Options and Real Options

1. THE TWO BASIC OPTIONS - PUT AND CALL

• A call (put) is the right to buy (sell) an asset.

• Most other options are just combinations of these.

• Options are “derivatives” and other derivatives may include options

• The price of an option is called a “premium” because options are equivalent to insurance and the price of insurance is called a premium.

2. For most of this lecture we will assume that the option

a. Can only be exercised at maturity (called European). An American option, which is the most common type should behave similarly because, in most cases, American options are not exercised until maturity. They are almost always worth more left unexercised so very few are exercised. If they are never exercised before expiration, there should be no difference in value between an American and European option.

b. Pays no dividends – most options aren’t dividend protected so dividends will affect price.

CALL OPTION CONTRACT

Definition: The right to purchase 100 shares of a security at a specified exercise price (Strike) during a specific period.

EXAMPLE: A January 60 call on Microsoft (at 7 1/2)

This means the call is good until the third Friday of January and gives the holder the right to purchase the stock from the writer at $60 / share for 100 shares.

Cost is $7.50 / share x 100 shares = $750 premium or option contract price.

PUT OPTION CONTRACT

Definition: The right to sell 100 shares of a security at a specified exercise price during a specific period.

EXAMPLE: A January 60 put on Microsoft (at 14 1/4)

This means the put is good until the third Friday of January and gives the holder the right to sell the stock to the writer for $60 / share for 100 shares.

Cost $14.25 / share x 100 shares = $1425 premium.

INTRINSIC AND TIME VALUE•AN OPTION'S INTRINSIC VALUE IS ITS VALUE IF IT WERE EXERCISED IMMEDIATELY.

•AN OPTION'S TIME VALUE IS ITS COST ABOVE ITS INTRINSIC VALUE.

Microsoft Stock Price = 53.50 at the time - October 1987

QUESTION: Which Microsoft option has greater intrinsic value? - put

QUESTION: Which Microsoft option has greater time value? – call

a. For the Call - Time Value = 7.50 (the full premium) Intrinsic Value = 0 (Stock price < exercise price).

b. For the Put - Time Value = $8.00 = 14. 50 - (60 - 53.50) Intrinsic Value = (60 - 53.50) = 6.50.

QUESTION: Which option is a better deal?

COMBINED OPTIONSstraddle - 1 call and 1 put

• strip - 1 call, 2 puts • strap - 2 calls, 1 put• money spread • time spread

REAL OPTIONS EXAMPLES

Call - option to buy another company or company's line.

Call - capital expenditures on R & D and marketing. Give an option to make further investments if promising.

Call - buy car at the end of the lease

Call - rain check at a grocery store

Put - abandonment

Put - agreement to buy company but only if loan losses are less than 50 million (WCIS).

Put - guarantees - government price supports - consider farmer's incentives

Note: The option pricing model can be used to price any asset for which we can obtain the required inputs. Although stock options are more common, many assets can be priced – often called “real” options.

Call - A stock is a call option on the value of the firm if there is debt in the capital structure. The value of the debt is the strike price. Shareholders exercise their option to own the firm if the firm's value exceeds the value of debt, otherwise, they default and give the firm to the debt-holders. This means that many stock options are actually options on options – pricing is more complex than Black-Scholes.

Firm value 10mm 3mmDebt value 5mm 3mmEQUITY value 5mm 0mm

This situation can also be consider from the bondholders perspective which involves a put option.

Put – Bondholders have the equivalent of a risk-free bond and a short position in a put on the company that is given to shareholders. If the value of the firm drops below 5mm, then shareholders put the company to the bondholders so that

Firm value 10mm 3mmRisk-free Debt value 5mm 5mmShort put value 0mm -2mmDEBT value 5mm 3mm

USE OPTIONS TO CUT UP PRICE DISTRIBUTIONS - NOW CALLED "FINANCIAL ENGINEERING"

PresentStock Price

Stock Price Distribution

PresentStock Price

Put Exercise Price

CallExercise Price

If you believe the stock will soon rise (fall) sharply, buy the right (left) tail by purchasing a call (put). If you believe that the stock will eitherrise or fall sharply, but don’t know which, buy both tails by purchasinga call and a put. This combination is called a straddle. Selling bothtails is called a strangle.

PresentStock Price

Put Exercise Price

If you are longterm bullish on a stock but wish to avoid any near-termsharp decline, buy the stock and a put. This is called a protective put.Of course, the put costs money so your net gains will be smaller.

One way to pay the additional costs of the put in a protective put is toalso sell a call of similar value. This is called a collar. Of course, byselling the call you give up some gains if the stock price rises above the call’s exercise price.

Financial engineering involves combining purchases and sales ofstocks, options or futures to capture only the parts of a stock pricedistribution that you want. The possibilities are endless. If you buy only the stock you own the whole price distribution.

PresentStock Price

Put Exercise Price

CallExercise Price

BLACK - SCHOLES MODEL

CRUCIAL INSIGHT - it is possible to replicate the payoff to an option by some investment strategy involving the underlying asset and lending or borrowing - like the stock of a leveraged company.

Therefore, we should be able to derive the value of an option from the asset price and the interest rate.

Other derivatives like credit default swaps and colateralized mortgage obligations are leveraged positions. Folks took on to many derivatives in the 1990s and 2000s, i.e., were too leveraged.

BLACK-SCHOLES MODEL - A NEARLY EXACT OPTION PRICING MODEL

C0 = P0N(d1) - E e-rt N(d2)

where Price of Stock = P0 Exercise price = ERisk free rate = rTime until expiration in years = tNormal distribution function = N( )Exponential function (base of natural log) = e

where:

where Standard deviation of stocks return = Natural log function = ln

Intuitively, the Black Sholes model can be explained as the discounted expected value of the cash flows at expiration of the option. At expiration, assuming P > E, we pay the exercise price E (a cash outflow) and receive the stock with the value P (a cash inflow). Before expiration, we attach probabilities to these events. The N(d) are probabilities.

dP E r t

t10

25

ln( / ) ( . )

dP E r t

td t2

02

1

5

ln( / ) ( . )

How Model Variables Impact Options Prices

1. Stock price – Positively related to call price.

Negatively related to put price.

2. Exercise price – Negatively related to call price.

Positively related to put price.

3. Return Stand. Deviation – Positively related to call price.

Positively related to put price.

4. Maturity – Positively related to call price.

Positively related to put price (usually).

5. Interest rate – Positively related to call price.

Negatively related to put price.

All but the interest rate effects are clear. To explain the effect of interest rate intuitively, consider the following.

From the call option formula, we have the discounted exercise price enter the equation with a negative sign. This reflects the fact that if we exercise we have to make a cash payment of the exercise price. If interest rates increase, the present value of that payment today falls, so that the call price must rise.

From the put-call parity formula, we see the opposite occurs since, for a put, we will be receiving the exercise price in the future in exchange for the stock if we exercise.

TO GET THE VALUE OF THE CALL, C0

EXAMPLE: ASSUME

Price of Stock P0 = 36Exercise price E = 40Risk free rate r = .05time period 3 mo. t = .25Std Dev of stock return s = .50

•Substitute into d1 and d2.

•Substitute d1, d2 and other variables in the main equation

C0 = 36N(-.25) - 40e-.05(.25)N(-.50)

•Look up in the normal table for d to get N(d).

here N(d1) = N(-.25) = .4013 and N(d2) = N(-.50) =.3085

•Substitute in the main equation

d1

236 40 05 5 50 25

50 2525

ln( / ) [. . (. ) ].

. ..

50.25.50.25.2

d

C e005 2536 4013 40 3085 2 26 (. ) (. ) .. (. )

USE PUT CALL PARITY FORMULA TO GET PUT PRICE

T0 = PUT PRICE

To see why this holds, look at the stock price distribution and how the put gives you the left tail of the distribution. Then see that shorting the stock and buying the call leaves you with the same left tail. Or see that payoff at time t=0 is equal on both sides no matter what price is.

EXAMPLE - use info above - you need the call price

= 2.26 - 36 + 39.5 = 5.76

When the chance of early exercise is relatively large, for example, for long maturity puts, put prices are derived using computerized numerical methods. The main reason is that if the underlying stock price were to fall to very low levels (think internet stocks) then it pays to exercise early. Time becomes a negative there because the most that you can gain by holding is for the stock to fall to zero, but by waiting, the stock still has unlimited upside. It is then best to exercise an American put early.

T C P Ee rt0 0 0

T e005 252 26 36 40 . . (. )

NET PRESENT VALUE RULE FOR PROJECT ACCEPTANCE MUST BE ADJUSTED IF OPTIONS ARE INVOLVED.

There are two types of options to consider for most projects

A. The call option to delay a project to the future when the project may have a larger NPV. A project that can be delayed effectively competes with itself in the future.

This call option is more valuable when a project can be delayed for a longer time (t), when a project’s (returns) are very risky (), and when interest rates (r) are high.

This could explain why it may be rational to delay a positive NPV project; Managers have often been criticized by governments for not investing in plant and equipment during recessions. Managers are not being indecisive or too risk-averse but simply evaluating projects based upon their option values which may be high during recessions.

The basic idea is that if you undertake a project now, you can’t undertake it in the future when it may have a higher NPV. The more likely a project could have a higher NPV in the future, the larger its option’s time value. If the project is accepted, its time value is lost.

Thus, time value must be considered in the project selection criteria. Thus instead of

NPVproject > 0

we use

NPVproject > time value of the option to delay > 0

Hence we should accept a project only when it has a relatively large NPV. A large NPV in options terms means that the market value or present value of the project’s cash flows greatly exceeds its exercise price (cost of the project). In other words - when its option is sufficiently “in the money” i.e., it has much intrinsic value.

B. When a project’s acceptance allows one to undertake additional projects in the future then we must make another adjustment to the NPV criteria above.

NPVproject + Value of option on extended projects > time value of option to delay

For example, if we delay building a new pentium chip-making plant it may be cheaper in the future, all else equal. However, if not building the plant means we may forfeit the opportunity to build the next generation chip, then this extra option must be considered.

Example: You have a project that requires a $20 million investment. You expect the project to provide cash flows with present value of $22 million. Assume the risk-free rate is 10% and the return standard deviation is .60. If you can delay the project for two years should you accept the project now or wait? What if the project gives us the option to make future investments where this option is worth $8 million? Assume that the investment remains $20 million whenever it is made and the present value of cash flows remains $22 million. Also assume that if you delay then you lose the option to make future investments. (You don’t have to present value $20 because it is discounted in the Options model).

P0 = 22 X = 20 = 0.6 t = 2 r = .10

Find Value of option to delay - Call option

d1 = [ ln(22/20) + (.10 + 0.5(0.36)2] / 0.6(2).5

= [0.095 + (.10 + 0.18)(2) ] / 0.849 = .77

N(d1) = N(.75) = 0.7734

d2 = .77 - 0.849 = -0.079

N(d2) = N(-.10) = 0.4602 Vc = P0 N(d1) - Xe- r t [N(d2)] = 22 (0.7734) - [20e-(.10)(2)](0.4602) = 17.015 - 16.37(0.4602) = 9.5

Time value = 9.5 - (22 - 20) = 7.5

Since NPV = (22 -20) = 2 < 7.5 then wait.

If the project gives us the option to make future investments but only if we invest now and this option is worth 8 then we would have

NPV + Option on Future Project = 2 + 8 = 10 > 7.5 - so now we would go ahead with the project.

A widely followed index of overall stock market volatility is the VIX - the standard deviation of the S&P 100 implied by one-month index option premiums. See www.cboe.com.