Topic 1 Introduction To Derivatives

1
Topic 1 Introduction To Derivatives
2
Basics

• This first lecture has four main goals
• Introduce you to the notion of risk and the role
  of derivatives in managing risk.
• Discuss some of the general terms such as
  short/long positions, bid-ask spread from
  finance that we need.
• Introduce you to three major classes of
  derivative securities.
• Forwards
• Futures
• Options
• Introduce you to the basic viewpoint needed to
  analyze these securities.
• Introduce you to the major traders of these
  instruments.

3
Basics

• Finance is the study of risk.
• How to measure it
• How to reduce it
• How to allocate it
• All finance problems ultimately boil down to
  three main questions
• What are the cash flows, and when do they occur?
• Who gets the cash flows?
• What is the appropriate discount rate for those
  cash flows?
• The difficulty, of course, is that normally none
  of those questions have an easy answer.

4
Basics

• As you know from other classes, we can generally
  classify risk as being diversifiable or
  non-diversifiable
• Diversifiable risk that is specific to a
  specific investment i.e. the risk that a single
  companys stock may go down (i.e. Enron). This is
  frequently called idiosyncratic risk.
• Non-diversifiable risk that is common to all
  investing in general and that cannot be reduced
  i.e. the risk that the entire stock market (or
  bond market, or real estate market) will crash.
  This is frequently called systematic risk.
• The market pays you for bearing
  non-diversifiable risk only not for bearing
  diversifiable risk.
• In general the more non-diversifiable risk that
  you bear, the greater the expected return to your
  investment(s).
• Many investors fail to properly diversify, and as
  a result bear more risk than they have to in
  order to earn a given level of expected return.

5
Basics

• In this sense, we can view the field of finance
  as being about two issues
• The elimination of diversifiable risk in
  portfolios
• The allocation of systematic (non-diversifiable)
  risk to those members of society that are most
  willing to bear it.
• Indeed, it is really this second function the
  allocation of systematic risk that drives rates
  of return.
• The expected rate of return is the price that
  the market pays investors for bearing systematic
  risk.

6
Basics

• A derivative (or derivative security) is a
  financial instrument whose value depends upon the
  value of other, more basic, underlying variables.
• Some common examples include things such as stock
  options, futures, and forwards.
• It can also extend to something like a
  reimbursement program for college credit.
  Consider that if your firm reimburses 100 of
  costs for an A, 75 of costs for a B, 50 for
  a C and 0 for anything less.

7
Basics

• Your right to claim this reimbursement, then is
  tied to the grade you earn. The value of that
  reimbursement plan, therefore, is derived from
  the grade you earn.
• We also say that the value is contingent upon the
  grade you earn. Thus, your claim for
  reimbursement is a contingent claim.
• The terms contingent claims and derivatives are
  used interchangeably.

8
Basics

• So why do we have derivatives and derivatives
  markets?
• Because they somehow allow investors to better
  control the level of risk that they bear.
• They can help eliminate idiosyncratic risk.
• They can decrease or increase the level of
  systematic risk.

9
A First Example

• There is a neat example from the bond-world of a
  derivative that is used to move non-diversifiable
  risk from one set of investors to another set
  that are, presumably, more willing to bear that
  risk.
• Disney wanted to open a theme park in Tokyo, but
  did not want to have the shareholders bear the
  risk of an earthquake destroying the park.
• They financed the park through the issuance of
  earthquake bonds.
• If an earthquake of at least 7.5 hit within 10 km
  of the park, the bonds did not have to be repaid,
  and there was a sliding scale for smaller quakes
  and for larger ones that were located further
  away from the park.

10
A First Example

• Normally this could have been handled in the
  insurance (and re-insurance) markets, but there
  would have been transaction costs involved. By
  placing the risk directly upon the bondholders
  Disney was able to avoid those transactions
  costs.
• Presumably the bondholders of the Disney bonds
  are basically the same investors that would have
  been holding the stock or bonds of the
  insurance/reinsurance companies.
• Although the risk of earthquake is not
  diversifiable to the park, it could be to Disney
  shareholders, so this does beg the question of
  why buy the insurance at all.
• This was not a free insurance. Disney paid
  LIBOR310 on the bond. If the earthquake
  provision was not it there, they would have paid
  a lower rate.

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A First Example

• This example illustrates an interesting notion
  that insurance contracts (for property insurance)
  are really derivatives!
• They allow the owner of the asset to sell the
  insured asset to the insurer in the event of a
  disaster.
• They are like put options (more on this later.)

12
Basics

• Positions In general if you are buying an asset
  be it a physical stock or bond, or the right to
  determine whether or not you will acquire the
  asset in the future (such as through an option or
  futures contract) you are said to be LONG the
  instrument.
• If you are giving up the asset, or giving up the
  right to determine whether or not you will own
  the asset in the future, you are said to be
  SHORT the instrument.
• In the stock and bond markets, if you short an
  asset, it means that you borrow it, sell the
  asset, and then later buy it back.
• In derivatives markets you generally do not have
  to borrow the instrument you can simply take a
  position (such as writing an option) that will
  require you to give up the asset or determination
  of ownership of the asset.
• Usually in derivatives markets the short is
  just the negative of the long position

13
Basics

• Commissions Virtually all transactions in the
  financial markets requires some form of
  commission payment.
• The size of the commission depends upon the
  relative position of the trader retail traders
  pay the most, institutional traders pay less,
  market makers pay the least (but still pay to the
  exchanges.)
• The larger the trade, the smaller the commission
  is in percentage terms.
• Bid-Ask spread Depending upon whether you are
  buying or selling an instrument, you will get
  different prices. If you wish to sell, you will
  get a BID quote, and if you wish to buy you
  will get an ASK quote.

14
Basics

• The difference between the bid and the ask can
  vary depending upon whether you are a retail,
  institutional, or broker trader it can also vary
  if you are placing very large trades.
• In general, however, the bid-ask spread is
  relatively constant for a given
  customer/position.
• The spread is roughly a constant percentage of
  the transaction, regardless of the scale unlike
  the commission.
• Especially in options trading, the bid-ask spread
  is a much bigger transaction cost than the
  commission.

15
Basics

• Here are some example stock bid-ask spreads from
  8/22/2006
• IBM Bid 78.77 Ask 78.79 0.025
• ATT Bid 30.59 Ask 30.60 0.033
• Microsoft Bid 25.73 Ask 25.74 0.039
• Here are some example option bid-ask spreads (All
  with good volume)
• IBM Oct 85 Call Bid 2.05 Ask 2.20 7.3171
• ATT Oct 15 Call Bid 0.50 Ask 0.55 10.000
• MSFT Oct 27.5 Bid 0.70 Ask 0.80. 14.285

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Basics

• The point of the preceding slide is to
  demonstrate that the bid-ask spread can be a huge
  factor in determining the profitability of a
  trade.
• Many of those option positions require at least a
  10 price movement before the trade is
  profitable.
• Many trading strategies that you see people
  propose (and that are frequently demonstrated
  using real data) are based upon using the
  average of the bid-ask spread. They usually lose
  their effectiveness when the bid-ask spread is
  considered.

17
Basics

• Market Efficiency We normally talk about
  financial markets as being efficient information
  processors.
• Markets efficiently incorporate all publicly
  available information into financial asset
  prices.
• The mechanism through which this is done is by
  investors buying/selling based upon their
  discovery and analysis of new information.
• The limiting factor in this is the transaction
  costs associated with the market.
• For this reason, it is better to say that
  financial markets are efficient to within
  transactions costs. Some financial economists say
  that financial markets are efficient to within
  the bid-ask spread.
• Now, to a large degree for this class we can
  ignore the bid-ask spread, but there are some
  points where it will be particularly relevant,
  and we will consider it then.

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Basics

• Before we begin to examine specific contracts, we
  need to consider two additional risks in the
  market
• Credit risk the risk that your trading partner
  might not honor their obligations.
• Familiar risk to anybody that has traded on ebay!
• Generally exchanges serve to mitigate this risk.
• Can also be mitigated by escrow accounts.
• Margin requirements are a form of escrow account.
• Liquidity risk the risk that when you need to
  buy or sell an instrument you may not be able to
  find a counterparty.
• Can be very common for outsiders in commodities
  markets.

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Basics

• So now we are going to begin examining the basic
  instruments of derivatives. In particular we will
  look at (tonight)
• Forwards
• Futures
• Options
• The purpose of our discussion tonight is to
  simply provide a basic understanding of the
  structure of the instruments and the basic
  reasons they might exist.
• We will have a more in-detail examination of
  their properties, and their pricing, in the weeks
  to come.

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Forward Contracts

• A forward contract is an agreement between two
  parties to buy or sell an asset at a certain
  future time for a certain future price.
• Forward contracts are normally not exchange
  traded.
• The party that agrees to buy the asset in the
  future is said to have the long position.
• The party that agrees to sell the asset in the
  future is said to have the short position.
• The specified future date for the exchange is
  known as the delivery (maturity) date.

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Forward Contracts

• The specified price for the sale is known as the
  delivery price, we will denote this as K.
• Note that K is set such that at initiation of the
  contract the value of the forward contract is 0.
  Thus, by design, no cash changes hands at time 0.
  The mechanics of how to do this we cover in
  later lectures.
• As time progresses the delivery price doesnt
  change, but the current spot (market) rate does.
  Thus, the contract gains (or loses) value over
  time.
• Consider the situation at the maturity date of
  the contract. If the spot price is higher than
  the delivery price, the long party can buy at K
  and immediately sell at the spot price ST, making
  a profit of (ST-K), whereas the short position
  could have sold the asset for ST, but is
  obligated to sell for K, earning a profit
  (negative) of (K-ST).

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Forward Contracts

• Example
• Lets say that you entered into a forward
  contract to buy wheat at 4.00/bushel, with
  delivery in December (thus K3.64.)
• Lets say that the delivery date was December 14
  and that on December 14th the market price of
  wheat is unlikely to be exactly 4.00/bushel, but
  that is the price at which you have agreed (via
  the forward contract) to buy your wheat.
• If the market price is greater than 4.00/bushel,
  you are pleased, because you are able to buy an
  asset for less than its market price.
• If, however, the market price is less than
  4.00/bushel, you are not pleased because you are
  paying more than the market price for the wheat.
• Indeed, we can determine your net payoff to the
  trade by applying the formula payoff ST K,
  since you gain an asset worth ST, but you have to
  pay K for it.
• We can graph the payoff function

23
Forward Contracts
24
Forward Contracts

• Example
• In this example you were the long party, but what
  about the short party?
• They have agreed to sell wheat to you for
  4.00/bushel on December 14.
• Their payoff is positive if the market price of
  wheat is less than 4.00/bushel they force you
  to pay more for the wheat than they could sell it
  for on the open market.
• Indeed, you could assume that what they do is buy
  it on the open market and then immediately
  deliver it to you in the forward contract.
• Their payoff is negative, however, if the market
  price of wheat is greater than 4.00/bushel.
• They could have sold the wheat for more than
  4.00/bushel had they not agreed to sell it to
  you.
• So their payoff function is the mirror image of
  your payoff function

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Forward Contracts
26
Forward Contracts

• Clearly the short position is just the mirror
  image of the long position, and, taken together
  the two positions cancel each other out

27
Forward Contracts
Short Position
Long Position
Net Position
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Futures Contracts

• A futures contract is similar to a forward
  contract in that it is an agreement between two
  parties to buy or sell an asset at a certain time
  for a certain price. Futures, however, are
  usually exchange traded and, to facilitate
  trading, are usually standardized contracts.
  This results in more institutional detail than is
  the case with forwards.
• The long and short party usually do not deal with
  each other directly or even know each other for
  that matter. The exchange acts as a
  clearinghouse. As far as the two sides are
  concerned they are entering into contracts with
  the exchange. In fact, the exchange guarantees
  performance of the contract regardless of whether
  the other party fails.

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Futures Contracts

• The largest futures exchanges are the Chicago
  Board of Trade (CBOT) and the Chicago Mercantile
  Exchange (CME).
• Futures are traded on a wide range of commodities
  and financial assets.
• Usually an exact delivery date is not specified,
  but rather a delivery range is specified. The
  short position has the option to choose when
  delivery is made. This is done to accommodate
  physical delivery issues.
• Harvest dates vary from year to year,
  transportation schedules change, etc.

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Futures Contracts

• The exchange will usually place restrictions and
  conditions on futures. These include
• Daily price (change) limits.
• For commodities, grade requirements.
• Delivery method and place.
• How the contract is quoted.
• Note however, that the basic payoffs are the same
  as for a forward contract.

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Options Contracts

• Options on stocks were first traded in 1973.
  That was the year the famous Black-Scholes
  formula was published, along with Mertons paper
  - a set of academic papers that literally started
  an industry.
• Options exist on virtually anything. Tonight we
  are going to focus on general options terminology
  for stocks. We will get into other types of
  options later in the class.
• There are two basic types of options
• A Call option is the right, but not the
  obligation, to buy the underlying asset by a
  certain date for a certain price.
• A Put option is the right, but not the
  obligation, to sell the underlying asset by a
  certain date for a certain price.
• Note that unlike a forward or futures contract,
  the holder of the options contract does not have
  to do anything - they have the option to do it or
  not.

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Options Contracts

• The date when the option expires is known as the
  exercise date, the expiration date, or the
  maturity date.
• The price at which the asset can be purchased or
  sold is known as the strike price.
• If an option is said to be European, it means
  that the holder of the option can buy or sell
  (depending on if it is a call or a put) only on
  the maturity date. If the option is said to be
  an American style option, the holder can exercise
  on any date up to and including the exercise
  date.
• An options contract is always costly to enter as
  the long party. The short party always is always
  paid to enter into the contract
• Looking at the payoff diagrams you can see why

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Options Contracts

• Lets say that you entered into a call option on
  IBM stock
• Today IBM is selling for roughly 78.80/share, so
  lets say you entered into a call option that
  would let you buy IBM stock in December at a
  price of 80/share.
• If in December the market price of IBM were
  greater than 80, you would exercise your option,
  and purchase the IBM share for 80.
• If, in December IBM stock were selling for less
  than 80/share, you could buy the stock for less
  by buying it in the open market, so you would not
  exercise your option.
• Thus your payoff to the option is 0 if the IBM
  stock is less than 80
• It is (ST-K) if IBM stock is worth more than 80
• Thus, your payoff diagram is

34
Options Contracts
T
35
Options Contracts

• What if you had the short position?
• Well, after you enter into the contract, you have
  granted the option to the long-party.
• If they want to exercise the option, you have to
  do so.
• Of course, they will only exercise the option
  when it is in there best interest to do so that
  is, when the strike price is lower than the
  market price of the stock.
• So if the stock price is less than the strike
  price (STltK), then the long party will just buy
  the stock in the market, and so the option will
  expire, and you will receive 0 at maturity.
• If the stock price is more than the strike price
  (STgtK), however, then the long party will
  exercise their option and you will have to sell
  them an asset that is worth ST for K.
• We can thus write your payoff as
• payoff min(0,ST-K),
• which has a graph that looks like

36
Options Contracts
37
Options Contracts

• This is obviously the mirror image of the long
  position.
• Notice, however, that at maturity, the short
  option position can NEVER have a positive payout
  the best that can happen is that they get 0.
• This is why the short option party always demands
  an up-front payment its the only payment they
  are going to receive. This payment is called the
  option premium or price.
• Once again, the two positions net out to zero

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Options Contracts
Long Call
Net Position
Short Call
39
Options Contracts

• Recall that a put option grants the long party
  the right to sell the underlying at price K.
• Returning to our IBM example, if K80, the long
  party will only elect to exercise the option if
  the price of the stock in the market is less than
  80, otherwise they would just sell it in the
  market.
• The payoff to the holder of the long put
  position, therefore is simply
• payoff max(0, K-ST)

40
Options Contracts
41
Options Contracts

• The short position again has granted the option
  to the long position. The short has to buy the
  stock at price K, when the long party wants them
  to do so. Of course the long party will only do
  this when the stock price is less than the strike
  price.
• Thus, the payoff function for the short put
  position is
• payoff min(0, ST-K)
• And the payoff diagram looks like

42
Options Contracts
43
Options Contracts

• Since the short put party can never receive a
  positive payout at maturity, they demand a
  payment up-front from the long party that is,
  they demand that the long party pay a premium to
  induce them to enter into the contract.
• Once again, the short and long positions net out
  to zero when one party wins, the other loses.

44
Options Contracts
Long Position
Net Position
Short Position
45
Options Contracts

• The standard options contract is for 100 units of
  the underlying. Thus if the option is selling
  for 5, you would have to enter into a contract
  for 100 of the underlying stock, and thus the
  cost of entering would be 500.
• For a European call, the payoff to the option is
• Max(0,ST-K)
• For a European put it is
• Max(0,K-ST)
• The short positions are just the negative of
  these
• Short call -Max(0,ST-K) Min(0,K-ST)
• Short put -Max(0,K-ST) Min(0,ST-K)

46
Options Contracts

• Traders frequently refer to an option as being
  in the money, out of the money or at the
  money.
• An in the money option means one where the
  price of the underlying is such that if the
  option were exercised immediately, the option
  holder would receive a payout.
• For a call option this means that StgtK
• For a put option this means that StltK
• An at the money option means one where the
  strike and exercise prices are the same.
• An out of the money option means one where the
  price of the underlying is such that if the
  option were exercised immediately, the option
  holder would NOT receive a payout.
• For a call option this means that StltK
• For a put option this means that StgtK.

47
Options Contracts
At the money
Out of the money
In the money
T
48
Options Contracts

• One interesting notion is to look at the payoff
  from just owning the stock its value is simply
  the value of the stock

49
Options Contracts
50
Options Contracts

• What is interesting is if we compare the payout
  from a portfolio containing a short put and a
  long call with the payout from just owning the
  stock

51
Options Contracts
52
Options Contracts

• Notice how the payoff to the options portfolio
  has the same shape and slope as the stock
  position just offset by some amount?
• This is hinting at one of the most important
  relationships in options theory Put-Call
  parity.
• It may be easier to see this if we examine the
  aggregate position of the options portfolio

53
Options Contracts
54
Options Contracts

• We will come back to put-call parity in a few
  weeks, but it is well worth keeping this diagram
  in mind.
• So who trades options contracts? Generally there
  are three types of options traders
• Hedgers - these are firms that face a business
  risk. They wish to get rid of this uncertainty
  using a derivative. For example, an airline
  might use a derivatives contract to hedge the
  risk that jet fuel prices might change. 
• Speculators - They want to take a bet (position)
  in the market and simply want to be in place to
  capture expected up or down movements.
• Arbitrageurs - They are looking for imperfections
  in the capital market.

55
Financial Engineering

• When we start examining the actual pricing of
  derivatives (next week), one of the fundamental
  ideas that we will use is the law of one price.
  
• Basically this says that if two portfolios offer
  the same cash flows in all potential states of
  the world, then the two portfolios must sell for
  the same price in the market regardless of the
  instruments contained in the portfolios.
• This is only true to within transactions costs,
  i.e. the bid-ask spread on each individual
  instrument.
• Sometimes one portfolio will have such lower
  transactions costs that the law will only
  approximately hold.

56
Financial Engineering

• Financial engineering is the notion that you can
  use a combination of assets and financial
  derivatives to construct cash flow streams that
  would otherwise be difficult or impossible to
  obtain.
• Financial engineering can be used to break
  apart a set of cash flows into component pieces
  that each have different risks and that can be
  sold to different investors.
• Collateralized Bond Obligations do this for
  junk bonds.
• Collateralized Mortgage Obligations do this for
  residential mortgages.
• Financial engineering can also be used to create
  cash flows streams that would otherwise be
  difficult to obtain.

57
Financial Engineering

• The Schwab/First Union equity-linked CD is a good
  example of financial engineering.
• When it was issued (in 1999), the stock market
  was (and had been) incredibly hot for several
  years.
• Many investors wanted to be in the market, but
  did not want to risk the market going down in
  value.
• The equity-linked CD was designed to meet this
  need.
• As we will demonstrate, an investor could roll
  their own version of this, but in doing so would
  have incurred significant transaction costs.
• Plus, many small investors (to whom this was
  targeted) probably could not get approval to
  trade options.

58
Financial Engineering

• The Contract
• An investor buys the CD (Certificate of Deposit)
  today, and then earns 70 of the simple rate of
  return on SP 500 index over the next 5.5 years.
• If the SP index ended up below the initial index
  level (so that the appreciation was negative),
  then the investor received their full initial
  investment back, but nothing else.
• Thus, the payoff to the CD was simply
• So lets say that you invested 10,000, and that
  in June of 1999 the index was 1300 (so that you
  were, in essence, buying 10,000/1,300 or 7.69
  units of the index).

59
Financial Engineering

• In 5.5 years your payoff will be based upon the
  index level. Potential index levels and payoffs
  include
• Index Simple Rate of Return Cash Received
• 1000 - 23.07 10,000
• 1200 - 7.69 10,000
• 1300 0.00 10,000
• 1400 7.69 10,538
• 1500 15.38 11,076
• 2000 53.85 13,769
• (Note that on 12/30/2004 the SP 500 was at
  1211.92!)
• The following chart demonstrates the payouts.

60
Financial Engineering
61
Financial Engineering

• Now, the first thing about that chart that you
  should notice is that it looks an awful lot like
  the shape of a call option, although the slope of
  the upward-sloping part is not as steep.
• This is our first indication that we may be able
  to decompose this into two simpler securities.
• Indeed, one way of decomposing this security
  would be to assume that we bought a bond that
  paid 10,000 at time 5.5, and that we bought 5.38
  call options with a strike of 1300 (70 of
  10,000/1300.)
• The next graph demonstrates this positions
  payoff.

62
Financial Engineering
63
Financial Engineering

• This position is ALSO identical to a position
  consisting of
• 10,000/1300 7.692 units of the index.
• 10,000/1300 7.692 put options on the index
  (K1300)
• (-(1-.7)10,000/1300 -2.30769) CALL options on
  the index.
• The reason for the short call options is because
  the CD only gives us 70 of the return on the
  index, so we have to sell back some of that
  return via the call option (note that we will
  earn a premium for this.)
• The following chart shows this

64
Financial Engineering
65
Financial Engineering

• Now, all three of these should sell for the same
  price but there will be some differences
  because of transactions costs.
• Really, this is why the Schwab equity-linked CD
  can work investors (retail investors) are
  willing to turn to the prepackaged asset to
  avoid transaction costs (and to avoid timing
  difficulties with unwinding their position.)
• Lets just think of this as a bond and .7 long
  call options for a moment.
• Clearly the call cannot be free, since the
  investor holds this option they must pay
  something for it. How much do they pay?
• The interest that they could have earned on this
  money had they invested in a traditional CD.
• At that time 5.5 year CDs were yielding 6, so
  the investor gives up 3,777 dollars in year
  5.5 dollars.

66
Financial Engineering

• The equity-linked CD is just one example of
  financial engineering the notion that investors
  are really just purchasing potential future cash
  flows and that any two sets of identical
  potential future cash flows must sell for the
  same price.
• This has led to a real revolution in finance, and
  we will discuss this idea throughout the
  semester.
• We will return to options pricing later in the
  semester. Next, we turn our attention to the
  futures/forwards markets and pricing.