Extensions of Option Pricing

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All the option pricing models we have described so far - the Binomial, the Black-Scholes and the jump process models - are designed to value options, with clearly defined exercise prices and maturities, on underlying assets that are traded. The options we encounter in investment analysis or valuation are often on real assets rather than financial assets, thus leading to them to be categorized as real options, can take much more complicated forms. In this section, we will consider some of these variations.

Capped and Barrier Options

With a simple call option, there is no specified upper limit on the profits that can be made by the buyer of the call. Asset prices, at least in theory, can keep going up, and the payoffs increase proportionately. In some call options, the buyer is entitled to profits up to a specified price but not above it. For instance, consider a call option with a strike price of K1 on an asset. In an unrestricted call option, the payoff on this option will increase as the underlying asset's price increases above K1. Assume, however, that if the price reaches K2, the payoff is capped at (K2 -K1). The payoff diagram on this option is shown in Figure 5.5:

Figure 5.5: Payoff on Capped Call

When the price of the asset exceeds K2, the payoff

Figure 5.5: Payoff on Capped Call

This option is called a capped call. Notice, also, that once the price reaches K2, there is no time premium associated with the option anymore and the option will therefore be exercised. Capped calls are part of a family of options called barrier options, where the payoff on and the life of the options are a function of whether the underlying asset price reaches a certain level during a specified period.

The value of a capped call will always be lower than the value of the same call without the payoff limit. A simple approximation of this value can be obtained by valuing the call twice, once with the given exercise price and once with the cap, and taking the difference in the two values. In the above example, then, the value of the call with an exercise price of K and a cap at K2 can be written as:

Value of Capped Call = Value of call (K=Kj) - Value of call (K=K2) Barrier options can take many forms. In a knockout option, an option ceases to exist if the underlying asset reaches a certain price. In the case of a call option, this knock-out price is usually set below the strike price and this option is called a down-and-out option. In the case of a put option, the knock-out price will be set above the exercise price and this option is called an up-and-out option. Like the capped call, these options will be worth less than their unrestricted counterparts. Many real options have limits on potential upside or knock-out provisions and ignoring these limits can result in the overstatement of the value of these options.

Compound Options

Some options derive their value not from an underlying asset but from other options. These options are called compound options. Compound options can take any of four forms - a call on a call, a put on a put, a call on a put and a put on a call. Geske (1979) developed the analytical formulation for valuing compound options by replacing the standard normal distribution used in a simple option model with a bivariate normal distribution in the calculation.

Consider, for instance, the option to expand a project that we will consider in the next section. While we will value this option using a simple option pricing model, in reality there could be multiple stages in expansion, with each stage representing an option for the following stage. In this case, we will undervalue the option by considering it as a simple rather than a compound option.

Notwithstanding this discussion, the valuation of compound options become progressively more difficult as we add more options to the chain. In this case, rather than wreck the valuation on the shoals of estimation error, it may be better to accept the conservative estimate that is provided with a simple valuation model as a floor on the value.

Rainbow Options

In a simple option, the uncertainty is about the price of the underlying asset. Some options are exposed to two or more sources of uncertainty and these options are rainbow options. Using the simple option pricing model to value such options can lead to biased estimates of value. As an example, we will consider an undeveloped oil reserve as an option, where the firm that owns the reserve has the right to develop the reserve. Here, there are two sources of uncertainty. The first is obviously the price of oil and the second is the quantity of oil that is in the reserve. To value this undeveloped reserve, we can make the simplifying assumption that we know the quantity of the reserves with certainty. In reality, however, uncertainty about the quantity will affect the value of this option and make the decision to exercise more difficult4. Conclusion

An option is an asset with payoffs which are contingent on the value of an underlying asset. A call option provides its holder with the right to buy the underlying asset at a fixed price, whereas a put option provides its holder with the right to sell at a fixed price, any time before the expiration of the option. The value of an option is determined by six variables - the current value of the underlying asset, the variance in this value, the strike price, life of the option, the riskless interest rate and the expected dividends on the asset. This is illustrated in both the Binomial and the Black-Scholes models, which value options by creating replicating portfolios composed of the underlying asset and riskless lending or borrowing. These models can be used to value assets that have option-like characteristics.

4 The analogy to a listed option on a stock is the case where you do not know what the stock price is with certainty when you exercise the option. The more uncertain you are about the stock price, the more margin for error you have to give yourself when you exercise the option to ensure that you are in fact earning a profit.

Questions and Short Problems: Chapter 5

1. The following are prices of options traded on Microsoft Corporation, which pays no dividends.

Call Put

K=85

K=90

K=85 K=90

1 month

2.75

1.00

4.50 7.50

3 month

4.00

2.75

5.75 9.00

6 month

7.75

6.00

8.00 12.00

The stock is trading at $83, and the annualized riskless rate is 3.8%. The standard deviation in ln stock prices (based upon historical data) is 30%.

a. Estimate the value of a three-month call, with a strike price of 85.

b. Using the inputs from the Black-Scholes model, specify how you would replicate this call.

c. What is the implied standard deviation in this call?

d. Assume now that you buy a call with a strike price of 85 and sell a call with a strike price of 90. Draw the payoff diagram on this position.

e. Using put-call parity, estimate the value of a three-month put with a strike price of 85.

2. You are trying to value three-month call and put options on Merck, with a strike price of 30. The stock is trading at $28.75, and expects to pay a quarterly dividend per share of $0.28 in two months. The annualized riskless interest rate is 3.6%, and the standard deviation in ln stock prices is 20%.

a. Estimate the value of the call and put options, using the Black-Scholes.

b. What effect does the expected dividend payment have on call values? on put values? Why?

3. There is the possibility that the options on Merck, described above, could be exercised early.

a. Use the pseudo-American call option technique to determine whether this will affect the value of the call.

b. Why does the possibility of early exercise exist? What types of options are most likely to be exercised early?

4. You have been provided the following information on a three-month call:

a. If you wanted to replicate buying this call, how much money would you need to borrow?

b. If you wanted to replicate buying this call, how many shares of stock would you need to buy?

5. Go Video, a manufacturer of video recorders, was trading at $4 per share in May 1994. There were 11 million shares outstanding. At the same time, it had 550,000 one-year warrants outstanding, with a strike price of $4.25. The stock has had a standard deviation (in ln stock prices) of 60%. The stock does not pay a dividend. The riskless rate is 5%.

a. Estimate the value of the warrants, ignoring dilution.

b. Estimate the value of the warrants, allowing for dilution.

c. Why does dilution reduce the value of the warrants.

6. You are trying to value a long term call option on the NYSE Composite Index, expiring in five years, with a strike price of 275. The index is currently at 250, and the annualized standard deviation in stock prices is 15%. The average dividend yield on the index is 3%, and is expected to remain unchanged over the next five years. The five-year treasury bond rate is 5%.

a. Estimate the value of the long term call option.

b. Estimate the value of a put option, with the same parameters.

c. What are the implicit assumptions you are making when you use the Black-Scholes model to value this option? Which of these assumptions are likely to be violated? What are the consequences for your valuation?

7. A new security on AT&T will entitle the investor to all dividends on AT&T over the next three years, limit upside potential to 20%, but also provide downside protection below 10%. AT&T stock is trading at $50, and three-year call and put options are traded on the exchange at the following prices

Call Options Put Options

K

1 year

3 year

1 year

3 year

45

$8.69

$13.34

$1.99

$3.55

50

$5.86

$10.89

$3.92

$5.40

55

$3.78

$8.82

$6.59

$7.63

60

$2.35

$7.11

$9.92

$10.23

How much would you be willing to pay for this security?

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