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Binomial Option PricingPowerPoint Presentation

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

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

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

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

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