🔎 Intro to Options

Call

Buying a call is a low-risk way to bet that a stock price will go up. We sometimes refer to this as a “long call”

You do this by locking in a low price that you can buy 100 shares of the stock for. This “locked-in” price is called the “strike price” or just K. For example, suppose you lock in a strike price of K=$100 on ABCD stock and the stock price rises to S=$128, then you can make money. Basically, you use your locked in price (from the call option) to purchase the shares for $100 and then you turn around and sell the shares for their market price of $128. In the process, you make a profit of $28 on each of the 100 shares. This $28 profit is known as the Intrinsic Value or Gain from the option. (Speaking loosely). In this case, we calculate it as S-K=$128-$100=$28. Remember this formula, S-K, because it will come up frequently!

If, on the contrary, the stock price for ABC corp ended up below $100, you would not want to exercise the option and purchase the shares for the locked in price. In this case, your option would have a gain/intrinsic value of $0.

Tying this together, if you exercise the call option, your gain will be S-K, ie $128-$100. If you don’t exercise your option, your gain will be $0. Your gain will either be S-K (if you exercise the option) or $0 if you don’t exercise the option. Because you get to choose whether you exercise the option or not, you will always choose the higher gain.

The gain you can get by exercising an option gives the option an intrinsic value. Therefore, we refer to this gain as the “Intrinsic Value” of the option.

In summary:

$IV_{call} = Max( S-K , \$0 )$

As a second example, suppose you lock in a strike price of K=$122.50 on XYZ shares. If the price rises up to $148.31, then you can buy the shares at $122.50 using the option and sell them for $148.31, netting a profit of S-K=$148.31-$122.50=$25.81 per share. With 100 shares per contract, you’ve made $2581 per contract of gain.

Clearly, you can make money with call options. But nothing is free. To buy a call option, you must pay a premium. The premium is determined by supply and demand, much like stock prices are determined by supply and demand. Your final profit will be the gain you make from the option minus the premium.

In other words, your final Profit/Loss will be the Gain you get minus the premium you pay:

$$\begin{align} \text{P/L} &= \text{Gain}-Premium \\ &= IV-Premium \\ &= Max( S-K , \$0 )-Premium \end{align}$$

Put

When you buy a Call, you lock in a price that you can buy shares for. When you buy a Put, you lock in a price that you can sell shares for.

Buying a put is a low-risk way to bet that a stock will go down. Buying a put is sometimes known as a “long put.”

Buying a put allows you to lock in a high price that you may sell 100 shares of stock for. For example, if you lock in a price of K=$100 on 100 shares of ABCD stock and stock price falls to S=$70, then you can make money. Specifically, you can buy 100 shares for ABCD stock for $70 each on the open market (possibly using borrowed money). Then you can turn around and sell the the same shares by exercising your put option contract for $100 each. The difference of $30=K-S between K=$100 and S=$70 is known as the Intrinsic Value of Gain from your put option (we are speaking slightly loosely, because IV is often calculated using the current stock price rather than the stock price at expiration, but we don’t care about that distinction here).

If, on the contrary, the stock price rose to $128, you would not exercise the option. Your gain/IV would be $0.

Tying this together, if you exercise the option, your gain is K-S. If you don’t, your gain is $0. Therefore, because you get to choose and always choose the best option:
IVput = max( K-S, $0)

As before, you’ll have to pay a premium in order to purchase the option.

Profit and Loss

Whether you buy a call or a put, you will have two cash flows: the gain and the premium that you pay.

When you buy an option, your final profit or loss will be:

P/L = Gain - Premium

The Gain is money you receive. You subtract the premium, because it is money you pay.

Let’s do some examples. First we will do Calls and then Puts.

It may help to have a mnemonic:
A call option allows you to “call” the stock to you and buy it for the strike price.
A put option allows you to “put” the stock out in the market and someone will buy it for the market price.

Long Call (Out of The Money)

✏️ Consider a Long K=$90 Call. Suppose S=$85 and Premium=$2.
a.) Will this option be exercised? What is the IV?
b.) What is your Per-Share Profit or Loss (P/L)?

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Short Answer:
a.) S<K, so Out of The Money, and WON’T be exercised.
b.) P/L = -$2.

Full Explanation: This Long Call allows you to buy shares of stock worth S=$85 for K=$90.

You could easily purchase the stock on the open market for $85, so exercising the option would mean overpaying by $5. We would think of this as a per-share $5 loss.

Clearly, you won’t exercise the option, so it will just expire worthless. Because you can’t profitably exercise the option, we say the option is “Out of The Money (Out of The Money)” and has no intrinsic value (IV=$0).

The only impact on your P/L is the $2 premium you paid. In hindsight, this money was wasted. P/L = -$2 (per share). Better luck next time!

Long Call (In The Money)

✏️ Consider a Long K=$65 Call. Suppose S=$72 and Premium=$1.
a.) Will this option be exercised? What is the Gain/IV?
b.) What is your Per-Share Profit or Loss (P/L)?

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Short Answer:
a.) S>K, so In The Money, and WILL be exercised.
b.) P/L = $6.

Full Explanation: This Long Call allows you to buy shares of stock worth S=$72 for K=$65. A mnemonic is that you can “Call” the shares to you and someone else (your counterparty) will have to sell them to you for K= $65.

That is a discount of $7, so this option is “In The Money (In The Money)” and you will exercise it. When you exercise it, you will gain $7. This $7 is the option’s intrinsic value.

(If you don’t want to buy more shares, you could easily exercise the option to buy them for K=$65 and then turn around and sell them on the market for S=$72, netting a $7 profit. You could even take out a short-term loan to do this.)

In addition to the $7 gain, you also paid a premium of $1. Therefore, your final P/L combines the $7 gain and the $1 loss: P/L = $7 - $1 = $6 (per share).

Long Put (Out of The Money)

✏️ Consider a Long K=$54 Put. Suppose S=$59 and Premium=$4.
a.) Will this option be exercised?
b.) What is your Per-Share Profit or Loss (P/L)?

✔ Click here to view answer

Short Answer:
a.) S>K, so Out of The Money, and WON’T be exercised.
b.) P/L = -$4.

Full Explanation: This Long Put allows you to sell shares of stock worth S=$59 for K=$54.

There is no way to make money by selling something worth S=$59 for only K=$54. Because you can’t exercise the option to make money, we say the option is “Out of The Money (Out of The Money)” and has no intrinsic value (IV=$0).

If you had shares you wanted to sell, you could easily sell them on the open market for $59. Exercising the option would mean being underpaid by $5, which we would think of as a $5 loss.

Clearly, you won’t exercise the option, so it will just expire unexercised.

The only impact on your P/L is the $4 premium you paid. In hindsight, this money was wasted. P/L = -$4 (per share). Better luck next time!

Long Put (In The Money)

✏️ Consider a Long K=$75 Put. Suppose S=$67 and Premium=$4.
a.) Will this option be exercised?
b.) What is your Per-Share Profit or Loss (P/L)?

✔ Click here to view answer

Short Answer:
a.) S<K, so In The Money, and WILL be exercised.
b.) P/L = $4.

Full Explanation: This Long Put allows you to sell shares of stock worth S=$67 for K=$75. $75 is a great price for something only worth $67. It’s a $8 premium.

A mnemonic is that you can “Put” the shares out on the market and someone else (your counterparty) will have to buy them for K=$75.

If you have shares you want to sell, you can exercise the option to sell them for $8 extra. If you don’t have shares you want to sell, you could easily borrow money and buy the shares for $67 on the open market. You then exercise your option to sell the same shares for $75, pocketing a $8 profit.

Either way, the option “In The Money (In The Money)”. You will exercise it to gain the option’s intrinsic value of $8.

In addition to the $8 gain, you also paid a premium of $4. Therefore, your final P/L combines the $8 gain and the $4 loss: P/L = $8 - $4 = $4 (per share).

Writing Calls and Puts

But who were you buying the shares from (with a call) or selling the shares to (with a put)?

To buy a call or a put, someone must sell the call or the put to you. They are referred to as the “option writer,” “option seller,” or the “Short Call”/“Short Put.” If you buy an option, the option writer is known as your “counterparty.” Conversely, you are their “Counterparty” (the counterparty is just the person on the opposite side of a derivative trade).

Your counterparty must do the opposite of what you do. If you have exercised a call option you purchased, you will “call” the shares to you and buy them. Your counterparty must sell them to you for the strike price.

If you have exercised a put option you own, you put the shares out in the market, and your counterparty must buy them.

✏️ Fill in the blanks. If you sell a call option, you might be forced to ________ the stock.

If you sell a put option, you might be forced to ________ the stock.

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If you sell a call option, you might be forced to __sell___ the stock.

If you sell a put option, you might be forced to __buy__ the stock.

Tip: to calculate P/L for an option seller, make sure you think about the question from the perspective of the option buyer first, because that is the only way to figure out if the option is exercised. What would your counterparty do?

Caveat: When you buy or sell an option on an exchange, counterparties are only assigned if the option buyer exercises the option. It’s harder to understand options if you think about this, so we will ignore this for now.

P/L for writing an option

When you buy an option, the P/L is IV - premium. It’s the reverse of this for when you sell an option. Specifically,

$$P/L_{sell an option} = Premium - IV$$

It’s like this because you receive the premium, but the IV is a loss for you, so it is subtracted.

Short Call (Out of the Money)

✏️ Consider a Short K=$44 Call. Suppose S=$42 and Premium=$2.
a.) Will this option be exercised?
b.) What is your Per-Share Profit or Loss (P/L)?

✔ Click here to view answer

Short Answer:
a.) S<K, so Out of The Money, and WON’T be exercised.
b.) P/L = $2.

Full Explanation:
You have sold an option that obligates you to sell shares of stock worth S=$42 for K=$44 if requested.

You can think of yourself as being matched with a “counterparty” who purchased the option you wrote/sold. Just as you have a Short Call, they have a Long Call. They decide whether to exercise the Long Call they purchased. If they do, you must sell them 100 shares of the underlying stock for K=$44.

From your counterparty’s perspective, there is no way to make money by paying $44 for shares only worth $42. Therefore, we say this option is “Out of The Money (Out of The Money)” and has no intrinsic value (IV=$0).

Given this, your counterparty won’t exercise the option. The only impact on your P/L is the $2 premium your counterparty paid you. P/L = $2 (per share).

Short Call (In the Money)

✏️ Consider a Short K=$53 Call. Suppose S=$56 and Premium=$3.
a.) Will this option be exercised?
b.) What is your Per-Share Profit or Loss (P/L)?

✔ Click here to view answer

Short Answer:
a.) S>K, so In The Money, and WILL be exercised.
b.) P/L = $0.

Full Explanation: You have sold an option that obligates you to sell shares of stock worth S=$56 for only K=$53 if requested.

You can think of yourself as being matched with a “counterparty” who purchased the option you wrote/sold. Just as you have a Short Call, they have a Long Call. They decide whether to exercise the Long Call they purchased. If they do, you must sell them 100 shares of the underlying stock for K=$53.

By purchasing your shares worth $56 for only $53 (a discount of $3), your counterparty can make $3 per share. Therefore, we say the option is “In The Money (In The Money)” and has an Intrinsic Value of $3 for its owner.

We’ll assume your counterparty will exercise the option. They will “Call” for your shares to purchase them for K=$53. This will cause a loss for you of $3. For example, if you own the shares, you will have to sell them for $3 less than you could get on the open market. Alternatively, if you don’t have the shares, you’ll have to buy them for $56 and sell them for $3 less. Either way, you are $3 poorer than you would be if your counterparty hadn’t exercised the option. (Correspondingly, they are $3 richer. The IV of the option is both your loss and their gain.)

Offsetting the $3 loss from selling the shares, you also received a premium of $3. Your final per-share premium is P/L = $3 - $3 = $0.

(For comparison, your counterparty’s final P/L is $0. As always, one party’s gain is the other party’s loss.)

The above example illustrates an important principle. In derivatives, the only way to make money is by taking someone else’s money (legally). One person’s gain is always their counterparty’s loss. One person’s loss is always their counterparty’s gain. You don’ t make money unless you take it from someone else.

The only entity that makes money on every transaction is the options exchange. This is why our readings for this week’s lecture are provided free by the options industry! They want you to trade so that they make money from you in fees.

Short Put (Out of The Money)

✏️ Consider a Short K=$81 Put. Suppose S=$86 and Premium=$1.
a.) Will this option be exercised?
b.) What is your Per-Share Profit or Loss (P/L)?

✔ Click here to view answer

Short Answer:
a.) S>K, so Out of The Money, and WON’T be exercised.
b.) P/L = $1.

Full Explanation: You have sold an option that obligates you to buy shares of stock worth S=$86 for K=$81 if requested.

You can think of yourself as being matched with a “counterparty” who purchased the option you wrote/sold. Just as you have a Short Put, they have a Long Put. They decide whether to exercise the Long Put they purchased. If they do, you must buy 100 shares of the underlying stock from them for K=$81.

There is no way for your counterparty to make money by selling you shares worth $86 for only $81. Therefore, we say this option is “Out of The Money (Out of The Money)” and has no intrinsic value (IV=$0).

Given this, your counterparty won’t exercise the option. The only impact on your P/L is the $1 premium your counterparty paid you. P/L = $1 (per share).

Short Put (In The Money)

✏️ Consider a Short K=$70 Put. Suppose S=$63 and Premium=$1.
a.) Will this option be exercised?
b.) What is your Per-Share Profit or Loss (P/L)?

✔ Click here to view answer

Short Answer:
a.) S<K, so In The Money, and WILL be exercised.
b.) P/L = -$6.

Full Explanation:
You have sold an option that obligates you to buy shares of stock worth S=$63 for K=$70 if requested.

You can think of yourself as being matched with a “counterparty” who purchased the option you wrote/sold. Just as you have a Short Put, they have a Long Put. They decide whether to exercise the Long Put they purchased. If they do, you must buy 100 shares of the underlying stock from them for K=$70.

By selling shares worth $63 for $70 (a premium of $7), your counterparty can make $7 per share (at your expense). Therefore, we say the option is “In The Money (In The Money)” and has an Intrinsic Value of $7 for its owner.

We’ll assume your counterparty will exercise the option. Your counterparty will “Put” the shares out there and you will have to buy them for $70. The shares are only worth $63, so you will overpay by $7. Just as your counterparty gains $7, you lose $7. The IV of the option is both their gain and your loss.

Offsetting your $7 loss from selling the shares, you have also already received a premium of $1 from your counterparty. Your final per-share premium is P/L = $1 - $7 = -$6.

(For comparison, your counterparty’s final P/L is $6. As always, one party’s gain is the other party’s loss.)