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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

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Binomial Option Pricing

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Binomial option pricing l.jpg

Binomial Option Pricing

Professor P. A. Spindt


A simple example l.jpg

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 example3 l.jpg

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 l.jpg

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 l.jpg

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


Portfolio value l.jpg

$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 portfolio7 l.jpg

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 l.jpg

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


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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 $13.33 - $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


A replicating portfolio14 l.jpg

$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 portfolio15 l.jpg

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 portfolio16 l.jpg

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.


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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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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


Example l.jpg

Example

TelMex Jul 45 143 CB 23/16 -5/16 472,703


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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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