Introduction to Derivatives

1
Introduction to Derivatives
2
Spot Market Transaction

• The purchase/sale of an asset for immediate
  delivery.
• Both the cash and asset change hands immediately
• The spot price (S) is the price at which this
  transaction for immediate delivery occurs

3
Forward Contracts

• Definition a binding agreement (obligation) to
  buy/sell an underlying asset in the future, at a
  price set today
• Futures contracts are the same as forwards in
  principle except for some institutional and
  (minor) pricing differences.
• A forward contract specifies at origination
• The features and quantity of the asset to be
  delivered
• The delivery logistics, such as time, date, and
  place
• The price the buyer will pay at the time of
  delivery
• Note that forward price (F) at initiation is
  typically set such that forward premium 0

4
Payoff diagram for forwards at expiration

• Long and short forward positions on the SR 500
  index with F1020

Long forward payoff (ST F0,T) Short forward
payoff (F0,T ST)
5
Settlement

• Cash settlement less costly and more practical
• Physical delivery often avoided due to
  significant costs
• Enter offsetting contract
• Credit risk of the counter party
• Major issue for over-the-counter (forward)
  contracts
• Credit check, collateral, bank letter of credit
• Less severe for exchange-traded contracts
• Exchange guarantees transactions, requires
  collateral

6
Swaps
7
Introduction to Swaps

• A swap is a contract calling for an exchange of
  payments, on one or more dates, determined by the
  difference in two prices
• A swap provides a means to hedge a stream of
  risky payments
• A single-payment swap is the same thing as a
  cash-settled forward contract

8
Evolution of Swaps

• Increase in exchange rate volatility (1972)
• increase in earnings volatility
• fluctuation in asset value due to exchange rate
  volatility
• The Solution Parallel Loans
• two firms simultaneously make financial loans to
  each other
• increasing use in the 1970s
• difficult to find partners
• Swaps start being written 1981 by banks to help
  firms conduct parallel loan transactions

9
Parallel loans

• Two firms with opposite exposure to something
• Foreign exchange rates, interest rates,
  volatility of an asset (equity, commodity,
  foreign currency)
• structure a loan to eliminate risk
• Currency parallel loan
• Firm B lends dollars to firm C
• Firm C lends foreign currency to firm B
• Interest parallel loan
• Firm B lends A to firm C, charges fixed rate
• Firm C lends A to firm B, charges floating rate

10
Parallel Loan Example

• Suppose a US firm and a UK firm want to swap
  foreign currency exposure (US firm has UK
  exposure, and vice-versa). Suppose US firm is
  receiving 100,000 per year.
• Assume S1.562 /, rus .08 and ruk .10
• US firms perspective
• borrow 379,079 and repay 100,000 per year for
  5 years
• lend 592,121 and receive 148,300 per year for
  5 years
• Cash flow today 379,079 (1.562 /) -
  592,121 0
• Cash flow at date t1 through t5
• Ct 148,300 - 100,000(St /)
• Ct 100,000 (1.483/ - St /)
• whose form should look familiar

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Parallel loan example continued

• What firm would enter such an agreement?
• A US exporter with 100,000 /year in revenue
• revenues at time period t parallel loan payoff
  hedged rev
• 100,000 (St/) 100,000 (1.483/ - St
  /) 148,300
• Problems with parallel loans
• default risk, impact on balance sheet, search
  costs
• Solution
• staple two contracts together to form a currency
  swap
• netting the payments on each of the dates
• First swap created in 1979 for IBM
• Swap market was largely developed by Chase
  Manhattan in 1981-82

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Swaps

• A swap is an agreement to exchange cash flows at
  a specified future times according to certain
  rules
• A swap may be viewed as a package of forward
  contracts
• Primary differences between swap and parallel
  loan

13
Call Options

• A non-binding contract (right but not an
  obligation) to buy an asset in the future, at a
  price set today
• Preserves the upside potential, while eliminating
  the unpleasant downside (for the buyer)
• The seller of a call option is obligated to
  deliver if asked

Today
Expiration date
or at buyers choosing
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Definitions and Terminology

• Strike (or exercise) price the amount paid by
  the option buyer for the asset if he/she decides
  to exercise
• Exercise the act of paying the strike price to
  buy the asset
• Expiration the date by which the option must be
  exercised or become worthless
• specifies when the option can be Exercise style
  exercised
• European-style can be exercised only at
  expiration date
• American-style can be exercised at any time
  before expiration

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Call Option Payoff and Profit

• Purchased Call
• Payoff Max 0, spot price at expiration
  strike price
• Profit Payoff future value of option premium
• Written Call
• Payoff max 0, spot price at expiration
  strike price
• Profit Payoff future value of option premium

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Call option payoffs at expiration
Long Call payoff max(0, ST-K) Short Call payoff
-1max(0, ST-K)
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Put Options

• A put option gives the owner the right but not
  the obligation to sell the underlying asset at a
  predetermined price during a predetermined time
  period
• The seller of a put option is obligated to buy if
  asked
• Payoff/profit of a purchased (i.e., long) put
• Payoff max 0, strike price spot price at
  expiration
• Profit Payoff future value of option premium
• Payoff/profit of a written (i.e., short) put
• Payoff max 0, strike price spot price at
  expiration
• Profit Payoff future value of option premium

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Put Payoffs at expiration
Long Put payoff max(0, K-ST) Short Put payoff
-1max(0, K-ST)
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The moneyness of an option

• In-the-money option
• positive payoff if exercised immediately
• At-the-money option
• zero payoff if exercised immediately
• Out-of-the money option
• negative payoff if exercised immediately

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Note any similarities?

• Derivatives, per se, neither create nor destroy
  wealth. They only transfer wealth.
• Forwards can be replicated with options

21
Products with options Any product/decision
whose payoff is dependent on the outcome of
something else is a derivative product. Any
product/decision whose payoff is dependent, and
one party has the right of refusal, is a
derivative product with embedded
options. Examples -equity linked
CDs -insurance contracts -warrants -bond
contracts with contingent payoffs
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Basic Insurance Strategies and Payoff Replications
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Insuring a Long Position Floors

• A put option is combined with a long position in
  the underlying asset
• Goal to insure against a fall in the price of
  the underlying asset

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Example

• You buy XOM stock at 60 per share
• You buy XOM put at K60 per share
• Note that the result looks like a call option

profit
60
0
XOM price
25
Insuring a Short Position Caps

• A call option is combined with a position in the
  underlying asset
• Goal to insure against an increase in the price
  of the underlying asset (when one has a short
  position in that asset)

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Example

• You short XOM stock at 60 per share
• You buy XOM call at K60 per share
• Note the result looks like a put option

profit
60
0
XOM price
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Selling Insurance

• Above strategies, we purchased insurance
• Strategies used to sell insurance
• Covered writing is writing an option when there
  is a corresponding position in the underlying
  asset
• Naked writing is writing an option when the
  writer does not have a position in the asset

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Example of a covered call

• You buy XOM stock at 60 per share
• You sell XOM call at K65 per share

profit
60
0
XOM price
Lower b/e point by premium of call option
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Example of a covered put

• You short XOM stock at 60 per share
• You sell XOM put at K55 per share

profit
60
0
XOM price
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Pricing Relationships
31
Synthetic Forwards

• A synthetic forward contract
• Buying a call and selling a put on the same
  underlying asset, with each option having the
  same strike price and time to expiration
  replicates a forward contract
• Differences between a synthetic long forward
  contract and the actual forward
• The forward contract has a zero premium, while
  the synthetic forward requires that we pay the
  net option premium
• With the forward contract, we pay the forward
  price, while with the synthetic forward we pay
  the strike price

32
Synthetic Forwards (contd)

• Note that we can create synthetic forwards with
  KltF or with KgtF.

K2
K1
F
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Forward-Option No-Arbitrage Condition

• Let Call (K, t) and Put (K, t) denote the
  premiums of options with strike price K and time
  t until expiration, and PV (F0,t ) is the present
  value of the forward price
• The net cost of buying the index using options
  must equal the net cost of buying the index using
  a forward contract
• PV (F 0,t) Call (K, t) Put (K, t) PV (K),
  or
• Call (K, t) Put (K, t) PV (F0,t K)

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Definition of arbitrage

• Arbitrage entails the creation of a zero net
  investment position that has positive payoff
• What is the return on such a position?

35
What if PV(F 0,t) gt Call(K, t) Put(K, t)
PV(K)?

• That is, suppose CP, but FgtK.
• Arbitrageur would note that forward market is
  expensive relative to the option market
• Buy low and sell high
• Enter to sell in forward market at time t
• Enter to buy with synthetic forward at time t (w/
  options)

Positive payoff locked-in today
36
Another synthetic forward
This replicates a long forward, right?
37
Forward-spot no-arbitrage condition

• suppose F 20, but S10 and r10
• I could borrow 10 and buy the asset, and then
  simultaneously enter short forward contract.
• At maturity,
• I sell at F20, and repay 11, earning a
  risk-free return with zero net investment.
• To eliminate such opportunities, it must be that
  F the cost of carry
• The cost of carry often includes dividends
  (bonds, currencies, equities), storage costs and
  convenience yields (commodities)
• Above the cost of carry is S(1r), therefore,
• F S(1r)

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Spot-Option No-Arbitrage Condition

• Recall
• PV (F 0,t) Call (K, t) Put (K, t) PV (K),
• And recall
• F S(1r), so that PV(F) S
• Then it must be that
• S Call (K, t) Put (K, t) PV (K),
• This is sometimes called put-call parity

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Spreads and Collars
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Definitions of Spreads and Collars

• An option spread is a position consisting of only
  calls or only puts, in which some options are
  purchased and some written
• Examples bull spread, bear spread, box spread
• A collar is typically the purchase of a put
  option and the sale of a call option with a
  higher strike price, with both options having the
  same underlying asset and having the same
  expiration date
• If call price put price then it is called a
  zero-cost collar

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Spreads

• A bull spread is a position, in which you buy a
  call and sell an otherwise identical call with a
  higher strike price
• It is a bet that the price of the underlying
  asset will increase
• Bull spreads can also be constructed using puts

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Example Bull Spread

• XOM July06 option prices, 1141am, 2/14/06
• Buy 1 Call at K55, Sell 1 Call at K65
• Cost 6.90 1.75 5.15
• assumes we buy at ask, sell at bid
• also each contract is on 100 shares, so total
  cost 515 commission

43
Bull spread example continued

• Outcome on July 21 (when options expire)
• If Sgt65, say 100
• Profit max0,100-55-max0,100-65-5.15
• Profit 45-35-5.154.85 (or 94 return)
• If Slt55, say 0
• Profit max0,0-55-max0,0-65-5.15
• Profit0-0-5.15 -5.15 (or -100)
• Not for the faint of heart

Should be (1r)
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Spreads (contd)

• A bear spread is a position in which one sells a
  call and buys an otherwise identical call with a
  higher strike price
• Note that bull and bear spreads can also be
  created with puts, but the cash flows are
  different through time.
• A box spread is accomplished by using options to
  create a synthetic long forward at one price and
  a synthetic short forward at a different price
• A box spread is a means of borrowing or lending
  money It has no stock price risk

45
Collars

• A collar represents a bet that the price of the
  underlying asset will decrease and resembles a
  short forward
• A zero-cost collar can be created when the
  premiums of the call and put exactly offset one
  another

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Example Zero-Cost Collar with long stock position

• Buy stock at 60, buy put (K55) and sell call
  (K65). Assume CP.

profit
0
Stock price
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Why use collars?

• Hedging
• Investor can hedge stock position without selling
  the stock
• Suppose executive is granted stock at S60 but
  cannot sell for 1 year.
• Purpose is to give her incentives
• Mergers Acquisitions
• Suppose your firm is offered shares of another
  firm as payment due in 180 days.

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Hedging stock grant

• Granted 1,000 shares of XOM at 95
• Hedge Buy 10 puts (K90) and sell 10 calls at
  K100