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# Binomial Option Pricing - PowerPoint PPT Presentation

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

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### Binomial Option Pricing

Professor P. A. Spindt

• 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%.

• Stock price dynamics:

t = now

t = now + 1 month

up state

\$40x(1+.25) = \$50

\$40

\$40x(1-.125) = \$35

down state

• 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

• 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

• 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

=

• 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

• Payoffs at maturity

• 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

• As the time interval shrinks toward zero, delta becomes the derivative.

• 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

• 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

=

• 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.

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%

• 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

\$50.10

\$48.53

\$47

\$47.31

\$45.83

\$44.68

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

Cuu =\$5.10

\$Cu

\$C0

Cud =\$2.31

\$Cd

Cdd =\$0

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

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

• Consider a call option with 4 months to run (t = .333 yrs) and

• n = 2 (the 2-period version of the binomial model)

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

• The price of a call with an exercise price of \$105 on a stock priced at \$108.25

• 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