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8/2/2019 Bop Ricing
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Binomial Option Pricing
Professor P. A. Spindt
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A simple example
A stock is currently priced at $40 pershare.
In 1 month, the stock price may go up by 25%, or
go down by 12.5%.
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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
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Call option
A call option on this stock has a strikeprice of $45
t=0 t=1
Stock Price=$40;
Call Value=$c
Stock Price=$50;
Call Value=$5
Stock Price=$35;
Call Value=$0
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A replicating portfolio
Consider a portfolio containing Dsharesof the stock and $B invested in risk-free
bonds. The present value (price) of this portfolio isDS + B = $40 D + B
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Portfolio value
t=0 t=1
$50 D + (1+r/12)B
$35 D + (1+r/12)B
$40 D + B
up state
down state
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A replicating portfolio
This portfolio will replicate the option ifwe can find a D and a B such that
$50 D + (1+r/12) B = $5
$35 D + (1+r/12) B = $0
and
Portfolio payoff = Option payoff
Up state
Down state
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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) = $11.62
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The replicating portfolio
Payoffs at maturity
up state dow n state
Stock Price 50.00$ 35.00$
1/3 Share 16.67$ 11.67$
Bond Repayment 11.67$ 11.67$
Net portfolio 5.00$ -$
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The replicating portfolio
Since the the replicating portfolio hasthe 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
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A general (1-period) formula
D Cu Cd
Su SdB
SuCd SdCu
1 r Su Sd
p r d
u d
c DS B pCu 1 p Cd
1 r
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An observation aboutD
As the time interval shrinks towardzero, delta becomes the derivative.
D Cu Cd
Su Sd
C
S
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Put option
What about a put option with a strikeprice of $45
t=0 t=1
Stock Price=$40;
Put Value=$p
Stock Price=$50;
Put Value=$0
Stock Price=$35;
Put Value=$10
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A replicating portfolio
t=0 t=1
$50 D + (1+r/12)B
$35 D + (1+r/12)B
$40 D + B
up state
down state
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A replicating portfolio
This portfolio will replicate the put ifwe can find a D and a B such that
$50 D + (1+r/12) B = $0
$35 D + (1+r/12) B = $10
and
Portfolio payoff = Option payoff
Up state
Down state
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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.
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Two Periods
Suppose two price changes are possibleduring the life of the option
At each change point, the stock may goup by Ru% or down by Rd%
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Two-Period Stock Price
DynamicsFor 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
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Two-Period Stock Price
Dynamics
$47
$48.53
$45.83
$50.10
$47.31
$44.68
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Terminal Call Values
$C0
$Cu
$Cd
Cuu =$5.10
Cud =$2.31
Cdd =$0
At expiration, a call with a strikeprice of $45 will be worth:
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Two Periods
The two-period Binomial model formulafor a European call is
Cp2CUU 2p(1 p)CUD (1 p)
2CDD
1 r 2
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ExampleTelMex Jul 45 143 CB 23/16 -5/16 47 2,703
Two Per iod Binomial Model
Call Option Price Calculator
Stock Price $47.00
Exercise Price $45.00Years to Maturity 0.08
Risk-free Rate (per annum) 5.00%
Ru 3.25%
Rd -2.50%
p 47.10%
Stock Value in Up Up State 50.10$
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$
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Estimating Ruand Rd
According to Rendleman and Barter you canestimate Ru and Rd from the mean and
standard deviation of a stocks returns
Ru exptn
tn 1
Rd exptn
tn 1
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Estimating Ruand Rd
In these formulas, t is the options time to expiration(expressed in years) and n is the number of intervalst is carved into
Ru exptn
tn 1
Rd exptn
tn 1
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For Example
Consider a call option with 4 months torun (t = .333 yrs) and
n = 2 (the 2-period version of thebinomial model)
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For Example
The price of a call with an exercise price of $105 on a stockpriced at $108.25
Two Period Binomial Model
Call Option Price Calculator
Stock Price $108.25
Exercise Price $105.00
Years to Maturi ty 0.33
Risk-free Rate (per annum) 7.00%
Ru 12.36%
Rd -6.79%
p 41.49%
Stock Value in Up Up State 136.66$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$
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Anders Consulting
Focusing on the Nov and Jan options,how do Black-Scholes prices compare
with the market prices listed in caseExhibit 2?
Hints:
The risk-free rate was 7.6% and the expectedreturn on stocks was 14%.
Historical Estimates: IBM = .24 & Pepsico = .38