ANSWERS TO ASSIGNMENT QUESTIONS
Chapter 1
1A
An investment in call options entails higher risks but can lead to higher returns. If the stock price stays at
$94, an investor who buys call options loses $9,400 whereas an investor who buys shares neither gains
nor loses anything. If the stock price rises to $120, the investor who buys call options gains
2000 × (120 − 95) − 9400 = $40, 600
An investor who buys shares gains
100 × (120 − 94) =$2, 600
The strategies are equally profitable if the stock price rises to a level, S, where
100 × ( S − 94)
= 2000( S − 95) − 9400
or,
S = 100
The option strategy is therefore more profitable if the stock price rises above $100.
1B
ImportCo should buy three-month call options on $10 million with a strike price of 1.2400. ExportCo
should buy three-month put options on $30 million with a strike price of 1.2000. In this case, the spot
foreign exchange rate is 1.22185 (the average of the bid and offer quotes in Table 1.1.), the (domestic)
risk-free rate is 2%, the foreign risk-free rate is 1%, the volatility is 12%, and the time to exercise is 0.25
years. Using the Equity_FX_Index_Futures_Options worksheet in the DerivaGem Options Calculator
select Currency as the underlying and Black−Scholes−European as the option type. The software shows
that a call with a strike price of 1.24 is worth 0.02245 and a put with a strike of 1.20 is worth 0.01816.
This means that the hedging would cost 0.022245 × 10,000,000 or about $222,450 for ImportCo and
0.01816 × 30,000,000 or about $544,800 for ExportCo.
1C
The trader has a long European call option with strike price K and a short European put option with
strike price K . Suppose the price of the underlying asset at the maturity of the option is ST . If ST > K ,
the call option is exercised by the investor and the put option expires worthless. The payoff from the
portfolio is then ST − K . If ST < K , the call option expires worthless and the put option is exercised
against the investor. The cost to the investor is K − ST . Alternatively, we can say that the payoff to the
investor in this case is ST − K (a negative amount). In all cases, the payoff is ST − K , the same as the
payoff from the forward contract. The trader’s position is equivalent to a forward contract with delivery
price K .
Suppose that F is the forward price. If K = F , the forward contract that is created has zero value.
Because the forward contract is equivalent to a long call and a short put, this shows that the price of a
call equals the price of a put when the strike price is F.
, Chapter 2
2A
The price goes up during the time the company holds the contract from 120 to 125 cents per
pound. Overall the company therefore takes a loss of 15,000 × 0.05 = $750. If the company is
classified as a hedger, this loss is realized in 2023, If it is classified as a speculator, it realizes a
loss of 15,000 × 0.20 = $3000 in 2021, a gain of 15,000 × 0.30 = $4,500 in 2022, and a loss of
15,000 × 0.15 = $2,250 in 2023.
2B
If the transactions are cleared bilaterally, the company has to provide collateral to Banks A, B,
and C of (in millions of dollars) 0, 15, and 25, respectively. The total collateral required is $40
million. If the transactions are cleared centrally, they are netted against each other and the
company’s total variation margin (in millions of dollars) is –20 + 15 + 25 or $20 million in total.
The total margin required (including the initial margin) is therefore $30 million.
2C Excel file
(a) For crude oil, the standard deviation of daily changes is $1.5777 per barrel or $1577.7 per
contract. The standard deviation of two-day price changes is $ 1577.7 × 2 = $2,231.2 per
contract.
(b) Margin for member = $2,231.2 × 2.326 = $5,190.6
(c) Worksheet shows that the margin would be wiped out 24 times or on 2.31% of the days.
This suggests that price changes have heavier tails than the normal distribution.
(d) Worksheet shows that there would be 157 margin calls and the client has an incentive to
default 9 times.
Chapter 3
3A (Excel file)
Denote xi and yi by the i -th observation on the change in the futures price and the change in the
spot price, respectively.
∑x i 0.96 ∑ yi =
= 1.30
∑x 2
i 2.4474 ∑ yi2 =
= 2.3594
∑x y =
i i 2.352
An estimate of σ F is
, 2.4474 0.962
− =
0.5116
9 10 × 9
An estimate of σ S is
2.3594 1.302
− =
0.4933
9 10 × 9
An estimate of ρ is
10 × 2.352 − 0.96 × 1.30
= 0.981
(10 × 2.4474 − 0.962 )(10 × 2.3594 − 1.302 )
The minimum variance hedge ratio is
σS 0.4933
ρ =.
0 981× =.
0 946
σF 0.5116
3B (Excel file)
a) The number of contracts the fund manager should short is
50, 000, 000
0.87 × = 138.20
1259 × 250
Rounding to the nearest whole number, 138 contracts should be shorted.
b) The following table shows the impact of the strategy. To illustrate the calculations in the
table, consider the first column. If the index in two months is 1,000, the futures price is
1000 × 1.0025. The gain on the short futures position is therefore
(1259 − 1002.50) × 250 × 138 =$8, 849, 250
The return on the index is 3 × = 0.5% in the form of dividend and
− = −20% in the form of capital gains. The total return on the index is
therefore −19.5% . The risk-free rate is 1% per two months. The return is therefore
−20.5% in excess of the risk-free rate. From the capital asset pricing model, we expect
the return on the portfolio to be 0.87 × −20.5% = −17.835% in excess of the risk-free
rate. The portfolio return is therefore −16.835% . The loss on the portfolio is
0.16835 × 50, 000, 000 or $8,417,500. When this is combined with the gain on the futures,
the total gain is $431,750.
, Index now 1250 1250 1250 1250 1250
Index Level in Two Months 1000 1100 1200 1300 1400
Return on Index in Two Months -0.20 -0.12 -0.04 0.04 0.12
Return on Index incl divs -0.195 -0.115 -0.035 0.045 0.125
Excess Return on Index -0.205 -0.125 -0.045 0.035 0.115
Excess Return on Portfolio -0.178 -0.109 -0.039 0.030 0.100
Return on Portfolio -0.168 -0.099 -0.029 0.040 0.110
Portfolio Gain -8,417,500 -4,937,500 -1,457,500 2,022,500 5,502,500
Futures Now 1259 1259 1259 1259 1259
Futures in Two Months 1002.50 1102.75 1203.00 1303.25 1403.50
Gain on Futures 8,849,250 5,390,625 1,932,000 -1,526,625 -4,985,250
Net Gain on Portfolio 431,750 453,125 474,500 495,875 517,250
3C
To hedge the February 2022 purchase, the company should take a long position in March 2022
contracts for the delivery of 800,000 pounds of copper. The total number of contracts required is
800, , 000 = 32 . Similarly, a long position in 32 September 2022 contracts is required to
hedge the August 2022 purchase. For the February 2023 purchase, the company could take a
long position in 32 September 2022 contracts and roll them into March 2023 contracts during
August 2022. (As an alternative, the company could hedge the February 2023 purchase by taking
a long position in 32 March 2022 contracts and rolling them into March 2023 contracts.) For the
August 2023 purchase, the company could take a long position in 32 September 2022 and roll
them into September 2023 contracts during August 2022.
The strategy is therefore as follows:
Oct. 2021: Enter into long position in 96 Sept. 2022 contracts
Enter into a long position in 32 Mar. 2022 contracts
Feb 2022: Close out 32 Mar. 2022 contracts
Aug 2022: Close out 96 Sept. 2022 contracts
Enter into long position in 32 Mar. 2023 contracts
Enter into long position in 32 Sept. 2023 contracts
Feb 2023: Close out 32 Mar. 2023 contracts
Aug 2023: Close out 32 Sept. 2023 contracts
With the market prices shown the company pays
