# LEC9 - A C C G 3 2 9 L e c tu r e 9 B in o m ia l O p tio n...

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1 Introduction to Continuous Time Processes 2009, Semester 1 Egon Kalotay A Simple Binomial Model • A stock price is currently \$20 • In three months it will be either \$22 or \$18 Stock Price = \$22 Stock Price = \$18 Stock price = \$20

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2 Stock Price = \$22 Option Price = \$1 Stock Price = \$18 Option Price = \$0 Stock price = \$20 Option Price=? A Call Option Figure 11.1, page 238 A 3-month call option on the stock has a strike price of 21. • Consider the Portfolio: long Δ shares short 1 call option • Portfolio is riskless when 22 Δ – 1 = 18 Δ or Δ = 0.25 22 Δ – 1 18 Δ Setting Up a Riskless Portfolio
3 Valuing the Portfolio (Risk-Free Rate is 12%) • The riskless portfolio is: long 0.25 shares short 1 call option • The value of the portfolio in 3 months is 22 × 0.25 – 1 = 4.50 • The value of the portfolio today is 4.5e – 0.12 × 0.25 = 4.3670 Valuing the Option • The portfolio that is long 0.25 shares short 1 option is worth 4.367 • The value of the shares is 5.000 (= 0.25 × 20 ) • The value of the option is therefore 0.633 (= 5.000 – 4.367 )

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4 Alternative: Payoff Replication Generalization (Figure 11.2, page 239) A derivative lasts for time T and is dependent on a stock S 0 u ƒ u S 0 d ƒ d S 0 ƒ
5 Generalization Cont’d • Consider the portfolio that is long Δ shares and short 1 derivative • The portfolio is riskless when S 0 u Δ ƒ u = S 0 d Δ ƒ d or d S u S f d u 0 0 ! ! = " ƒ S 0 u Δ ƒ u S 0 d Δ ƒ d Generalization Cont’d • Value of the portfolio at time T is S 0 u Δ ƒ u • Value of the portfolio today is ( S 0 u Δ – ƒ u ) e rT • Another expression for the portfolio value today is S 0 Δ f • Hence ƒ = S 0 Δ – ( S 0 u Δ ƒ u ) e rT

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6 Generalization Cont’d • Substituting for Δ we obtain ƒ = [ p ƒ u + (1 – p d ] e rT where p e d u d rT = ! ! p as a Probability • It is natural to interpret p and 1- p as probabilities of up and down movements • The value of a derivative is then its expected payoff in a risk-neutral world discounted at the risk-free rate S 0 u ƒ u S 0 d ƒ d S 0 ƒ p ( 1 )
7 Risk-neutral Valuation • When the probability of an up and down movements are p and 1- p the expected stock price at time T is S 0 e rT • This shows that the stock price earns the risk-free

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LEC9 - A C C G 3 2 9 L e c tu r e 9 B in o m ia l O p tio n...

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