Binomial Tree: Option Pricing We begin understanding option valuation by considering a simple case - a...

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Binomial Tree: Option Pricing We begin understanding option valuation by considering a simple case - a stock price can take only two possible values at option expiration: The stock will either increase to a given higher price or decrease to a given lower price. 120 10 100 C 00 0 90 Stock Price Call Option Value X = 110 Binomial Tree: Option Pricing Value the Call Option via an Alternative Portfolio Buy 1 share of stock at $100 Borrow $81.82 (10% Interest Rate) Net outlay $18.18 Payoff Value of Stock 90 120 Repay loan - 90 - 90 Net Payoff 0 30 18.18 30 Payoff Structure is exactly 3 times the Call Hence 3C $18.18 or C = $6.06 Option Payoff Payoff of the option is C+=max(S+-X,0) - Assume uS>X, do How to replicate? Can we pick H and B such that the portfolio has the same payoffs as the option? Equate payoffs in both states u = Hux S + B(1 + r) = uS - X =Hxdx So + B(1+r) = 0 2 equations in two unknowns Binomial Tree: Option Pricing We begin understanding option valuation by considering a simple case - a stock price can take only two possible values at option expiration: The stock will either increase to a given higher price or decrease to a given lower price. 120 10 100 C 00 0 90 Stock Price Call Option Value X = 110 Binomial Tree: Option Pricing Value the Call Option via an Alternative Portfolio Buy 1 share of stock at $100 Borrow $81.82 (10% Interest Rate) Net outlay $18.18 Payoff Value of Stock 90 120 Repay loan - 90 - 90 Net Payoff 0 30 18.18 30 Payoff Structure is exactly 3 times the Call Hence 3C $18.18 or C = $6.06 Option Payoff Payoff of the option is C+=max(S+-X,0) - Assume uS>X, do How to replicate? Can we pick H and B such that the portfolio has the same payoffs as the option? Equate payoffs in both states u = Hux S + B(1 + r) = uS - X =Hxdx So + B(1+r) = 0 2 equations in two unknowns