EuropeanCall<-function (S0,X,r,tau,sigma,M){
  ##compute constants
  f7<-1 
  dt<-tau/M
  v<-exp(-r*dt)
  u<-exp(sigma*sqrt(dt)) 
  d<-1/u
  p<-(exp(r*dt)-d)/(u-d)
  ##initialise asset prices at maturity ( period M)
  S<-numeric(M+1)
  S[f7+0]<-S0*d^M
    for(j in 1:M){
      S [f7+j]<-S[f7+j-1]*u/d
    }
  ##initialise option values at maturity ( period M )
  C<-pmax(S-X,0)
  ##step back through the tree
  for (i in seq(M-1,0,by=-1)){
    C<-v*(p*C[(1+f7):(i+1+f7)]+(1-p)*C[(0+f7):(i+f7)])
    }
  return(C)
  }