make_spot_table = function(S0, u, d, n_times){
    ##
    ## make_spot_table(62, 1.05943, 0.9439, 5) # duplicates the spot grid in Figure 12.7 on page 332.
    ##
    n_steps = n_times+1 ## = number of columns = number of time units + one column for the initial spot value
    n_rows = n_steps

    Spot = matrix(NA, nrow=n_rows, ncol=n_steps)
    Spot[1, 1] = S0
    for( jj in 2:n_steps ){
        last_ii = jj ## the last row we will compute for in column jj
        for( ii in 1:last_ii ){
            if( ii == 1 ){
                Spot[ii, jj] = u * Spot[ii, jj-1]
            }else{
                Spot[ii, jj] = d * Spot[ii-1, jj-1]
            }
        }
    }

    ## Give this table descriptive column names:
    ##
    colnames(Spot) = paste0('t_', seq(0, n_times))

    return(Spot)
}



evaluate_call_option = function(SSpot, K, q, R, type='European'){
    ##
    ## Working backwards evaluate a call option.
    ##
    n_steps = dim(SSpot)[2] # = number of columns = number of time units + one column for the initial spot value
    n_rows = n_steps

    V = matrix(NA, nrow=n_rows, ncol=n_steps)
    for( jj in seq(n_rows, 1, -1) ){
        for( ii in 1:jj ){
            if( jj == n_steps ){ ## the final column/timestep
                V[ii, jj] = max( c(SSpot[ii, jj]-K, 0) )
            }else{
                if( type=='European' ){
                    V[ii, jj] = (q*V[ii, jj+1] + (1-q)*V[ii+1, jj+1])/R
                }
                if( type=='American' ){
                    pv_option = (q*V[ii, jj+1] + (1-q) *V[ii+1, jj+1])/R
                    early_exercise_value = max( c(SSpot[ii, jj]-K, 0) )
                    V[ii, jj] = max( c(pv_option, early_exercise_value) )
                }
            }
        }
    }
    return(V)
}



evaluate_put_option = function(SSpot, K, q, R, type='European'){
    ##
    ## Working backwards evaluate a put option.
    ##
    n_steps = dim(SSpot)[2] # = number of columns = number of time units + one column for the initial spot_value
    n_rows = n_steps

    V = matrix(NA, nrow=n_rows, ncol=n_steps)
    for( jj in seq(n_rows, 1, -1) ){
        for( ii in 1:jj ){
            if( jj == n_steps ){ ## the final column/timestep
                V[ii, jj] = max( c(K-SSpot[ii, jj], 0) )
            }else{
                if( type=='European' ){
                    V[ii, jj] = (q*V[ii, jj+1] + (1-q)*V[ii+1, jj+1])/R
                }
                if( type=='American' ){
                    pv_option = (q*V[ii, jj+1] + (1-q)*V[ii+1, jj+1])/R
                    early_exercise_value = max( c(K-SSpot[ii, jj], 0) )
                    V[ii, jj] = max( c(pv_option, early_exercise_value) )
                }
            }
        }
    }
    return(V)
}