Exercise 1 (5 points)
The stock index Alfa stands at 1500. The 3-month interest rate is equal to 4% and the expected
dividend yield is equal to 1.8% (both with continuous compounding).
a) Calculate the theoretical price of an index futures with 3 months maturity.
b) Suppose that each index point is worth 5 €. Find out the algebraic sign of the position and the
number of contracts that a portfolio manager must trade to minimize the variance of a stock
portfolio whose value is 5.000.000 € and whose beta is equal to 0.85. Suppose the hedge is
made with 3-month futures.
Solution
a)
The theoretical futures price can be calculated as
F  S * e r  q   1500 * e 0.040.018*0.25  1508.273
b) The value of each contract is equal to 1508.273 * 5  7541.365
In order to hedge the long position in shares the portfolio manager has to sell index futures. The
number of contracts to sell is equal to 5,000,000 * 0.85 / 7541.365  563.56 . We can round this
number to 564 contracts.
Exercise 2 (10 points)
Consider a short strangle constructed with two European options on the Toronto Stock exchange
index, with September 2006 maturity. The index futures with 2006 maturity currently stands at
1700, and the two options used for the short strangle are a put option strike 1600 (with premium
equal to 50) and a call option strike 1750 (with premium equal to 70).
a) Draw the profit/loss graph at maturity, with the hypothesis that the option premia are paid at
maturity.
b) What is the algebraic sign of Delta of the position at inception? Give reasons.
c) Add to the position described the purchase of two call options strike 1800. What is the algebraic
sign of the Gamma of the total position of four option contracts when the futures stands at
1700? Give reasons.
d) What is the algebraic sign of the Vega of the total position (four contracts) when the futures
stands at 1900?
e) Using the put-call parity show that an ATM forward call option and an ATM forward put option
on the same underlying asset have the same premium.
Solution
a) The following graph represents the profit/loss profile at maturity of a short straddle.
150
Profit/loss
100
50
0
1300
1400
1500
1600
1700
1800
1900
2000
2100
-50
-100
Index Futures
b) The sale of a call option represents a negative Delta strategy and the sale of a put represents a
positive Delta strategy. Both the options are out of the money but, given that the strike price of
the call is slightly closer to the underlying asset price, we can deduce that the call is less out of
the money than the put and therefore its Delta is somewhat higher in absolute value. As a
consequence, this is a slightly negative Delta position.
c) The two initial positions (short put, short call) are Gamma negative, whereas the purchase of
two call options strike 1800 is a positive Gamma strategy. Given that the Gamma is at
maximum levels when the option is at the money, we can deduce that the sale of the put strike
2
1600 can perfectly balance the purchase of one of the two call options strike 1800 bought (in
terms of Gamma). The remaining two positions can be easily analysed: the call strike 1750 sold
is closer to be ATM than the call strike 1800 purchased. Its Gamma is therefore higher (in
absolute value) and the total position is negative Gamma.
d) The two initial positions (short put, short call) are Vega negative, whereas the purchase of two
call options strike 1800 is a positive Vega strategy. Given that the Vega is at its maximum
levels when the option is at the money, if the futures stands at 1900 the two calls strike 1800
purchased will have higher Vega (in absolute value), and therefore prevails on the other two
options. As a consequence, the total position is positive Vega.
e) The out-call parity for a standard European options on an underlying asset that does not pay
dividends states that
c-p=S-Ke-rt
where c and p are the call and put premia, S is the underlying asset price, K is the strike price, r
is the risk-free interest rate and t is the option maturity.
If the option is at the money forward the following must be true: K=Sert. Therefore, the right
hand side of the equation above must be equal to zero (S-Ke-rt= S- Serte-rt =S-S=0). Thus, we
have that c=p.
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Exercise 3 (10 points)
A trader must assess several options on Lambda shares with 3-month maturity. The share does not
pay dividends and currently quotes at 40 €. Based on his estimation of the future volatility of the
return on Lambda shares, and knowing that the risk-free interest rate – with continuous
compounding – is equal to 6%, the trader builds the following binomial tree with 3 periods (the
time interval between two consecutive knots is one month):
72.88
59.67
48.86
40.00
48.86
40.00
32.75
32.75
26.81
21.95
Based on the data above:
a) calculate the value of a 3-month European call option strike 50 on Lambda shares;
b) calculate the Delta of such European call option strike 50 at t=2 (that is, at one month to
maturity), in case the Lambda shares quote at 59.67 (see the bold knot in the binomial tree
above);
c) calculate the value at t=0 of a 3-month Asian option that pays the difference between the
average price of Lambda shares observed at the end of each month and 50, in case the difference
is a positive value, and zero otherwise;
d) calculate the Delta of the Asian option, in case at t=2 we are in the bold knot of the tree;
e) calculate the implied volatility used by the trader to build the binomial tree and compute its
parameters.
Solution
a) The option value can be calculated as the present value of the expected payoff at maturity based
on the risk-neutral probabilities, using the risk-free rate to discount the payoff. In this case, we
know that a European call pays a strictly positive payoff when three ups are verified, and a final
price of 72.88 is obtained. Given p the risk-neutral probability of an upside movement, we have
e rt  d
p
ud
where r = 6%, t = 1/12, d =32.75/40 = 0.819 and u = 48.86/40 = 1.221 (u and d can be derived
from the tree). p is therefore equal to 0.4626.
Given that the probability to obtain 3 consecutive ups obtaining a final price of 72.88 is equal to
p3, the option value can be calculated as follows:
c  e  rt p 3  (72.88  50)  e 0.06 / 4  (0.4626) 3  22.88  2.232
b) The Delta is simply
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
C uuu  C uud
Su  Su d
3
2

22.88  0
 0.952
72.88  48.86
c) In this case we have to consider, using a similar reasoning, when the average price will be
higher that 50 at maturity. For example, in case the path is up-up-up the average price is
(48.86+59.67+72.88)/3=60.47
72.88
60.47-50=10.47
48.86
52.46-50=2.46
59.67
48.86
40.00
40.00
32.75
32.75
26.81
21.95
whereas in case the path is up-up-down the average price is equal to
(48.86+59.67+48.86)/3=52.46
All the other paths lead to an average price that is lower than 50, and therefore the option is
worthless. As a consequence, the value of the option is given by (always according to the riskneutral valuation)


c  e  rt p 3  10.47  p 2 1  p   2.46  1.300
Note that while the probability to have a final price of 48.86 is equal to 3p2∙(1-p), given that it is
sufficient that the path is one down and two ups disregarding the order of these price
movements, in the calculation of the Asian option payoff we have to consider the only path
where not only the final price is 48.86 (and this is not significant) but also the average price
during the path is above 50.
d) The Delta can be calculated as in the case of a standard Asian option. Therefore we have

C uuu  C uud
Su  Su d
3
2

10.47  2.46
 0.333
72.88  48.86
e) Since u  e t , from u and t we can derive the volatility used by the trader to build the
binomial tree. We have that

ln( u )
t
 69.28%
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Exercise 4 (6 points)
Consider the stock Lambda whose spot and forward (3 months maturity) prices are:
Spot price
Bid
Ask
40,00
40,10
Forward price on the market
(3 months)
40,25
40,35
The forward interest rates, with continuous compounding, are equal to 3% (bid) and 3.10% (ask).
a) Derive the no-arbitrage boundaries for the 3-month forward quotation of Lambda shares,
verifying the existence (or non existence) of arbitrage opportunities (suppose there are no
fees connected to the repo market).
b) Based on both the theoretical forward prices and on the market prices, verify the possible
existence of arbitrage opportunities for price-taker operators willing to:
e1) buy forward (3 months) the stock Lambda, or
e2) sell forward (3 months) the stock Lambda.
Solution
a) The two operations needed (cash and carry and reverse cash and carry) to find out the non
arbitrage boundaries are the following:
Cash and carry: borrow cash at 3.10%, buy the share at 40.10 and carry to maturity. This creates a
synthetic forward ask price of
40.10 * e 0.0310.25  40.412
Reverse cash and carry: sell the share at 40.00 and invest the cash obtained at 3.00% till maturity.
This creates a synthetic forward bid price of
40.00 * e 0.030.25  40.301
Given that the market prices (40.25 – 40.35) are both outside the non arbitrage boundaries there are
NO arbitrage possibilities. In fact, a price taker operator could only buy (synthetically) at 40.412
and sell to the market at 40.25 or sell (synthetically) at 40.301 and buy from the market at 40.35.
b) Although there are no arbitrage opportunities, a price taker operator willing to sell forward
would be better off doing it through a reverse cash and carry (receiving 40.301) than through
the market (receiving the market bid price of 40.25). Vice versa, an operator willing to buy
forward would be better off buying from the market at 40.35 instead of carrying out a cashand-carry strategy at a price of 40.412.
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