Calisthenics Assignment Part #1. Models of Commodity Price This assignment requires that you implement a binomial model for the copper price, and that you implement a monte carlo simulation of the copper price. To do this will require looking at textbooks you may have used in 434 or 437 or other resources. References Richard A Brealey and Stewart C. Myers, Principles of Corporate Finance, various editions, discuss the binomial method for simulating stock prices in the material on valuing options; in the 7th edition see Chapter 21, section 21.2. John C. Hull, Options, Futures & Other Derivatives, provides a fuller introduction to binomial trees and also a good cookbook methodology for constructing simulations in spreadsheets; in the 4th edition see Chapter 9 and also 10, esp. section 10.4 on Monte Carlo simulation; see also Chapter 12, sections 12.1 and 12.7 Robert McDonald, Derivatives Markets, also provides a full introduction to binomial trees and simulations; in the 1st edition see Chapter 10 for binomial trees, but not he starts with the “forward tree” method when it is more common to use the Cox-Ross-Rubenstein method which he describes in Chapter 11 section 3; see Chapter 19 for Monte Carlo simulation. Binomial Assignment (1) Construct a spreadsheet simulation models of copper prices. Implement the binomial model with T = 2 years and using a two-step tree, one step for each year. Set the expected rate of appreciation in the price to 10%, the annual volatility to 28%, the risk-free rate to 5%, and the convenience yield to zero and copper spot price to $2.65/pound. • What are the risk-neutral probabilities? What are the actual probabilities? Is the risk-neutral probability of an up move higher or lower than the actual probability, and why? • What is the expected price of copper in two years? What is the expected price using the risk-neutral probabilities? • What is the value of a contract which delivers one unit of copper in two years? Do this once by discounting the future value of the copper delivery using the risk-neutral methodology – i.e., using risk neutral probabilities and the risk-neutral discount rate. Do this a second time discounting the future value of the copper delivery using a risk-adjusted discount rate – i.e., using the actual probabilities and some discount rate greater than the risk-neutral discount rate. What risk-adjusted discount rate gives you the same answer? • What should be the price of a forward contract delivering one unit of copper 2 years from now? • Now find the value of an option to buy copper two years from now at a strike price $2.50/pound. First, use the risk-neutral approach – risk-neutral probabilities and a risk-free discount rate. Second, use the risk-adjusted discounting approach, using the risk-adjusted discount rate derived earlier. Compare the results. Third, suggest an alternative risk-adjusted discounting approach that yields the same answer as the risk-neutral approach. (2) What if you add convenience yield? Adjust your spreadsheet and redo all of the calculations above. Consider several cases: (a) convenience yield is positive, greater than the risk-free rate; (b) convenience yield is equal to the risk free rate; (c) convenience yield is negative, which can be thought of as storage cost. What is the highest rate for the convenience yield that can make sense? (3) Build the binomial model for T=10 • • • • • • • • Build the 10-step binomial tree for copper price Draw the tree with actual probabilities of reaching the given node Calculate the expected price of T=1..10 and plot it Calculate the expected value of returns over different time horizons starting at t=0 and going from time 1 to time 10. How does it change with the horizon? Calculate the volatility of these returns. How does it change with the horizon? How does the single period expected return change through time and as a function of the level of the copper price? How does the volatility of the single period return change through time and as a function of the level of the copper price? Calculate the price of an option to buy copper with exercise price $2.50/pound and maturity T=10. Compare the calculated price with price of the two-year option calculated using two-step binomial tree. Why they are different? Page 2 • • • • Calculate the expected value of returns over different time horizons starting at t=0 and going from time 1 to time 10. How does it change with the horizon? Calculate the volatility of these returns. How does it change with the horizon? How does the single period expected return change through time and as a function of the level of the copper price? How does the volatility of the single period return change through time and as a function of the level of the copper price? Monte Carlo Assignment (4) Monte-Carlo Simulations. Repeat the exercise in (3), but using Monte Carlo simulation using risk-neutral probabilities. Redo the simulation using the true probabilities. Page 3