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DERIVATIVES: ANALYSIS AND VALUATION
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 ( )3 = 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

Intrinsic Value = (Stock Price – Exercise Price) x Number of Share
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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