Variance Swap Replication with Nonzero Rates and Dividends

Summary

The document asks whether a variance swap can be replicated explicitly with vanilla options under the Black–Scholes model when both the risk-free rate and dividend yield are nonzero. It contrasts this request with a familiar replication formula derived for the zero-rate, zero-dividend case.

No derivation or answer is provided, so the document does not establish whether a corresponding explicit formula exists or how it would be obtained. Its value is as a focused question about how financing and dividend assumptions affect option-based variance replication. The request also distinguishes an explicit formula from a replication approach attributed to Carr and Madan, but supplies no details with which to assess either method.

Key ideas

• The document asks whether vanilla options can explicitly replicate a variance swap when rates and dividends are nonzero.
• It identifies a formula for the zero-rate, zero-dividend Black–Scholes case as a point of comparison.
• It does not provide a derivation, answer, or evidence about the nonzero-rate case.
• The question distinguishes an explicit replication formula from the Carr–Madan approach.

Tags

# 50921


# Explicit formula replication of variance swap using vanilla option under black and scholes model with nonzero risk-free rate and nonzero dividend












I didn't find the formula for the following portfolio (variance swap replication) with nonzero risk-free rate and nonzero dividend under black and scholes model :

(1)

I found formula and proof only with risk-free rate and dividend equal to zero under black and scholes :

(2)

An explicit formula exist (as (2)) for nonzero risk-free rate and nonzero dividend ? If yes, what is the result ? (Carr-Madan is not an explicit formula)

Thanks

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