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

Question

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

How does your dividend look like, proportional or discrete or with dividend yield? — Gordon, Jan 30 '20 at 13:52
@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
Can you please give the result ? I am skeptical, I never saw it. — , Jan 30 '20 at 15:08
Daneel Olivaw, KeSchn, skoestlmeier, byouness, Gordon.... My post is closed but I have not an answer. — , Jan 30 '20 at 15:28

score 0 · Answer 1 · answered Jan 30 '20 at 14:16

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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.

answered Jan 30 '20 at 14:16

Valometrics.com

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

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

1 Answers1