7 Valuation of options

7 Valuation of options - Option Pricing Approaches...

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FIN 819 Option Pricing Approaches Valuation of options
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FIN 819 Today’s plan Review of what we have learned about options We discuss two ways of valuing options Binomial tree (two states) Simple idea Risk-neutral valuation The Black-Scholes formula (infinite number of states) Understanding the intuition How to apply this formula
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FIN 819 What have we learned in the last lecture? Options Financial and real options European and American options Rights to exercise and obligations to deliver the underlying asset Position diagrams Draw position diagrams for a given portfolio Given position diagrams, figure out the portfolio No arbitrage argument Put-call parity
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FIN 819 The basic idea behind the binomial tree approach Suppose we want to value a call option on IBM with a strike price of K and maturity T. We let C(K,T) be the value of this call option. Remember C(K,T) is the price for the call or present value of the call option. Let the current price of IBM is S and there are two states when the call option matures: up and down. If the state is up, the stock price for IBM is S u ; if the state is down, the price of IBM is S d .
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FIN 819 The stock price now and at maturity S uS dS S u S d Now maturity If we define: u = S u /S and d = S d /S. Then we have S u =uS and S d =dS S Now maturity
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FIN 819 The risk free security The price now and at maturity R f R f 1 now maturity Here R f =1+r f
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FIN 819 The call option payoff C u =Max(uS-K,0) C d =Max(dS-K,0) C(K,T) Now maturity
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FIN 819 Now form a replicating portfolio A portfolio is called the replicating portfolio of an option if the portfolio and the option have exactly the same payoff in each state of future. By using no arbitrage argument, the cost or price of the replication portfolio is the same as the value of the option.
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FIN 819 Now form a replicating portfolio (continue) Since we have three securities for investment: the stock of IBM, the risk- free security, and the call option, how can we form this portfolio to figure out the price of the call option on IBM?
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FIN 819 Now form a replicating portfolio (continue) Suppose we buy Δ shares of stock and borrow B dollars from the bank to form a portfolio. What is the payoff for the this portfolio for each state when the option matures? What is the cost of this portfolio? How can we make sure that this portfolio is the replicating portfolio of the option?
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FIN 819 How can we get a replicating portfolio? Look at the payoffs for the option and the portfolio Option Portfolio C(K,T) Cd=max(dS-K,0) Cu=max(uS-K,0) ΔdS+BRf ΔuS+BRf B+ΔS now Maturity Now maturity
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FIN 819 Form a replicating portfolio From the payoffs in the previous slide for the call option and the portfolio, to make sure that the portfolio is the replicating portfolio of the option, the option and the portfolio must have exactly the same payoff in each state at the expiration date.
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7 Valuation of options - Option Pricing Approaches...

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