1. Chap 5 Numerical Methods for Simple Options

2.  NPV is forced to treat future courses of action as mutually exclusive, ROA can combine them into a single value with a decision rule for choosing among them.  We explore the effect of dividend payments by the underlying risky asset on American call options such as a deferral or an expansion option.  They would never be exercised early unless the underlying risky asset dropped in value, for example, when it pays a dividend.

6. Methodology for modeling the stochastic process of the underlying asset  A normal distribution is symmetric and has values in the left tail that go to minus infinity.  The multiplicative process, the additive process is also recombining.  To preserve the recombining property of the event trees that describe the evolution of the price of the underlying through time, we usually assume that dividends paid are proportional to value in a multiplicative tree, and are constant and additive in an additive tree.

9.  The multiplicative process and constant for the additive process.  Also, the limiting distributions ( for the predividend values ) are still lognormal for the multiplicative process and normal for the additive process.  Finally, the obvious result of paying dividends is that the values at the end branches are lower than they were in the no-dividend case.  As the project generates free cash flows that can be dividended out, the remaining value is reduced by the amount of the free cash flow ( dividend ).

10. Abandonment options valued indirectly  If one has the right, but not be the obligation, to rid oneself of a risky asset at a fixed (predetermined) price, it is called an abandonment option.  Abandonment options are important in research and development, in exploration and development of natural resources,  Abandonment option analysis not only provides an estimate of the value of optimal abandonment, but it also indicates when abandonment should be implemented.

12.  In merger and acquisition situations, abandonment is equivalent to being able to bail out of an investment at a floor price (900)– the estimated exercise price of the abandonment option.

20.  The risk-adjusted rate of return (RAR) varies depending on where we are in the decision tree because the riskless of outcomes changes as well.  Recall that this is the major reason that decision-tree analysis does not work.  DTA inappropriately assumes a constant discount rate.

27. Valuing the option to contract (shrink) a project  The right to sell off some capacity, thereby shrinking the scale of operations, is an American put option that we call the option to contract.  But introduce an option to contract the scale of operations (and therefore its value) by 50 percent by selling assets (plant and equipment) worth $450 after taxes. An alternative that is similar, would be to scale back operations and sublet equipment and facilities to another company.

30. Value of the option to expand ( when the underlying does not pay dividends )  If projects turn out better than expected, it is often desirable to invest in expanding them.  The extra investment is the exercise price of an expansion option – an American call.  The value tree for the underlying shifts upward to reflect the benefits of the expansion.  And introduce the option to expand the scale of operations, at an expanse of $100, for a benefit that increases the value of operations by 10 percent

32. Valuing combinations of simple options 1.An American put, the option to abandon for a salvage value of $900. 2.A contraction option, also an American put, to shrink the scale of operations by 50 percent by selling assets for $450. 3.The option to expand the scale of the project by 10 percent at a cost of $100, an American call.

36.  While it is true that the simple sum of the values of the three separate options is not equal to the value of their combination, we must look more carefully and note that the option to contract was never used in the combination – in other words it was valueless in the combination because it was dominated by the other two options.  Therefore, the value of the combination, $6.51 is equal to the simple sum of the values of the (undominated) options, namely the abandonment option and the option to expand.  The sum of their values is $6.49, within rounding error, the same as the value of the combination.

37. Net present value and real options analysis handle mutually exclusive alternatives differently  The cash flow of a project right now is $100.  At the end of year 1, it goes up by 20 percent to $120 or down by 16.67 percent to $83.3 with similar percentage changes in year 2.  At the end of the second year, the company has two mutually exclusive alternatives.

39.  In one branch of the decision tree, it can spend $700 to lock in its annual level of cash flows forever.  In a mutually exclusive branch, it can spend an additional $120 to test-market a new version of the product (thereby forgoing a year of cash flows) to find out, as of year 4 that, with 50-50 probability, the perpetual cash flows will be either 50 percent higher, or percent lower.  Then, in year 4, the company can, at a cost of $700, lock in the perpetual cash flows or it can abandon the project.

40.  Additionally, the cost of capital is 10 percent, the risk-free rate is 10 percent, and the initial outlay required to commence the project is $400.

41.  Discounting these expected cash flows at 10 percent, adding the initial cash inflow of $100, and subtracting the initial investment of $400 results in an NPV of $312.55.

50.  At time zero the present value of the project is $1,895, it can move up or down by 20 percent, its expected present value at the end of the first year is $2,085, and we have assumed a 10 percent discount rate.  Using these facts, we can solve for the objective probability of the up movements as follows :