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Carr Madan formula for Laplace distribution
# Carr Madan formula i<-complex(real=0,imag=1) char<-function(v,r,b) { mu<-r+log(1-b^2); return(exp(i*mu*v)/(1+b^2*v^2)) } # characteristic function of the Laplace distribution # char<-function(v,r,b) { mu<-r-b^2/2; return(exp(i*mu*v-b^2*v^2/2)) } # characteristic function of the normal distribution r<-0 b<-0.2 N<-4096; d<-0.25 ; a<-1.5 f<-2*pi/(N*d) c<-N*f/2 v<-d*(0:(N-1)) k<--c+f*(0:(N-1)) weighting<-(3+(-1)^(1:N)-c(1,rep(0,N-1)))/3 rho<-exp(-r)*char(v-(a+1)*i,r,b)/(a^2+a+i*(2*a+1)*v-v^2) integr<-rho*exp(i*c*v)*d*weighting call<-Re(exp(-a*k)*fft(integr)/pi) # comparison with the exact formula callexact <- function(r,S0,vol,K) { b<-vol/sqrt(2) d<-(log(S0/K)+r+log(1-b^2))/b if (d>0) { return(S0-K*exp(-r)+(b/(2*(1+b)))*K*exp(-r-d)) } else { return((b/(2*(1-b)))*K*exp(-r+d)) } } range<-exp(k[1600:2400]) y<-c() for (l in range) { y<-c(y,callexact(r,1,b*sqrt(2),l))} z<-rbind(call[1600:2400],y) par(mfrow=c(1,2)) matplot(range,t(z),type='l') plot(range,log(abs(z[1,]-z[2,]))/log(10),type='l')
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