An underlying asset has a current value of $100. Over the next interval (a year)

Question

An underlying asset has a current value of $100. Over the next interval

(a year), the value may change to either 150 or 20. The risk-free rate over

the interval is 5%.

(a) Value a call option on this asset assuming a strike price of 120.

(b) The option above actually represents a potential investment opportu-

nity for XYZ, Inc. XYZ is considering investing in some technology

(some ARM switches), which will give them the ability (after 1 year)

to decide whether to invest in an even newer, potentially more effi-

cient technology called ARMOUR switching; the payoffs above are

the relevant payoffs for ARMOUR. That is, the initial investment

required for ARMOUR is represented (in millions of dollars) by the

strike price of 120 in part a. Similarly, the potential payoffs in AR-

MOUR are represented by the 150 (million) (on the upside) and 20

(million) (on the downside). Clearly, this second investment is risky;

in fact, you have calculated its traditional NPV and it is negative.

You also know that the only way to obtain ARMOUR technology is

to first invest in ARM switches.

You believe that the ARMOUR switching technology has a great deal

of potential, but after calculating the traditional NPV of ARM (the

first investment required), you’re discouraged by its negative NPV of

-10 million dollars.

Given the information you have about ARMOUR, your answer to

part a above and your intuition about option valuation, can you sell

ARM to your supervisor? How?

If new information comes in about ARMOUR which increases the

risk of its cash flows and reduces its NPV, would this change your

mind about its benefits? If so, how?

Explanation / Answer

Current Price (S0) = 100 ; Possible prices after 1 year (S1) = 150 or 20 ; Call Strike Price = 120 ad risk free rate = 5%

Let us build a portfolio of X units of underlying asset and short 1 Call option. The current cost will be = 100X - 1C (since we have sold call we will receieve premium).

After1 year, the value will be either:

For no arbitrage to be satisfied, both these pay offs shouls equal:

150X - 30 = 20X or X = 0.23

Hence value of this portfolio after 1 year = 150 * 0.23 - 30 = 20* 0.23 = 4.62

Now the cost to build this portfolio should be present value of 4.62 or we can write:

100 * 0.23 - 1C = 4.62 * e-5% ; hence C = 18.69

Part b. As we can see that the value of the option to invest in ARMOUR atfer 1 year is 18.69 million today . Given that the option value is more than the negative NPV given at 10 million, this investment can be considered since there will be a positive expected pay off . However of the NPV reduces further then we will have compare the NPV with the option value. As long as the negative NPV is less than 18.69 million it can be considered otherwise the project should be abandoned.

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