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

Asked Jan 30 '20 at 15:25

Active Jan 30 '20 at 15:25

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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

asked Jan 30 '20 at 15:25

$S_t$ and $S_c$ are values of the asset at different times. That formula cannot be that of a price, a price is given at a time $c$ for example, if $c<t$. Is it the final payoff of your variance swap? – Daneel Olivaw Jan 30 '20 at 22:12
1

See this. – Daneel Olivaw Jan 30 '20 at 22:29

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

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