Variance Swap Replication with Dividends and Interest Rates

Summary

The document addresses whether the vanilla option portfolio used to replicate a variance swap still applies when the underlying pays dividends and the risk-free rate is nonzero. Its answer says the replication formula remains valid, provided the forward price reflects the dividends. Calls and puts should then be priced with the Black–Scholes framework using that forward, across the relevant strikes.

The explanation offers a practical adjustment to the inputs rather than deriving the formula or presenting a proof. It gives no numerical example, assumptions beyond the Black–Scholes setting, or treatment of implementation details such as discrete dividends, strike coverage, or discretization of the replication integral. Readers should therefore treat it as a concise pointer to how rates and dividends enter the calculation, not as a complete derivation of variance swap replication.

Key ideas

• The variance swap replication formula is stated to remain valid when rates and dividends are nonzero.
• Account for dividends when calculating the underlying forward price.
• Use forward-based Black–Scholes call and put prices across strikes.

Tags

# Answer by Valometrics.com (score 0)


# 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 :

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

An explicit formula exist for nonzero risk-free rate and nonzero dividend ? If yes, what is the result ?

Thanks

## Answer by Valometrics.com (score 0)

https://quant.stackexchange.com/a/50919

The first formula remains valid for paying dividends stocks. You should at first compute the forward that takes into account dividends then calculate forward black scholes calls and puts prices for different strikes with that forward.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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