Chapter 12

1. Chapter 12 DERIVATIVES: ANALYSIS AND VALUATION

2. 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?

3. 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?

4. 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)

5. 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

6. 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.

7. 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

8. 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)

9. 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.

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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 (+)

16. 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!

17. 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

18. 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

19. 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

20. 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

21. 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

22. Option-like Securities • Several types of securities contain embedded options: • Callable and Putable Bonds • Warrants • Convertible Securities

23. 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

24. 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

25. 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

26. 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

27. 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

28. 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

29. 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

30. 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

31. 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

32. 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

33. 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

34. 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