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1Option Valuation: Binomial TreeHull: Ch. 12FINS 3635 Week 9Option Value - Binomial tree1Pricing options prior to maturityHow do we know what c0or c1should be?ProfitcFINS 3635 Week 9Option Value - Binomial tree2X1X2STC01Pricing with no arbitrageAssume absence of arbitrage opportunityFirst, establish a portfolio of stock and option such that there is no uncertainty about the value of the portfolio in the next periodFINS 3635 Week 9Option Value - Binomial tree3portfolio 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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2Pricing Options (Binomial Trees)A stock price is currently $20In three months it will be either $22 or $18FINS 3635 Week 9Option Value - Binomial tree4Stock Price = $22Stock Price = $18Stock price = $20Stock Price = $22Option Payoff = $1Stock price = $20A Call Option (Figure 12.1)A 3-month call option on the stock has a strike price of 21. FINS 3635 Week 9Option Value - Binomial tree5Stock Price = $18Option Payoff = $0Option Price=?How can we value it now?Pricing a Call Option with no arbitrage opportunitySuppose we write 1 call option and buy ∆shares of the stockCan the portfolio be riskless?Riskless means that the values of the portfolio areFINS 3635 Week 9Option Value - Binomial tree6Riskless 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.