Investments: Derivatives

1. Investments:Derivatives Professor Scott Hoover Business Administration 365

2. Call option • The buyer • Pays for the call option up front • Has the right to buy the underlying asset… • …on some specified later date • …for a specified price (the “strike price” or “exercise price”). • The seller (aka, “writer”) • Receives money up front. • Has the obligation to sell the underlying asset if the buyer so chooses • Note: • American options: may be exercised early • European options: may only be exercised on the expiration date.

3. Put option • The buyer • Pays for the put option up front • Has the right to sell the underlying asset… • …on some specified future date • …for a specified price (the “strike price” or “exercise price”). • The seller (aka, “writer”) • Receives money up front. • Has the obligation to buy the underlying asset if the buyer so chooses • Forwards/Futures • No money exchanged up front • Agree to do a transaction… • …on some pre-specified future date • …for a pre-specified price

4. Why do we care? • Employee stock options (ESOs) • …reduce the value of outstanding stock, so we must value the ESOs in order to value the stock. • Hedging • Derivatives allow us to hedge the risk associated with investments. • Portfolio shaping • Derivatives allow us to structure the payoff on our portfolio to best take advantage of our beliefs. • To be successful, … • our views must differ from those of the market. • the market must correct prior to expiration of the portfolio elements.

5. Payoff/Profit Diagrams and Shaping Portfolios • <See spreadsheet> • Note: • We typically concentrate on payoff diagrams. Why? • Profit diagrams ignore the opportunity cost of the initial investment (i.e., they ignore the time value of money)

6. Valuing Derivatives • Based on finding replicating portfolios • identical payoffs  identical values • Notation • C  value of call option • P value of put option • S price of the underlying asset • X strike (exercise) price of option • R continuously-compounded risk-free • T time to maturity •   volatility of the underlying asset’s returns.

7. The Black-Scholes Model

8. In the equation… • N(.) is the cumulative normal distribution. • N(.) is the area under the standard normal curve up to the point indicated in parentheses. • No closed-form equation for N(.), so must use tables or Excel.

9. Characteristics of call option values • <see Black-Scholes spreadsheet> • C is … • …increasing in  • …decreasing in X • …increasing in Rf • …increasing in S • …increasing in T • For American calls… • call will never be exercised early in the absence of dividends • will be exercised immediately prior to a dividend if the dividend is large enough

10. Put-Call Parity • Suppose we short sell the stock, buy a call option, and invest in the risk-free asset. What would the portfolio payoff look like? • See spreadsheet • Note that if we invest the PV of X in the risk-free, our portfolio replicates the payoffs of a put option! put must have same cost as portfolio • Put-call parity: -S + C + X/eRT = P

11. Characteristics of put option values • <see Black-Scholes spreadsheet> • P is … • …increasing in  • …increasing in X • …decreasing in Rf • …decreasing in S • …ambiguous in T • For American puts… • put may be exercised early if the put is sufficiently in the money • put will never be exercised immediately prior to a dividend payment

12. What else can we learn? • implied volatility the volatility that makes the B-S value equal to the current price • gives the market’s expectation of market volatility • The VIX (ticker) gives the 30-day expectation (see chart) • put-call ratio ratio of the volume of puts traded to the volume of calls traded • gives an indication of the direction of expected volatility • see chart

13. Valuing Employee Stock Options (ESOs) • Recall that … • Value of the firm = Debt + Preferred stock + Common stock + Current and expected ESOs • We can infer the value of common stock using • Common stock = Value of the firm- Debt - Preferred Stock - Current and expected ESOs

14. How do ESOs differ from European call options? • May be exercised early • When employees leave a company, they often must exercise or forfeit their ESOs. •  Some are exercised early, even in the absence of dividends • Some are forfeited • When exercised… • # of shares outstanding increases • company gets a tax deduction for the difference between the stock price and the exercise price.

15. To use Black-Scholes, we must adjust for these differences • Value of ESO = Value of call option×(1-F)×(1-T)× • F ≡ expected forfeiture rate • T ≡ tax rate •  ≡ dilution factor •  = (# shares before exercise)/(# shares after exercise) • See text for more details • Note that an alternative, widely-used approach is to use diluted shares instead of valuing the ESOs directly. • This is necessarily incorrect, but commonly done.

16. Fortunately, the company must provide ample information about the ESOs in the annual report (10-K). • See spreadsheet example.

17. The Forward-Spot Relationship • What price should a forward contract have? • example: • A firm needs 500 barrels of oil in one year • The firm wants to pay for the oil today and receive it in one year. There are two riskless ways to achieve this. • 1. Buy the oil today and store it. • Requires insurance • 2. Invest today and enter into a forward contract. • What is the cost of doing this today? • Need to invest in a risk-free security today that pays us precisely the forward price in one year.

18. Information: • Oil sells for $28 per barrel today. • Storage of oil costs $0.80 per barrel per year. • Insurance for the oil costs $0.24 per barrel per year. • The risk-free rate is 8%, continuously compounded.

19. Consider the two approaches • Approach #1 • We purchase the oil for $28500 = $14,000 today. • Storage costs $0.80500 = $400 • Insurance costs $0.24500 = $120 • Total cost today = $14,520 • Approach #2 • We invest F/e0.081 today. • We receive F in one year • We use the proceeds (F) to cover our obligation on the forward contract. • We receive the oil.

20. These two approaches are equivalent. • We invest money today and receive the oil in one year. • The investments are risk-free. • If there is no arbitrage, the approaches must cost the same today. •  F/e0.081 = $14,520  F = $15,729 • Thus, the forward contract must trade at $15,729 to avoid arbitrage. • This concept is called the Forward-Spot Relationship. • F/eRT=S0 + Other costs/benefits • The costs include whatever it takes to make the purchase/storage option risk-free. • In developing it, we form a replicating portfolio, much like we did with put-call parity.