bs_vega<-function(spot, strike, v, r, rf, t){
r_c=log(1+r*t)/t
rf_c=log(1+rf*t)/t
d1=(log(spot/strike)+(r_c-rf_c+v^2/2)*t)/(v*sqrt(t))
vega=spot*sqrt(t)*exp(-rf_c-d1^2/2)/sqrt(2*pi)
return(vega)
}
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