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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 :

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

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

Thanks

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  • $\begingroup$ Alternatively: Variance replication using options. $\endgroup$ – Daneel Olivaw Jan 29 '20 at 17:21
  • $\begingroup$ It is not an explicit formula $\endgroup$ – user44204 Jan 30 '20 at 11:13
  • $\begingroup$ How does your dividend look like, proportional or discrete or with dividend yield? $\endgroup$ – Gordon Jan 30 '20 at 13:52
  • $\begingroup$ Like a constant, deterministic and continuous $\endgroup$ – user44204 Jan 30 '20 at 14:05
  • $\begingroup$ @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? $\endgroup$ – Daneel Olivaw Jan 30 '20 at 14:30
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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.

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  • $\begingroup$ Why the first formula is valid with dividend ? $\endgroup$ – user44204 Jan 30 '20 at 14:34
  • $\begingroup$ It is my question how I can compute ? Thanks $\endgroup$ – user44204 Jan 30 '20 at 14:37