2. 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%.
3. 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
4. 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
5. 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
7. 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
8. 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) = $11.62
9. The replicating portfolio
Payoffs at maturity
up state down state
Stock Price 50.00$ 35.00$
1/3 Share 16.67$ 11.67$
Bond Repayment 11.67$ 11.67$
Net portfolio 5.00$ -$
10. 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
11. 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
12. An observation about ∆
As the time interval shrinks toward
zero, delta becomes the derivative.
∆ =
Cu − Cd
Su − Sd
→
∂C
∂S
13. 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
15. 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
16. 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.
17. 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%
18. 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
21. Two Periods
The two-period Binomial model formula
for a European call is
C =
p2
CUU + 2p(1− p)CUD + (1− p)2
CDD
1+ r( )2
22. Example
TelMex 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 $47.00
Exercise Price $45.00
Years to Maturity 0.08
Risk-free Rate (per annum) 5.00%
Ru 3.25%
Rd -2.50%
p 47.10%
Stock Value in Up Up State 50.10$
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$
23. 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
µt
n + σ t
n( )−1
Rd = exp
µt
n − σ t
n( )−1
24. 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
µt
n + σ t
n( )−1
Rd = exp
µt
n − σ t
n( )−1
25. 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)
26. For Example
If the stock’s expected annual return is
14% and its volatility is 23%, then
Ru = exp .14 × .33
2 + .23 .33
2( )−1 = .1236
Rd = exp .14 × .33
2 − .23 .33
2( )−1 = −.0679
27. 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 $108.25
Exercise Price $105.00
Years to Maturity 0.33
Risk-free Rate (per annum) 7.00%
Ru 12.36%
Rd -6.79%
p 41.49%
Stock Value in Up Up State 136.66$
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$
28. 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 wasThe risk-free rate was 7.6%7.6% and the expectedand the expected
return on stocks wasreturn on stocks was 14%14%..
Historical Estimates:Historical Estimates: σσIBMIBM = .24= .24 && σσPepsicoPepsico = .38= .38
