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Introduction to Derivatives Energy in a Carbon Concerned Economy HEC Certificate 2013

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Instructor:

Prof. Christophe Pérignon

Deloitte – SociétéGénérale Chair in Energy and Finance

HEC Paris

Office 29, W2 building

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.

Prof. Christophe Pérignon, HEC Paris

Energy in a Carbon Concerned EconomyHEC Certificate 2013

- 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)

- 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

Source: Bank for International Settlement (BIS)

Source: Bank for International Settlement (BIS)

- 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?

- 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:

- Study Etude MEDEF-HEC 2012

http://appli9.hec.fr/hec-medef/doc/MEDEF-2012-rapport-final.pdf

Etude MEDEF-HEC 2012: Financial Hedging vs. OperationalHedging

Prof. Christophe Pérignon, HEC Paris

Energy in a Carbon Concerned EconomyHEC Certificate 2013

- 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)

- 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)

S0 = $1,810.1 F0(Nov 2011) = $1,803.2

Source: www.kitco.com Source: www.cmegroup.com

Quotesretrieved on September 7, 2010

- 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)mn

limm–›∞ A(1+Rm / m)mn = Ae r n

where ris the continuously compounded interest rate per annum

- $100 grows to $100er×Tat time T
- $100 received at time Tdiscounts to $100e-r×Tat time zero
- A risky cash-flow of $X received at time T discounts to $Xe-k×Tat time zero, where k = r + p and p is the risk premium

- 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

- 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

- 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

- 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 pre-specified 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

- Proportion of the exposure that should optimally be hedged is:
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.

- 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
- The size of one heating oil contract is 42,000 gallons
- Optimal number of contracts:

- 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?

- 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

- 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?

- 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 S0exp(r T) = $1,275

For any investment asset that provides no

income and has no storage costs

F0 = S0erT

Immediate arbitrage opportunity if:

F0 > S0erT short the Futures, long the asset

F0 < S0erT long the Futures, short sell the asset

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

- 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

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 = (S0e-qT)erT= S0e(r - q)T

Storage costs can be treated as negative income:

F0=(S0+U )erT

where U is the present value of the storage costs

Alternatively F0=S0e(r+u)T

where u is the storage cost per unit time as a

percent of the asset value

- 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 F0S0ecT
- The convenience yield, y, is the benefit provided when owning a physical commodity.
- It is defined as:
F0 = S0 e(c–y )T

Source: www.theoildrum.com Source: Quarterly Bulletin, Bank of England, 2006

Prof. Christophe Pérignon, HEC Paris

Energy in a Carbon Concerned EconomyHEC Certificate 2013

- 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

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)

- 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

Su

ƒu

S

ƒ

S d

ƒd

- An option maturing in T years written on a stock that is currently worth

where u is a constant > 1

: option price in the upper state

where d is a constant < 1

: option price in the lower state

S u D – ƒu

- Consider the portfolio that is D shares and short one option
- The portfolio is riskless when S u D –ƒu= S d D –ƒd or

S d D – ƒd

- Value of the portfolio at time Tis:
S uD – ƒu or S dD – ƒd

- Value of the portfolio today is:
(S uD – ƒu )e–rT

- Another expression for the portfolio value today is SD – f
- Hence the option price today is:
f = S D – (S uD – ƒu)e–rT

- Substituting for D we obtain:
f = [ p ƒu + (1 – p )ƒd ]e–rT

where

24.2

3.2

D

- 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

22

B

19.8

0.0

20

1.2823

2.0257

E

A

18

C

0.0

16.2

0.0

F

Assumptions

- The stock price follows
where

- 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

- 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

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

Prof. Christophe Pérignon, HEC Paris

Energy in a Carbon Concerned EconomyHEC Certificate 2013

- 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

---------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

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

(Source: http://ftalphaville.ft.com/tag/cds/)

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%

Exchange Traded Fund (ETF)

Physical ETF vs. Synthetic ETF

Prof. Christophe Pérignon, HEC Paris

Energy in a Carbon Concerned EconomyHEC Certificate 2013

- 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

- Time 0, investor pays $5,000
- Time T, investor receives:
- $5,000*(1+0.75*(Stock Index Return)) or $5,000
- if Stock Index Return > 0 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

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))

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

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)

End of Year 3

EURCHF remains at 1.50

Average coupon: 2.50%

Coupon = 2.50%

EURCHF

drops to 1.20

Coupon = 12.50%

Average coupon: 11.00%

Coupon = 30.50%

EURCHF

drops to 1.02

Average coupon: 26.30%

Maturity = 20 years

- 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