5_Binomial_trees.pdf

# The value of the portfolio at tt is 22 025 1 45 or 18

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The value of the portfolio at t=T is 22 × 0.25 – 1 =\$ 4.5 or 18 x 0.25 = \$ 4.5 l Suppose that annual interest rate (continuously compounded) is 12%. l The value (=cost) of the portfolio at t=0 is 4.5 e – 0.12 × (3/12) = \$ 4.3670 7

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Fin330 Step 3: Compute the option price l Cost of the portfolio at t=0 is \$ 4.3670: l The value of the shares is 0.25 × 20=\$5 ; l Thus, the value of the option c must satisfy: 5-c = \$4.3670 c = \$0.633 l The obtained call price c precludes arbitrage. l What could investors do if f > 0.633 ? l What could investors do if f < 0.633 ? 8
Generalization l u>1 , d<1 l ƒ u - option price in case of an up movement. l ƒ d - option price in case of a down movement. S 0 u ƒ u S 0 d ƒ d S 0 ƒ 9 Up Move Down Move Fin330

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Generalization: Step 1 l Value of a portfolio that is long Δ shares and short 1 option: l The portfolio is riskless when S 0 u Δ c u = S 0 d Δ c d or Fin330 10 Δ = c u c d S 0 u S 0 d S 0 u Δ c u S 0 d Δ c d Up Move Down Move
Generalization: Step 2 and 3 l Value of portfolio at t = T is S 0 u Δ c u or S 0 d Δ c d l Value of portfolio at t=0 is (S 0 u Δ – c u )e –rT . l This must be equal to S 0 Δ f. l Hence, we have c = S 0 Δ - ( S 0 u Δ - c u )e –rT . l Substituting for Δ : 11 Fin330

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Overview 1. One-step binomial model: 1. Option pricing using no-arbitrage argument 2. Option pricing using risk-neutral probabilities 2. Two-step binomial model 3. Towards the Black-Scholes-Merton formula Fin330 12
p as a probability l It is natural to interpret p and 1 p as the probabilities of up and down movements .

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• Spring '16
• Chang
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