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Introduction to Binomial Trees

Introduction to Binomial Trees. Chapter 12. A Simple Binomial Model Note: Initial discussion is for options on assets that pay no income, e.g. a non-dividend paying stock. A stock price is currently $20 In three months it will be either $22 or $18. Stock Price = $22. Stock price = $20.

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Introduction to Binomial Trees

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  1. Introduction to Binomial Trees Chapter 12

  2. A Simple Binomial ModelNote: Initial discussion is for options on assets that pay no income, e.g. a non-dividend paying stock. • A stock price is currently $20 • In three months it will be either $22 or $18 Stock Price = $22 Stock price = $20 Stock Price = $18

  3. A Call Option A 3-month call option on the stock has a strike price of 21. Stock Price = $22 Option Price = $1 Stock price = $20 Option Price=? Stock Price = $18 Option Price = $0

  4. 22D – 1 18D Setting Up a Riskless Portfolio • Consider the Portfolio: long D shares short 1 call option • Portfolio is riskless when 22D – 1 = 18D or D = 0.25

  5. 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 = 18 ´ 0.25 • The value of the portfolio today is 4.5e – 0.12´0.25 = 4.3670

  6. 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 )

  7. Su ƒu S ƒ Sd ƒd Generalization A derivative lasts for time T and is dependent on a stock (Su is Sxu, Sd is Sxd versus subscripts ƒu, ƒd)

  8. Generalization(continued) • Consider the portfolio that is long D shares and short 1 derivative • The portfolio is riskless when SuD – ƒu = SdD – ƒd or SuD – ƒu SdD – ƒd

  9. Significance of delta If write one option (call or put) on one share, delta is the number of shares you must own to form a riskless portfolio. Delta equals the (partial) derivative of the option price with respect to the underlying asset price.

  10. Generalization(continued) • Value of the portfolio at time Tis SuD – ƒu • Value of the portfolio today is (SuD – ƒu )e–rT • Another expression for the portfolio value today is SD – f • Hence ƒ = SD – (SuD – ƒu)e–rT

  11. Generalization(continued) • Substituting for D we obtain ƒ = [ p ƒu + (1 – p )ƒd ]e–rT where (note: d < erT < u)

  12. Su ƒu S ƒ Sd ƒd Risk-Neutral Valuation • ƒ = [ p ƒu + (1 – p )ƒd ]e-rT • The variables p and (1– p ) can be interpreted as the risk-neutral probabilities of up and down movements • The value of a derivative is its expected payoff in a risk-neutral world discounted at the risk-free rate p (1– p )

  13. Irrelevance of Stock’s Expected Return When we are valuing an option in terms of the underlying stock the expected return on the stock is irrelevant. The expected return on the stock provides no information additional to S, the initial value of the stock.

  14. Implication of Risk-Neutrality In a risk-neutral market, the expected total return (comprised of the income component and the capital gains component) on any risk asset equals the risk-free rate.

  15. Original Example Revisited Su = 22 ƒu = 1 p • Since p is a risk-neutral probability 20e0.12 ´0.25 = 22p + 18(1 – p ); p = 0.6523 • Alternatively, we can use the formula S ƒ Sd = 18 ƒd = 0 (1– p )

  16. Su = 22 ƒu = 1 0.6523 S ƒ Sd = 18 ƒd = 0 0.3477 Valuing the Option The value of the option is e–0.12´0.25 [0.6523´1 + 0.3477´0] = 0.633

  17. 24.2 22 19.8 20 18 16.2 A Two-Step Example:half year time horizon • Step=3 months, RNP constant • K=21, r=12%, u=1.1, d=.9

  18. Valuing a European Call Option 24.2 3.2 D 22 • Value at node B = e–0.12´0.25(0.6523´3.2 + 0.3477´0) = 2.0257 • Value at node A = e–0.12´0.25(0.6523´2.0257 + 0.3477´0) = 1.2823 B 19.8 0.0 20 1.2823 2.0257 A E 18 C 0.0 16.2 0.0 F

  19. 72 0 D 60 B 48 4 50 4.1923 1.4147 A E 40 C 9.4636 32 20 F A European Put Option Example (2 year expiry) K = 52, Dt = 1yr r = 5% u=1.2 d=.8

  20. 72 0 D 60 B 48 4 50 5.0894 1.4147 A E 40 C 12.0 32 20 F What Happens When an Put is American (premature exercise at end of 1st year if S=40)

  21. Delta • Delta (D) is the ratio of the change in the price of a stock option to the change in the price of the underlying stock • The value of D varies from node to node: Dynamic (not static) hedging is required

  22. The Probability of an Up MoveWhat happens when the underlying asset is not a non-dividend paying stock?

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