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### Chapter 12

DERIVATIVES: ANALYSIS AND VALUATION

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?

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?

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)

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 (1+rf)T

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.

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 (1+rf +SI)T

- Incorporating ownership benefits lost with a futures position, especially dividends(d):

F0,T = S0 (1+rf +SI -d)T

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)

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 (1+rf +-d)T

- If it does not hold, construct a risk-free position to take advantage of the situation.

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

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

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

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

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

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 (+)

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!

Black-Scholes Option Pricing Formula

P0 = PS[N(d1)] - X[e-rt][N(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

Black-Scholes Option Pricing Formula

P0 = PS[N(d1)] - X[e-rt][N(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

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

Option Valuation Terminology

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

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

Option-like Securities

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

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

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

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

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

Convertible Securities

- Allows the holder to convert one type of security into a stipulated amount of another type (usually common stock) at the investor’s 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

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

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

Convertible Securities

Convertible Bonds

- Value of a convertible as a bond
- Determine the bond’s value as if it had no conversion feature
- This is the convertible’s 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

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

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

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

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

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