# Replication of variance swap using vanilla option under black and scholes model with nonzero risk-free rate and nonzero dividend [duplicate]

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

• Alternatively: Variance replication using options. – Daneel Olivaw Jan 29 '20 at 17:21
• It is not an explicit formula – user44204 Jan 30 '20 at 11:13
• How does your dividend look like, proportional or discrete or with dividend yield? – Gordon Jan 30 '20 at 13:52
• Like a constant, deterministic and continuous – user44204 Jan 30 '20 at 14:05
• @ThomasArpe you mean something like: $S_t=S_c\exp\{(r-q-\sigma^2/2)(t-c)+\sigma W_t\}$, where $q$ is the continuous dividend yield? – Daneel Olivaw Jan 30 '20 at 14:30