1a. Answer
The
current strike price of $80 will be split into two call options using 2for1
stock split rule. Hence, strike price for each call option will be $80/2 = $40.
1b. Answer
i.
For
a 20% stock Dividend
No of Shares

Strike Price

Maturity

Stock Dividend


200

80

6 Months

20%


Option Contact

Exercise Price


240

66.67


Hence,
the option contract will become one to buy 200 x 1.2 = 240 shares with an
exercise price 80/1.20 = 66.67.
ii.
For
a 20% cash dividend
There
will be no effect for 20% cash dividend, because the terms of an options
contract are not normally adjusted for cash dividends.
2a. Answer
i.
If
stock price volatility increases
This will result in an increase in
value of a call option, based on rule of “The greater the expected volatility,
the higher the option value”.
ii.
If
the time expiration increases
This will also result in an increase in
value of call option, based on the rule of “The longer the time until
expiration, the higher the option price”.
2b. Answer
i.
For
an European Call Option
European call option gives a right to the option holder to purchase one share
stock for a certain price. The option itself cannot be worth more than the
stock.
The upper bound for
European Call Option is: C ≤ S_{0},
C ≤ S_{0}
In case this
condition does not hold, we can make profit by selling an option and buying a
stock.
ii.
For
an European Put Option
European put option gives the holder the right to sell one share of stock for
the strike price. The put option cannot be worth more than the strike price.
The upper bound for
European Put Option is: P ≤
K, P ≤ K
However, since European
option can be exercised at maturity only, the European put option cannot be
worth more than the present value of the strike price.
3a. Answer
Portfolio
A: one
European call option plus an amount of cash equal to.
Portfolio
B: one
share of underlying stock
After years portfolio A
will yield an amount of cash equal to.
If, after years, the stock price is above, the call option in portfolio
A will be exercised, the share sold and the portfolio will be worth. Otherwise, after years,, the option is not exercised
and the portfolio will be worth .
Hence after T years portfolio A is
worth max(, and since portfolio
B is always worth after years, the initial value of portfolio A must
be no less that the initial value of portfolio B, which is just .
3b. Answer
Portfolio
C: one
European put option plus one share of underlying stock.
Portfolio
D: an
amount of cash equal to.
After years portfolio D will be worth.
If, after years,, then the put option in
portfolio C will be exercised; the share is sold for and the portfolio will be worth . Otherwise, if after years,, the option is not exercised and
the portfolio will be worth .
So after years portfolio C is worth and the initial value of portfolio C must be
no less that the initial value of portfolio D which is just.
3c.
Answer
Portfolio
A: one
European call option plus an amount of cash equal to.
Portfolio
C: one
European put option plus one share.
After T years they are both worth () where ST is the stock price after
T years.
These two portfolios must have identical initial
values, hence the putcall parity equation is:
4a.
Answer
A long
straddle position is entered into simply by buying a call option and a put
option with the same strike price and the same expiration month. The maximum
profit potential on a long straddle is unlimited. The maximum risk for a long
straddle will only be realized if the position is held until option expiration
and the underlying security closes exactly at the strike price for the options.
4b.
Answer
Breakeven price for call option will be = Strike
Price + Cost of the call option = $20 + $5 = $25, and Breakeven price for put
option will be = Strike Price  Cost of the put option = $20  $3 = $22
4c.
Answer
Profit and loss diagram for a long straddle looks
like this:
5a. Answer
The delta of a European futures call
option is usually defined as the rate of change of the option price with
respect to the futures price (not the spot price).
In this case:
                      
    
Hence,
             
             
             
             0.8538
5b. Answer
The Value of riskfree portfolio at
the end of six months is
         
                 
             
              0.3023
             
             x
             
              = 0.2875
5c. Answer
The Value of riskfree portfolio today
is
         
                 
             
              0.3023
             
             x
             
              = 0.30199
5d.
Answer
The value of
call option today is
             
              $80 x
             
              = $80 x 0.30199 = $24.16
6a. Answer
Stock Price = $80
Strike Price = $85
R_{f }= 0.1
Volatility_{ = }0.1
There are two six months period hence T= 0.5 for
each period, The binomial tree will be as following:
$88.88
$88
$80 $88.88
_{ } $72
$71.28
6b. Answer:
Yes, this could be exercised early at $88 in first
period of 6 months, because the only difference between evaluating American and
European option is that when pulling back for an American option, we need to
always compare the pulled back value and the exercise value. If the pulled back
value is greater than the exercise value, continue moving backwards along the
tree using this value. And, if the pulled back value is less than the exercise
value, then exercise is optimal and we need to use this value to move backwards
along the tree.
7 Answer:
Information


Stock Price now (P)

80

Exercise Price of Option (EX)

75

Number of periods to Exercise in years (t)

0.25

Compounded RiskFree Interest Rate (rf)

8.00%

Standard Deviation (annualized s)

20.00%

Equation Solution


Present Value of Exercise Price (PV(EX))

73.5149

s*t^.5

0.1000

d1

0.8954

d2

0.7954

Delta N(d1) Normal Cumulative Density
Function

0.8147

Bank Loan N(d2)*PV(EX)

57.8419

Value of Call

7.3348

8 Answer:
Information


Stock Price now (P)

80

Exercise Price of Option (EX)

75

Number of periods to Exercise in years (t)

0.25

Compounded RiskFree Interest Rate (rf)

8.00%

Standard Deviation (annualized s)

20.00%

Equation Solution


Present Value of Exercise Price (PV(EX))

73.5149

s*t^.5

0.1000

d1

0.8954

d2

0.7954

Delta N(d1) Normal Cumulative Density
Function

0.8147

Bank Loan N(d2)*PV(EX)

57.8419

Value of Put

0.8497



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