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Law & Valuation
4.4 Options - Characteristics and Valuation

4.4.1 Option Fundamentals

How does an option work? And how is it valued? Here are some fundamentals.

A game. Let's play a game. I will give you $5 if you flip a coin twice and it comes up heads both times. If not, I pay nothing. To play my game, you must pay me $1.00 per play. We can play as many times as you'd like. Are you interested?

I have just offered you an option. If you play and things turn out well (two heads) you can claim $5. If you play and things don't turn out well (tails on either flip), you walk away. What is the value of the option?

Coin flips
Probability
Return
Expected return
T-T
25%
0
0
T-H
25%
0
0
H-T
25%
0
0
H-H
25%
$5.00
$1.25
Total
$1.25

The option is worth $1.25. Assuming that you are risk-neutral, that you have no better use for your time, that the coin is not loaded, and that I will be good on my promise to pay, the $1.00 price for the option is less than its value. You should make money. On average, you should make $0.25 per option.

A definition. What is a financial option? It's a bet on a financial asset -- a contract under which one person promises to sell (or buy) a designated financial asset at a specified price at a specified time, if the other person exercises the option to buy (or sell). For the person who buys the option (pays the "premium"), it's a relatively cheap way to take advantage of --

  • anticipated price increases in the financial asset ("call" option -- holder can buy the asset)
  • anticipated price decreases in the financial asset ("put" option -- holder can sell the asset).

Consider an example. Suppose you hear good things about XYZ Biotech Company, whose stock is currently trading at $20. At the same time, you don't want to buy shares of the company. Instead, you buy a call option that allows you to purchase 1,000 shares of XYZ at a strike price of $22 anytime within the next 6 months (an "American" option).

  • Scenario 1: Days after buying the call option, you read that XYZ has developed a vaccine for the AIDS virus. In addition, regulatory approval of the drug is fast-tracked, and it will soon hit the market. Shares of XYZ skyrocket to $100/share. You would simply exercise your call option, buy the shares from the option writer for $22, sell the stock at the new price, and pocket a nice profit. (Actually, in developed stock option markets the option writer would close the contract in cash - the difference between the option's exercise price and the stock's current market price.)
  • Scenario 2: Days after buying the call option, you read that XYZ announces that its AIDS vaccine has flunked clinical trials and will certainly not win FDA approval. Even worse, XYZ had thrown all of its resources into this project, leaving the company with no other projects. Soon, the company declares bankruptcy. You would simply not exercise your option to buy the stock and let it expire. While you are out the premium you paid for the option, this is much better than if you had bought the stock outright for $20/share.

An option is a financial contract that represents the right, but not the obligation, to buy or sell a specified amount of an underlying security at a specified price at a specified time. Think of an option as a "lottery ticket" - cash in the ticket if your number comes up, toss the ticket if not.

INSERT OPTION QUOTES FROM WSJ

Options can be written by the company that itself has issued the underlying asset, such as common stock. If the option is written (issued) by the company it is called a warrant. With options, the contract is between two removed individuals with no involvement of the company.

Option Nomenclature

American Style Option Option contract that can be exercised at any time before it expires.
European Style Option Option contract that can only be exercised at the expiration date. Curiously, most options traded on US exchanges are of this type.
Capped-style Option Option with established profit cap. Capped options are automatically exercised when the underlying security closes at the preset cap price.
Option writer The seller of an option. This person receives premium, and is obligated to buy or sell the underlying security at a specified price, if called on to do so.
Call Option Option contract giving the holder the right, not the obligation, to buy a 100 shares of a particular stock, stock index, or futures at a specified price within a specified time
Put Option Contract that grants the right, not the obligation, to sell a specific number of shares at a specific price by a specific date. In essence, the option holder has the right to "put" the stock on the option writer if the stock price dips below the strike price.
Strike Price Price at which the stock or commodity underlying a call or put option can be purchased or sold over the specified period
Deep-in-the-Money For a call, the strike price of the option is well below the current price of the underlying instrument. For a put, the strike price of the option is well above the current price of the underlying instrument.
In-the-Money Option Refers to an option that if exercised would generate a profit. For a call, the strike price would be less than the market price, and for a put, the market price would be less than the strike price.
At-the-Money Option Option with a strike price equal, or close to equal, the market price of the underlying security.
On-the-Money Option Not in-the-money or out-of-the money, but exactly in the middle. In other words, the underlying security is trading at the strike price of the option.
Out-of-the-Money Option Call option is out of the money if the strike price exceeds the price of the underlying security. The opposite is true for a put option.
Option premium Price of the option; This is what the option holder pays for the right and the option writer gets for whatever the option grants.
Extrinsic Value Price of an option less its intrinsic value. For example, an option that is out-of-the-money consists of nothing but time value, or extrinsic value.
Intrinsic Value Amount by which an option is in-the-money. For a call, this is the underlying asset's price less the strike price; for a put, this is the strike price less the underlying asset's price.
Naked position Refers to a securities position that is not hedged from market risk. A naked position occurs when an investor writes a call or put without having a corresponding position on the underlying security.
Covered Option Option contract backed by the shares underlying the option. This is the opposite of a naked option.
Uncovered option Option contract not backed by the shares underlying the option. Also called a naked option.
Zero-sum game A game in which the losses of the losers are matched by the gains of the winners. Options trading is so called because for every trader holding a profitable contract there is another holding a losing contract for the same amount.

Simple option valuation. Let's assume that you are interested in selling a European "call" option on stock that is currently trading at $50. It will be exercisable in 6 months with an exercise price of $50. How much should you sell (or write) the option for?

Assume that stock will go to in one of two directions - it will rise to $60 or it will fall to $45.

Outcomes
Expected return
$60
($10)
$45
0

This means if the stock price rises, you'll lose $10 for each option you write. But if it falls, you're golden. You'll lose nothing since the option holder will not exercise (no reason to buy stock at $50 when the market price is $45) and you'll pocket the payment you receive for the option.

To value the option, let's imagine that you "hedge" your position by buying stock. In this way, you will own the stock if the options you sell are exercised when the stock price rises. And if the stock price declines, you'll have received the payments for the options to offset the price decline. Either way, you'll want to be in the same financial position.

Here's the tricky question:How many shares must you buy and how many options must you sell to be hedged? This can be calculated by computing the ratio of the spread of option value to spread of stock value --

Ratio of --

"option spread" of possible ovalue of options at exercise date ($10 or $0)

"stock spread" of possible stock values at exercise date ($60 or $45)

Ratio =

option spread
stock spread
  = [$10 - $0]
  [$60 - $45]
  = 2/3

This means that writing three call options (going short) and buying two shares of stock (going long) will be financially identical.

Stock price at end of period
Stock value (long)
Call value
(short)
Combined
(hedged)
$60
2($60) = $120
3(-$10)
$90
$45
2($45) = $90
3($0)
$90

At the end of the six months, you'll have $90 whichever way the stock price goes.

Now we can value the option - Co. All we need to know is the cost of capital - let's say 4% (assuming an efficient market that recognizes your financial position is risk-free). Remember that to buy two shares of stock (price $50) and write three calls, you'll have to borrow $100 minus the payment you receive for the options (3Co), and you won't get your money back for six months.

Cash
(t = 0)
Cost of capital
End value
(t = 6 mo)
$100 - 3Co
(1 + .04)
$90

We can solve this equation:

$100 - 3Co
=
$90/(1.04)
3Co
=
$100 - $90/(1.04)
3Co
=
$100 - $86.54
3Co
=
$13.46
Co
=
$4.49

You should charge $4.49/option. If the option buyer offers to pay less, you should refuse - you'll lose money since you'll end up spending more than your expected return ($90). if the option buyer offers more, take it. You'll make money. For example, if the option buyer offers $5.00/option, you'll make $1.60 by buying two shares and selling three options:

Option price
Buy 2 shares +
sell 3 options

End
(t = 6 mo)
$3
(100 - 9)*(1 + .04)
$94.64
$4.49
(100 - 13.46)*(1 + .04)
$90.00
$5
(100 - 15)*(1 + .04)
$88.40

Compare to Black-Scholes

This methodology is essentially the same used for the Black-Scholes Options Pricing Model. The option value can be computed by calculating the ratio of calls (short) to shares (long) and then take into account the effect of borrowing to create the hedge.

(# longVs - #shortCo) = Hedge position/(1 + rf)

Solve for Co:

#shortCo
=
 # longVs - Hedge position/(1 + rf)
Co
=
 [# long/#short]Vs -
[Hedge position/#short]/(1 + rf)
Co
=
 [ratio]Vs - [adjusted loan]

In the Black-Scholes equation given in the next section N(d1) represents the hedge ratio and N(d2)/ert represents the loan to achieve the hedged position.

Notice some things:

  • The value of the option is a function of --
    • short-term interest
    • time to expiration
    • variance rate of return on the stock
  • It does not depend on the expected return of the stock.

Avery Wiener Katx, "The Efficient Design of Option Contracts: Principles and Applications" SSRN Paper 512146 (March 2004)

ABSTRACT: The law of contracts has often treated options quite differently from other contractual transactions; for example, the characterization of a transaction as an option contract calls forth specially required formalities, but on the other hand often has the effect of releasing parties from doctrinal limitations on their contractual freedom, such as the duty to mitigate damages or the rule that holds excessively high liquidated damages void as penalties. Such differential treatment is challenging to explain from a functional viewpoint, in part because all contracts resemble options to the extent they are enforceable in terms of monetary damages, and in part because contracts that are nominally structured as explicit options can be close economic substitutes for contracts that are nominally structured as unconditional.

This essay sets out a theoretical account of the efficient design of option contracts - one that explains how contracting parties should strike the balance among option premium, option life, and exercise price, in order to maximize the expected surplus from their transaction. It shows that the tradeoffs between these various aspects of option contracts can affect the parties incentives to acquire and disclose information, to invest in relation-specific investments, and to take efficient precautions against the event of breach. It then goes on to develop an organizing framework for private parties choosing whether and how to structure their contractual arrangements as options, and for policymakers choosing whether or how to regulate such private choices. In short, the appropriate balance between option premium, option life, and exercise price will depend on the relative importance that the one attaches to these various dimensions of incentives.

 
4.4 Options - Characteristics and Valuation

©2003 Professor Alan R. Palmiter
This page was last updated on: March 2, 2005