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Introduction to Binomial Trees Chapter 11

Introduction to Binomial Trees Chapter 11. 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 = $20. Stock Price = $18. A Call Option. A 3-month call option on the stock has a strike price of 21. .

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Introduction to Binomial Trees Chapter 11

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  1. Introduction toBinomial TreesChapter 11

  2. 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 = $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 • 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. S0 u ƒu S0 ƒ S0d ƒd Generalization • A derivative lasts for time T and is dependent on a stock

  8. Generalization(continued) • Consider the portfolio that is long D shares and short 1 derivative • The portfolio is riskless when S0uD – ƒu = S0dD – ƒd or S0 uD – ƒu S0– f S0dD – ƒd

  9. Generalization(continued) • Value of the portfolio at time Tis S0uD – ƒu • Value of the portfolio today is (S0uD – ƒu )e–rT • Another expression for the portfolio value today is S0D – f • Hence ƒ = S0D – (S0uD – ƒu)e–rT

  10. Generalization(continued) • Substituting for D we obtain ƒ = [ p ƒu + (1 – p )ƒd ]e–rT where

  11. S0u ƒu p S0 ƒ S0d ƒd (1– p ) 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

  12. 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

  13. Original Example Revisited S0u = 22 ƒu = 1 p • Since p is a risk-neutral probability: 20 = [22p + 18(1 – p )] e-0.12 ´0.25; Solve for p: p = 0.6523 • Alternatively, we can use the formula S0 ƒ S0d = 18 ƒd = 0 (1– p )

  14. S0u = 22 ƒu = 1 0.6523 S0 ƒ S0d = 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

  15. Risk-Neutral vs Real World • In risk-neutral world anything earns the risk-free rate of return and the probabilities of price movements are different from those in the real world. • Assume the stock from previous slide earns an expected return µ = 16%. Then, by definition: Real world probability

  16. Risk-Neutral vs Real World • In the case of the call option it means that the expected option payoff is $0.7041 and, hence, the expected option return (implied from that of the stock) is 42.58%. Here is the algebra:

  17. 24.2 22 19.8 20 18 16.2 A Two-Step Example • Each time step is 3 months

  18. Valuing a 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 Put Option Example; K=52r = .05, T=2 years

  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 the Put from Previous Slide is American At each node choose either the continuation value or the exercise value, whichever is higher.

  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 • Think of D as the number of shares that one needs to sell short to hedge one long call option. The change in the value of this portfolio from node to node is zero: • C - D S = (1-0) – 0.25(22-18) = 0 • Delta changes over time which gives rise to dynamic hedging strategies

  22. Computation of Delta Su=25, fu=5 • In practice, compute delta as follows: • Delta is then: • Call Delta is positive. What about puts? Sd=15, fd=0

  23. Choosing u and d in practice One way of matching the volatility is to set where s is the volatility and dt is the length of the time step. This is the approach used by Cox, Ross, and Rubinstein

  24. Example • A stock price is currently $25. The standard deviation of the stock return is 20% per year and the risk-free rate is 10% per year (continuous). What is the value of the derivative that pays $ in t=T=2 months? (We’ll use one period tree for simplicity)

  25. Value a Forward • Given the data on the previous slide value a two-month forward on the stock. Delivery price K is $20. • From before you remember that • f0 = S0 – Ke-rT = 25 – 20e-0.1x2/12 = 5.33 • Using risk neutral valuation: • fu = Su – K = 27.127 – 20 = 7.127 • fd = Sd – K = 23.040 – 20 = 3.040 • f0 = [p fu + (1-p) fd)] e-rT = = [0.5824x7.127+0.4176x3.040]e-0.1x2/12 = 5.33

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