Chapter 9_Example_7
A Real Call Options Value with a Binomial Lattice Model
A portfolio is one which there are more than two investment assets in an investment project.
A binomial lattice model is that an event has only two outcomes branched out at the node
and some of those outcomes are merged at the next nodes.
A Pascal Triangle
A Replicating Portfolio

A replicating portfolio must be constructed with a stock and bond/a option itself, which is regarded to mimic an option value and provides a
risk-free return.

In order to construct the replicating portfolio, we need to collect information on the following parameters.
i) an upward ratio of “u” by which a stock price will increase at the next time.
ii) a downward ratio of “d” by which a stock price will decrease at the next time, which is obtained with the reciprocal of the upward ratio once it is
determined.
iii) a volatility of a stock price of σ
iv) a time interval dissecting a maturity time.
Then, the upward ratio is determined by the following equation.
u  e

t
t
v) A hedge ratio of Д which is a number
of stock
d 
e shares purchased

vi) A risk-free return of “rf”
1
u
vii) An amount of money of “B” which is borrowed or invested with a risk-free return.
A Replicating Portfolio
B(1+rf)
ДuS0
+
ДS0
B
=
B(1+rf)
ДdS0
Stock
Cu=Max(St-K,0)
Bond
C
Cd=Max(St-K,0)
Call Option
A Construction of Two Simultaneous
Equation with The Replicating Portfolio
 i) when the stock price goes up, then the first equation is
uS 0  B (1  rf )  Cu  Max[ S t  K , 0]
 Ii)when the stock price goes down, then the second equation is
dS 0  B (1  rf )  Cd  Max[ St  K , 0]
 When you solve the equations with respect to B and Д, then you get
 
Cu  Cd
S 0 (u  d )
B 
Cu  uS 0
(1  r f )
(1)

uCd  dCu
(1  r f )(u  d )
(2)
 Finally, we will obtain the call option value as follows:
C  S 0  B
(3)
A Risk Neutral Probability
 When you plug Equation (1) and (2) into Equation (3), then you will get
 Then, the equation of “C” is rewritten as follows:
C 
1
1  rf
 qCu
 (1  q )Cd
where ,
Rd
ud
u  R (u  d )  (d  R)
Rd

 1
 1 q
ud
ud
ud
q

A General Binomial Lattice Model
u
u
d
u
u
d
ДS0
d
u
d
d
u
d
Black-Scholes Model
 It was proposed in 1973 by Fisher Black and Robert Scholes.
 Unlike a binomial lattice model, it is a continuous model in which a stock price is
assumed to continuously vary over time.
 Some of the assumption for the model.
i) (riskless rate) The rate of return on the riskless asset is constant and thus called the riskfree interest rate.
ii) (random walk) The instantaneous log return of stock price is an infinitesimal random
walk with drift; more precisely, it is a geometric Brownian motion, and we will assume its
drift and volatility is constant (if they are time-varying, we can deduce a suitably
modified Black–Scholes formula quite simply, as long as the volatility is not random).
iii) The stock does not pay a dividend.
iv) There is no transaction fee.
v) A stock may be infinitesimally divided and money can be borrowed with a short-term
interest rate.
vi) A short sale is allowed.
A B-S Equation
C  S 0 N ( d1 )  e
 rf T

KN ( d 2 )

2
S0


ln 
 rf  
T

K
2

d1 
 T
d 2  d1  
T
The Value of N()
Example for B-S Model
A Relationship Between A Financial
Option and Real Options
Example 9.7
실물콜옵션을 이용한 투자의사결정체계
초기 투자금액이 2억 원이고 3년 동안 현금흐름이 아래 표와 같이 발생하는 투자프로젝트를 고려해
보기로 하자.
수요량
확률
연간 현금흐름
기대 이상
0.25
2억5,000만
보통
0.30
1억
기대 이하
0.45
3,500만
위험보정할인율은 10%이고 무위험 할인율은 6%이라고 한다. 수요량이 기대 이상이거나 보통일
경우 이 투자프로젝트를 1년 연기해서 실행하는 것이 경제적으로 바람직한지를 실물옵션가치 결
정이론을 사용해서 분석해 보아라.
A Decision with a Traditional NPV
Expected Annual Cash Flow
Expected P.V of Cash Flows
Expected NPV
A Decision with the Real Options Concept
 What will happen to the Investment Project if one year is delayed?
 If the sales volume is less than expected, the company will not undertake
the investment project under consideration.
 As a result, we have the following cash flows ignoring those regarding the
event of a less-expected sales volume.
현금흐름
수요량
0
기대이상
－
－200,000
250,000
250,000
250,000
376,510
25%
보통
－
－200,000
100,000
100,000
100,000
37,400
30%
기대이하
－
0
0
0
0
0.000
45%
1
2
3
4
확률
NPV
NPVHE  200, 000( P / F , 6%,1)  250, 000( P / A,10%,3)( P / F ,10%,1)  376,510
NPVME  200, 000( P / F , 6%,1)  100, 000( P / A,10%,3)( P / F ,10%,1)  37, 400
NPVLE  0
 The Expected NPV with One year Delayed.
E[ NPV1_ yr _ delayed ]  0.25(376,510)  0.30(37, 400)  0.45(0)  105,350
ROP  E[ NPV1_ yr _ delayed ]  NPV  105,350  69, 200  36,510
Care should be exercised here that E[NPV] is a net present project value
involving an option value as indicated before.
A Decision with a Binomial Lattice Model

You first determine volatility, σ, like variance

To do this work needs the following formula.
 
   2

T
ln 
  1




T



T
μT: the present value of the project at the maturity date of T: 269,000
σT: the variance of the investment project at the maturity date of T
A Variance of the cash flows from year 2 to 4.
i) A Case with HE: 250,000(P/A,10%,3)=622,000 with p=0.25
ii) A Case with ME: 100,000(P/A,10%,3)=249,000 with p=0.30
iii) A Case with LE: 35,000(P/A,10%,3)=87,000 with p=0.40
Var[CF ]  [622, 0002 (0.25)  249, 0002 (0.30)  87, 000 2 (0.45)]  (269,3502 )  (214,8902 )
 Now we are ready to determine the variance of the present value of the cash flow.
1
1 
 1
Var[ PV ]  (214,8902 )  2  4  6   (332,0062 )
 1.06 1.06 1.06 
 Then, finally we can calculate the volatility of the project.
 
  332, 008  2

ln  
 1 
269,
350




  0.55
3
 It is now time to determine the up- and down- rate of the project value after 1 year.
u  e 0.55
1
 1.73
d  1 / 1.73  0.58
1.73
269,350
0.58
465,975
156,223
Real Call Options Value
(1  0.06)  0.58
p 
 0.42
1.73  0.58
0.42(265, 975)
C 
 105, 386
1.06
The ROV with a B-S Model
S 0  269, 350
  55%
rf  6%
K  200, 000
d1

 269, 350 
0.552
ln 

0.06

2
200, 000 



0.55 1
d 2  0.925  0.55
 (1)
 0.925
1  0.375
C  269, 350 N (0.925)  200, 000e 0.06 (1) N (0.375)
=99,836
A difference in the ROV between a binomial lattice model and a B-S model is 5,550
which is just 2.8% in terms of a strike price .