# Binomial trees

A very popular technique of pricing a stock option involves constructing a binomial trees.To understand the binomial tree first we understand the option pricing.An option’s price also referred to as the premium is priced per share.The seller is paid the premium is priced per share.The seller is paid the premium ,giving the buyer the right granted by the option.

Binomial model
Binomial tree is a graphical representation of possible intrinsic values that an option may take at different time periods.one step binomial model suppose that a stock  price is currently \$20.We make a simplifying assumption that at the end of three month.The stock price will be either 22 or 18 .This mean that the option will have one of two values at the end of three month.If stock price turns out to be \$22 the value of the option will be \$1.If stock price turns out to be \$22 the value of the option will be \$1.If the stock price turns out to be \$18 the value of the option is zero.

We then argue that because the portfolio has no risk the return it earns must equal the risk free interest rate.This enables us to work out the cost of selling up the portfolio and therefore the options price because there are two securities and only two possible out comes.
Consider a portfolio consisting in Δ  share of the stock and short position in one call option .We calculate the value of Δ that makes the portfolio riskless .If the stock price moves up from \$20 to \$22 .The value of the share is 22  Δ   and the value of the option is 1.So that the total value of portfolio is 22 Δ -1.If the stock price moves down from \$20 to \$18 ,the value of the share is 18 Δ  and the value of the option is zero.So that the total value of the portfolio is 18Δ

22Δ-1=18Δ  o r Δ=0.25

A riskless portfolio is therefore
Long  0.25 shares
short  1 option
If the stock price moves up to \$22,the value of the portfolio is \$22  × 0.25-  1=4.5
If the stock price moves down to \$18,the value of the portfolio is  18 × 0.25=4.5
Regardless of whether the stock price moves up or down ,the value of the portfolio is always 4.5 the end of the life of the option.

Two step binomial  model
Now we look at a two step binomial tree .Now we start to understand suppose that the stock price starts at\$20 and in each of two time steps may go up by 10% or down by 10%.we are supposing that each time step in three month long and the risk free interest rate is 12% per annum.We consider an option with a strick price of \$21

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The objective of the analysis is to calculate the option price at the initial node of the tree.The option prices at the final nodes of the tree are easily calculated.They are the pay off from the option at node D the stock price is 24.2-21=3.2  at nodes E and F the option is out of the money and its value is zero.
At node C the option price is zero,because node C leads to either node E or node F and at both nodes the option price is zero.Calculating the option price at node B by using the notation u=1.1,d=0.9 ,r=0.12 and T=0.25 so that P=0.6523 gives value of the option at node B as.

${e}^{–0.12×3/12}\left(0.6523×3.2+0.3477×0\right)=2.0257$

Calcuation of node A we know that the value of the option B is 2.0257and C it is zero.

${e}^{–0.12×3/12}\left(0.6523×2.0257+0.3477×0=1.2823\phantom{\rule{0ex}{0ex}}$

The value of the option is \$1.2823

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