Introduction to Derivatives
Energy in a Carbon Concerned Economy
HEC Certificate 2013
Instructor:
Prof. Christophe Pérignon
Deloitte – Société Générale Chair in Energy and Finance
HEC Paris
Office 29, W2 building
perignon@hec.fr
www.hec.fr/perignon
Course Objective:
The objective of this 3-hour course is to provide an overview of the
derivatives securities, with special emphasis on the energy and
commodity markets. We discuss the different classes of derivatives, as
well as the main pricing methods.
1-1
Introduction to Derivatives
Topic 1:
Introduction
Prof. Christophe Pérignon, HEC Paris
Energy in a Carbon Concerned Economy
HEC Certificate 2013
1-2
The Nature of Derivatives
• A derivative is a financial asset whose value depends on
the value of another asset, called underlying asset
• Examples of derivatives include Futures, Forwards,
Options, Swaps, Credit Derivatives, Structured Products
• Derivatives, while seemingly new, have been used for
thousands years
* Aristotle, 350 BC (Olive)
* Netherlands, 1600s (Tulips)
* USA, 1800s (Grains, Cotton)
* Spectacular growth since 1970’s
• Increase in volatility + Black-Scholes model (1973)
1-3
Examples of Underlying Assets
•
•
•
•
•
•
•
Stocks
Bonds
Exchange rates
Interest rates
Commodities/metals
Energy
Number of bankruptcies
among a group of
companies
• Pool of mortgages
• Temperature, quantity of
rain/snow
• Real-estate price index
• Loss caused by an
earthquake/hurricane
• Dividends
• Volatility
• Derivatives
• etc
1-4
Trading Activity
Notional Principal ($bio)
Number of Contracts (mio)
70000
180
60000
160
140
50000
120
40000
100
30000
80
60
20
2012
2010
2009
2008
2007
2006
2005
2004
0
2011
Options
2003
2012
2010
2009
2008
2007
2006
2005
2004
2003
2002
2001
2000
0
2011
Options
Futures
2002
10000
40
2001
Futures
2000
20000
Source: Bank for International Settlement (BIS)
1-5
Trading Activity (II)
Notional Amounts Outstanding ($bio)
800000
700000
600000
Others
CDS
Commodities
500000
Equity
400000
Interest rate
300000
Foreign exchange
200000
100000
0
2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012
Source: Bank for International Settlement (BIS)
1-6
Ways Derivatives are Used
• To hedge risks (reducing the risk)
• To speculate (betting on the future direction of
the market)
• To lock in an arbitrage profit (taking advantage of
a mispricing)
Net effect for society?
1-7
Risk Management in Practice
Survey of International Evidence on Financial Derivatives Usage by
Bartram, Brown and Fehle (2009, http://ssrn.com/abstract=424883):
•
7,319 non-financial firms from 50 countries
•
60% of the firms use derivatives
 45% FX risk / 35% Interest rate risk / 10% Commodity price risk
Hedging Increases Firm Value:
1-8
The Risk Management Policy of French Firms
Study Etude MEDEF-HEC 2012
http://appli9.hec.fr/hec-medef/doc/MEDEF-2012-rapport-final.pdf
5
4.5
4
3.5
3
Risque de récession
Coût de financement
Prix des matières
premières
Risque de change
1-9
The Risk Management Policy of French Firms (II)
Etude MEDEF-HEC 2012: Financial Hedging vs. Operational Hedging
4
3
2
1
Faire payer en Fournisseurs
euro ses en dehors de
clients hors la zone euro
zone euro
Forwards
Futures
Employés en Unités de
dehors de la production en
zone euro dehors de la
zone euro
Swaps
Dettes en
devises
étrangères
Payer des
fournisseurs
de la zone
euro en
devises
étrangères
COFACE
Options
1-10
Introduction to Derivatives
Topic 2:
Futures and Forwards
Prof. Christophe Pérignon, HEC Paris
Energy in a Carbon Concerned Economy
HEC Certificate 2013
2-1
Futures Contracts
• A FUTURES contract is an agreement to buy or sell
an asset at a certain time in the future for a certain
price
• By contrast in a SPOT contract there is an
agreement to buy or sell an asset immediately
• The party that has agreed to buy has a LONG
position (initial cash-flow = 0)
• The party that has agreed to sell has a SHORT
position (initial cash-flow = 0)
2-2
Futures Contracts (II)
• The FUTURES PRICE (F0) for a particular contract is the
price at which you agree to buy or sell
• It is determined by supply and demand in the same way as
a spot price
• Terminal cash flow for LONG position: ST - F0
• Terminal cash flow for SHORT position: F0 - ST
Futures are traded on organized exchanges:
• Chicago Board of Trade, Chicago Mercantile Exch. (USA)
• Montreal Exchange (Canada)
• EURONEXT.LIFFE (Europe)
• Eurex (Europe)
• TIFFE (Japan)
2-3
Example: Gold
Sept 07, 2011
(10.26 NY Time)
S0 = $1,810.1
Source: www.kitco.com
Oct 2011
$1,797.0
Nov 2011
$1,803.2
Dec 2011
$1,801.4
Dec 2012
$1,812.0
F0 (Nov 2011) = $1,803.2
Source: www.cmegroup.com
2-4
Quotes retrieved on September 7, 2010
2-5
Sep-18
Apr-18
Nov-17
Jun-17
Jan-17
Aug-16
Mar-16
Oct-15
May-15
Dec-14
Jul-14
Feb-14
Sep-13
Apr-13
Nov-12
Jun-12
Jan-12
Aug-11
Mar-11
Oct-10
Jul-12
May-12
Mar-12
Jan-12
Nov-11
Sep-11
Jul-11
May-11
Mar-11
Jan-11
Nov-10
Sep-10
CME Copper Futures Prices
3.52
3.51
3.5
3.49
3.48
3.47
3.46
3.45
CME Natural Gas Futures Prices
7
6.5
6
5.5
5
4.5
4
3.5
Measuring Interest Rates
•
•
•
•
A: Amount invested
n: Investment period in years
Rm: Interest rate per annum
m: Compounding frequency
For any n and m, the terminal value of an investment A at rate Rm is:
A(1+Rm / m)mn
limm –›∞ A(1+Rm / m)mn = A e r  n
where r is the continuously compounded interest rate per annum
• $100 grows to $100er×T at time T
• $100 received at time T discounts to $100e-r×T at time zero
• A risky cash-flow of $X received at time T discounts to $Xe-k×T at time
zero, where k = r + p and p is the risk premium
2-6
Forward Contracts
• Forward contracts are similar to futures except that they
trade on the over-the-counter market (not on exchanges)
• Forward contracts are popular on currencies and interest
rates
2-7
Market Organization
• Derivatives Exchanges vs. Over-the-Counter (OTC)
• Standardized vs. Tailor-Made Products
– Underlying asset
– Size of the position
– Delivery date
– Delivery location
– Market Makers and Liquidity
– Default risk and Collateral
2-8
Contract Specifications: Futures on CAC40 Index
Contract
CONTRAT À TERME FERME SUR L’INDICE CAC 40
(FCE)
Underlying Asset
CAC 40 stock index, made of 40 French blue chip
companies, computed by Euronext Paris SA, released
every 30 seconds (value of 1000 on Dec. 31, 1987)
Notional
Value of the index × 10 €
Minimum Tick
0,5 index point (5 €)
Maximum Price
Fluctuation
+/- 200 points with respect to last closing price.
As soon as the futures price exceed this limit, trading is
suspended
Maturity Date
Third Friday of the month at 4PM
Liquidation
Settled in Cash. The terminal value of the index is the
average value of the index between 3:40 and 4:00PM
(41 observations).
Margin
Margin requirement is 225 points per contract
Margin is reduced for trading on spread (long and short
positions on contracts with different maturities)
Transaction Cost
Trading Fee (Euronext Paris) : 0,14 €
Clearing Fee (LCH.Clearnet) : 0,13 €
2-9
Default Risk and Margins
• Two investors agree to trade an asset in the future
• One investor may:
– regret and leave
– not have the financial resources
Margins and Daily Settlement
2-10
Margins
• A margin is cash (or liquid securities) deposited by an
investor with his broker
• The balance in the margin account is adjusted to reflect
daily gains or losses: “Daily Settlement” or “Marking to
Market”
• If the balance on the margin account falls below a prespecified level called maintenance margin, the investor
receives a margin call
• If the investor is unable to meet a margin call, the position
is closed
• Margins minimize the possibility of a loss through a default
on a contract
2-11
Hedging with Futures: Theory
• Proportion of the exposure that should optimally be
hedged is:
sS
h r
sF
*
sS is the standard deviation of DS, the change in the spot
price during the hedging period,
sF is the standard deviation of DF, the change in the
futures price during the hedging period
r is the coefficient of correlation between DS and DF.
2-12
Hedging with Futures: Example
• Airline will purchase 2 million gallons of jet fuel in one
month and hedges using heating oil futures
• From historical data sF =0.0313, sS =0.0263, and r=
0.928
0.0263
*
h  0.928 
 0.7777
0.0313
• The size of one heating oil contract is 42,000 gallons
• Optimal number of contracts:
= 0.7777 × 2,000,000 42,000 = 37.03
2-13
Pricing Futures
• Suppose that:
– The spot price of gold is $1,250
– The quoted 1-year futures price of gold is
$1,300
– The 1-year US$ interest rate is 1.98% per
annum
– No income or storage costs for gold
• Is there an arbitrage opportunity?
2-14
• NOW
– Borrow $1,250 from the bank
– Buy gold at $1,250
– Short position in a futures contract
• IN ONE YEAR
– Sell gold at $1,300 (the futures price)
– reimburse 1,250  exp(0.0198) = $1,275
ARBITRAGE PROFIT = $25 
NOTE THAT ARBITRAGE PROFIT AS LONG AS
S0  exp(r T) < F0
2-15
Pricing Futures (II)
• Suppose that:
– The spot price of gold is $1,250
– The quoted 1-year futures price of gold is
$1,265
– The 1-year US$ interest rate is 1.98% per
annum
– No income or storage costs for gold
• Is there an arbitrage opportunity?
2-16
• NOW
– Short sell gold and receive $1,250
– Make a $1,250 deposit at the bank
– Long position in a futures contract
• IN ONE YEAR
– Buy gold at $1,265 (the futures price)
– Terminal value on the bank account 1,250  exp(0.0198)
= $1,275
ARBITRAGE PROFIT = $10 
NOTE THAT ARBITRAGE PROFIT AS LONG AS
S0  exp(r T) > F0
Therefore F0 has to be equal to S0  exp(r T) = $1,275
2-17
Futures Pricing (III)
For any investment asset that provides no
income and has no storage costs
F0 = S0erT
Immediate arbitrage opportunity if:
F0 > S0erT
F0 < S0erT
 short the Futures, long the asset
 long the Futures, short sell the asset
2-18
When an Investment Asset Provides
a Known Dollar Income
Consider a Futures on a bond
S0 = $900, F0 = $850
Tbond = 5 years, Tfutures = 1 year
Coupon in 6 months: $40
Coupon in 12 months: $40
r(6 months) = 1%, r(12 months) = 2%
• NOW
– Borrow $900 (39.80 for 6 m and 860.20 for 12 m)
– Buy 1 bond at $900
– Short position in the Futures
2-19
• IN 6 MONTHS
– Receive first coupon and reimburse $40
• IN 12 MONTHS
– Receive second coupon $40
– Sell the bond at $850 (futures price)
– Reimburse $860.20  exp(0.02) = 877.58
ARBITRAGE PROFIT = $12.42 
TO PREVENT AN ARBITRAGE PROFIT:
I2 + F0 – [S0 – I1exp(-r6m  0.5)] exp(r12m  1) = 0
F0 = [S0 – I1exp(-r6m  0.5) – I2exp(-r12m  1)] exp(r12m  1)
F0 = (S0 – I) exp(r  T) where I is the PV of all future incomes
2-20
When an Investment Asset Provides a
Known Yield
Yields: Income expressed as a % of asset price, usually
measured by continuous compounding per year, and denoted
by q
Yields work just like interest rates
e.g. Final value after T years of S0 dollars invested in an asset
generating a yield q is S0 eqT
Intuitively, we have:
with cash income: F0 = (S0 - I)erT
with yield: F0 = (S0 e-qT)erT = S0 e(r - q)T
2-21
Accounting for Storage Costs
Storage costs can be treated as negative income:
F0  (S0+U )erT
where U is the present value of the storage costs
Alternatively F0  S0 e(r+u )T
where u is the storage cost per unit time as a
percent of the asset value
2-22
Cost of Carry
• The cost of carry, c, is the storage cost plus the interest
costs less the income earned
• For an investment asset F0 = S0ecT
• For a consumption asset F0  S0ecT
• The convenience yield, y, is the benefit provided when
owning a physical commodity.
• It is defined as:
F0 = S0 e(c–y )T
2-23
Examples
Source: www.theoildrum.com
Source: Quarterly Bulletin, Bank of England, 2006
2-24
Introduction to Derivatives
Topic 3:
Options
Prof. Christophe Pérignon, HEC Paris
Energy in a Carbon Concerned Economy
HEC Certificate 2013
3-1
1. Definitions
• A call option is an option to buy a certain asset by a
certain date for a certain price (the strike price K)
• A put option is an option to sell a certain asset by a certain
date for a certain price (the strike price K)
• An American option can be exercised at any time during
its life. Early exercise is possible.
• A European option can be exercised only at maturity
• ITM, ATM, OTM
3-2
Example: Cisco Options (CBOE quotes)
From NASDAQ :
Option Cash Flows on the Expiration Date
• Cash flow at time T of a long call : Max(0, ST - K)
• Cash flow at time T of a long put : Max(0, K - ST)
3-3
2. Relation Between European Call and
Put Prices (c and p)
• Consider the following portfolios:
Portfolio A : European call on a stock +
present value of the strike price in cash (Ke -rT )
Portfolio B : European put on the stock +
the stock
• Both are worth Max(ST , K ) at the maturity of the
options
• They must therefore be worth the same today:
c + Ke -rT = p + S0
3-4
3. The Binomial Model of Cox, Ross
and Rubinstein
• An option maturing in T years written on a
stock that is currently worth
S
ƒ
Su
ƒu
S d
ƒd
where u is a constant > 1
: option price in the upper state
where d is a constant < 1
: option price in the lower state
3-5
• Consider the portfolio that is D shares
and short one option
S  u  D – ƒu
S  d  D – ƒd
• The portfolio is riskless when S  u  D –
ƒu = S  d  D – ƒd or
ƒu  f d
D
S u  S d
3-6
• Value of the portfolio at time T is:
S  u  D – ƒu or S  d  D – ƒd
• Value of the portfolio today is:
(S  u  D – ƒu )e–rT
• Another expression for the portfolio value today is S  D – f
• Hence the option price today is:
f = S  D – (S  u  D – ƒu )e–rT
• Substituting for D we obtain:
f = [ p ƒu + (1 – p )ƒd ]e–rT
e rT  d
where p 
ud
3-7
Example: Call (K=21, T=0.5, r=0.12)
D 24.2
22
20
1.2823
A
3.2
B
2.0257
18
0.0
E
19.8
0.0
C
16.2
F 0.0
• Value at node B
= e–0.12×0.25(0.6523×3.2 + 0.3477×0) = 2.0257
• Value at node A
= e–0.12×0.25(0.6523×2.0257 + 0.3477×0)
= 1.2823
3-8
4. The Black-Scholes Model
Assumptions
• The stock price follows dS  S dt  sS dz
where dz   dt and  is  (0,1)
• Short selling of securities is permitted
• No transaction costs or taxes
• Securities are perfectly divisible
• No dividends during the life of the option
• Absence of arbitrage
• Trading is continuous
• Risk-free interest rate is constant
3-9
Concept Underlying Black-Scholes
• The option price and the stock price depend on the
same underlying source of uncertainty: f = f(S)
• We can form a portfolio consisting of the stock and the
option which eliminates this source of uncertainty
• The portfolio is instantaneously riskless and must
instantaneously earn the risk-free rate
• This leads to the Black-Scholes differential equation
3-10
The Black-Scholes Formulas
cBS  S 0 N ( d1 )  K e  rT N ( d 2 )
pBS  K e
 rT
N ( d 2 )  S 0 N ( d1 )
ln( S 0 / K )  ( r  s 2 / 2 )T
where d1 
s T
ln( S 0 / K )  ( r  s 2 / 2 )T
d2 
 d1  s T
s T
where N(x) is the probability that a normally distributed
variable with a mean of zero and a standard deviation of
1 is less than x
3-11
Introduction to Derivatives
Topic 4:
Swaps
Prof. Christophe Pérignon, HEC Paris
Energy in a Carbon Concerned Economy
HEC Certificate 2013
4-1
1. Interest Rate Swaps
•
•
•
•
Consider a 3-year interest rate swap initiated on 5
March 2011 between Microsoft and Intel.
Microsoft agrees to pay to Intel an interest rate of
5% per annum on a notional principal of $100
million.
In return, Intel agrees to pay Microsoft the 6-month
LIBOR on the same notional principal.
Payments are to be exchanged every 6 months, and
the 5% interest rate is quoted with semi-annual
compounding.
5%
Intel
MSFT
LIBOR
4-2
Microsoft Cash Flows
---------Millions of Dollars--------LIBOR FLOATING
FIXED
Net
Date
Rate
Cash Flow Cash Flow Cash Flow
Mar. 5, 2011
4.2%
Sep. 5, 2011
4.8%
+2.10
–2.50
–0.40
Mar. 5, 2012
5.3%
+2.40
–2.50
–0.10
Sep. 5, 2012
5.5%
+2.65
–2.50
+0.15
Mar. 5, 2013
5.6%
+2.75
–2.50
+0.25
Sep. 5, 2013
5.9%
+2.80
–2.50
+0.30
Mar. 5, 2014
6.4%
+2.95
–2.50
+0.45
4-3
2. Credit Default Swaps (CDS)
Payment if default
by reference entity
Default
protection buyer
Default
protection seller
CDS spread
• Provides insurance against the risk of default by a
particular company
• The buyer has the right to sell bonds issued by the
company for their face value when a credit event occurs
• The buyer of the CDS makes periodic payments to the
seller until the end of the life of the CDS or a credit event
occurs
4-4
(Source: http://ftalphaville.ft.com/tag/cds/)
4-5
Implied Probability of Default
1-year CDS contract on firm i, with CDS spread = S/year
p = default probability
R = recovery rate
Protection buyer: fixed payment = S
Protection seller: contingent payment = (1-R)p
S is set so that the value of the swap is 0:
S = (1-R)p or p = S / (1-R)
If S = 500bp and R = 0.25: p = 6.6%
If S = 500bp and R = 0: p = S = 5%
4-6
4-7
3. Total Return Swap
Exchange Traded Fund (ETF)
 Physical ETF vs. Synthetic ETF
4-8
Introduction to Derivatives
Topic 5:
Structured Products
Prof. Christophe Pérignon, HEC Paris
Energy in a Carbon Concerned Economy
HEC Certificate 2013
5-1
1. Capital Protected Products (CPP)
• Structured Products are financial securities based on
positions in one or several underlying assets and in one or
several derivatives written on the assets
• Sold by banks as a package since the 80’s in the US and
90’s in Europe
• Attractive features: Upside participation; Leverage effect;
Limited or no downside risk; No margin requirements
• Extremely popular among individual investors
• Underlying asset: equity, fixed income
• Fancy names: Bonus, Diamant, Perles, Protein, Speeder,
Turbo, Wave, etc
• Traded on exchanges (e.g. EURONEXT) or secondary
market organized by issuing banks
5-2
Example of CPP
• Time 0, investor pays $5,000
• Time T, investor receives:
$5,000*(1+0.75*(Stock Index Return))
if Stock Index Return > 0
or
$5,000
if Stock Index Return ≤ 0
If stock index return is +10%:
With structured product: CFT = 5,000 * (1 + 0.075) = 5,375
If direct investment in stocks: CFT = 5,000 * (1 + 0.1) = 5,500
If stock index return is -10%:
With structured product: CFT = 5,000
If direct investment in stocks: CFT = 5,000 * (1 - 0.1) = 4,500
5-3
Example of CPP: Pricing
Cash-flow at time T = CFT ; Stock Index Value = S
CFT = 5,000 + 5,000 * 0.75 * Max( (ST – S0) / S0 ; 0)
CFT = 5,000 + 5,000 * 0.75 * (1/ S0) * Max( ST – S0 ; 0)
Suppose S0 = 10,000. Then
CFT = 5,000 + 5,000 * 0.75 * (1 / 10,000) * Max( ST – 10,000 ; 0)
CFT = 5,000 + (3/8) * Payoff ATM call
Theoretical (Fair) Value of this structured product = V0
V0 = PV(5,000) + (3/8) * ATM call price
This security is fairly priced if and only if V0 = $5,000
Bank makes a profit if ATM call price < (8/3) * (5,000-PV(5,000))
5-4
2. Structure Debt
• Massive use of structured loans by European local
governments (municipalities, regions) during the past
decade
• Three features: long maturity; fixed/low interest rate for the
first years; adjustable rate that depends on a given index (FX
rate, interest rate, slope of the swap curve, inflation)
• Problem: When volatility increases, interest rate explodes
(>20% per annum, termed “toxic”)
• Widespread: Thousands of local authorities contaminated in
Austria, Belgium, France (20% of outstanding debt),
Germany, Greece, Italy, Norway, Portugal, US, etc
5-5
5-6
An Example of Toxic Loan
The City of Saint-Remy is being proposed by its bank a standard vanilla loan:
• Notional:
EUR20m
• Maturity:
20 years
• Coupon:
4.50%, annual
Or, an FX linked loan, with same notional and maturity:
• Coupon:
Y1-3:
2.50%
Y4-20:
2.50% + Max(1.30 – EURCHF, 0), uncapped
The city is selling a put option on EURCHF with a strike at 1.30
The put is OTM as EURCHF is currently at 1.50
Vanilla loan coupon:
4.50%
(1) Option pay-off if out of the money:
-2.00%
(receives annual premium)
(2) Option pay-off if in the money:
-2.00% + (1.30-EURCHF)
(pays option pay-off)
5-7
Potential Scenari
End of Year 3
EURCHF
remains at
1.50
EURCHF
drops to 1.20
Coupon = 2.50%
Coupon = 12.50%
Average
coupon: 2.50%
Average
coupon: 11.00%
Coupon = 30.50%
Average
coupon: 26.30%
EURCHF
drops to 1.02
Maturity = 20 years
5-8
Why Do Local Governments Use Toxic
Debt?
Pérignon and Vallée (2013) show that:
• Politicians use toxic loans to hide debt, especially when
the local government is highly indebted
• Politicians running in politically contested areas are
more inclined to use toxic loans
• Toxic transactions are more frequent shortly before
elections than after them
• politicians are more likely to enter into toxic loans if some
of their neighbors have done so recently (herding)
• Source: http://ssrn.com/abstract=1898965
5-9