Why Forward Starting Options Use Process Dynamics Instead of Copulas

Summary

The note distinguishes dependence modeling for spread options from pricing forward starting options. A copula can join the marginal distributions of two different assets observed at the same time, where their relationship is not already specified by a shared process. For a single asset observed at two dates, the joint distribution comes from the underlying process used to price other claims, such as Asian and barrier options.

That process supplies the time consistency needed across claims and forward start dates, so choosing a separate copula would not fit the stated framework. Once the option is priced, its value can be expressed as a Black Scholes implied forward volatility and used to examine the forward volatility skew. The answer cautions that local volatility alone does not provide sensible views of that skew, pointing instead to stochastic volatility models. The note gives a conceptual explanation rather than a numerical example or comparison of models.

Key ideas

• Copulas can link marginal distributions for different assets observed at the same time.
• For one asset across two dates, the chosen process determines the joint distribution.
• A consistent process supports pricing forward starting options alongside other path dependent claims.
• Implied forward volatility can be used to examine the forward volatility skew.
• Stochastic volatility models are suggested for more sensible views of the skew.

Tags

# Why do we not use copula for forward starting options?


# Why do we not use copula for forward starting options?












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?

## Answer by ir7 (score 5, accepted)

https://quant.stackexchange.com/a/55921

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.