Financial and Real Options

Abstract

Financial derivatives constitute a large proportion of trading in world financial markets, due to their unbeatable features. Derivatives are popular in fact for their flexibility and combinability.

Problems

• 1. Consider a stock meant to pay a dividend of 0.10 € per share in 3 months and 9 months from now. With no other dividend payments within the next 12 months, and the interest rate is 5 % with continuous compounding.

  • (a) Calculate the 12-month forward price of the stock if the current market price is 20 € per share.

• 2. Using the data in exercise 1, assume an investor entering a forward contract to buy 100 shares of the stock in 12 months

  • (a) What is the value of the forward contract today?

  • (b) What is its value in 6 months, provided the stock is then traded at 18 € per share?

• 3. Assume the current stock price is 27 €, and a 3-month call with a strike price of 28 € costs 2.65 €. You have 6200 € to invest.

  • (a) Describe two alternative speculation strategies, one in the stock and the other in the option on the stock.

  • (b) What are the potential gains and losses from each strategy?

• 4. Prove that the price of a call option satisfies the BSM PDE

• 5. Prove that the solutions for the BSM Greeks satisfy the PDE.

• 6. Companies A and B have been offered the following rates per annum on a 5-year loan of 20,000,000 €:

  	Fixed rate (%)	Floating rate
  Company A	12.0	Libor + 0.1 %
  Company B	13.4	Libor + 0.6 %

  Company A requires a floating-rate loan while company B requires a fixed-rate loan.

  • (a) Design a swap contract netting the 0.1 % per annum to the intermediary bank, and that will appear equally attractive to both companies.

• 7. Consider a binomial world in which a stock, over the next year, can go up in value by 20 % (by a subjective probability of 55 %) or down by 10 % (by a subjective probability of 45 %). The stock is currently trading at 10 €. The risk free return is 5 %. Consider a call that expires in 1 year, with a strike price of 11 €.

  • (a) What is the value of the call option?

  • (b) If the call option was trading for 0.32 €, can you find an arbitrage opportunity?

  • (c) If the call option was trading for 0.61 €, can you find an arbitrage opportunity?

• 8. Consider a European call option, which is written on a stock whose current value is 8 €, with the strike price 9 € and expiration date in 1 month. Assume for simplicity that 1 month later, when the option can be exercised, the stock price can either appreciate to 10 € or depreciate to 6 €. Assume that the risk-free interest rate is zero.

  • (a) What is the current price of the call option?

• 9. Suppose that the strike price of an American call option on a non-dividend-paying stock grows at some constant rate g.

  • (a) Show that if g is less than the risk-free rate, r, it is never optimal to exercise the call early.

• 10. You would like to speculate on a rise in the price of a certain stock. The current stock price is 29 €, and a 3-month call with a strike price of 30 € costs 2.90 €. You have 5800 ¤ to invest. Identify two alternative investment strategies, one in the stock and the other in an option on the stock. What are the potential gains and losses from each?

• 11. Consider a binomial world in which a stock, over the next year, can go up in value by 50 % (subjective probability of 60 %) or down by 33.33 % (subjective probability of 40 %). The stock is currently trading at 50 €. The risk-free return is 5 %.

  • (a) What is the value of the call option that expires in 1 year with a strike price of 55 €?

  • (b) Calculate the expected return from the stock

  • (c) Is the expected return on the call option higher or lower?

• 12. For a European put option on a stock with 1 year to expiry, the stock price is 50, the strike is 50, risk-free interest rate is 4 %, dividend is 4 % and the price of the option is 6.22. Determine the implied volatility of the stock.

• 13. Prove that, using this binomial tree, we can find the price of any contingent claim. In other words, any contract paying h(S ₁) where h is any function of S ₁.

• 14. Consider options on two different assets (asset 1 and asset 2). Would you prefer to have two different options, one on each asset (with the same strike price K), or would you prefer to own an option on the portfolio of the two assets with a strike of 2K?

• 15. A company is considering investing in a project. The present value (PV) of future discounted expected cash flows is either 12,000,000 € in growth or 3,000,000 € in recession next year. The objective probability the market will go up is 45 %. The appropriate risk-adjusted cost of capital is 18 %. The initial capital investment required at time 0 is 7,500,000 €. Risk-free rate is 2.5 % per year.

  • (a) Determine the PV of the project at time 0.

  • (b) Determine the NPV of the project at time 0.

  • (c) Should the company invest in this project?

  • (d) The company can abandon the project and liquidate its original capital investment for 55 % of the original value. It can also expand operations, which will result in twice the original PV. To expand the company will have to make an additional capital expenditure of 3,500,000 €. Determine the real option analysis value of this project with exibility.

• 16. Consider this multi-stage project. The first stage requires an initial investment outlay of ${I}_0=17,500,000 {\textsf{C}{=}}$, and expected cash inows over 2 years of

  $$ {C}_1=3,500,000 {\textsf{C}{=}} {C}_2=6,500,000 {\textsf{C}{=}} $$

  The second stage will become available in year 4. It will require an additional investment of

  $$ {I}_2=125,000,000 {\textsf{C}{=}} $$

  The expected cash inflows over the subsequent 2 years are

  $$ {C}_3=37,500,000 {\textsf{C}{=}} {C}_4=75,000,000 {\textsf{C}{=}} $$

  The cost of capital for both stages is 16 %. The risk-free rate is 3 %. The volatility of the second stage’s project value is 48 %.

  • (a) Explain how Real Options Analysis would characterize this project with exibility using the language of Financial Options Analysis. Be specific with numbers.

  • (b) Use a binomial lattice to determine the real option analysis value of this project with exibility.

• 17. The present value of a project without exibility is 50 million. The project pays no dividends. The project without exibility follows a binomial lattice with $u=1.25$ and $d=0.8$. Risk-free rate is 3 %. The required investment is

  • 6,000,000 € at time 0

  • 18,000,000 € at time 1

  • 27,500,000 € at time 2.

  Investments must be made to continue the project for the upcoming year. At any time the company has the option to default on these planned investments at which point the project is terminated.

  • (a) Determine the real option analysis value of this project.

  • (b) Describe the optimal stopping policy in words.

  • (c) Determine the value of the installment option.

• 18. A company owns a parcel of land on which it can build either an 4-unit or a 8-unit condominium. The current real-estate price for a 1-unit condominium is 250,000 €. Next year the price will either rise to 225,000 € or decline to 180,000 €. The chance the market will move favorably is 55 %. The construction cost is flat over time at 155,000 € per unit for a 4-unit building or 190,000 thousand euros for an 8-unit building. Assume construction is instantaneous. Rent covers operating expenses, so no free cash flow is generated. Risk-free rate is 2.5 %. What is the value of the land if the company has a 1 year delay option?