Bo pricing

Binomial Option Pricing

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


� 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


� 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$ -$


� 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 − Cd

Su − Sd

B =SuCd − SdCu

1 + r( ) Su − Sd( )

p =r − d

u − 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 − Cd

Su − 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


t=0 t=1

$50 ∆ + (1+r/12)B

$35 ∆ + (1+r/12)B

$40 ∆ + B

up state

down state


� 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


� 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)2 CDD

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$

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

Ru = exp µtn + σ t

n( )−1

Rd = exp µtn − 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 = exp µtn + σ t

n( )−1

Rd = exp µtn − 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

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