# Bivariate risk neutral distribution through copula

I want to build a bivariate risk-neutral distribution from two liquid assets (A and B) through the use of a copula. As A and B are liquid, I have the marginal distributions from the market. All I have to do is to build a copula in order to relate both assets. I want to do this with a non-parametric copula estimation method.

For this purpose I am willing to use the method given in this paper (1), which requires the time series (from A and B) as input. But there is an issue: the real world time series from A and B are under the real-world measure, while I want to estimate the bivariate distribution under the risk-neutral measure, thus I cannot use the real world time series from A and B as input. Would you have any idea on how to tackle this problem.

(1): Estimating copula densities through wavelets. Genest, C., Masiello, E., Tribouley, K. Elsevier, pp. 170-181, 2009.

• The risk neutral measure is not fully specified just by the prices of the two assets. Either you need observations of asset/derivative prices sensitive to the joint distribution (such as basket or index options) or you need to make assumptions (such as choosing the entropy maximal measure).
– g g
Jan 22 '18 at 20:04
• @gg that is exactly my point: I have no information on "asset/derivative prices sensitive to the joint distribution". I want to price those, and that is why I am building a coupla. I want to use historical time series data in order to get some information on the relation between A and B and use this as input to estimate a copula (precisely, to link the market univariate risk neutral distributions of A and B). The issue is that I cannot use raw historical time series data, as this is under the real world measure and my copula wil be under risk-neutral measure. Jan 22 '18 at 20:12
• Well it depends on the assumptions you are willing to make. If you consider GBMs (which I know you don't do here but it's just for the sake of the argument) Girsanov tells you that the linear dependence structure tying the individual driving Brownian motions (Gaussian copula) will stay the same under Q. So in that case information under P can be used to infer the dependence structure under Q. Or as g g states it well, nothing prevents you from estimating the bivariate distribution under P and risk-neutralising it by minimising KL divergence subject to some constraints(e.g. match forwards) Jan 23 '18 at 7:52
• @Quantuple thank you for your answer. It is much clearer now. Could you please give a step by step on how to "risk-neutralising it by minimising KL divergence subject to some constraints(e.g. match forwards)". I did not understand this part. Maybe through an answer to my question... Jan 23 '18 at 10:53
• No problem Pierre, you can find additional details on the entropy method here: emanuelderman.com/media/strike_adjusted_spread.pdf. I must say that I never got astounding results with that approach. In general, I prefer the more practical approach of calibrating the dependence structure under Q to reproduce e.g. terminal correlations observed empirically. But this requires choosing a pricing model and you seem to be looking for a model-free result. Anyway, that's not an easy problem so good luck. Jan 23 '18 at 11:03