Risk Management & Real Options Interlude The Link to Financial Options and Black-Scholes Stefan Scholtes Judge Institute of Management University of Cambridge MPhil Course

2 September 2004 © Scholtes 2004Page 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”)

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

2 September 2004 © Scholtes 2004Page 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

2 September 2004 © Scholtes 2004Page 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

2 September 2004 © Scholtes 2004Page 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

2 September 2004 © Scholtes 2004Page 7 Example x Call option at strike price 4 Bankaccount Stockprice = “Price up” “Price down” ? All moves are triggered by the same flip of the coin: Price up or Price down

2 September 2004 © Scholtes 2004Page 8 Example x = /3 * 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 1/3 * - “Price up” “Price down” Call option at strike price 4 Bankaccount Stockprice

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

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

2 September 2004 © Scholtes 2004Page 11 Example x = /3 * 1/3 * - “Price up” “Price down” Call option at strike price 4 Bankaccount Stockprice

2 September 2004 © Scholtes 2004Page 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 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”

2 September 2004 © Scholtes 2004Page 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

2 September 2004 © Scholtes 2004Page 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

2 September 2004 © Scholtes 2004Page 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

2 September 2004 © Scholtes 2004Page 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 correct value 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?