Risk Management & Real Options Interlude The Link to Financial Options and Black-Scholes

1. Risk Management & Real OptionsInterludeThe Link to Financial Options and Black-Scholes Stefan Scholtes Judge Institute of Management University of Cambridge MPhil Course 2004-05

2. A financial option is… • A right but not an obligation • To buy (“call”) or sell (“put”) • A market-valued asset (“underlying asset”) • At a fixed price (“strike price”) • At some fixed time in the future (“European”) or during a fixed time span (“American”) © Scholtes 2004

3. Value of a European call option Value of the call at time of exercise Stock price - strike price Decision: Don’t exercise Decision: Exercise Stock price Strike price © Scholtes 2004

4. What is the difference to real options? • FOs are purely financial contracts, i.e., a bet on changing values of the underlying asset • At exercise money changes hands but nothing material (“real”) happens • FOs are traded in markets • There exists a market price (law of one price) • FOs have short time horizons • Used to hedge risks • E.g. a Put on a stock price hedges the owner of the stock against low stock prices • As stock falls, value of put option rises © Scholtes 2004

5. The Black Scholes Model • Key question: What’s the “correct” market price of a financial option? • Nobel Prize-winning answer given by Black, Scholes and Merton in 70ies • Black-Scholes formula • There are many finance people (in academia) who believe that the “right” way of valuing a real option is the Black-Scholes valuation model © Scholtes 2004

6. How does the B-S model work? • The B-S model assumes that the underlying asset value follows a “geometric Brownian motion” • Another way of saying that the returns have log-normal distributions • Underlying asset value can be modelled in a spreadsheet by a lattice • The B-S model values the option, using the “consistent valuation” of chance nodes that we had used in the R&D option valuation © Scholtes 2004

7. Example 1 0 Call option at strike price 4 ? x = 1 1 All moves are triggered by the same flip of the coin: Price up or Price down Bank account 1 5 2 Stock price 3 “Price up” “Price down” © Scholtes 2004

8. Example 5 1 1 2 1 0 Call option at strike price 4 3 1 2/3 * - x 1/3 * = 1 1 All moves are triggered by the same flip of the coin Investing $ 1 in stock and borrowing $ 2/3 from the bank fully REPLICATES the call payoffs To buy this REPLICATING PORTFOLIO I need £ 1/3 – that’s the price of the call Bank account 1 5 2 Stock price 3 “Price up” “Price down” © Scholtes 2004

9. The general case: Binomial lattice model All chance nodes follow THE SAME underlying uncertainty: The price of the asset moves up or down uS Price of asset moves up or down S dS (1+r) Risk-free investment r=one-period risk-free rate 1 (1+r) Cu Value of the option on the stock price C=? Cd © Scholtes 2004

10. Computing the one-period B-S value The consistent value for C can be computed as where © Scholtes 2004

11. Example 5 1 1 2 1 0 Call option at strike price 4 3 1 2/3 * - x 1/3 * = 1 1 Bank account 1 5 2 Stock price 3 “Price up” “Price down” © Scholtes 2004

12. Multi-period models • Financial option is valued by • Dividing the time to maturity into a number of periods • Spanning out the lattice for the underlying asset value • Applying backwards induction, as discussed before, to value the option • The B-S price is the theoretical price one obtains as the number of periods goes to infinity • 10-15 periods is normally sufficient for good accuracy • There is a closed form solution for European options, called the Black-Scholes formula • It also applies to American call options without dividend payments • Spreadsheet example of a lattice valuation of an American call can be found in “BlackScholesOptionsPricing.xls” © Scholtes 2004

13. Hedging and the Non-Arbitrage argument • The key to financial options valuations is hedging • Buying the option and selling the replicating portfolio (or vice versa) has zero future cash flows, no matter what, because they have the same payoffs in every state of nature (at least in the model) • If the option was cheaper than the replicating portfolio, one could make risk-less profits (“arbitrage profits”) by buying the option and selling the replicating portfolio • and vice versa if the replicating portfolio was cheaper • Only price that would make both, the replicating portfolio and the option tradable is the price of the replicating portfolio, which is the Black-Scholes price • This is called the “non-arbitrage” argument for the options price © Scholtes 2004

14. Hedging in a real options situation • What’s the value of a 10-year lease on a mine? • Extraction rate 10,000 ounces / year • Extraction cost £250 / ounce • Risk-free interest 5% • Company discount rate 10% • Current gold price £260 / ounce • Growth rate of gold price 2.5% For those who are interested: This is worked out in the spreadsheet GoldMine.xls © Scholtes 2004

15. Summary • Financial options analysis has been instrumental in raising awareness in the value of real options analysis • Largely responsible for real options lingo • Financial options techniques are valuable to deal with market uncertainties • Equivalent to consistent chance node valuation • Blind-folded application of financial options techniques is dangerous • Hybrid approach to deal with technical and market risk separately is preferable and can give hugely different results © Scholtes 2004

16. Real versus financial options • Most important difference between financial and real options: Financial options are “priced” Real options are “valued” • Typically, real options analysis needs to help us make a decision, not to find the correct price! • But: there are situations where we will have to name a “price” – e.g. bidding • Biggest drawback of Black-Scholes: It is often “sold” as a black-box “…give me the volatility and I give you the correctvalue of your real option…” • People don’t focus enough on the need to tell a good story with the model • B-S is like telling a story in a foreign language; it may well be a great story but what good is it if no-one is willing to listen? © Scholtes 2004