Question 1: As an options trader, you constantly analyse the share market, looking for
profitable opportunities. Currently you are unsure of the general equity market direction, but
you anticipate very high volatility over the next few months. Shares in company ABC trade at
$30 and has the following six-month European options available:
Strike price Option class Option premium
(a)
$25
Call
$6.07
$25
Put
$0.09
$30
Call
$2.31
$30
Put
$1.14
$35
Call
$0.53
$35
Put
$4.16
Construct a straddle from the options listed above. What position (i.e., long or short)
will you take?
Construct a long straddle.
Long a call with strike of 30 and premium of $2.31.
Long a put with strike of 30 and premium of $1.14.
(b)
What is the premium of the long straddle? Construct a table showing the net payoff of
the straddle.
Premium = 2.31 + 1.14 = $3.45.
Stock price
S < 30
S > 30
Long call
0
S – 30
Long put
30 – S
0
Premium
-3.45
-3.45
Net payoff
26.55 – S
S – 33.45
(c)
Consider an alternative strategy: long a call with strike of $35 and long a put with strike
of $25. What is (are) the difference(s) between this new strategy and long straddle?
The premium is lower: 0.53 + 0.09 = $0.62.
Larger price moves are required to make money.
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Question 2: Discuss the differences between long covered call and short naked call.
Long covered call involves selling a call on stock the investor owns. Thus, this strategy is very
conservative; the investor receives the premium income from writing the call. If the call is
exercised, the stock is called away from the investor; thus, the investor has limited his or her
upside potential.
Short naked call is a very risky strategy. The investor sells a call on a stock the investor does
not own. If the price of the stock increases, the option will be exercised by the call owner and
the investor must go into the open market and buy the stock at the prevailing market price.
Theoretically, the price to which the stock can increase is unlimited; thus, the investor's
potential loss is unlimited.
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Question 4: Why would you prefer both Delta and Gamma hedge?
Delta hedge ensures the value of the portfolio is not sensitive to changes in the underlying.
Gamma hedge provides additional hedging accuracy in the sense that it ensures delta is not
very sensitive to changes in the underlying so that rebalancing of the hedge does not need to
be frequent.
Question 5: A stock is trading at 55. A call option on the stock with a maturity of three months
is trading at 3.3 and has a delta of 0.7. If the stock price increases by 2 (i.e., 55 to 57), will the
new call price more than 4.7. Explain.
With a delta of 0.7, a $2 increase in the stock price would lead to approximately $1.4 in the
call price. Since the option is a convex function of the stock price, the true option price change
will be more than $1.4, hence the new price call will be more than 3.3+1.4=$4.7.
Question 6: With the aid of appropriate profit and loss profiles, explain how you can create a
synthetic short position in a stock by combining positions in call options and put options on the
stock. Outline the circumstances under which you might wish to create a synthetic short
position in a stock.
Synthetic short
=
Long put
+
Short call
Percent Profit or Loss
10
8
6
4
2
0
-1
-4
-6
-8
-10
-10 -8
0
0
-6
-4
-2
0
2
4
6
8
10
-10 -8
-6
-4
-2
0
2
4
6
8
10
-10
-8
-6
-4
-2
0
2
4
6
Percent Change in Spot
There can be an advantage to taking the synthetic short position if it is cheaper to construct
than the actual short position in the underlying or if there are institutional restrictions whish
prevent a short position being taken for a specific time horizon.
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8
10
It is also possible to create a synthetic short position in a stock by selling a call, buying a put,
and borrowing the strike price at the risk-free rate until expiration – as per the Put-Call Parity
theorem.
Question 7: You have the option to invest $100 in one of the two alternatives:

Invest in stock that is currently trading at $100 per share.

Buy three-month 100-strike call option on above stock at $4 each.
You expect the stock price to appreciate with a maximum price of $110 after three months.
What are the maximum returns in percentage on investment that you can generate by using
stock and option, respectively?
Invest all in stock:
$100 can buy 1 share
Return = (110 - 100)/100 = 10%
Invest all in options:
$100 can buy 100/4 = 25 calls
Gross payoff for 25 options = 25*(110-100) = 250
Return = (250 - 100)/100 = 150%
Or
Gross payoff for 1 option = 110 - 100 = 10
Return = (10 – 4)/4 = 150%
The return for 1 option is the same as the return for 25 options.
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Question 8: The ABC stock currently stands at 5000 and has a volatility of 20% per annum.
The risk-free rate of interest is 4.25% per annum and the stock provides a dividend yield of
2.5% per annum, both with continuous compounding. Using the Black‐Scholes model,
calculate the value of six-month European call and put with an exercise price of 4900.
Step 1: adjust share price to reflect the dividend payment.
S = 5000*e^(-0.025*0.5) = 4937.8890
Step 2: input adjusted share price of 4937.8890 into BSM model.
d1 = (ln(4937.8890 /4900) + (4.25%+0.5*0.2^2)*0.5) / (0.2*sq(0.5)) = 0.2754
d2 = 0.2754 - 0.2*sq(0.5) = 0.1340
N(d1) = N(0.2754) = N(0.27) + 0.54*[N(0.28)-N(0.27)] = 0.6064 + 0.54*[0.6103-0.6064] =
0.6085
N(d2) = N(0.1340) = N(0.13) + 0.40*[N(0.14)-N(0.13)]= 0.5517 + 0.40*[0.5557-0.5517] =
0.5533
Note: it is also fine if you keep 2 decimal points. For example, N(d2) = N(0.1340) ≈ N(0.13)=
0.5517
N(-d1) = N(-0.2754) = 1- N(0.2754) = 1 - 0.6085= 0.3915
N(-d2) = N(-0.1340) = 1- N(0.1340) = 1 - 0.5533= 0.4467
C = S*N(d1) − e^(-rt)*K*N(d2) = 4937.8890*0.6085− e^(-4.25%*0.5)*4900*0.5533 =
350.5400
P = e^(-rt)*K*N(−d2) – S*N(−d1) = e^(-4.25%*0.5)*4900*0.4467 - 4937.8890*0.3915 =
209.6245
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