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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%.
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
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 − 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
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
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