Futures, Swaps & Other Derivatives
• Who we are
• Who you are: name sheet
• This week’s lectures
– First half
• syllabus and course outline
• why we should be here
• basic finance principles -- reminder
– Second half: Part I -- Forwards
Who I am
• Michel Robe
• (202) 418-4459
• gwfutures@yahoo.com
• Web page
Syllabus
• Prerequisites
• Course Objectives
• Materials
• Grading
• 50% (Exam) + 2*20% (GA) + 10% (CP)
• Groups
• Honor Code
1
This Course within Finance
• Finance
• corporate finance
• investment & portfolio management
• market microstructure
• This course’s main themes
• pricing (microstructure & investments)
• hedging (corporate, banking & investments)
• all-in cost of capital (corporate)
Markets and Instruments
• Types of Markets
• direct search vs. broker vs. dealer vs. auction
• Instruments analyzed
• Forwards –Forex, Commodities
• Interest Rate Derivatives
» Money Market; Fixed-income capital markets (bonds)
• Equity Market Derivatives (Index futures)
• Commodity Market Derivatives
• Swaps
Why Should We Be Here?
• Trading in financial assets is huge
• stock market (example ) vs. derivatives market
– daily global FX deriv. turnover = $1.9 trn (2004 triennal BIS survey)
» was 1.21 trn/day (2001) & $1.49 trillion/day (1998)
– trading volume for Eurodollar futures alone > $1.25 trillion/day (2004)
– notional on OTC interest-rate derivatives > $204 trillion (mid-2005)
• 90+% of large corporations use derivatives (mostly fx)
• Importance of theory
• computerization
• experience vs. new situations (importance of reference points)
• know the theory to know its limits
2
Course Outline
• 4 Parts
• Reading Packet
• Cases
Introduction
Basic Finance Principles
Time is
Money
Options have
value, always
Self Interest
Transaction?
2 parties
Market
Efficiency
Risk Aversion Diversification Marginal
Analysis
3
Derivative Securities
• Definition
•
•
•
•
A derivative security or derivative
is a financial instrument
whose value depends on
the values of other more basic underlying variables
• Examples
• Options
• Forward & Futures
• Swaps
Terminology
• Number of parties
• 2 (buyer & seller) + intermediaries (sometimes)
• The party that has agreed to:
– BUY
• has what is termed a LONG position
– SELL
• has what is termed a SHORT position
Terminology 2
• Types of Traders
• Hedgers
» want to reduce risk of existing assets or liabilities
• Speculators
» are willing to take risk based on their forecasts
» try to exploit price movements
• Arbitrageurs
» use risk-free trading strategies
» to exploit asset mis-pricings
• Investors
» use futures to gain exposure (e.g., commodity index funds)
4
Ways Derivatives are Used
• To invest or speculate
• To hedge risks
• To infer views
• about the future direction of the market or about risk
• To lock in arbitrage profits
• To change the nature
• of a liability
• of an investment
Options
• A call is an option
• A put is an option
to BUY
to SELL
a certain asset
a certain asset
by a certain date
by a certain date
for a certain price
for a certain price
that is fixed today
that is fixed today
Forwards and Futures
• Forward contracts
• basic idea & market participants
• links between spot and futures prices
» forward as predictor of future spot prices
» spot-futures parity theorem
• Futures contracts
• market microstructure
» participants, major contracts, exchanges
• differences with forwards (contracts & prices)
5
Forwards & Futures vs. Options
Customized Standardized
right
OTC option exchangetraded option
obligation forward
futures
Forward Fundamentals
• Definition
• contract calling for delivery of a given asset
• at a given future date, at a price agreed-upon today
• no money changes hands today (caveat)
• Market microstructure
• OTC market
• Market participants
• hedgers-traders-arbitrageurs
• speculators
Forward Fundamentals 2
• Market participants
• traders-arbitrageurs
• hedgers
» try to avoid impact of price movements
» short hedgers: have (or expect) long position, go short
» long hedgers : have short underlying position, go long
• speculators
» try to profit from price movements
6
Forward Fundamentals 3
• Taxation
• hedging vs. speculation
» ordinary income vs. capital income
• Risks borne by parties
• volatility of underlying asset price
• default
» why?
• Solution
• currencies:
• most other assets:
forwards
futures
Futures
• Fundamentals
– participants, major contracts, exchanges
• Differences w/ forward contracts
(main ones)
– exchange-traded (not OTC)
– standardized
– marking-to-market / risk control
• Differences b/ forward & futures prices
– in theory
– in practice / arbitrage
Part I: Forwards
7
Forwards: Fundamentals
• Definition
• contract calling for delivery of a given asset
• at a given future date, at a price agreed-upon today
• no money changes hands today (caveat)
• Market microstructure
• OTC market
• Market participants
» hedgers-traders-arbitrageurs
» speculators
• Forward quotes
Forwards 2
• Market participants
• traders-arbitrageurs
• hedgers
» try to avoid impact of price movements
» short hedgers: have long underlying position, go short
» long hedgers : have short underlying position, go long
• speculators
» try to profit from price movements
Forwards 3
• Forward quotes (OTC)
• bid vs. ask
» bid = price at which market maker buys from customers
» ask = price at which market maker sells to customers
• outright forward vs. “ swap rate” (FX market)
» spot
» “swap rate ”
» outright forward
1.6275 - 1.6299 $ / 1£
33 -
46
1.6308 - 1.6345 $ / 1£
8
Forwards 4
• Forward quotes (OTC)
• outright forward vs. “ swap rate” (forward premium)
» spot
1.6275 - 1.6299 $ / 1£
» “swap rate ”
» outright forward
33 46
1.6308 - 1.6345 $ / 1£
• outright forward vs. “ swap rate” (forward discount)
» spot
1.6275 - 1.6299 $ / 1£
» “swap rate ”
» outright forward
45 -
33
1.6230 - 1.6266 $ / 1£
Forwards 5
• Swap rate & B-A spread
• observation
» subtract swap if discount, add if premium
» why? (size of B-A spread)
• explanations
» risk?
» liquidity (market depth)?
» others?
Forwards 6
• Regulation?
• Risks
• volatility of underlying asset price
• default
» why?
• Solution
• currencies vs. most other assets (futures)
9
Forwards 7
• Payoff at maturity (or expiration or delivery)
• What matters?
– long position vs. short position
– hedged position vs. naked position
» long hedge vs. short hedge
• Regrets, anyone?
Profit from a
LONG Naked Forward Position
Profit
Price of Underlying
at Expiration (ST )
Profit from a
SHORT Naked Forward Position
Profit
Price of Underlying
at Expiration (S T)
10
Options vs. Forwards
• A forward contract
• An option contract
gives the holder the
gives the holder the
OBLIGATION to
RIGHT (but not the
buy or sell at a
obligation) to buy or
certain price
sell at a certain price
Option Payoffs at Maturity Payoffs
• Summary:
Payoff
X = Strike price;
ST = Price of underlying asset at maturity
Payoff
X
X
ST
Payoff
ST
Payoff
X
X
ST
ST
Forwards: Pricing
• Relationship to current spot
• forward-spot parity for domestic assets
• forward-spot parity for currencies -- IRP
• compounding & asset type
• Relationship to expected future spot
• forward parity for domestic assets
» expectations theory vs. contango vs. (normal ) backwardation
• forward parity for currencies
» uncovered IRP
• theory vs. practice
11
Forward Price
• Forward price -- Definitions
• delivery price
» price at which the underlying asset will be delivered
» agreed upon at time forward is entered into
• forward price
» delivery price that would make the contract have 0 value
» changes during life of contract (but, who cares…)
• forward price = delivery price
» when contract is created
Links between Forward & Spot Prices
• Forward -Spot Parity
• relationship between fwd price & current spot price
• basic idea:
» (1) borrow today
» (2) buy an asset
» (3) go short forward on the asset
» the position should have zero risk & zero cost
» hence, the rate of return should be zero
• basic idea vs. complications (dividends, seasonality)
Links between Forward & Spot Prices 2
• Domestic-asset sure -dividend case
•
t
T
• (1) borrow cash
S0
- S0 (1+r)
• (2) buy asset
•
•
- S0
+ asset
get cash dividends
+ D = S0 d
• (3) short asset forward
•
• total
+ F0
- asset
0
• hence: 0 = - S (1+r) + S
0
0
+ asset
d + F0
- S0 (1+r) + S0 d + F0
and thus
F0 = S0 (1 + r - d)
12
Links between Forward & Spot Prices 3
• Cost-of-Carry relationship
= Forward-Spot Parity = Futures-Spot Parity
• idea
» buy forward
vs.
“buy spot and carry ”
» F0
vs.
S0 (1 + r - d)
• examples of cost of carry
– stock:
» r-d
( no storage cost, dividend rate d earned )
» F0 vs. [S0 -D ’](1 + r)
» F0 vs. S0 (1 + r) - D
( PV of dividend known to be D’)
(future dividend known to be D)
– commodity :
» r+u
( storage cost born at rate u)
» F0 vs. [S0 +U ’](1 + r) ( PV of storage cost known to be U’)
Links between Forward & Spot Prices 6
• Dividends
• problem
» varies over time (May vs. other periods)
• solution
» theory (no-arbitrage bands) vs. practice (triple witching)
• Generalizations
• multiperiod: F 0 = S0 (1 + r - d)T
• commodities: F 0 = S0 (1 + r + u)T --- really?
» seasonal variations (e.g., harvests)
» carrying costs vs. storage costs vs. convenience yield
Links between Forward & Spot Prices 7
• Commodities
• financial assets
» we must have F0 = S0 (1 + r - d)
• commodities as consumption assets
– basic idea
» some commodities are not held as financial assets
» having the commodity may provide benefits (shortages)
– situation #1
» F0 > S0 (1 + r + u)
» arbitrage
– situation #2
» F0 < S0 (1 + r + u)
» convenience yield y : defined by F0 = S0 (1 + r + u - y)
13
Links between Forward & Spot Prices 4
• Domestic-asset sure -dividend case (continuous time):
t
•
S0 / ed
- S / ed
• (1) borrow
• (2) buy asset
•
•
•
T
- ( S0 / ed ) e r
0
+ (fraction of asset)
& reinvest continuous
dividends
+ asset
• (3) short asset forward
•
• total
• hence:
+ F0
- asset
- [ S0 e r / ed ] + F0
0
0 = - [ S0 e r / / ed ] + F0
and thus
F0 = S0 e (r-d)
Links between Forward & Spot Prices
• Gold futures example
• Assume delivery in 1 year
• Let’s not forget anything:
» dividends? −> lease rate
» Any convenience yield? unlikely
• F = S (1 + r + u - d)
» suppose S = $600/ounce, r =5%, d = 1% and u=0%
– > Equilibrium F = 600 x (1 + 0.05 - 0.01) = 624
Links between Forward & Spot Prices 5
• Discrete time vs. continuous version
• discrete
• continuous
F0 = S0 (1 + r - d)
F = S e (r-d)
0
0
• Source of the difference
• how much is hedged?
• how are the dividends reinvested?
• Does it matter?
• Size of difference vs. reasonable character of reinvestment hyp.
• Domestic assets vs. currencies
14
Links between Forward & Spot Prices 8
• Foreign currency case
•
t
T
• (1) borrow local
S0
- S0 (1+r)
• (2) buy 1 FX unit
•
•
- S0
+ 1 FX
receive interest
+ 1 FX
+ r* FX
• (3) short currency forward
•
+ F0 (1+r*)
- (1+r*) FX
• total
- S0 (1+r) + F0 (1+r*)
0
• hence: 0 = F (1+r*) - S (1+r)
0
0
and thus
F0 = S0 (1+r)/(1+r*)
Links between Forward & Spot Prices 9
• Interest rate parity theorem
• = Forward-Spot Parity for currencies
» basic idea
ft,T = st
1+i T
360
1 +i* T
360
or
i - i* T
ft,T - st
360
=
st
1 +i* T
360
» key difference with “formula” for domestic assets?
» examples
Links between Forward & Spot Prices 10
• Question
•
•
•
•
The 3-month interest rate in Denmark is currently 3.5%.
Meanwhile, the equivalent interest rate in England is 6.5%.
All rates are annualized.
What should be the annualized 3-month forward discount or
premium at which the Danish krone will sell against the
pound?
15
Links between Forward & Spot Prices 11
• Answer
• the Danish krone (DKr)
•
should sell at a premium
•
against the pound
•
approximately equal to
• the interest rate differential between the two countries
*
T
i£ - i DKr
6.5% - 3.5% 90
f t, T - st
360 =
360 = 0.7435%
=
st
*
T
1 + i DKr
1 + 3.5% 90
360
360
Links between Forward & Spot Prices 12
• Forward & expected future spot
• convergence property (maturity only)
• expectation hypothesis
» no uncertainty
» risk neutrality
• normal backwardation
» forward price < expected future spot price
» idea: natural hedgers = producers
• contango
» forward price > expected future spot price
Links between Forward & Spot Prices 13
• Forward & expected future spot
• Forward Parity
ft,T = E s t+T | IM
t
• Uncovered IRP(Uncovered Interest Rate Parity)
E s t+T = st
1+i T
360
1 + i* T
360
16
Links between Forward & Spot Prices 14
• Forward & expected future spot (general case)
– speculators choose between
» PV of the forward price, known for sure: F0 .e -r
» PV of the future spot price, risky:
E[ST].e -k
• General case: F 0 = E[S T] e(r-k) < E[S T]
» Why? k > r if underlying has positive systematic risk
• Special case: F 0 = E[S T]
» if speculators are risk neutral or
» if the underlying asset is uncorrelated with the market
Links between Forward & Spot Prices 15
• Comparison
– Forward & current spot
• discrete time:
F 0 = S0 (1 + r - d)
• continuous time:
F 0 = S0 e(r-d)
– Forward & expected future spot
• discrete time:
F 0 = E[S T] (1 + r - k)
• continuous time:
F 0 = E[S T] e(r-k)
Parity Conditions
• Forward Parity
• speculative efficiency
» hypothesis vs. arbitrage
ft,T = E st+T | IM
t
» is efficiency consistent with a risk premium?
ft,T = E[st+T | IM
t ] + RPt,T
» Buser, Karolyi & Sanders (JFI 96): short-term premia
17
Parity Conditions 2
• Uncovered IRP
1+i T
360
1 + i* T
360
• CIRP
f t,T = st
• + FP
f t,T = E st+T | I M
t
• = UIRP
E st+T = s t
1+i T
360
1 + i* T
360
Parity Conditions 3
• Uncovered IRP
• UIRP
• Usefulness
i - i* T
E st+T - st
360
=
st
1 + i* T
360
– empirical evidence: depends on horizon
» Meredith and Chinn (NBER; IMF SP 2004)
Valuing Forward Positions
• Forward price
– delivery price
– price at which the underlying asset will be delivered
– agreed upon at time forward is entered into
– forward price
– delivery price that would make the contract have 0 value
– changes during life of contract (who should care?)
– forward price = delivery price
– when contract is created
18
Valuing Forward Positions 2
• Valuing an existing forward position
– enter into position at K (delivery price, i.e., initial fwd price)
– forward price moves from K to F0
• value of a long: f = (F0 – K) e-rT
» long at 5c/bushel, fwd price goes up to 6cpb, I “made” 1cpb
» but this difference is only realized at maturity −> PV it!
• value of a short: f = (K – F0 ) e-rT
» if the fwd price goes up, the value of a short position falls
Part I extra: Forwards
Real Life Forward-Spot Parity
• Arbitrage
• so far:
bid and asked rates are the same
• in reality
» bid-asked spreads on the markets (or brokerage fees)
» borrowing costs more than depositing
• consequence
• equalities vs. bounds
19
“Real-life” IRP -- Transaction Costs 2
• No-arbitrage condition 1
F bt ,T ≤
S at (1+ i a
T
S bt (1+ i b
T
360
T
( 1+ i*b
)
360
)
• No-arbitrage condition 2
F at ,T ≥
360
T
(1+ i *a
)
360
)
“Real-life” IRP -- Example with Costs 1
Question.
You are a trader of JP Morgan allowed to do arbitrage. From a phone call to a trader at
Daiwa Bank, you learn that Daiwa will:
lend and borrow ¥ at 0.5%-0.625% for 6 months (annualized rates)
lend and borrow $ at 5.375%-5.5% for 6 months (annualized rates)
buy and sell ¥ spot at 100.00-50 ¥/1$
buy and sell ¥ 6-month forward at 97.00-50 ¥/1$
a. Can you make money out of Daiwa?.
“Real-life” IRP -- Example with Costs 2
Answer
At first sight, it appears that the gain or loss will be very small, since covered IRP
roughly holds: the $ is selling at about a 2.5% 6-month forward discount, which is about
the 6-month interest rate differential between Japan and the US.
When looking closely at the numbers, though, we see an arbitrage opportunity.
To see this, construct the forward rate implied by the interest rate differential and the spot
rate:
f=s
1+i T
1+0.055
360 = (0.01 $ / 1¥)
1+i * T
1+0.005
360
180
360 = 0.010249 $ / 1¥
180
360
Now compare this with the 6-month forward rate quoted directly by Daiwa: 0.010256 $ /
1¥.
20
“Real-life” IRP -- Example with Costs 3
Clearly, you will want to buy low and sell high, i.e.:
sell to Daiwa ¥ 6-month outright forward at 97.50¥/1$.
The other sides of the transaction are:
borrow $ at 5.5%,
buy ¥ spot with the borrowed dollars at 100¥/1$,
and invest the ¥ at the rate of 0.5%.
“Real-life” IRP -- Example with Costs 4
cash-flows today
a.
b.
+ 1$
- 1$
+ 100¥
- 100¥
d.
e.
total
none
none
___________
0
cash-flows in 6 months
(borrow 1$)
(exchange for ¥)
(invest ¥)
(sell ¥ forward)
- 1.0275$ (loan & interest)
none
none
+ 100.25 ¥
- 100.25 ¥
+ 1.0282 $
_________________
+ 0.0007 $
For every dollar borrowed, the gains are 0.0007$, i.e., a 0.07% profit margin.
“Real-life” IRP -- Example with Costs 5
Question
Suppose that you are a trader of JP Morgan allowed to do arbitrage. From a phone call
to a trader at Daiwa Bank, you learn that Daiwa will:
lend and borrow ¥ at 0.5%-0.625% for 6 months (annualized rates)
lend and borrow $ at 5.375%-5.5% for 6 months (annualized rates)
buy and sell ¥ spot at 100.00-50 ¥/1$
buy and sell ¥ 6-month forward at 97.00-50 ¥/1$
b. Suppose you must borrow $1m from Daiwa for JP Morgan (e.g., to carry out some
unrelated investment strategy). What would your total borrowing cost be?
21
“Real-life” IRP -- Example with Costs 6
Answer
Your borrowing choices are the following:
(1) either borrow $ from Daiwa at 5.5%: the total $ c o s t i n 6 m o n t h s w o u l d b e $ 2 7 , 5 0 0 .
(2) or create a similar pattern of cash-flows, borrowing in ¥, converting the ¥ into $, and
locking in the $ cost of the ¥ loan through a forward contract. Here, the cost would be
as follows:
- you need $1m today, hence you borrow ¥100,500,000 and sell them spot for $1m
(i.e., you buy $1m at the asked price of ¥100.50/1$)
- in 6 months, you will need to pay back ¥100,814,063; you can lock in today the $
cost of this repayment by buying $1,039,320 6-month forward. The total $ cost
would be: $39,320. The operation I have just described is called a swap.
Bottom Line: Since borrowing directly in $ is cheaper ($27,000 vs. $39,320), you
should borrow $.
Real Life Forward-Spot Parity
• Derivations
• in the following pages, we derive the no-arbitrage
bounds in the presence of bid-asked spreads
• Not exam material
• those pages are here only for students interested in
more theoretical aspects of the relationships used in
class; no one will be tested on them
IRP without costs
–
t
• invest domestically:
$:
t+T
• invest abroad
$:
•
FX: 1/S t
•
FX: - 1/S t
•
•
$ : (1 + i
-1
T
360
)
-1
FX:
FX: $:
1
St
1
St
Ft,T
St
( 1+ i*
( 1+ i*
T
360
T
( 1+ i*
)
360
T
)
)
360
22
IRP without costs 2
• Net cost
• domestic:
$1
• foreign:
$1
• Risk
• similar
• Return equality condition
T
(1+ i
)=
360
Ft,T
T
( 1+ i*
St
)
360
IRP without costs 3
–
t
• (1) borrow
t+T
$ : − (1+ i
$1
• (2a) buy spot
$:
•
FX: 1/S t
• (2b) invest FX
FX: - 1/S t
T
)
360
-1
FX:
• (2c) short FX forward
1
St
FX: $:
( 1+ i*
1
St
T
360
( 1+ i*
Ft,T
T
360
( 1+ i*
St
)
)
T
)
360
IRP without costs 4
• Net flows
• t:
FX: 0; $: 0
• t+T:
FX: 0; $ : − (1 + i
T
)+
360
Ft ,T
( 1+ i*
St
T
)
360
• No-arbitrage condition
(1+ i
T
360
)=
Ft ,T
St
T
( 1+ i*
T
360
)
Ft,T − St (i − i* ) 360
St
=
(1 + i*
T
)
360
23
“Real-life” IRP -- Transaction Costs
–
t
• (1) borrow
t+T
$ : − ( 1+ i a
$1
• (2a) buy FX spot
$: -1
a
FX :1/ S t
• (2b) invest FX
FX: -1/ S at
FX:
T
360
)
1
T
( 1+ i*b
)
360
S ta
1
T
( 1+ i *b
)
360
S at
F bt ,T
T
$:
( 1+ i *b
)
360
S at
• (2c) short FX forward
FX: -
“Real-life” IRP -- Transaction Costs 2
• Net flows
• t:
FX: 0; $: 0
• t+T:
FX: 0; $ : − (1+ i
a
Fb
T
t ,T
)+
( 1+ i *
b 360 )
360
S ta
T
• No-arbitrage condition
(1+ ia
T
360
)≥
Fb
t,T
T
( 1+ i *
b 360)
Sa
t
F bt ,T ≤
S ta(1 + ia
T
360
T
( 1+ i*b
)
360
)
“Real-life” IRP -- Transaction Costs 3
–
t
t+T
• (1) borrow
FX 1
T
FX : − (1+ i*a
)
360
• (2a) buy $ spot
$ : S bt
FX: -1
• (2b) invest $
• (2c) buy FX forward
$ : − S bt
T
$:Sb
)
t (1+ i b
360
T
$ : - S bt( 1+ ib
)
360
T
S bt
FX:
( 1+ i b
)
360
F at,T
24
“Real-life” IRP -- Transaction Costs 4
• Net flows
• t:
FX: 0; $: 0
• t+T:
b
T
T
S
$: 0; FX : − (1+ i*a
) + t (1 + i b
)
360 F a
360
t ,T
• No-arbitrage condition
b
T
T
S
(1+ i *a
) ≥ t (1+ ib
)
360 F a
360
t ,T
F at ,T ≥
S bt (1+ i b
T
360
T
(1+ i *a
)
360
)
25