Options and futures market #5
Transcript of Options and futures market #5
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Options and Futures Markets
Class #5
30.4.2015
Prof Rafi Eldor
Mr. Eitan Zeevi
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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
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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
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Option
C
10
0
Stock
100
110
90
Solution
Binomial Model
Values of Stock and Call Option – End of Period
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Hedge Ratio
090 *10110 * xhxh
5.020
10
100)9.01.1(
010 *
h
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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
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( 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:
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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%
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Binomial Model Formula
R = 1+ r
RCddu
RuCu
du
dRC /
Call option value can also be calculated directly
from the below formula:
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Call Option Value
Call option value:
C = (0.75 x 10 + 0.25 x 0) / 1.05 = 7.14
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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
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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?
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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
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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
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38.210005.1
10014.7
)1(
S
r
XCP
Put Option Value
Put option value can be derived by:
1. Calculating Call option value
And
2. Using Put Call Parity formula
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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)
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Put Option Value
38.205.1
10
4
310
4
1
1
RPd
du
RuPu
du
dRP
Put option value:
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SummaryCase 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
147051
10
9011
90051.
...
..
C 382
051
110
4
10
4
3.
.
P
5010020
10.
.
h
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הערות למודל הבינומי
לא נזקקים להסתברויות למצבי העולם בכדי .לחלץ את מחיר האופציה
בנוסחה הינם ההסתברויות Cd-וCuמקדמי .למימוש מצבי העולם בעולם ניטרלי לסיכון
.הינה יחסיתDOWN-וUPההגדרה של מצבי העולם . יתכן כי מחיר הנכס בשני מצבי העולם
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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
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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
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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
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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
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Example 1
38.205.1/104
10
4
3
14.705.1/109.01.1
9.005.1
P
C
100
110
90
C(100)
10
0
P(100)
0
10
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Example 2
14.705.1/208.02.1
05.12.1100
9.1105.1/208.02.1
8.005.1100
P
C
100
120
80
C(100)
20
0P(100)
0
20
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Summary
911051
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
147051
20
8021
05121.
...
..
P
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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
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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?
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European Option
100EC
Cu
100110
9099
121
81
Cd
21
0
0
100EP
EPu
EPd
0
1
19
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Call Option Value
1505.1
214
3
0
Cu
Cu
Therefore
71.1005.1
154
3
)100( eC
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Summary – Call Option
15051
121
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
0Cd
95011020
21.
.
hu 0hd
75010020
15.
.
h
7110051
115
4
3.
.
C
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Put Option Value
23.505.1
194
11
4
3
23.005.1
194
11
4
3
E
E
Pd
Pu
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Put Option Value
4.105.1
238.04
323.5
4
1
100
EP
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Summary – Put Option
230051
1
4
1.
.
Pu
u= 1.10
d= 0.90
X= 100
r= 5%
045011020
1.
.
hu 1
9020
18
.hd 250
10020
5.
.
h
41051
1
4
23803
4
235.
.
..
P
S= 121
110
100 99
90
81
P= 0
0.24
1.42 1
5.24
19
235051
1
4
19
4
3.
.
Pd
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American Option Value
Now lets check for American Options
Is there a difference from European Option?
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American Option Value
10PdFor American Options
55.205.1
104
1238.0
4
3
)100(
AP
There is no difference for a Call option as early exercise is not recommended
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Summary – American option
230051
1
4
1.
.
Pu
u= 1.10
d= 0.90
X= 100
r= 5%
045011020
1.
.
hu 1
9020
18
.hd 490
10020
24010.
.
.
h
552051
1
4
23803
4
10.
.
.
P
10Pd
S= 121
110
100 99
90
81
P= 0
0.24
2.55 1
10.00
19
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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)
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Call Option Value
05.1
4
15
05.1
54
3
)105( C
S=100
110
90
C=105
5
0
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Put Option Value
05.1
4
15
05.1
154
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
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Summary
573051
15
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
573051
115
20
050.
..
.
P
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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
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Solution #1
8.901.1
15
5.02
01.12)25(
05.501.1
0155.02
5.001.1
)25(
P
C
S 20=
40
10C(25)
15
0
P(25)
0
15
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Summary
055011
115
502
50011.
..
..
C 89
011
115
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
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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)
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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)
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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)
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Solution #2
Underlying asset price
S=100
110
90
99
121
81
133.1
108.9
89.1
72.9
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Solution #2
Call option price
C(100)
Cu
Cd
Cud
Cuu
Cdd
Cuuu=33.1
Cuud=8.9
Cddu=0
Cddd=0
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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
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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
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Question 3.1
100LIP100LIC
S=100110
9099
121
81
133.1
108.9
89.1
72.9
Calculate the value of: and
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Question 3.2
S=100110
9099
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
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Question 3.3
S=100110
9099
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)
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Question 3.4
S=100110
9099
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)
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S=100110
9099
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
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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%
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Generalization
rCppCC
rCppCC
rCppCC
dddddudd
dduduudu
duuuuuuu
/1
/1
/1
dudrpwhere
rCppCC
rCppCC
rCppCC
du
dddud
duuuu
//
/1
/1
/1
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33223
222
222
/11313
112
112
rCpCppCppCp
C
rCpCppCpC
rCpCppCpC
ddddduduuuuu
ddddduduud
dduduuuuuu
Generalization