(1) Summarize the importance of options, and demonstrate how they can dramatically affect value.
(2) Identify "hidden" options in business situations.
(3) Describe the dramatic effect a contingency can have on a situation.
(4) Calculate payoffs under various possible outcomes.
(5) Cite the determinants of the value of an option.
(6) Describe how changes in those determinants affect the value of an option.

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(1) A call option is the right to buy an asset while a put option is the right to sell an asset.
  (a) The asset on which an option is written is known as the underlying asset.
  (b) The strike price is the price at which the optionholder may buy or sell the underlying asset when the option is exercised.
  (c) An option that is not exercised will expire on its expiration date.
  (d) An option is in-the-money if exercising the option will provide a monetary gain.
  (e) An option is out-of-the-money if exercising the option will not provide a monetary gain.

(2) The amount of monetary gain for an in-the-money option is called the intrinsic (or exercise) value of the option.
(a) For a call option, the intrinsic value of an asset is its current price minus its strike price.
(b) For a put option, the intrinsic value of an asset is its strike price minus its current price.

(3) An American option is an option that can be exercised at any time prior to its expiration, while a European option can only be exercise on the day of expiration.

(4) An option is a contingency claim because it is only exercised if particular conditions occur.

(5) The value of an option is the maximum of its intrinsic value and zero.

(6) Ignoring transaction costs, the break even point of an option is its strike price.

(7) An option contract involves a zero sum game when exercised. The gain (loss) accruing to the buyer is offset by the loss (gain) by the seller of the contract.

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(1) When you buy an asset and sell a call option, you lower your net investment but put a limit on your upside potential.

(2) An owner of an asset who purchases a put option on its asset, in essence, is paying someone to take the bad outcomes if they occur. It is like an insurance policy where the insurance company pays for damages (e.g., bad outcomes).

(3) Puts and calls are mirror images of one another when plotting the exercise value of an option versus the value of the underlying asset over time.

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(1) The time premium of an option is the extra value (above the exercise value) provided by having control of when to exercise.
  (a) The time premium allows the optionholder to claim good outcomes and avoid bad ones.
  (b) It decreases as the option approaches expiration.
  (c) It is determined by three factors: time until expiration, risk of the underlying asset, and market riskless return.
    (i) A greater time period increases the probability that an option will be favorably exercised.
   (ii) Greater volatility in the value of the underlying asset increases the probability of the option being exercised (and exercised at a greater profit).
   (iii) The higher the riskless return, then the less is the present value of the strike price if the option is exercised. This increases the value of a call option to the buyer (who pays less) but decreases the value of a put option to the buyer (who receives less).

(2) The exercise value is determined by two factors: the underlying asset's current market value and the option's strike price.

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(1) The largest time premium for an option occurs when the underlying assets value equals the strike price. This is because the asset's value could go either way causing greater uncertainty.

(2) An option's time premium decreases as the option becomes more in- or out-for-the-money because there is less uncertainty whether the option will expire in- or out -of -the-money.

(3) A European option is never worth more than an American option which has the flexibility of being exercised earlier)

(4) An option's time premium is generally positive. An exception is high-coupon bond selling above par value but which will be worth par value at maturity.

(5) It is generally better to sell an option prior to expiration than to exercise it.

(6) The further an option is out-of-the-money, the less it is worth (because the probability of exercising is lower).

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(1) Insurance is a put option because the value of the investment (e.g., policy) becomes greater when the value of the asset (e.g., your health fails).

(2) Real estate developers purchase call options on land (that increase in value as the plans to develop the land materialize).

(3) A convertible security is a "straight" security (e.g., one that pays a fixed rate of income) plus a call option on shares of the firm's common stock.

(4) A warrant is a long-term call option on a stock.
  (a) When attached to bonds, the package is like a convertible bond, except the two parts are independent and can be bought and sold separately.
  (b) Warrants are sometimes traded on the stock exchanges.
  (c) The maturity is 10 years or longer.

(5) Many corporate bonds include a call provision that allows the firm to redeem the bond for a preset amount prior to maturity.

(6) In addition to the Chicago Board Options Exchange, standardized puts and calls are also traded on the American and Philadelphia Stock Exchanges, among others.

(7) The Wall Street Journal lists publicly traded options including stock index options, interest rate options, commodity options, and currency options.

(8) Whenever a claim is contingent on particular outcomes, there is probably a hidden option involved. For example, investing in stock has a "hidden" tax option that involves selling the asset before January 1 to achieve a tax benefit.

(9) When stockholders pay an interest payment they exercise a call option that allows them to pay themselves certain "residuals."

(10) Limited liability means that the stockholders hold a put (or default) option with a strike price of zero. When they exercise their option, they default by selling their equity value for zero.

(11) Home owner's have a call option to refinance their house at a lower rate by buying back their house.

(12) A production process with relatively more variable costs and less fixed cost provides a hidden option to reduce total cost if production ever have to be suspended temporarily.

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(1) A simple model of option valuation has four steps.
  (a) Compute the probabilities of possible price changes on the basis of what an investor can earn on the riskless asset.
  (b) Calculate the possible exercise values at expiration.
  (c) Determine the expected outcomes as the probability-weighted average of the outcomes.
  (d) Compute the present value of the expected outcome by discounting at the riskless return.

(2) Assume that $100,000 worth of land has two possible values next year: $120,000 or $94,138 (returns are thus 20% or -5.862%) and the riskless rate is 5% per year. What is the option worth?
  (a) For step one, we have: 5% = (Prob)(20%) + (1-Prob)(-5.862%) where Prob is the probability of an increase; solving, we get: Prob = 0.42.
  (b) For step two, we have: Exercise value of call option = max[(market value-$110,000; 0] = max ($120,000 - $110,000; 0) = max[$10,000; 0] = $10,000.
  (c) For step three, we have: the expected outcome = (0.42)($10,000) + (0.58)(0) = $4,200.
  (d) For step four, we have: the present value of outcome = $4,200/(1.05) = $4,000 which is what the option is worth.

(3) The value of an option on a portfolio of assets is always less than or equal to the value of a portfolio of comparable options on the individual assets. This is because the portfolio of options allows you to claim every good outcome individually and not the collective good (where bad outcomes can cancel out good outcomes).

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(A1) Define the terms option, call option, and put option.

An option is the right, without obligation, to buy or sell an asset. A call option is an option to buy, and a put option is an option to sell. With a call option you hope the asset will increase in value, while with a put option you hope the asset will decrease in value. [NOTE. For a call option the following relationships between five determinants and the value of the call option hold: stock price (P₀): +; exercise or strike price (X): -; time to expiration (t): +; risk-free rate (r_f): +; and, variability of stock's return (standard deviation): +. For a put option the following relationships between five determinants and the value of the call option hold: stock price (P₀): -; exercise price (X): +; time to expiration (t): either (typically +); risk-free rate (r_f): -; and, variability of stock's return (standard deviation): +.]

(A2) Define the terms strike price, exercise value, in-the-money, and out-of-the-money.

The strike price is the price at which the holder may buy or sell the asset when the option is exercised. The exercise value is the amount of advantage an in-the-money option provides over buying or selling the underlying asset in the market. A call option is in-the-money when the price of the underlying asset exceeds the strike price. That is, a call is in-the-money when there is an advantage to exercising over buying the asset in the open market. A call option is out-of-the-money when the price of the underlying asset is below the strike price (no advantage to exercise). A put option is in-the-money when the price of the underlying asset is below the strike price. A put option is out-of-the-money when the price of the underlying asset is above the strike price.

(A3) Cite three situations that involve "hidden" options.

Three situations involving "hidden" options are mortgage refunding options, tax-time options, and capital investment options such as the ability to postpone, expand, or abandon an investment project. (Two additional situations are the following. First, home owner's have a call option to refinance their house at a lower rate by buying back their house. Second, a production process with relatively more variable costs and less fixed cost provides a hidden option to reduce total cost should production ever have to be suspended temporarily.)

(A4) In your own words, explain how limited liability makes shares of common stock like an option.

Limited liability makes shares of common stock like a call option on the firm's assets because the shareholders only have to pay the debt (exercise the call) when firm value exceeds the required debt payment. If the value of the firm is less than the required debt payment, the shareholders default on the debt (fail to exercise the call) and the debtholders will own the assets of the firm.

(A5) Explain how auto insurance can be viewed as a put option.

Auto insurance can be viewed as a put option because when an auto is stolen or seriously damaged, the owner can exercise his option to "sell" the auto to the insurance company for the insured amount. The cost of this put is the periodic payment of the insurance premium and any deductible amount that may have to be paid.

(A6) Why would an American call option traded in an efficient capital market never be worth less than its exercise value?

An American call option traded in an efficient capital market would never be worth less than its exercise value because an American option can be exercised at any time during its life and, if it were ever worth less than its exercise value, a person could purchase the option and immediately exercise it to earn a riskless arbitrage profit.

(B1) Tom Smith purchased a building in uptown Indianapolis for $50,000. Right after this, he sold a one-year European call option on the building, with a strike price of $54,000, for $2,200 to Sarah Smyth. What are Tom's and Sarah's outcomes, in terms of the land's possible value when the option expires?

Tom's net investment (cost) in the building is $50,000 - $2,200 = $47,800. The outcomes for Tom and Sarah (in $ thousands) are:

Bldg   Tom's       Tom's    Sarah's     Sarah's
Value  Gain(loss)  Return   Gain(loss)  Return
              *          **       #           ##
 $65     $6.20     12.97%     $8.80     400.00%
  60      6.20     12.97       3.80     172.73%
  55      6.20     12.97      -1.20     -54.55%
  45     -2.80     -5.86      -2.20    -100.00%
  50      2.20      4.60      -2.20    -100.00%
  40     -7.80    -16.30      -2.20    -100.00%
  35    -12.80    -26.78      -2.20    -100.00%

Row One:
* Net CF = Min(Bldg Value; $54K) - Cost = $54K - $47.8K = $6.2K (K=thousands)
** Return = Net CF/Cost = $6.2/$47.8 = 0.1297 = 12.97%
# Net CF = Revenue - Cost = max($65K - $54; 0) - $2.2K = $8.8K
## Return = Net CF/Cost = $8.8K/$2.2K = 400.00%

Row Two:
* Net CF = Min(Bldg Value; $54K) - Cost = $54K - $47.8K = $6.2K
** Return = Net CF/Cost = $6.2/$47.8 = 0.1297 = 12.97%
# Net CF = Revenue - Cost = max($60K - $54; 0) - $2.2K = $3.8K
## Return = Net CF/Cost = $3.8K/$2.2K = 172.73%

Row Three:
* Net CF = Min(Bldg Value; $54K) - Cost = $54K - $47.8K = $6.2K
** Return = Net CF/Cost = $6.2/$47.8 = 0.1297 = 12.97%
# Net CF = Revenue - Cost = max($55K - $54; 0) - $2.2K = -$1.2K
## Return = Net CF/Cost = -$1.2K/$2.2K = -54.55%

Row Four:
* Net CF = Min(Bldg Value; $54K) - Cost = $50K - $47.8K = $2.2K
** Return = Net CF/Cost = $2.2/$47.8 = 0.0460 = 4.60%
# Net CF = Revenue - Cost = max($50K - $54; 0) - $2.2K = -$2.2K
## Return = Net CF/Cost = -$2.2K/$2.2K = -100.00%

Row Five:
* Net CF = Min(Bldg Value; $54K) - Cost = $45K - $47.8K = -$2.8K
** Return = Net CF/Cost = -$2.2/$47.8 = -0.0586 = -5.86%
# Net CF = Revenue - Cost = max($45K - $54; 0) - $2.2K = -$2.2K
## Return = Net CF/Cost = -$2.2K/$2.2K = -100.00%

Row Six:
* Net CF = Min(Bldg Value; $54K) - Cost = $40K - $47.8K = -$7.8K
** Return = Net CF/Cost = -$7.8/$47.8 = -0.1630 = -16.30%
# Net CF = Revenue - Cost = max($40K - $54; 0) - $2.2K = -$2.2K
## Return = Net CF/Cost = -$2.2K/$2.2K = -100.00%

Row Seven:
* Net CF = Min(Bldg Value; $54K) - Cost = $35K - $47.8K = -$12.8K
** Return = Net CF/Cost = -$12.8/$47.8 = -0.1630 = -26.78%
# Net CF = Revenue - Cost = max($35K - $54; 0) - $2.2K = -$2.2K
## Return = Net CF/Cost = -$2.2K/$2.2K = -100.00%

(B11) Suppose you can buy a call option on a business that is currently worth $10,000,000 but will be worth either 25% more or 15% less 1 year from today. The option's strike price is $11,000,000, and the riskless return is 6% per year. What is this call option worth today?

There are four steps. For step one, we compute the probabilities of the 2 outcomes if the return on the business must equal the riskless return of 6%. We have: 6% = (Prob)(25%) + (1-Prob)(-15%) where Prob is the probability of the first outcome or of an increase. Multiplying out and rearranging, we have: Prob = 52.5%.

For step two, we have: exercise value of call option = max[(market value - strike price; 0] where the strike price is $11M and the current business value is $10M. Inserting these values, we have: max(1.25*$10M - $11M; $0M) = max($1.5M; $0M). Thus, the two possible outcomes are $1.5M and $0M.

For step three, we determine the expected outcomes as the probability-weighted average of the outcomes. We have: the expected value of the outcome = 0*(0.475) + 1.5M*(0.525) = $787,500.

For step four, we compute the value of the call option as the present value of the outcome's expected value. This value = ($787,500)/1.06 = $742,924.53.

(C5) The Gibson Greetings example referred to a contingent amount P. The formula for P was:

where the 2-year yield is the yield to maturity (YTM) on a 2-year Treasure note (expressed in decimal form), and the 30-year T price is the market price of a 30-year Treasury bond (expressed as a percentage of face amount). Assume that (1) the 30-year Treasury bond has a coupon rate of 7%, (2) there is a flat term structure so that the 2-year yield and the 30-year yield are always equal (they vary together), and (3) bonds make semiannual coupon payments. (a) Calculate the 30-year T price, assuming its yield is 6%. (b) Calculate P, assuming yields are 6%. (c) Calculate the bank's principal obligation when the yield is 6%. (d) Calculate the bank's principal obligation when the yield is 8%. (e) Why does the bank's principal obligation decrease when yields increase? (f) Explain why this swap can be described as a put option.

(a) Calculate the 30-year T price, assuming its yield is 6%. [Hint: Future value of principal (FV) = $1,000; CPN = 0.07*$1,000 = $70 (thus, CPN/2 = $35); r = 6% (thus, r/2 = 3%); and, N = 30 years (thus, 2N = 60 periods).]

(b) Calculate P, assuming yields are 6% where principal is $30M.

(c) Calculate the bank's principal obligation when the yield is 6%.

As stated in the text, the bank's principal obligation is the smaller of $0.6M or P. So the bank's obligation is $26,159,597.

(d) Calculate the bank's principal obligation when the yield is 8%.

For the contingent amount P, the 30-year T price must be expressed as a percentage of face value expressed in 100s; thus P = 88.688.

Now calculate P, assuming yields are 8% where principal is $30M.

(e) Why does the bank's principal obligation decrease when yields increase?

The first term in the numerator of P increases when yield increase while the second term (the 30-year T price) decreases. The result of this is to increase the fraction being subtracted from 1 inside the first brackets in P (e.g., inside [ ]). Thus, P must decrease when rates increase.

(f) Explain why this swap can be described as a put option.

When interest rates are at or below an implicit strike price (the strike price can be found by setting P = $30.6M and solving for the 2-year yield) the bank must pay $30.6M to Gibson. When rates rise above the implicit strike price, the bank saves by having to pay Gibson less. In other words, when interest rates rise above the implicit strike price, the bank has the right to "sell" Gibson back smaller and smaller amounts of principal.

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