options and futures market #5

אלדורלפרופסור רפי הזכויות שמורות כל ©

Options and Futures Markets

Class #5

30.4.2015

Prof Rafi Eldor

Mr. Eitan Zeevi

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

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

Option

Solution

Binomial Model

Values of Stock and Call Option – End of Period

Hedge Ratio

090 *10110 * xhxh

100)9.01.1(

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

( 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:

Call Option Value

(Investment t0) X ( 1+ Interest Rate) = 45

(50-C) = 45/1.05 = 42.86

Therefore

C = 50-42.86 = 7.14

Interest Rate = 5%

Binomial Model Formula

R = 1+ r

Call option value can also be calculated directly

from the below formula:

Call Option Value

Call option value:

C = (0.75 x 10 + 0.25 x 0) / 1.05 = 7.14

Replicating Portfolio

OptionStock

C(100)

50-42.86

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?

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

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

38.210005.1

10014.7

Put Option Value

Put option value can be derived by:

1. Calculating Call option value

2. Using Put Call Parity formula

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)

Put Option Value

38.205.1

Put option value:

SummaryCase A

u= 1.10 S=

d= 0.90 110

X= 100

r= 5% 100

7.14 2.38

147051

90051.

5010020

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

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

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

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

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

Example 1

38.205.1/104

14.705.1/109.01.1

9.005.1

C(100)

P(100)

Example 2

14.705.1/208.02.1

05.12.1100

9.1105.1/208.02.1

8.005.1100

C(100)

0P(100)

Summary

911051

80051.

Case B

u= 1.20 S=

d= 0.80 120

X= 100

r= 5% 100

11.90 7.14

147051

05121.

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

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?

European Option

100110

Call Option Value

1505.1

Therefore

71.1005.1

)100( eC

Summary – Call Option

u= 1.10

d= 0.90

X= 100

S= 121

100 99

10.71 0

95011020

hu 0hd

75010020

7110051

Put Option Value

23.505.1

23.005.1

Put Option Value

4.105.1

238.04

Summary – Put Option

230051

u= 1.10

d= 0.90

X= 100

045011020

.hd 250

S= 121

100 99

1.42 1

235051

American Option Value

Now lets check for American Options

Is there a difference from European Option?

American Option Value

10PdFor American Options

55.205.1

1238.0

There is no difference for a Call option as early exercise is not recommended

Summary – American option

230051

u= 1.10

d= 0.90

X= 100

045011020

.hd 490

24010.

552051

S= 121

100 99

2.55 1

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)

Call Option Value

)105( C

Put Option Value

)105( P

P(105)

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

Summary

573051

u= 1.10 S=

d= 0.90 110

X= 105

r= 5% 100

3.57 3.57

573051

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

Solution #1

8.901.1

01.12)25(

05.501.1

0155.02

5.001.1

10C(25)

Summary

055011

50011.

u= 2.00 S=

d= 0.50 40

r= 1% 20

5.05 9.80

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)

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)

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)

Solution #2

Underlying asset price

Solution #2

Call option price

C(100)

Cuuu=33.1

Cuud=8.9

Cddu=0

Cddd=0

Solution #2

Cuu = (0.75* 33.1 +0.25*8.9)/ 1.05=25.76

Cud = 0.75*8.9 / 1.05=6.36

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

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

Question 3.1

100LIP100LIC

S=100110

Calculate the value of: and

Question 3.2

S=100110

What is the value of an Asian Call and Put

option?

Using the average prices of the second and third steps

Question 3.3

S=100110

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)

Question 3.4

S=100110

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)

S=100110

Question 3.5

Calculate the value of a portfolio with Lookback

Call and Lookback Put with the same expiration date

Summary – 3 Steps

S= 133.1

110 108.9

100 99

90 89.1

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

Generalization

rCppCC

dddddudd

dduduudu

duuuuuuu

dudrpwhere

rCppCC

/11313

rCpCppCppCp

rCpCppCpC

ddddduduuuuu

ddddduduud

dduduuuuuu

Generalization

options and futures market #5

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