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## Binomial Option Pricing

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**Binomial Option Pricing**Professor P. A. Spindt**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%.**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**Call option**• A call option on this stock has a strike price of $45 t=0 t=1 Stock Price=$50; Call Value=$5 Stock Price=$40; Call Value=$c Stock Price=$35; Call Value=$0**A replicating portfolio**• Consider a portfolio containing Dshares of the stock and $B invested in risk-free bonds. • The present value (price) of this portfolio is DS + B = $40 D + B**$50 D + (1+r/12)B**$40 D + B $35 D + (1+r/12)B Portfolio value t=0 t=1 up state down state**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 =**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/(3x1.00417) = $11.62**The replicating portfolio**• Payoffs at maturity**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 $13.33 - $11.62 = $1.71. • Therefore, c = $1.71**An observation about D**• As the time interval shrinks toward zero, delta becomes the derivative.**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**$50 D + (1+r/12)B**$40 D + B $35 D + (1+r/12)B A replicating portfolio t=0 t=1 up state down state**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 =**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/(3x1.00417) = $33.19.**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%**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**Two-Period Stock Price Dynamics**$50.10 $48.53 $47 $47.31 $45.83 $44.68**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**Two Periods**The two-period Binomial model formula for a European call is**Example**TelMex Jul 45 143 CB 23/16 -5/16 472,703**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**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**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)**For Example**• If the stock’s expected annual return is 14% and its volatility is 23%, then**For Example**• The price of a call with an exercise price of $105 on a stock priced at $108.25**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