Binomial Option Pricing

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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

t0
t1
Stock Price50 Call Value5
Stock Price40 Call Valuec
Stock Price35 Call Value0
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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
t0
t1
up state
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 (1r/12) B 5
and
Down state
35 D (1r/12) B 0
Portfolio payoff

Option payoff
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The replicating portfolio

• Solution
• D 1/3
• B -35/(3(1r/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

t0
t1
Stock Price50 Put Value0
Stock Price40 Put Valuep
Stock Price35 Put Value10
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A replicating portfolio
t0
t1
up state
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 (1r/12) B 0
and
Down state
35 D (1r/12) B 10
Portfolio payoff

Option payoff
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The replicating portfolio

• Solution
• D -2/3
• B 100/(3(1r/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

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Example
TelMex Jul 45 143 CB 23/16 -5/16 47 2,703
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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 stocks returns

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Estimating Ru and Rd

• In these formulas, t is the options 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 stocks 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