1. Venture Capital and the Finance of Innovation [Course number] Professor [Name ] [School Name] Chapter 13 Option Pricing

2. Option Definitions  A European call option gives the holder the right to buy an asset at a preset exercise price on an expiration date.  For a put option, just substitute “sell” for “buy”.  “Strike price” is a synonym for exercise price.  For an American option, exercise can occur anytime on or before the expiration date.

3. Option “Moneyness”  If the exercise price is higher, the same, or lower than the current stock price, then we say that a call option is “out-of-the-money”, “at-the-money”, or “in-the-money”, respectively.  For put options, reverse this ordering (e.g., put options are out-of-the-money when the exercise price is lower than the current stock price.)

4. Call Option X=100 C T (X;T) = Max(V T – X, 0) = C 1 (100;1) = Max(V 1 -100,0)

5. Put Option P T (X;T) = Max(X –V T, 0) = C 1 (100;1) = Max(100 –V 1,0)

6. Example Suppose that Bigco is currently trading for $100 per share. We are offered a European call option to purchase one share with an expiration date in one year. We know that on the expiration date Bigco stock will sell for either $120 per share (a “good day”: probability = 80%) or for $80 per share (a “bad day”: probability = 20%). No other prices are possible. The stock will not pay any dividends during the year. The riskless interest rate is zero, so a bond can be purchased (or sold) for a face value of $100 and have a certain payoff of $100 in one year. Stocks, bonds, and options can all be bought or sold, long and short, without any transactions costs.

7. Black-Scholes Assumptions  “Perfect Markets” that are  Open all the time  Allow assets to be traded in any quantity  No taxes or transactions costs  No remaining arbitrage opportunities  Technical assumptions about the statistical properties of stock and bond returns. (Most famously, log normal stock returns).

8. Black-Scholes Formula where S = current stock (or “underlying asset” price X = Exercise (or “strike”) price T = Time to expiration (in years) R = annualized riskfree rate (continuously compounded = log) σ = volatility (annualized standard deviation of log returns) N(d) = cumulative normal distribution evaluated at d

9. EXHIBIT 13.7 – FIVE-YEAR COMPOUND RETURNS FOR A LOGNORMAL DISTRIBUTION

10. Random-Expiration (RE) Options  Continuous-time probability (q) of forced expiration.  Expected holding period = H = 1/q

11. RE Call Formula You do not need to memorize this!