nu = 0.12
sigma = 0.2
delta_t = 3/12

p = 1/2 + (1/2) / sqrt( sigma^2 / (nu^2 * delta_t) + 1 )
ln_u = sqrt( sigma^2 * delta_t + ( nu * delta_t )^2 )
u = exp(ln_u)
ln_d = - sqrt( sigma^2 * delta_t + ( nu * delta_t )^2 )
d = exp(ln_d)

print(sprintf('p= %f; u= %f; d= %f', p, u, d))


## Build the grid that represents the possible outcomes at each point in time to plot:
##
reductions = c(d, u) ## how much does our price change

n_steps = 4

seg_x0 = c()
seg_y0 = c()
seg_x1 = c()
seg_y1 = c()

current_nodes = c(100) ## the initial price

for( yi in 1:n_steps ){
    new_current_nodes = c()

    for( pti in 1:length(current_nodes) ){
        xx = yi-1
        yy = current_nodes[pti]

        ## Create a segment from (xx, yy) -> (xx+1, yy*reduction):
        ##
        for( rai in 1:length(reductions) ){
            xxp = xx+1
            yyp = yy*reductions[rai]

            new_current_nodes = c(new_current_nodes, yyp)

            seg_x0 = c(seg_x0, xx)
            seg_y0 = c(seg_y0, yy)
            seg_x1 = c(seg_x1, xxp)
            seg_y1 = c(seg_y1, yyp)
        }
    }

    current_nodes = new_current_nodes
}
##print(cbind(seg_x0, seg_y0, seg_x1, seg_y1))

##postscript('../../WriteUp/Graphics/Chapter11/chap_11_ex_1_lattice.eps', onefile=FALSE, horizontal=FALSE)
plot(NULL, xlim=c(0, max(c(seg_x0, seg_x1))+0.5), ylim=c(min(c(seg_y0, seg_y1)), max(c(seg_y0, seg_y1))), xlab='timestep', ylab='price', xaxt='n')
segments(seg_x0, seg_y0, seg_x1, seg_y1)
axis(1, at=0:4, labels=c('0','1','2','3','4'))
final_state = sort(unique(seg_y1[seg_x1==4]), decreasing=T)
points(rep(4, length(final_state)), final_state, type='p', pch=19)
grid()
##dev.off()

print('Final price can be one of:')
print(round(final_state, 2))