FINS3635 Week 9 - Option Prices_binomial model (3)

# FINS3635 Week 9 - Option Prices_binomial model (3) - Option...

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1 Option Valuation: Binomial Tree Hull: Ch. 12 FINS 3635 Week 9 Option Value - Binomial tree 1 Pricing options prior to maturity How do we know what c 0 or c 1 should be? Profit c FINS 3635 Week 9 Option Value - Binomial tree 2 X 1 X 2 S T C 0 1 Pricing with no arbitrage Assume absence of arbitrage opportunity First, establish a portfolio of stock and option such that there is no uncertainty about the value of the portfolio in the next period FINS 3635 Week 9 Option Value - Binomial tree 3 portfolio in the next period Because the portfolio has no risk it must earn the risk free rate of return. From this we deduce the value of the portfolio today, and hence the value of the option today.

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2 Pricing Options (Binomial Trees) A stock price is currently \$20 In three months it will be either \$22 or \$18 FINS 3635 Week 9 Option Value - Binomial tree 4 Stock Price = \$22 Stock Price = \$18 Stock price = \$20 Stock Price = \$22 Option Payoff = \$1 Stock price = \$20 A Call Option ( Figure 12.1) A 3-month call option on the stock has a strike price of 21. FINS 3635 Week 9 Option Value - Binomial tree 5 Stock Price = \$18 Option Payoff = \$0 Option Price=? How can we value it now? Pricing a Call Option with no arbitrage opportunity Suppose we write 1 call option and buy shares of the stock Can the portfolio be riskless? Riskless means that the values of the portfolio are FINS 3635 Week 9 Option Value - Binomial tree 6 Riskless means that the values of the portfolio are equal in any state, that is 22 - 1 = 18 Requires that = 0.25, e.g., for each written option, 0.25 share stocks are needed to form a risk-free portfolio.