As I was studying volatility derivatives I made some charts that represent some key features of replication. Say variance swap has a payoff function \(f=(\sigma - K_{VOL}) \), which means that \(K_{VOL}\) will most likely be the forward volatility close to implied. To replicate this theory goes deep into maths and log-contracts that are not even traded on the market, however the idea is simple, buy a portfolio of options with equally distributed strike prices and weight them by reciprocal of squared strikes, i.e.\( 1 \big/ K^2 \). Go for liquid ones, i.e. out of the money puts and out of the money calls. Then volatility or in this case variance is dependent on *VEGA* sensitivity of portfolio. The following graph gives an idea of how it is done. The code is included below:

X-Y - spot/time to maturity, Z - Vega \( \left(\frac{\partial C}{\partial \sigma}\right) \).

vega <- function(S0=100, K=100, r=0.05, d=0.05, tau=1, vol=0.1){ d_1 <- (log(S0/K)+((r-d)+vol^2/2)*tau)/(vol*sqrt(tau)); S0*sqrt(tau)*(1/sqrt(2*pi)*exp(-d_1^2/2))*exp(-d*tau)} vegaK <- function(K, t){ sapply(t, function(x){vega(S0=50:200, K=K, tau=x)/K^2}) } my3d_plot <- function(mat, x, y, col, bgcol="slatergray", material="lines", add=FALSE, ...){ require(rgl); if( missing(x) ){ x <- 1:nrow(mat) }; if( missing(y) ){ y <- 1:ncol(mat) } if( missing(col) ){ col <- rev(rep(heat.colors(ncol(mat),alpha=0),each=nrow(mat))) } if( !add ){ open3d(); material3d(color=bgcol) } persp3d(x, y, mat, col = col, alpha=1, front=material, back=material, add=add, ...)} mat <- lapply(seq(60,180,15), vegaK, t=seq(1,0,length=20)) mat$sum <- Reduce("+", mat) my3d_plot(mat$sum, bgcol="white", col="grey", aspect=c(4, 3, 2)) sapply(mat[1:9], my3d_plot, col="green", add=TRUE)