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Why do we use copulas for spread options but do not use them to correlate random variables across time, such as in the forward starting option?

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One is exploring forward volatility of a price of a single asset (joint distributions from within a process), the other explores correlation of two prices at the same time for two different underlyings (glueing, otherwise unrelated, marginal distributions).

Forward starting options depend on the joint distribution of the (already chosen and used to price other types of options, say Asians and continuous barriers) underlying process at two different times. The process provides consistency for pricing forward starting options for various pairs of times (and consitency with the rest of exotics depending on that process). There is no room for copula games. Once priced, one can obtain a Black-Scholes-implied forward volatility, giving a view of the forward volatility skew (one needs to explore stochastic volatility models to get sensible views, local volatility models are not sufficient).

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  • $\begingroup$ OK you're saying that while one can theoretically use a copula approach, the fact that you need to mark the parameter for different time combinations makes it more useful to take a specifying the dynamics approach than specifying the joint dbn approach, although both are in theory equivalent. Risk managing in the dynamics approach is much simpler as you just have 1 exotic parameter to manage. Is this correct? $\endgroup$ – Arshdeep Singh Duggal Jul 26 at 12:59
  • $\begingroup$ I agree with your thought on consistent risk monitoring and hedging. But I wouldn’t use the word ‘simpler’. If anything, as the underlying process needs to evolve to keep up with such exotics (say from standard local volatility to stochastic local volatility processes), the complexity increases and the mess could get bigger (conceptually and implementation-wise). $\endgroup$ – ir7 Jul 26 at 14:06
  • $\begingroup$ I can always assign a copula correlation between the 2 random variables so that the price is the same as that generated by a stochastic vol process. A 'distributional specification' and a 'dynamic specification' approach to the same problem are not straightforwardly comparable to me. They are 2 different sides of the same coin. So its difficult for me to understand that one is 'better' than the other, since I can come up with either that is consistent with the other. $\endgroup$ – Arshdeep Singh Duggal Jul 26 at 16:34
  • $\begingroup$ I agree. I'm not trying to downplay the usefulness of copula or its complementary nature. It may even be the only practical choice in some cases (in particular when the underlying is already complex, say interest rate curves, not just single price/rate, and adding stochastic volatility to an already complex, perhaps multifactor, dynamics can get pretty challenging - still desired, but not easy to implement and manage.) $\endgroup$ – ir7 Jul 26 at 17:00

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