DERIVATIVES: ANALYSIS AND VALUATION

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Chapter 12

• DERIVATIVES ANALYSIS AND VALUATION

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Chapter 12 Questions

• How are spot and futures prices related?
• What is basis risk?
• What is program trading and stock index
  arbitrage? How can futures be used to hedge or
  speculate on changes in yield curve spreads and
  credit quality spreads?
• Why would investors want to invest in an option
  on a futures contract?

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Chapter 12 Questions

• What factors influence the price of an option?
• How does one use the Black-Scholes option-pricing
  model?
• Why are the terms delta,theta, vega, rho, and
  gamma important to option investors?
• How do option-like features affect the price of
  bonds?

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Futures Valuation Issues

• Cost of Carry Model
• Suppose that you needed some commodity in three
  months. You have at least the following two
  options
• Purchase the commodity now at the current spot
  market price (S0) and carry the commodity for 3
  months
• Buy a futures contract for delivery of the
  commodity in 3 months for the current futures
  price (F0,3)

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Futures Valuation Issues

• Cost of Carry Model
• The futures prices and spot prices must be
  related to one another in order for there to be
  no arbitrage opportunities for investors.
• If the carrying cost only amounts to forgone
  interest at a risk-free rate (rf) for T time
  periods, then the following relationship must
  hold
• F0,T S0 (1rf)T

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Futures Valuation Issues

• Cost of Carry Model Example Suppose that you can
  buy gold in the spot market for 300. The
  monthly risk-free is .25. You need the gold in
  three months.
• What should be the current futures price?
• F0,T 300 (1.0025)3 302.26
• What if the futures price is 305?
• You have a risk-less profit opportunity. Buy
  gold at 300, sell futures at 305. In three
  months, delivery the gold, pay the known
  interest, pocket the difference.

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Futures Valuations Issues

• Similar futures-spot price relationships can be
  derived when there are market imperfections
  involved with carrying the commodity or financial
  asset
• Incorporating storage and insurance costs as a
  percentage of contract value (SI)
• F0,T S0 (1rf SI)T
• Incorporating ownership benefits lost with a
  futures position, especially dividends(d)
• F0,T S0 (1rf SI -d)T

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Futures Valuation Issues

• Basis
• Basis is the difference between the spot and
  futures prices.
• For a contract expiring at time T, the basis at
  time t is
• Bt,T St Ft,T
• Over time, the spot and futures prices converge,
  and basis becomes zero at expiration
• Between time t and expiration, basis can change
  as the difference between spot and futures prices
  vary (known as basis risk)

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Advanced Applications of Financial Futures

• Stock Index Arbitrage
• An example of a program trading strategy designed
  to take advantage of temporarily mis-pricing of
  securities
• Monitor the parity condition
• F0,T S0 (1rf -d)T
• If it does not hold, construct a risk-free
  position to take advantage of the situation.

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Advanced Applications of Financial Futures

• T-Bond/T-Note Futures Spread
• Note over bond (NOB) spread
• Strategies based on speculating the changing
  slope of the yield curve

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Options on Futures

• Also known as Futures Options
• Options on Stock Index Futures
• Gives the owner the right to buy (call) or sell
  (put) a stock futures contract
• Options on Treasury Bond Futures
• Gives the owner the right to buy (call) or sell
  (put) a Treasury bond futures contract

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Options on Futures

• Why would they be attractive?
• If exercised, it would seem to have been better
  to simply buy a futures contract instead (no
  option premium to pay)
• One primary advantage can be found when looking
  at all the potential price movements
• Futures contracts used for hedging offset
  portfolio value changes thus, advantageous price
  movements for a portfolio are offset by the
  futures position
• Options give the right (but not the obligation)
  to purchase the futures contract thus, favorable
  price movements will be offset only by the option
  premium rather than by a corresponding loss on
  the futures position

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Valuation of Options

• Factors influencing the value of a call option
• Stock price ()
• For a given exercise price, the higher the stock
  price, the greater the intrinsic value of the
  option (or at least the closer to being
  in-the-money)
• Exercise price (-)
• The lower the price at which you can buy, the
  more value
• Time to expiration ()
• The longer the time to expiration, the more
  likely the option will be valuable

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Valuation of Options

• Factors influencing the value of a call option
• Interest rate ()
• Options involve less money to invest, lower
  opportunity costs
• Volatility of underlying stock price ()
• The greater the volatility of the underlying
  stock, the more likely that the option position
  will be valuable

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Valuation of Options

• Factors influencing the value of a put option
• The same listed, but different directions for
  several items.
• Stock price (-)
• Exercise price ()
• Time to expiration ()
• Interest rate (-)
• Volatility of underlying stock price ()

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Black-Scholes Option Pricing Model

• Model for determining the value of American call
  options
• This work warranted the awarding of the 1997
  Nobel Prize in Economics!

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Black-Scholes Option Pricing Formula

• P0 PSN(d1) - Xe-rtN(d2)
• where
• P0 market value of call option
• PS current market price of underlying stock
• N(d1) cumulative density function of d1 as
  defined later
• X exercise price of call option
• r current annualized market interest rate for
  prime commercial paper
• t time remaining before expiration (in years)
• N(d2) cumulative density function of d2 as
  defined later

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Black-Scholes Option Pricing Formula

• P0 PSN(d1) - Xe-rtN(d2)
• The cumulative density functions are defined as

Where ln(PS/X) natural logarithm of (Ps/X) S
standard deviation of annual rate of return on
underlying stock
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Using the Black-Scholes Formula

• Besides mathematical values, there are five
  inputs needed to use this model
• Current stock price (Ps)
• Exercise price (X)
• Market interest rate (r)
• Time to expiration (t)
• Standard deviation of annual returns (s)
• Of these, only the last in not observable
• Also, using the put/call parity, we can value put
  options as well after calculating call value

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Option Valuation Terminology

• Delta
• The sensitivity of an options price to the price
  of the underlying security
• Positive for calls, negative for puts
• Theta
• Measures how the option premium changes as
  expiration approaches

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Option Valuation Terminology

• Vega
• The sensitivity of the option premium to the
  price volatility (s) of the underlying security
• Rho
• Measures the sensitivity of the option premium to
  changes in interest rates
• Gamma
• Measures the sensitivity of delta to changes in
  the underlying security price

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Option-like Securities

• Several types of securities contain embedded
  options
• Callable and Putable Bonds
• Warrants
• Convertible Securities

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Callable and Putable Bonds

• Callable Bonds contain a call provision
• The issuer has the option of buying the bonds
  back at the call (exercise) price rather than
  having to wait until maturity
• Attractive option for issuers if interest rates
  fall, since they can purchase back old bonds and
  refinance (refunding) with new, lower interest
  bonds
• Typically will trade at no more than the call
  price, since call becomes likely at that point

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Callable and Putable Bonds

• Putable Bonds contain a put provision
• Investors may resell the bonds back to the issuer
  prior to maturity at the put (exercise) price,
  often par value
• Puts can generally be exercised only when
  designated events take place

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Warrants

• Warrant is an option to buy a stated number of
  shares of common stock at a specified price at
  any time during the life of the warrant
• Similar to a call option, but usually with a much
  longer life
• Issued by the company whose stock the warrant is
  for

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Warrants

• Intrinsic value is the difference between the
  market price of the common stock and the warrant
  exercise price
• Intrinsic Value (Stock Price Exercise Price)
  x Number of Share
• Speculative value is the value of the warrant
  above its intrinsic value
• Like other options, the value is higher than
  intrinsic value, except at maturity

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Convertible Securities

• Allows the holder to convert one type of security
  into a stipulated amount of another type (usually
  common stock) at the investors discretion
• With convertible securities, value depends both
  on the value of the original asset and the value
  if conversion takes place
• Value cannot fall below the greater of the two
  values

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Convertible Securities

• Convertible Bonds
• Advantages to issuing firms
• Lower interest rate on debt
• Debt represents potential common stock
• Advantages to investors
• Upside potential of common stock
• Downside protection of a bond

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Convertible Securities

• Convertible bonds
• Conversion ratio number of shares obtained if
  converted
• Conversion price Face Value/Number of shares
• Valuation of convertible bonds
• Combination value of stock and bond
• Two step process to determine minimum value

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Convertible Securities

• Convertible Bonds
• Value of a convertible as a bond
• Determine the bonds value as if it had no
  conversion feature
• This is the convertibles investment value or
  floor value
• Value of a convertible as stock
• Compute the value of the common stock received on
  conversion
• This is the conversion value

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Convertible Securities

• Convertible Bonds
• Minimum Value Max (Bond Value, Conversion
  Value)
• Like other options, including embedded options,
  they typically only sell at their minimum,
  intrinsic value only at maturity.
• Conversion Premium (Market Price Minimum
  Value)/Minimum Value

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Convertible Securities

• Convertible Bonds
• Conversion Parity Price Market Price/Conversion
  Ratio
• An risk-free profit opportunity would exist if
  the price of the convertible below this price,
  since immediate conversion of the bond and then
  selling the stock would yield a profit
• Payback
• How long it takes the higher-interest income from
  the convertible bond (compared to the stock
  dividend) to make up for the conversion premium

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Convertible Securities

• Convertible Preferred Stock
• Combination of preferred stock and common stock
• Common characteristics
• Cumulative but not participating dividends
• No sinking fund or purchase fund
• Fixed conversion rate
• Waiting period not required before conversion
• Conversion privilege does not expire
• Usually issued in connection with mergers

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Convertible Securities

• Convertible Preferred Stock
• Value as preferred stock
• Value as common stock, given the conversion rate
• Parity relationships imply that the value has to
  be higher than the maximum of the two values