Project – Financial Instruments
Chapter 1
Some definitions
To long: to buy
To short: to sell
Call option: an option to buy (you can always back out)
Put option: ab option to sell (you can always back out)
Over-the-counter-market: a market via telephone and computer
Forward and futures contracts
Forward: over-the-country
Futures: popular on currencies and interests (organized stock exchange).
The biggest difference between those two is that the futures contact is more risky.
European: only at maturity
American: at any time
Arbitrage
If there is continuous compounding (less than one year) you use the following formula:
PV =S0∗e rT
If : PV>FV no arbitrage is possible
PV<FV arbitrage is possible
If there is discrete compounding (more than one year) you use the following formula:
F=S ( 1+ r )t
Arbitrage example with discrete compounding
Suppose that:
The spot price of oil is US$70
The quoted 1-year futures price of oil is US$80
The 1-year US$ interest rate is 5% per annum
The storage costs of oil are 2% per annum
Is there an arbitrage opportunity?
70∗( 1+ 0.05−0.02 )=72.1
72.1 < 80: there is arbitrage opportunity if you long.
1
,Project – Financial Instruments
Chapter 2
A margin is cash or marketable securities deposited by an investor with his or her broker.
Example
An investor takes a short position in 2 May 6 th crude oil futures contract on February 8 th. The current
price is $2.75 and the margin requirement is $25.000 per contract. Fill in the gaps
Day Futures P Daily gain Cumulative Margin Margin call
gain account
balance
2.75 20.000
08-04 2.80 (2500) (b) 0
18-04 2.60 2000 27500 (c)
20-04 2.85 (a) 7500 (d) 12500 (e)
24-04 3.00 (7500) (12500) 8060 0
−7500
a) =−0.15 →−0.15+3=2.85
50000
b) 2.75−2.80=−0.05−0.05∗50000=−2500
c) 2.75−2.60=0.15 → 0.15∗50000+ 20000=27500
d) 2.60−2.85=−0.25→−.25∗50000 →20000−12500=7500
e) 20000−12500=7500
2
,Project – Financial Instruments
Chapter 3
Hedging
Long: when you know you will buy an asset in the future and want to lock in the price.
Short: when you know you will sell an asset in the future and want to lock in the price.
For hedging Against hedging
Companies should focus on minimizing risk Shareholders are usually well diversified and can
and arise profits from market variables make their own decisions
May increase risk to hedge if competitors do not
explaining where there is a loss on the hedge and
a gain on underlying asset can be difficult
The hedge ratio is calculated by the following formula:
S
h
F
If a negative value arises, it means that you should long the futures contracts because of the negative
correlation. Negative assets are able to hedge against each other. The -h becomes possible.
Hedging the portfolio is done by the following formula:
β∗current portfolio
amount of contracts you need ¿ hedge=
futures∗index
In order to reduce the beta, you should long. In order to increase it, you should short.
When you want to calculate the total profit / loss, you calculate this by using the CAMP formula:
E ( rp )=rf + βp [ E ( rm )−rf ]
profit=current value∗CAMP%
3
, Project – Financial Instruments
Example hedging
Suppose S&P 500 index currently stands at 1,000. The 3-month Futures price of S&P 500 is 1,020. One
futures contract on S&P 500 index has a value of $250 times the index. An investor holds a portfolio
with the current value of $2.04 million, and the beta of portfolio is 1.5. Suppose further that the risk-
free rate is 4% per annum (thus 1% for 3 months).
a) What position in futures contracts on S&P 500 is necessary to hedge the portfolio?
2.04 m
1.5∗ ( 250∗1000 )=12.24
The futures contracts should be shorted because they are overpriced. 1020>1000. Short 12 contracts.
b) Following the above question, suppose the investor has optimally hedged his portfolio. What
would be the total profit or loss of his hedged portfolio if S&P 500 index changes to 1,100,
and the futures price of S&P 500 becomes 1,120 after 3 months?
1000
=10 %11 %−1.5 ( 10 %−1 % ) =14.5 %
1100
2.04 m∗0.145=295800( 1020−1120 ) 250∗12=−300.000
295800−300000=−4200
His loss is 4200.
Chapter 5
4
Chapter 1
Some definitions
To long: to buy
To short: to sell
Call option: an option to buy (you can always back out)
Put option: ab option to sell (you can always back out)
Over-the-counter-market: a market via telephone and computer
Forward and futures contracts
Forward: over-the-country
Futures: popular on currencies and interests (organized stock exchange).
The biggest difference between those two is that the futures contact is more risky.
European: only at maturity
American: at any time
Arbitrage
If there is continuous compounding (less than one year) you use the following formula:
PV =S0∗e rT
If : PV>FV no arbitrage is possible
PV<FV arbitrage is possible
If there is discrete compounding (more than one year) you use the following formula:
F=S ( 1+ r )t
Arbitrage example with discrete compounding
Suppose that:
The spot price of oil is US$70
The quoted 1-year futures price of oil is US$80
The 1-year US$ interest rate is 5% per annum
The storage costs of oil are 2% per annum
Is there an arbitrage opportunity?
70∗( 1+ 0.05−0.02 )=72.1
72.1 < 80: there is arbitrage opportunity if you long.
1
,Project – Financial Instruments
Chapter 2
A margin is cash or marketable securities deposited by an investor with his or her broker.
Example
An investor takes a short position in 2 May 6 th crude oil futures contract on February 8 th. The current
price is $2.75 and the margin requirement is $25.000 per contract. Fill in the gaps
Day Futures P Daily gain Cumulative Margin Margin call
gain account
balance
2.75 20.000
08-04 2.80 (2500) (b) 0
18-04 2.60 2000 27500 (c)
20-04 2.85 (a) 7500 (d) 12500 (e)
24-04 3.00 (7500) (12500) 8060 0
−7500
a) =−0.15 →−0.15+3=2.85
50000
b) 2.75−2.80=−0.05−0.05∗50000=−2500
c) 2.75−2.60=0.15 → 0.15∗50000+ 20000=27500
d) 2.60−2.85=−0.25→−.25∗50000 →20000−12500=7500
e) 20000−12500=7500
2
,Project – Financial Instruments
Chapter 3
Hedging
Long: when you know you will buy an asset in the future and want to lock in the price.
Short: when you know you will sell an asset in the future and want to lock in the price.
For hedging Against hedging
Companies should focus on minimizing risk Shareholders are usually well diversified and can
and arise profits from market variables make their own decisions
May increase risk to hedge if competitors do not
explaining where there is a loss on the hedge and
a gain on underlying asset can be difficult
The hedge ratio is calculated by the following formula:
S
h
F
If a negative value arises, it means that you should long the futures contracts because of the negative
correlation. Negative assets are able to hedge against each other. The -h becomes possible.
Hedging the portfolio is done by the following formula:
β∗current portfolio
amount of contracts you need ¿ hedge=
futures∗index
In order to reduce the beta, you should long. In order to increase it, you should short.
When you want to calculate the total profit / loss, you calculate this by using the CAMP formula:
E ( rp )=rf + βp [ E ( rm )−rf ]
profit=current value∗CAMP%
3
, Project – Financial Instruments
Example hedging
Suppose S&P 500 index currently stands at 1,000. The 3-month Futures price of S&P 500 is 1,020. One
futures contract on S&P 500 index has a value of $250 times the index. An investor holds a portfolio
with the current value of $2.04 million, and the beta of portfolio is 1.5. Suppose further that the risk-
free rate is 4% per annum (thus 1% for 3 months).
a) What position in futures contracts on S&P 500 is necessary to hedge the portfolio?
2.04 m
1.5∗ ( 250∗1000 )=12.24
The futures contracts should be shorted because they are overpriced. 1020>1000. Short 12 contracts.
b) Following the above question, suppose the investor has optimally hedged his portfolio. What
would be the total profit or loss of his hedged portfolio if S&P 500 index changes to 1,100,
and the futures price of S&P 500 becomes 1,120 after 3 months?
1000
=10 %11 %−1.5 ( 10 %−1 % ) =14.5 %
1100
2.04 m∗0.145=295800( 1020−1120 ) 250∗12=−300.000
295800−300000=−4200
His loss is 4200.
Chapter 5
4
