《金融期货与期权》（英文版） Chapter 28 Real options

28.1 Chapter 28 Real options Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 28.1 Chapter 28 Real Options

28.2 An alternative to the npv rule for Capital Investments Define stochastic processes for the key underlying variables and use risk- neutral valuation This approach(known as the real options approach) is likely to do a better job at valuing growth options, abandonment options, etc than NPV Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 28.2 An Alternative to the NPV Rule for Capital Investments • Define stochastic processes for the key underlying variables and use risk￾neutral valuation • This approach (known as the real options approach) is likely to do a better job at valuing growth options, abandonment options, etc than NPV

28.3 The Problem with using NPV to Value options Consider the example from Chapter 10 Stock Price $22 Stock price $20 Stock Price=$18 Suppose that the expected return required by investors in the real world on the stock is 16%. What discount rate should we use to value an option with strike price $21? Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 28.3 The Problem with using NPV to Value Options • Consider the example from Chapter 10 • Suppose that the expected return required by investors in the real world on the stock is 16%. What discount rate should we use to value an option with strike price $21? Stock Price = $22 Stock price = $20 Stock Price=$18

28.4 Correct Discount Rates are Counter-Intuitive Correct discount rate for a call option is 42.6% Correct discount rate for a put option is 52.5% Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 28.4 Correct Discount Rates are Counter-Intuitive • Correct discount rate for a call option is 42.6% • Correct discount rate for a put option is –52.5%

28.5 General Approach to valuation We can value any asset dependent on a variable e by Reducing the expected growth rate of e by ns where n is the market price of e-risk and s is the volatility of 0 Assuming that all investors are risk-neutral Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 28.5 General Approach to Valuation • We can value any asset dependent on a variable q by – Reducing the expected growth rate of q by ls where l is the market price of q-risk and s is the volatility of q – Assuming that all investors are risk-neutral

28.6 Extension to Many Underlying Variables When there are several underlying variable 0, We reduce the growth rate of each one by its market price of risk times its volatility and then behave as though the world is risk-neutral Note that the variables do not have to be prices of traded securities Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 28.6 Extension to Many Underlying Variables • When there are several underlying variable qi we reduce the growth rate of each one by its market price of risk times its volatility and then behave as though the world is risk-neutral • Note that the variables do not have to be prices of traded securities

Estimating the market price or a7 Risk(equation 28.7, page 665) The market price of risk of a variable is given by 入=+(μm-r) where p is the instantane ous correlation between percentage changes in variable and returns on the market; om is the volatility of the market s return; um is the expected return on the market and r is the short -term risk-free rate Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 28.7 Estimating the Market Price of Risk (equation 28.7, page 665) risk -free rate return on the market; and i s the short -term of the market' s return; i s the expected and returns on the market; i s the volatility between percentage changes i n variable where i s the instantane ous correlation The market price of risk of a variable i s given by r r m m m m     −   l = ( )

28.8 Schwartz and moon have applied the real options ApproachtovaluingAmazon.com They estimated stochastic processes for the company's sales revenue and its revenue growth rate They estimated the market prices of risk and other key parameters (cost of goods sold as a percent of sales, variable expenses as a percent of sales, fixed expenses, etc.) They used Monte Carlo simulation to generate different scenarios in a risk-neutral world The stock price is the present value of the net cash flows discounted at the risk-free rate Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 28.8 Schwartz and Moon Have Applied the Real Options Approach to Valuing Amazon.com • They estimated stochastic processes for the company’s sales revenue and its revenue growth rate. • They estimated the market prices of risk and other key parameters (cost of goods sold as a percent of sales, variable expenses as a percent of sales, fixed expenses, etc.) • They used Monte Carlo simulation to generate different scenarios in a risk-neutral world. • The stock price is the present value of the net cash flows discounted at the risk-free rate

28.9 Commodity Prices Futures prices can be used to define the process followed by a commodity price n a risk-neutral world We can build in mean reversion and use a process for constructing trinomial trees that is analogous to that used for interest rates in Chapter 23 Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 28.9 Commodity Prices • Futures prices can be used to define the process followed by a commodity price in a risk-neutral world. • We can build in mean reversion and use a process for constructing trinomial trees that is analogous to that used for interest rates in Chapter 23

28.10 Example (page 671) a company has to decide whether to invest $15 million to obtain 6 million barrels of oil at the rate of 2 million barrels per year for three years. The fixed operating costs are $6 million per year and the variable costs are $17 per barrel. The spot price of oil $20 per barrel and 1, 2, and 3-year futures prices are $22, $23, and $24, respectively. The risk-free rate is 10% per annum for all maturities Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 28.10 Example (page 671) A company has to decide whether to invest $15 million to obtain 6 million barrels of oil at the rate of 2 million barrels per year for three years. The fixed operating costs are $6 million per year and the variable costs are $17 per barrel. The spot price of oil $20 per barrel and 1, 2, and 3-year futures prices are $22, $23, and $24, respectively. The risk-free rate is 10% per annum for all maturities