is backward looking, I want to calibrate to what the market is implying for future i.e. extract the correlation from market prices. Jun 9 at 12:41