# Binomial Model

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How can I use binomial model to answer to the following questions?

Consider the stock with the current price of $20. It pays no dividends.

1) Maturity: in 4 months

Strike price:$20

Volatility = 30% per annum

Risk-free rate: 10%

What's the value of European call option?

2) What would be the value of option in 1), if you expect volatility over the next 4 month to be 20%?

3) Given the information in 1) and the expectation declared in 2), what option priced position should we take on?

4) Given that volatility = 30% per annum, what would be the replicating portfolio for the option?

5) What's the price of European put using put-call parity and volatility = 30% per annum?

What's the price of American put using put-call parity and volatility = 30% per annum?

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The expert examines binomial models for current price.

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1) To be able to solve this problem, and the subsequent ones, we must first set up the problem with the data we are given. We have the following situation:

The binomial tree shown above can be used to calculate the price of the option today. We will use the following terminology:

C = Value of the call option today

X = Exercise price = $20

T = Time period for which the option is active = 4 months = 4/12 years

R = Risk-free interest rate = 10%

S = Price of the stock today = $20

Su = Price of the stock after one year in the up state

Sd = Price of the stock after one year in the down state

Cu = Value of the option after one year in the up state

Cd = Value of the option after one year in the down state

u = Stock return in the up state

d = Stock return in the down state

Pr = Risk-neutral probability of a stock movement in the up state

(1 - Pr) = Risk-neutral probability of a stock movement in the down state

We will use the following formulas using binomial risk-neutral option pricing:

(1)

(2)

First we calculate the stock returns in both states:

Su = S*1.30 = ...

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