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 exercise day
• 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

• American options
• Recall American options allow for exercise before
  the exercise date
• An American option will never be exercised early
  if it can be sold
• Let Ao be the price of an American call that pays
  no dividends that has the same specifications as
  our European call
• Thus A0 ? C 0? Max(S - Xe-rf(T),0)

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Option contracts and pricing

• American calls that pay dividends may be
  exercised early
• An American call option that pays dividends may
  be exercised early if the drop in the stock price
  due to the payment of dividends is large enough
  to offset the potential gain from holding the
  option until maturity.

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Option contracts and pricing

• American calls that pay dividends may be
  exercised early
• If we exercise before the dividend we get
  S(td)-X
• If we hold the minimum price after the dividend
  will be S(td)-D-Xe-rf(T-td)
• Thus, thus if the minimum price after the
  dividend is still larger than the amount we get
  by exercising before the dividend no early
  exercise would occur

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Option contracts and pricing

• Exact condition for no early exercise if there is
  only one dividend payment
• D lt X(1 - e-r(T-tn))

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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

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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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Option contracts and pricing

• Strategies for pricing American Call Options with
  Dividends
• Blacks approximation
• Find the maximum of
• c(Sd, T, X) Black-Scholes option price using
  stock price net of any discounted future
  dividends
• c(S, td, X) Black-Scholes option price using
  time before first dividend

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Option contracts and pricing

• Strategies for pricing American call options with
  dividends
• American option pricing model
• An American option pricing model exists, however
  it does not have a closed form solution.
• It is much more complicated and more difficult to
  use than the Blacks approximation

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Option contracts and pricing

• Problems with the Black-Scholes Model
• Assumptions
• Volatility is not constant over the live of the
  option
• Stock return generating process is not correct

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Option contracts and pricing

• Problems with the Black-Scholes model
• Volatility estimation You need an estimate of
  expected future volatility over the life of the
  option
• Historical
• Implied volatility
• Statistical modeling

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Option contracts and pricing

• Problems with the Black-Scholes model
• Consistent mis-pricing
• The Black-Scholes model has trouble pricing
  options that are deep in-the-money or deep
  out-of-the-money