#Binomial option pricing (2)

#We can expand the tree to more than 3rd depth.

#U = exp(volatility)

#D = exp(-volatility)

#p = 0.5 (We have the equal chance of making or losing money)

#Risk free rate = 0.02 => exp(0.02)

#For those who are not familiar with data structure, I deliberately used just array.

#I'll make a new code for those who are familiar with tree data structure

library(igraph)

#Define the variable

depth<-3 #How many steps (tree depth) do you want to make

rate <-0.02 #Risk Free rate

volatility <- 0.35

exercise_price <- 35

stock_price <- 47.5

upside_probability<-(exp(rate)-exp(-volatility))/(exp(volatility)-exp(-volatility))

rate <- exp(rate)

total_node<-2^depth-1 #Total number of node

G <- graph.tree(n=total_node, children=2) #I'll make a graph whose nodes are 7, and each node has two children

stock_tree <- (1:total_node)

stock_tree[1]<-stock_price

tree_traverse <- 2^(depth-1) -1

for(i in 1:tree_traverse) {

#We are going to use mathematical trick to represent tree.

stock_tree[i*2] <- stock_tree[i] * exp(volatility)

stock_tree[i*2 1] <- stock_tree[i] * exp(-volatility)

}

V(G)$name <- round(stock_tree) #Name of the tree

lay <- layout.reingold.tilford(G) #It's tree. You can try other shape with other layout options

plot(G, layout=lay, vertex.size=15, edge.arrow.size=0.1) #Draw the tree.

#As opposed to the stock price, the option pricing starts out with end nodes (bottom nodes)

#I already explained the logic. Just follow it from one by one.

option_price<-(1:total_node)

bottom_node<-tree_traverse 1

#In order to value the option, we need to calculate bottom line first.

for(i in bottom_node:total_node) {

after_option <- stock_tree[i] - exercise_price

if( after_option >0 ) {

option_price[i] <- after_option

} else {

option_price[i] <- 0

}

}

#Discount it back to current time while considering the probabilty of up and down

for(i in tree_traverse:1) {

option_price[i]<-upside_probability*option_price[i*2]

option_price[i]<-option_price[i] (1-upside_probability)*option_price[i*2 1]

option_price[i]<-option_price[i]/rate

}

V(G)$name <- round(option_price)

plot(G, layout=lay, vertex.size=15, edge.arrow.size=0.1)