BinomialOption

# BinomialOption - Single period binomial option pricing...

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Unformatted text preview: Single period binomial option pricing model JHT Kim October 30, 2009 1/19 Introduction The binomial option pricing model is discussed for the single period When the stock price follows the single period binomial model, the option price is determined based on the CF replication This is the cornerstone of the option pricing theory Later we discuss some results on the continuous model as well 2/19 Assumptions Suppose the current stock price (at t = 0) is S At t = 1, the stock price will be either go up to S u or down to S d (This can be interpreted as two possible economy states) The risk free rate is r f The goal is to obtain the current price of an option that has a pay off of C u if thestock goes up, and C d if the stock goes down S 1 C 3/19 Note When the stock goes up, the bond earns at r f and the option pays C u When the stock goes down, the bond earns at r f and the option pays C d The option can be any type (call or put) in this setup We look for C , the current price of the option We replicate the option payoff with a portfolio consisting of stock and bond 4/19 Replicating portfolio To construct such a portfolio, let us buy Δ shares of stock and B dollars worth of risk-free bond ( Δ and B can be fractions and negatives) We need to find Δ and B by matching the outcomes of the option at t = 1 At t = 1, the payoffs thus should satisfy the following system of two equations: Δ S u + B (1 + r f ) = C u Δ S d + B (1 + r f ) = C d LHS represents the portfolio payoff, and RHS represents the option payoff 5/19 The solution of the system is: Δ = C u- C d S u- S d and B = (1 + r f )- 1 C d S u- C u S d S u- S d With this Δ and B , the portfolio that consists of “Δ shares of stock and B dollars worth of risk-free bond” replicates the option payoff at t = 1 This means that the option and the replicating portfolio have the same payoff at t = 1, no matter what the economy state will be. Therefore the current value of the replicating portfolio should be the same as the current price of the option; otherwise there will be an arbitrage opportunity (Law of one price) Note that the current value (or time 0 price) of the portfolio is, by the original proposal, Δ · S + B 6/19 Therefore the option price is C = Δ S + B = C u- C d S u- S d S + (1 + r f )- 1 C d...
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BinomialOption - Single period binomial option pricing...

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