Options and futures market #5

1
©‫כל‬‫שמורות‬ ‫הזכויות‬‫רפי‬ ‫לפרופסור‬‫אלדור‬
Options and Futures Markets
Class #5
30.4.2015
Prof Rafi Eldor
Mr. Eitan Zeevi

2
The Binomial model was first introduced by w.
Sharpe and formalized by Cox, Ross and
Rubinstein in 1979. (“Option pricing: A
simplified approach”)
Binomial Model
The Binomial model is an option valuation
model, the model provides an easy
explanation to option pricing
Option pricing is different from bond and
equity pricing

3
Binomial Model
An Example
Assumptions:
 One Step Binomial Tree
 Current stock price is $100
 Interest Rate 5%
 Stock price can move up to $110 or move down to $90
Calculate the value of a Call option with Strike of $100 at
the beginning of the period

4
Option
C
10
0
Stock
100
110
90
Solution
Binomial Model
Values of Stock and Call Option – End of Period

5
Hedge Ratio
090*10110*  xhxh
5.0
20
10
100)9.01.1(
010
* 


h

6
Risk Neutral Portfolio
The value of the portfolio at the end of the period:
0.5 x 110 – 10 = 0.5 x 90 – 0 = 45
Therefor, for each Call option we sell, we need to
buy 0.5 X Stock

7
( h*S –C ) x (1 + r ) = 45
Risk Neutral Portfolio
Under the assumption of arbitrage free market and
future stock prices are known (either $110 or $90),
the following formula must exists:

8
Call Option Value
(Investment t0) X ( 1+ Interest Rate) = 45
Or
(50-C) = 45/1.05 = 42.86
Therefore
C = 50-42.86 = 7.14
Interest Rate = 5%

9
Binomial Model Formula
R = 1+ r
 
 
 
 
RCd
du
Ru
Cu
du
dR
C /












Call option value can also be calculated directly
from the below formula:

10
Call Option Value
Call option value:
C = (0.75 x 10 + 0.25 x 0) / 1.05 = 7.14

11
Replicating Portfolio
OptionStock
100
110
90
Bond
50
55
45
100
105
105
42.86
45
45
C(100)
10
0
10
0
50-42.86

12
Arbitrage Profit
An example of Arbitrage profit:
Assume the same parameters as the previous
example, but in this case:
- Call Option price is 10 ILS
Please check if arbitrage profit is possible?

13
Arbitrage Table
Action Cash Flow S(T)=110 S(T)=90
Sell Call 10+ 10- 0
Buy 0.5
Stock
50- 55+ 45+
Take a
Loan
40+ 42- 42-
Total 0 3+ 3+
Cash Flow at Expiration Date

14
Put Option Value
Assumptions are the same as previous example
(Call Option)
Calculate the value of Put option with Strike of
$100 at the beginning of the period

15
38.2100
05.1
100
14.7
)1(


 S
r
X
CP
Put Option Value
Put option value can be derived by:
1. Calculating Call option value
And
2. Using Put Call Parity formula

16
Put Option Value
Direct Calculation
Put option value can also be calculated by
creating a risk neutral portfolio,
In this case we need to Buy both the Stock
and the Option as the correlation between
the prices is negative
(In the previous example for Call option we had
to buy one and sell the other one)

17
Put Option Value
 
 
 
 
38.2
05.1
1
0
4
3
10
4
1
1

















R
Pd
du
Ru
Pu
du
dR
P
Put option value:

18
Summary
Case A
u= 1.10 S=
d= 0.90 110
X= 100
r= 5% 100
90
C= P=
10 0
7.14 2.38
0 10
147
051
10
9011
90051
.
...
..








C 382
051
1
10
4
1
0
4
3
.
.






P
50
10020
10
.
.







h

19
‫הבינומי‬ ‫למודל‬ ‫הערות‬
‫בכדי‬ ‫העולם‬ ‫למצבי‬ ‫להסתברויות‬ ‫נזקקים‬ ‫לא‬
‫האופציה‬ ‫מחיר‬ ‫את‬ ‫לחלץ‬.
‫מקדמי‬Cu‫ו‬-Cd‫ההסתברויות‬ ‫הינם‬ ‫בנוסחה‬
‫לסיכון‬ ‫ניטרלי‬ ‫בעולם‬ ‫העולם‬ ‫מצבי‬ ‫למימוש‬.
‫העולם‬ ‫מצבי‬ ‫של‬ ‫ההגדרה‬UP‫ו‬-DOWN‫יחסית‬ ‫הינה‬.
‫העולם‬ ‫מצבי‬ ‫בשני‬ ‫הנכס‬ ‫מחיר‬ ‫כי‬ ‫יתכן‬.

20
Two Steps Binomial Tree
We can extend the analysis to a two steps
binomial tree
First we need to calculate the second step as
shown in previous example, then we can
calculate the first step using Cu and Cd from
earlier calculation
The procedure is to work back through the tree
from the end to the beginning

21
American Option
As an option with American expiration style
can be exercised at any time during the option
life, the calculation is different,
In this case we need to compare the value of
the portfolio at the first step to the present
value of the second step

22
Multi Steps Binomial Model
We will use the same methodology for
calculation, the procedure is to work back
through the tree from the end to the
beginning
When the number of periods is infinite the
price of the option will be similar to Black and
Schools model

23
Example 1
dS=90 , uS=110 , r=5% , S=100
dS=80 , uS=120 , r=5% , S=100
- One step tree
- Exercise price for Call and Put = 100
Show that an increase in volatility will have a
positive affect on both Call and Put prices

24
Example 1
38.205.1/10
4
1
0
4
3
14.705.1/10
9.01.1
9.005.1
















P
C
100
110
90
C(100)
10
0
P(100)
0
10

25
Example 2
 
  14.705.1/20
8.02.1
05.12.1
100
9.1105.1/20
8.02.1
8.005.1
100


















P
C
100
120
80
C(100)
20
0
P(100)
0
20

26
Summary
911
051
20
8021
80051
.
...
..








C
Case B
u= 1.20 S=
d= 0.80 120
X= 100
r= 5% 100
80
C= P=
20 0
11.90 7.14
0 20
147
051
20
8021
05121
.
...
..








P

27
Conclusion
When uncertainty increases (volatility is higher) the
value of the Call option will be higher,
Call option price changes from 7.14 -> 11.9
And also the price of the Put options will be higher,
Put option price changes from 2.38 -> 7.14

28
Binomial Model – 2 Steps
- 2 Steps
- r = 5%, Stock = 100
- For each period u = 1.1, d = 0.9
1. What is the value of Call and Put Option?
2. Compare these values to 1 step
3. Does the hedge ration change?

29
European Option
 100E
C
Cu
100
110
90
99
121
81
Cd
21
0
0
 100E
P
E
Pu
E
Pd
0
1
19

30
Call Option Value
15
05.1
21
4
3
0


Cu
Cu
Therefore
71.10
05.1
15
4
3
)100( e
C

31
Summary – Call Option
15
051
1
21
4
3







.
Cu
u= 1.10
d= 0.90
X= 100
r= 5%
S= 121
110
100 99
90
81
C= 21
15.00
10.71 0
-
0
0Cd
950
11020
21
.
.







hu 0hd
750
10020
15
.
.







h
7110
051
1
15
4
3
.
.






C

32
Put Option Value
23.5
05.1
19
4
1
1
4
3
23.0
05.1
19
4
1
1
4
3






E
E
Pd
Pu

33
Put Option Value
  4.1
05.1
238.0
4
3
23.5
4
1
100 

E
P

34
Summary – Put Option
230
051
1
4
1
.
.






Pu
u= 1.10
d= 0.90
X= 100
r= 5%
0450
11020
1
.
.







hu 1
9020
18



.
hd 250
10020
5
.
.







h
41
051
1
4
23803
4
235
.
.
..





 
P
S= 121
110
100 99
90
81
P= 0
0.24
1.42 1
5.24
19
235
051
1
4
19
4
3
.
.






Pd

35
American Option Value
Now lets check for American Options
Is there a difference from European Option?

36
American Option Value
10PdFor American Options
55.2
05.1
10
4
1
238.0
4
3
)100( 

A
P
There is no difference for a Call option as early
exercise is not recommended

37
Summary – American option
230
051
1
4
1
.
.






Pu
u= 1.10
d= 0.90
X= 100
r= 5%
0450
11020
1
.
.







hu 1
9020
18



.
hd 490
10020
24010
.
.
.








h
552
051
1
4
23803
4
10
.
.
.





 
P
10Pd
S= 121
110
100 99
90
81
P= 0
0.24
2.55 1
10.00
19

38
Change is Strike Price
- 1 Step
- r = 5%, Stock = 100
- For each period u = 1.1, d = 0.9
- Exercise price for Call and Put = 105
Calculate the value of a Call and Put options
and compare to the values of the options with
strike 100 (from previous example)

39
Call Option Value
05.1
4
15
05.1
5
4
3
)105( C
S=100
110
90
C=105
5
0

40
Put Option Value
05.1
4
15
05.1
15
4
1
)105( P
S=100
110
90
P(105)
0
15
We can see that if Strike price is equal to the future
price of the underlying asset, the price of a Call
options = price of Put option

41
Summary
573
051
1
5
20
150
.
..
.






C
u= 1.10 S=
d= 0.90 110
X= 105
r= 5% 100
90
C= P=
5 0
3.57 3.57
0 15
573
051
1
15
20
050
.
..
.






P

42
Question #1
- 1 Step
- Stock Price = $20, can move up to $40 or
down to $10
- r = 10%
- Call and Put exercise price = $25
- Calculate the value of a Call and Put options
- Check if there is an arbitrage in case C(25) = 3
- Check if there is an arbitrage in case P(25) = 11

43
Solution #1
8.9
01.1
15
5.02
01.12
)25(
05.5
01.1
015
5.02
5.001.1
)25(









P
C
S 20=
40
10
C(25)
15
0
P(25)
0
15

44
Summary
055
011
1
15
502
50011
.
..
..









C 89
011
1
15
502
0112
.
..
.









P
u= 2.00 S=
d= 0.50 40
X= 25
r= 1% 20
10
C= P=
15 0
5.05 9.80
0 15

45
Arbitrage Table #1
Action Cash
Flow
S(T)=10 S(T)=40
Buy Call -3 0 15
Sell ½ Stock 10+ -5 -20
Deposit -7 +7.07 +7.07
Total 0 +2.07 +2.07
3=If C(25)

46
Arbitrage Table #1
Action Cash
Flow
S(T)=10 S(T)=40
Sell Put +11 -15 0
Sell ½ Stock 10+ -5 -20
Loan -21 +21.21 +21.21
Total 0 +1.21 +1.21
11=If P(25)

47
Question #2
Calculate the value of Call and Put options
- 3 Steps
- Stock = 100
- Strike = 100
- r=0.05
- d = 0.9, u = 1.1 (constant for all steps)

48
Solution #2
Underlying asset price
S=100
110
90
99
121
81
133.1
108.9
89.1
72.9

49
Solution #2
Call option price
C(100)
Cu
Cd
Cud
Cuu
Cdd
Cuuu=33.1
Cuud=8.9
Cddu=0
Cddd=0

50
Solution #2
Cuu = (0.75* 33.1 +0.25*8.9)/ 1.05=25.76
Cud = 0.75*8.9 / 1.05=6.36
Cdd=0
Cu = (0.75*25.76 + 0.25*6.35) / 1.05=19.91
Cd = 0.75*6.35 / 1.05=4.53
     3.1505.1/53.425.091.1975.0 C
Call options value

51
Question 3
 100,100, 32  SSOMAXCLI
 32 100,100, SSOMAXPLI 
- 3 Steps
- S = 100, r = 5%
- X = 100
- Price can move up or down 10%
- Price of Call option and Put option can be
locked after the second step

52
Question 3.1
100LIP100LIC
S=100
110
90
99
121
81
133.1
108.9
89.1
72.9
Calculate the value of: and

53
Question 3.2
S=100
110
90
99
121
81
133.1
108.9
89.1
72.9
What is the value of an Asian Call and Put
option?
Using the average prices of the second and third
steps

54
Question 3.3
S=100
110
90
99
121
81
133.1
108.9
89.1
72.9
What is the value of a Call (100) with Knock-Out
trigger at $100?
(The option no longer exists is the price of the
stock is below $100)

55
Question 3.4
S=100
110
90
99
121
81
133.1
108.9
89.1
72.9
What is the value of a Put (100) with Knock-Out
trigger at $100?
(The option no longer exists is the price of the
stock is above $100)

56
S=100
110
90
99
121
81
133.1
108.9
89.1
72.9
Question 3.5
Calculate the value of a portfolio with Lookback
Call and Lookback Put with the same expiration
date

57
Summary – 3 Steps
S= 133.1
121
110 108.9
100 99
90 89.1
81
72.9
C= 33.1 P= 0
25.76 -
19.91 8.9 0.62 0
15.31 6.357 1.69 2.60
4.541 0 5.24 10.9
0 14.24
0 27.1
u= 1.10
d= 0.90
X= 100
r= 5%

58
Generalization
  
  
   rCppCC
rCppCC
rCppCC
dddddudd
dduduudu
duuuuuuu
/1
/1
/1



  
  
  
    dudrpwhere
rCppCC
rCppCC
rCppCC
du
dddud
duuuu




//
/1
/1
/1


59
    
    
       33223
222
222
/11313
112
112
rCpCppCppCp
C
rCpCppCpC
rCpCppCpC
ddddduduuuuu
ddddduduud
dduduuuuuu




Generalization

Options and futures market #5

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Options and futures market #5