After completing this reading, you should be able to:

- Describe the types, positions variations, and typical underlying assets of options.
- Differentiate between the two types of options.
- Explain how a trader can make profits trading with options.
- Understand the concepts of payoffs, intrinsic value and time value of options.
- Explain the three cycles of trade of stock options.
- Explain the specification of exchange-traded stock option contracts, including that of non-standard products.
- Explain how dividends and stock splits can impact the terms of a stock option.
- Describe how trading, commissions, margin requirements, and exercise typically work for exchange-traded options.
- Define and describe warrants, convertible bonds and employee stock options.

## Types, Positions Variations, and Typical Underlying Assets

The buyer of an option has the right but not the obligation to exercise the option. The maximum loss to the buyer is equal to the premium paid for the option. Note that premiums are paid by a trader in order to obtain the right to buy or sell an underlying asset at a certain price in the future. On the other hand, the potential gains are theoretically infinite.

To the seller (writer), however, the maximum gain is limited to the premium received after writing the option. The potential loss is unlimited.

Some of the symbols used to represent relevant factors when dealing in options include:

\(X\) = strike price

\({ S }_{ t }\) = Price of the underlying asset at time \(t\)

\({ C }_{ t }\) = the market value of a call at time \(t\)

\({ P }_{ t }\) = the market value of put option at time \(t\)

\(t\) = the time to maturity/expiration of the option

### Call Options

A call option gives the owner/holder/buyer the right but not the obligation to buy the underlying stock at a given price on expiry. The buyer is said to hold a long position in the contract, while the seller is said to hold the short position.

When the stock price is less than or equal to the stock price at maturity, the buyer cannot exercise the option because the payoff would be zero. If the stock price is higher than the exercise price at maturity, the long will most likely exercise the option. The payoff of the call will be equal to the difference between the market price and the strike price \(\left( { S }_{ t }-X \right) \).

### Put Options

A put option gives the holder/buyer the right but not the obligation to sell the underlying stock at a specified price. At expiration, the buyer will only benefit if the prevailing market price is less than the exercise/strike price. The payoff is equal to \(\left( { X-S }_{ t } \right) \). If the stock stays at \(X\) or above, the payoff will be zero.

## Types of Options

Options that can be exercised at any time, during and before their maturity/expiration period are known as **American options**. Those that can only be exercised on the expiration/maturity date are known as **European options**. Most exchange-traded options are American options, while most options traded in over the counter markets are European options. Since they are only tradable at maturity, European options can easily be analyzed using the Black-Scholes-Merton model. Numerical procedures, for example, binomial trees, are used to analyze American options.

The **exercise date** is the date that has been specified in the contract upon which the contract matures (maturity date). The pre-determined future price at which the underlying asset will be sold at is known as the **exercise (or the strike) price**.

## Moneyness of Options

Assume that options were to be exercised today. The option will be said to be:

- in the money if it gives a positive payoff,
- out of the money if it gives a negative payoff,
- and at the money if the payoff is zero.

A call (put) option is said to be in the money if the strike price is lesser (greater) than the asset price at the time of maturity of the contract. If the strike price is greater (lesser) than the asset price at the time of maturity of the contract, a call (put) option is said to be out of the money. When the asset price equals the strike price at the time of maturity, an option is said to be at the money.

#### Example of the Moneyness of an Option

A great way to visualize this concept is with a graph. Let’s say we have a call option on AAPL with a strike price of USD 150. Whenever the price of the underlying (AAPL stock) is above USD, the option is in the money:

Note that for a put option with a strike price of USD 150, it would be the exact opposite – the option would be out of the money anytime the underlying stock is above USD 150.

## Profits on Call Options

#### Example 1: P&L on European Calls

Assume that a trader buys an out of the money European call option with a strike price of $50 for an asset that is currently selling at $40. The option expires in 6 months and has its premium is $5. What are the trader’s profits/losses if the price of the underlying asset at maturity is (i) $40 and (ii) $60?

$$

\begin{array}{|l|l|}

\hline

\textbf{For a current asset price of \$40} & \textbf{For a current asset price of \$60} \\

\hline

\begin{array}{l} \text{The option will not be} \\ \text{exercised.} \\ \text{} \\ \text{The trader therefore incurs a} \\ \text{loss of \$5, the premium paid} \\ \text{to secure the option.} \\ \text{} \\ \text{The seller of the call option} \\ \text{earns a profit of \$5.} \end{array} & \begin{array}{l} \text{The option will be exercised.} \\ \text{The trader will buy the asset at} \\ \text{\$50 and then sell it at \$60.} \\ \text{} \\ \text{The trader will, as a result,} \\ \text{make a profit of \$60 (current} \\ \text{price of the asset) – \$50 (strike} \\ \text{price) – \$5 (premium paid) = \$5} \\ \text{} \\ \text{The seller of the option will} \\ \text{suffer a loss of \$5.} \end{array} \\

\hline

\end{array}

$$

### Net Loss on Call Options

Sometimes a trader may get a net loss by exercising an option. This is in cases where the current price of the underlying asset is between the strike price and the strike price plus the premium paid. In the above illustration, a trader will suffer a net loss if the current price of the asset is between $50 and $(50+5) = $55.

#### Example 2: Net Loss on European Calls

Suppose that the current market price of the asset in example 1 is $53. What is the trader’s net loss?

The trader will lose $5 if he chooses not to exercise the option.

If the option is exercised, the trader gets a negative profit: $53 – $50 – $5 = -$2.

In as much as the trader suffers a loss by exercising the option, the loss is smaller as compared to the loss obtained if the option is not exercised. In such a scenario, the profits made by the call option seller will also be less than $5.

## Profits on Put Options

#### Example 3: P&L on European Puts

Assume that a trader buys an out of the money European * put *option with a strike price of $50 for an asset that is currently selling at $40. The option expires in 6 months and has its premium is $5. What is the trader’s profits/losses if the price of the underlying asset at maturity is (i) $40 and (ii) $60?

$$

\begin{array}{|l|l|}

\hline

\textbf{For a current asset price of \$40} & \textbf{For a current asset price of \$60} \\

\hline

\begin{array}{l} \text{The option will be exercised.} \\ \text{The trader’s profit will be the} \\ \text{\$10 difference minus the \$5} \\ \text{premium.} \\ \text{} \\ \text{Since it’s a zero-sum game, the} \\ \text{option seller will lose \$5.} \end{array} & \begin{array}{l} \text{The option will not be} \\ \text{exercised.} \\ \text{} \\ \text{The option buyer will lose the} \\ \text{\$5 premium while the option} \\ \text{seller will gain the \$5 premium} \\ \text{paid upfront.} \end{array} \\

\hline

\end{array}

$$

### Net Loss on Put Options

The trader suffers a net loss if the current market price of the asset is between the strike price minus the premium paid and the strike price, in this case, $45 and $50. For example, if the underlying price at expiration is $49, the put buyer will make $1 from the option being in the money and lose $5 from the option premium paid upfront, for a net loss of $4. This is still better than losing the full option premium of $5!

## Payoffs

Denoting the price of the asset at maturity as \({ S }_{ t }\) and the strike price as \(X\), the payoffs from option positions are as shown below.

Long Call: \(max( { S }_{ t }-X,0) \)

Short Call: \(-max( { S }_{ t }-X,0) \) = \(min( X-{ S }_{ t },0) \)

Long Put: \(max( X-{ S }_{ t },0) \)

Short Put: \(-max( X-{ S }_{ t },0) \) = \(min( { S }_{ t }-X,0) \)

## Intrinsic Value and Time Value

The **intrinsic value** is the value of the option if the option were to be exercised immediately. It is the same mathematical formula as if the payoff of the option was today.

Long Call: \(max( { S }_{ t }-X,0) \)

Long Put: \(max( X-{ S }_{ t },0) \)

The **time value** of an option is the difference between the option premium and the intrinsic value:

\(\text{Option premium}=\text{Time value}+\text{Intrinsic value}\)

## Exchange-Traded Options on Stocks

Options traded in exchanges are American-style options. The largest exchange in the world is the Chicago Board Options Exchange (CBOE). Traders with a short position are randomly allocated (assigned) traders with a long position. A single option contract is the right to trade 100 shares.

### Maturity of Stock Options

The CBOE has three cycles of trade (maturity dates):

- Jan Cycle: January, April, July, October
- Feb Cycle: February, May, August, November
- March Cycle: March, June, September, December

The CBOE equally offers weekly options (short-term options) and LEAPS (Long Term Equity Anticipation Securities). LEAPS are simply publicly traded options contracts with expiration dates that are longer than one year.

### Strike Prices

The CBOE sets strike prices in different multiples.

$$

\begin{array}{c|c}

\begin{array}{c} \text{Current price of the underlying} \\ \text{asset (\$)} \end{array} & \begin{array}{c} \text{The strike price will be a} \\ \text{multiple of:} \end{array} \\

\hline

\text{5-25} & \text{2.5} \\

\text{25-200} & \text{5} \\

\text{larger than 200} & \text{10} \\

\end{array}

$$

The strike prices of an underlying asset are the three closest prices to the current price of an underlying asset.

#### Example: Quoted Strike Prices

Assume the price of an underlying asset is $20. The strike prices of the asset that will be listed will be the three closest prices to the current price that are multiples of 2.5. In this case, it would be $17.5, $20 and $22.5.

If the price of the asset moves below $17.5, options with a strike price of $15 will start trading. Conversely, if the price of the asset moves above $22.5, options with a strike price of $25 will start trading. These rules ensure that a lot of options are available for trade.

Options of the same type form a class. Options of a class having a specific maturity date and strike price form an option series.

## The Effect of Dividends and Stock Splits

### Stock dividends

Instead of paying cash, a stock dividend involves issuing extra shares to shareholders. For example, if a firm announces a 2% stock dividend, then for every 100 shares held, shareholders will receive 2 more shares.

Exchange-traded options are not usually adjusted for cash dividends. In other words, when a cash dividend occurs, there are no adjustments to the terms of the option contract.

### Stock splits

A stock split involves increasing the total number of shares outstanding by issuing more shares to shareholders at a specified ratio. For example, a 2-for-1 stock split means a shareholder will be awarded one more share for every two shares held.

If a stock has a b-for-a stock split, the share price will be reduced by a factor of (a/b). However, this is a theoretical assumption. In reality, the post-split share price can be different. The number of shares will increase by a multiple of (b/a).

The terms of exchange-traded options contracts are adjusted to reflect expected changes in a stock price arising from a stock split.

## Non-standard Products

They include

**Flexible exchange (FLEX) OPTION:**These are exchange-traded options on stock indices, but there’s a lot more flexibility. The strike price and expiration dates can be altered if the trading parties so wish.**ETF options:**These are American-style options that are settled by delivering the underlying shares rather than cash.**Weekly options:**These are short-term options with a maturity period of roughly 7 days. They are created on a Thursday, with the expiration date being the Friday of the next week.**Binary options:**Binary options have a fixed payoff in case the option is ITM at expiration.**Credit event binary options (CEBOS):**The CEBOs payoff is triggered when the reference entity suffers a credit event before the option’s expiration date.**Deep out-of-the-money (DOOM) options:**They are designed to only be ITM in the event of a large down price movement in the underlying asset.

## Market Makers, Trading Commissions, Closeout, and Margin Requirements

Most options exchanges use **market makers** to facilitate trading. The market maker will quote bids and offer prices.

A **commission** refers to the fee charged by a broker as a reward for their efforts in facilitating a transaction. Commission costs depend on the size of the trade as well as on the type of broker involved. They reduce the investor’s returns.

Options can be **closed out** by taking an offsetting position just like future markets.

To prevent the options markets from being influenced by just the investors, the CBOE imposes position and exercise limits on traded options:

**Position limit**refers to the number of contracts an investor can hold on the same side of the market. Long calls and short puts are on one side of the market while short calls and long puts are on one side of the market.**Exercise limit**is the number of contracts that can be exercised within five working days.

In options trading, the term “**margin**” refers to the collateral deposited by the option writer as a form of guarantee that they will honor their contractual obligations. Margin requirements differ from one broker to another and also depend on the nature of the underlying asset. As a general rule, options that mature before 9 months cannot be purchased on margin. Those that mature after 9 months can be purchased by borrowing up to 25% of the purchase price.

## Warrants, Convertibles, and Employee Stock Options

**Warrants**are call options that are issued by a corporation.**Convertible bonds**are bonds that can be converted to equity using a ratio that has been pre-determined.**Employee stock option**s are granted by a corporation to its employees.

When the holder of the above three securities exercises his/her right to obtain shares in the company, the company issues more stock.

## Question

What is the intrinsic value of a put option if the strike price is $63 and the prevailing market price of the underlying is $78?

- $15
- -$15
- $7.5
- $0

The correct answer is **D**.

Intrinsic value of a put option = \(max\left( 0,{ X-S }_{ t } \right) \)

$$ =max(0,63-78) =max(0,-15) = 0 $$