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:

$40

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

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

t = now t = now + 1 month

up state

down state

Call option

A call option on this stock has a strike price of $45

t=0

t=1

Stock Price=$40;

Call Value=$c

Stock Price=$50;

Call Value=$5

Stock Price=$35;

Call Value=$0

A replicating portfolio

Consider a portfolio containing shares of the stock and $B invested in risk-free bonds. The present value (price) of this

portfolio is S + B = $40 + B

Portfolio value

t=0

t=1

$50 + (1+r/12)B

$35+ (1+r/12)B

$40 + B

up state

down state

A replicating portfolio

This portfolio will replicate the option if we can find a and a B such that $50 + (1+r/12) B =

$5

$35 + (1+r/12) B = $0

and

Portfolio payoff = Option payoff

Up state

Down state

The replicating portfolio

Solution: = 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

up state down stateStock Price 50.00$ 35.00$ 1/3 Share 16.67$ 11.67$ Bond Repayment 11.67$ 11.67$ Net portfolio 5.00$ -$

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

A general (1-period) formula

=Cu −CdSu − Sd

B=SuCd −SdCu

1 + r( ) Su −Sd( )

p =r −du−d

c =S+ B=pCu + 1−p( )Cd

1+ r

An observation about

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

=Cu −CdSu − Sd

→∂C

∂S

Put option

What about a put option with a strike price of $45

t=0

t=1

Stock Price=$40;

Put Value=$p

Stock Price=$50;

Put Value=$0

Stock Price=$35;

Put Value=$10

A replicating portfolio

t=0

t=1

$50 + (1+r/12)B

$35+ (1+r/12)B

$40 + B

up state

down state

A replicating portfolio

This portfolio will replicate the put if we can find a and a B such that

$50 + (1+r/12) B = $0

$35 + (1+r/12) B = $10

and

Portfolio payoff = Option payoff

Up state

Down state

The replicating portfolio

Solution: = -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

$47

$48.53

$45.83

$50.10

$47.31

$44.68

Terminal Call Values

$C0

$Cu

$Cd

Cuu =$5.10

Cud =$2.31

Cdd =$0

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

Two Periods

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

C =p2CUU + 2p(1−p)CUD + (1−p)2CDD

1+ r( )2

ExampleTelMex Jul 45 143 CB 23/16 -5/16 47 2,703TelMex Jul 45 143 CB 23/16 -5/16 47 2,703

Two Period Binomial Model Call Option Price Calculator

Stock Price $47.00Exercise Price $45.00Years to Maturity 0.08Risk-free Rate (per annum) 5.00%Ru 3.25%Rd -2.50%p 47.10%Stock Value in Up Up State 50.10$ Call Value in Up Up State 5.10$ Stock Value in Down Up State 47.31$ Call Value in Down Up State 2.31$ Stock Value in Down Down State 44.68$ Call Value in Down Down State -$ Call Value 2.28$

Two Period Binomial Model Call Option Price Calculator

Stock Price $47.00Exercise Price $45.00Years to Maturity 0.08Risk-free Rate (per annum) 5.00%Ru 3.25%Rd -2.50%p 47.10%Stock Value in Up Up State 50.10$ Call Value in Up Up State 5.10$ Stock Value in Down Up State 47.31$ Call Value in Down Up State 2.31$ Stock Value in Down Down State 44.68$ Call Value in Down Down State -$ Call Value 2.28$

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 returnsRu =expμt

n +σ tn( )−1

Rd =expμtn −σ t

n( )−1

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

Ru =expμtn +σ t

n( )−1

Rd =expμtn −σ t

n( )−1

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

Ru =exp.14 ×.332 + .23 .33

2( )−1=.1236

Rd =exp.14 ×.332 −.23 .33

2( )−1=−.0679

For Example

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

Two Period Binomial Model Call Option Price Calculator

Stock Price $108.25Exercise Price $105.00Years to Maturity 0.33Risk-free Rate (per annum) 7.00%Ru 12.36%Rd -6.79%p 41.49%Stock Value in Up Up State 136.66$ Call Value in Up Up State 31.66$ Stock Value in Down Up State 113.37$ Call Value in Down Up State 8.37$ Stock Value in Down Down State 94.05$ Call Value in Down Down State -$ Call Value 9.30$

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:Hints: The risk-free rate was The risk-free rate was 7.6%7.6% and the expected and the expected

return on stocks was return on stocks was 14%14%..

Historical Estimates: Historical Estimates: σσIBMIBM = .24 = .24 & & σσPepsicoPepsico = .38= .38