Basic concepts of derivative instruments

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Basic concepts of derivative instruments

• What is a derivative instrument
• A derivative instrument is a contract whose value
  is derived from some underlying asset

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Basic concepts of derivative instruments

• .Why people use derivative instruments?
• 1. Leverage
• 2. Hedging
• 3. Substitutability
• 4. Financial Engineering
• 5. Information
• 6. Taxes and regulation

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Basic concepts of derivative instruments

• What types of contracts are used?
• Options Give the holder the right to buy (call
  option) or sell (put option) an asset at a fixed
  price some time in the future
• When an option is purchases a premium is paid

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Basic concepts of derivative instruments

• What types of contracts are used?
• Futures An obligation to buy or sell an asset
  at a specified price some time in the future
• No premiums are paid when you initiate a futures
  contract. However, a margin account must be
  established.
• Marking-to-market Futures contracts are
  re-written to reflect current futures price

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Basic concepts of derivative instruments

• What types of contracts are used?
• Swaps A swap involves the exchange of a set of
  cash flows in a predetermined manner
• Interest rate swap Exchange fixed rate interest
  payments for floating rate interest payments
• Currency swap Exchange fixed rate payments on
  loans denominated in different currencies

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Basic concepts of derivative instruments

• Different types of derivative contracts
• Generally, derivative contracts can be broken
  into four classes
• 1. Equity derivatives
• 2. Interest rate derivatives
• 3. Currency derivatives
• 4. Agricultural/Commodity derivatives

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Basic concepts of derivative instruments

• Positions
• Derivatives are contracts. Hence, derivatives
  are referred to as a zero-sum game.
• Long position You own the contract. You win if
  the value of the contract increases
• Short position You sold (wrote) the contract.
  You win if the value of the contract decreases.

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Options contracts and pricing

• Two kinds of options contracts
• Call options Give the holder the right to buy
  the underlying asset at a fixed price
• Put options Give the holder the right to sell
  the underlying asset at a fixed price
• Note Option contracts give the holder the
  right, not obligation to buy or sell

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Options contracts and pricing

• The value of put and call contracts
• Notation
• T option maturity
• S price of the underlying asset
• X exercise price of the contract
• C value of a call contract with exercise price
  X
• P value of a put contract with exercise price X
• Co initial price of the call option
• Po initial price of the put option

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Options contracts and pricing

• The value of put and call contracts
• In-the-money vs. Out-of-the-money

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Option contracts and pricing

• Pay off to call options
• The payoff of a long position in a call option at
  time T is Max ( S - X, 0), less the cost of the
  option. If S gt X then the call option is
  exercised and if S lt X the call option remains
  unexercised. The payoff of a short position in
  a call option (write a call) at time T is
  Min( X - S, 0), plus the initial price of
  the option.

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Option contracts and pricing

• Payoff to put options
• The payoff of a long position in a put option at
  time T is Max ( X -S, 0), less the cost of the
  option. If S lt X then the put option is
  exercised and if S gt X the put option remains
  unexercised. The payoff of a short position in a
  put option (write a put) at time T is
  Min(S - X, 0), plus the cost of the put option

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Option contracts and pricing

• Option strategies
• 1. Betting on volatility
• If you perceive that there will be more (or less)
  volatility in a particular asset than the market
  expects you can establish option positions to
  gamble on your intuition.
• Straddle Long call and long put, same exercise,
  same maturity. Strategy wins if volatility is
  higher than anticipated.

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Option contracts and pricing

• Option strategies
• 1. Betting on volatility
• Shorting the straddle Short call, short put,
  same exercise, same maturity. Strategy wins if
  volatility is lower than anticipated.
• Strangle Same as the straddle except the call
  and put have different exercise prices
• Butterfly spread Combines a short straddle and
  a strangle. Allows for profit in a narrow
  region. Strategy wins if volatility is lower
  than anticipated.

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Option contracts and pricing

• Option strategies
• 2. Betting on price movements
• Bull spread If you perceive that prices for a
  particular asset will increase, a bull spread
  allows you to gamble on upside potential while
  protecting against downside risk.
• The bull spread involves buying a call at a
  certain strike price and then selling a call on
  the same asset with a higher strike price.

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Option contracts and pricing

• Option strategies
• 2. Betting on price movements
• Bear spread If you perceive that prices for a
  particular asset will fall, the bear spread
  allows you gamble on the downside while
  protecting against the upside.
• The bear spread involves buying a call at a
  certain strike price and selling a call with a
  lower strike price.

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Option contracts and pricing

• Arbitrage The existence of a riskless profit
  opportunity with no investment.
• In finance, we require that the no-arbitrage
  principle holds. The no-arbitrage principle
  states that any rational price for a financial
  instrument must exclude arbitrage opportunities.

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Option contracts and pricing

• Factors affecting the price of an option
• The value of an option can be thought of as being
  comprised of two parts
• Intrinsic value (S-X)
• Time to maturity

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Option contracts and pricing

• Factors affecting the price of an option
• Stock price
• Strike price
• Volatility of underlying asset
• Time to maturity
• Risk-free rate

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Option contracts and pricing

• Bounds on call option prices
• Upper bound for European call option
• Co ? S -- The call price can never be more
  than the price of the stock

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Option contracts and pricing

• Bounds on call option prices
• Lower bound for a European call option
• To see where the lower bound comes from, assume
  that I have two portfolios. In portfolio A, I
  purchase a call option for Co and I invest
  Xe-rf(T) in a risk-free asset. In portfolio B, I
  buy the stock for So.

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Option contracts and pricing

• Bounds on call options prices

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Option contracts and pricing

• Put-Call Parity
• Put-call parity gives us a fixed relationship
  between put prices, call prices, and stock
  prices. This allows us to be able to price put
  contracts using call prices and it allows us to
  determine if there are inefficiencies in the
  options market.

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Option contracts and pricing

• Put-Call Parity
• To develop the put-call parity relationship
  suppose that we have the following portfolio
  invest in one share of stock, one put option, and
  write one call option. Both options are written
  on the share of stock we own and they both have
  the same maturity date and the same exercise
  price.

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Option contracts and pricing

• Put-Call Parity

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Option contracts and pricing

• So, no matter what state of nature occurs, the
  portfolio is worth X. Thus, the payoff from the
  portfolio is risk free. Thus, the price of this
  portfolio, when we buy it at time 0, should just
  be the discounted value of X.

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Option contracts and pricing

• Binomial Option Pricing Model for one Period
• The basic concept of the binomial option pricing
  model is that only two things will happen to the
  price of the stock, it will either go up or it
  will go down, hence, binomial option pricing
  model. Note, this is a simple one-period model.
  At the end of one time period, the option will
  expire.

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Option contracts and pricing

• Example
• Suppose we have the following
• S20, X21, rf .10,
• u upward movement in stock price 1.2
• d downward movement in stock price .67

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Option contracts and pricing

• Binomial option pricing model

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Option contracts and pricing

• Binomial option pricing model
• Risk-less hedge
• We can create a risk-less hedge by investing in
  the stock and going short in the options.
• To be riskless, the hedge must be created such
  that the value of the hedged portfolio is the
  same whether or not the stock price goes up or
  down. Thus, uS - m (Cu) dS - m (Cd)

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Option contracts and pricing

• Binomial option pricing model
• Risk-less hedge

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Option contracts and pricing

• Binomial option pricing model
• Payoff to riskless hedge
• uS - m (Cu) 1.220 - 3.53(3) 13.40
• dS - m (Cd) .6720 - 3.53(0) 13.40

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Option contracts and pricing

• Binomial option pricing model
• Since the payoff is riskless, the price of the
  contract now must is just the discounted value of
  the constant payoff.

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Option contracts and pricing

• Binomial option pricing model
• From the previous equation, solving for C0 and
  substituting m gives
• d 0.82232 Co 2.21

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Option contracts and pricing

• General features of all option pricing strategies
  gained from the binomial model
• Must make some assumption about how stock prices
  are generated
• All pricing models use the concept of the
  risk-less hedge to develop and equilibrium option
  price

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Option contracts and pricing

• Black-Scholes Option Pricing Model
• The model most widely used to price options is
  the Black-Scholes option pricing model. It is
  one of the best pricing models in finance.

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Option contracts and pricing

• Inputs to the Black-Scholes option pricing model

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Option contracts and pricing

• 1. Where does Black-Scholes come from?
• Black and Scholes assume that stock prices follow
  a geometric Brownian motion process
• Again, the concept of the risk-less hedge is
  employed. However, this time the hedge is
  assumed to be maintained continuously. Thus,
  changes in the hedge portfolio should be
  constant.
• Using this framework Black and Scholes use some
  complicated math to get their formula.

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Option contracts and pricing

• 2. Example
• S 50, X 45, rf .06, ?2 .20, t 3 months

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Stock as a Call Option

• Consider a firm that owes X to the bondholders
  at the end of the year.
• Let S be the cash flow of the firm at the end of
  the year.

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Stock as a Call Option
Payoff to Stockholders
0
S
X
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Stock as a Call Option

• Bondholders own the firm.
• They have sold a call option (with exercise price
  of X) on the firms assets to the stockholders.
• If asset (firm) value gt X at maturity,
  stockholders exercise the call option and buy
  the firm after paying the bondholders X.
• If asset (firm) value lt X at maturity, the
  stockholders walk away (do not exercise their
  call option), and bondholders keep the firm.

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Stockholders Own a Put Option

• Stockholders own the firm.
• They owe the bondholders X.
• They have also purchased a put option on the firm
  from the bondholders.
• The put option has an exercise price of X.

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Stockholders Own a Put Option

• The put will be exercised by the stockholders if
  the asset (firm) value is less than X (i.e. if
  S lt X).
• By exercising the put, they will sell the firm to
  the bondholders for X.
• Since stockholders owe X to the bondholders, and
  bondholders buy the firm for X, the debt is
  simply cancelled.

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Risky Debt

• Bondholders hold a risky bond.
• There is a possibility of default (if S lt X)
• Since bondholders have sold a put option to the
  stockholders,

Value of risky bond
Value of default-free bond
- Value of put option
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The Two Views

• Bondholders own the firm.
• Stockholders own a call option on the firms
  assets, sold to them by the bondholders.

• Stockholders own the firm.
• Stockholders owe X to the bondholders.
• Stockholders own a put option on the firms
  assets, sold to them by the bondholders.

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Put Call Parity
Value of put option on the firm
Value of the firm
Value of call option on the firm
Value of default-free bond
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Why do people use derivatives

• Speculation
• Hedging Hedging is the more interesting of the
  two reasons for the use of derivative contracts.
  Corporation face risks in the form of changes in
  interest rates, changes in exchange rates, and
  changes in input (commodity ) prices. Thus, it
  is not surprising that most corporation cite
  hedging as the reason that they use derivative
  contracts.

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Why would a company choose to hedge

• Lets first consider what the value of the firm
  is

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Why would a company choose to hedge

• It is easy to see from this equation that
  expected cash flows must be increased or the
  firms discount rate must be decreased in order
  for the value of the firm to rise.
• It is somewhat unclear how hedging can affect a
  firms discount rate (or cost of capital).
• For now, will focus on how hedging can increase
  our expected cash flows.

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Why would a company choose to hedge

• Hopefully, we remember that Modigliani and Miller
  (MM) showed us, in a perfect world, a firms
  financing choice will not affect the value of the
  firm.
• This also applies with hedging. Under the MM
  assumptions, hedging should have no effect on the
  value of the firm, since financing choice does
  not affect cash flows.
• Also, if we believe in portfolio theory, the
  riskiness of the firm is not important to
  investors since they can diversify on their own.

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Why would a company choose to hedge

• In the real-world, there are taxes, bankruptcy
  costs, financing costs, and conflicts between
  different types of investors. It is in these
  market imperfections that a firm can increases
  its value by increasing its expected cash flows.

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Why would a company choose to hedge

• 1. Taxes Hedging can reduce a firms tax
  burden if the firm has a convex tax function.
  The best way to see this is through a simple
  example.
• Suppose that we have a firm that has a 50 chance
  of earning 10 million and a 50 chance of
  earning 100 million. Suppose that the firm
  faces a convex tax schedule such that 10million
  is taxed at 20, 55 million is taxed at 25, and
  100 million is taxed at 40.

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Why would a company choose to hedge

• 2. Reducing the probability of bankruptcy If a
  firm hedges its value, there is less of a chance
  that it will earn an income that would be
  insufficient to meet it obligations.
• a. Direct costs
• b. Indirect costs
• c. Reduction of agency costs

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Why would a company choose to hedge

• 3. Reducing the cost of financing If a firm
  can maintain a more constant cash flow stream, it
  is more likely that the firm will have cash on
  hand in order to finance projects.
• Going into the capital markets for financing is
  extremely expensive and not always feasible.
• Thus, by hedging and keeping cash flows constant,
  a firm has a higher probability of having the
  cash available to invest in positive NPV projects.