1. Venture Capital and the Finance of Innovation [Course number] Professor [Name ] [School Name] Chapter 22 Binomial Trees

2. Call Option in a Decision Tree

3. A 3-Step Binomial Tree

4. Multi-step trees To solve 1)Compute the key inputs: u, d, and p. 2)Build the base tree by solving forwards using the up and down movements. 3) Build the option tree by solving backwards using the risk-neutral probabilities.

5. Base Tree

6. Option Tree

7. Cox, Ross, and Rubinstein (CRR) Model  Time length = t  Annualized volatility = σ  Riskless interest rate = r  Then u = d = 1/u = p = (e rt – d) / (u – d)

8. Joe’s Problem Assume  Starting stock price = S = $100  Strike Price = X = $50  Volatility = σ = 60%  Riskfree rate = r = 5%  Time to expiration = T = 1  3-step tree → N = 3 and t = 1/3

9. Joe’s Problem, Base Tree

10. Joe’s Problem, Option Tree

11. Example 1 Drugco is a publicly traded biotechnology company with several drugs in development, but no products on the market. To raise capital for the development of Newdrug, Drugco enters a strategic alliance with Bigco. In return for marketing rights for Newdrug, Bigco will pay for clinical trials and will give Drugco up-front and milestone payments. Bigco also agrees to make an equity investment in Drugco, purchasing ten million shares at the market price of $10 per share and also receiving warrants to purchase an additional 10 million shares. (The market price of $10 includes the market reaction to the Bigco alliance.) These warrants can either be exercised in exactly two years at a strike price of $20 per share or in exactly five years, with a strike price of $50 per share. Drugco does not pay dividends and has no plans (or cash) to do so for at least the next five years. The expected volatility of Drugco stock is 60 percent per year. Problem What is the value of Bigco’s warrants?

12. Dividends Consider again Joe’s three-step problem. Now, let’s add a dividend in the second-to-last period (eight months into the year) that is equal to 10 percent of the stock value. Now, if Joe decides to exercise after eight months, he would receive the whole value of the stock (including the 10 percent dividend). If, instead, he decides to wait until the full year is over, then the 10 percent dividend gets paid out after eight months, and the stock price falls by 10 percent before making an up or down move in the last period.

13. Joe’s Problem, Base Tree, with Dividends

14. Joe’s Problem, Option Tree, with Dividends

15. Example 2 Fuelco is considering a consumer application for their patented fuel-cell technology. They have already completed several R&D projects with this technology, so they have eliminated the technical risk for this new project. To begin producing and marketing to the consumer market would require a new investment of $200M. At the present time, Fuelco estimates that the completed project would have a present value of $400M (i.e., if Fuelco spent $200M to initiate the project, they believe they could spin off the initiated project for $400M). Fuelco can delay starting the project for up to five years, during which time they expect this value of the project to fluctuate, with an annual volatility of 90 percent. Once initiated, the project is expected to generate annual cash flows equal to 10 percent of its value. Thus, if Fuelco delays the project, they will forego these cash flows. After five years, some important Fuelco patents will expire, and they will not longer have the option to profitably enter this new market. If Fuelco does not enter the market, then Project C has no salvage value. Problem What is the NPV of Project C?

16. Fuelco’s problem: Project C with dividends

17. Appendix: CRR model details These three conditions can be solved to obtain the CRR equations.