1. 1 Investments: Derivatives Professor Scott Hoover Business Administration 365

2. 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. 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. 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. 5  Payoff/Profit Diagrams and Shaping Portfolios 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. 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. 7 The Black-Scholes Model

8. 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. 9 Characteristics of call option values   C is … …increasing in  …decreasing in X …increasing in R f …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. 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/e RT = P

11. 11  Characteristics of put option values P is …  …increasing in   …increasing in X  …decreasing in R f  …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. 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. 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. 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. 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. 16  Fortunately, the company must provide ample information about the ESOs in the annual report (10-K).  See spreadsheet example.

17. 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. 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. 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/e 0.08  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. 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/e 0.08  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/e RT =S 0 + 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.