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**Binomial Option Pricing**

Professor P. A. Spindt

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**A simple example A stock is currently priced at $40 per share.**

In 1 month, the stock price may go up by 25%, or go down by 12.5%.

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**A simple example Stock price dynamics: t = now t = now + 1 month**

up state $40x(1+.25) = $50 $40 $40x(1-.125) = $35 down state

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**Call option A call option on this stock has a strike price of $45 t=0**

Stock Price=$50; Call Value=$5 Stock Price=$40; Call Value=$c Stock Price=$35; Call Value=$0

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**A replicating portfolio**

Consider a portfolio containing D shares of the stock and $B invested in risk-free bonds. The present value (price) of this portfolio is DS + B = $40 D + B

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**Portfolio value t=0 t=1 up state down state $50 D + (1+r/12)B**

$40 D + B down state

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**A replicating portfolio**

This portfolio will replicate the option if we can find a D and a B such that Up state $50 D + (1+r/12) B = $5 and Down state $35 D + (1+r/12) B = $0 Portfolio payoff = Option payoff

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**The replicating portfolio**

Solution: D = 1/3 B = -35/(3(1+r/12)). Eg, if r = 5%, then the portfolio contains 1/3 share of stock (current value $40/3 = $13.33) partially financed by borrowing $35/(3x ) = $11.62

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**The replicating portfolio**

Payoffs at maturity

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**The replicating portfolio**

Since the the replicating portfolio has the same payoff in all states as the call, the two must also have the same price. The present value (price) of the replicating portfolio is $ $11.62 = $1.71. Therefore, c = $1.71

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**A general (1-period) formula**

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An observation about D As the time interval shrinks toward zero, delta becomes the derivative.

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**Put option What about a put option with a strike price of $45 t=0 t=1**

Stock Price=$50; Put Value=$0 Stock Price=$40; Put Value=$p Stock Price=$35; Put Value=$10

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**A replicating portfolio**

up state $50 D + (1+r/12)B $35 D + (1+r/12)B $40 D + B down state

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**A replicating portfolio**

This portfolio will replicate the put if we can find a D and a B such that Up state $50 D + (1+r/12) B = $0 and Down state $35 D + (1+r/12) B = $10 Portfolio payoff = Option payoff

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**The replicating portfolio**

Solution: D = -2/3 B = 100/(3(1+r/12)). Eg, if r = 5%, then the portfolio contains short 2/3 share of stock (current value $40x2/3 = $26.66) lending $100/(3x ) = $33.19.

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Two Periods Suppose two price changes are possible during the life of the option At each change point, the stock may go up by Ru% or down by Rd%

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**Two-Period Stock Price Dynamics**

For example, suppose that in each of two periods, a stocks price may rise by 3.25% or fall by 2.5% The stock is currently trading at $47 At the end of two periods it may be worth as much as $50.10 or as little as $44.68

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**Two-Period Stock Price Dynamics**

$50.10 $48.53 $47 $47.31 $45.83 $44.68

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**At expiration, a call with a strike price of $45 will be worth:**

Terminal Call Values At expiration, a call with a strike price of $45 will be worth: Cuu =$5.10 $Cu $C0 Cud =$2.31 $Cd Cdd =$0

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Two Periods The two-period Binomial model formula for a European call is

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Example TelMex Jul CB 23/16 -5/ ,703

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Estimating Ru and Rd According to Rendleman and Barter you can estimate Ru and Rd from the mean and standard deviation of a stock’s returns

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Estimating Ru and Rd In these formulas, t is the option’s time to expiration (expressed in years) and n is the number of intervals t is carved into

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For Example Consider a call option with 4 months to run (t = .333 yrs) and n = 2 (the 2-period version of the binomial model)

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For Example If the stock’s expected annual return is 14% and its volatility is 23%, then

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For Example The price of a call with an exercise price of $105 on a stock priced at $108.25

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Anders Consulting Focusing on the Nov and Jan options, how do Black-Scholes prices compare with the market prices listed in case Exhibit 2? Hints: The risk-free rate was 7.6% and the expected return on stocks was 14%. Historical Estimates: sIBM = .24 & sPepsico = .38

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