Derivatives Part 1

If you’ve read the article “Derivatives – Introduction,” which covers basic concepts, I recommend studying the payoff function graphs and reviewing the attached examples. I know this can be tedious for some, but understanding this material will give you a better chance of trading successfully.

Standard simplified graphs of the payout function at the option’s exercise or expiration date.

Analyze the payout (profit/loss) function graph for an options transaction in detail, paying attention to its specific characteristics and terminology used.

Note!

The horizontal numerical axis only accepts positive values and does not represent time. The current share price increases to the right – the “blue circle,” also known as the “spot price.”

The vertical numerical axis displays both positive and negative values, representing the payout function, which is the difference between the share’s spot price and the option’s strike price (the “brown circle”).

The price of share S may fluctuate during the period until the option expires, while the strike price of option X remains fixed (set at the time the position was taken – the transaction was completed, like the option premium paid or received, which represents the initial investment).

Therefore, the payoff function graph displays a partially diagonal line at a 45-degree angle and a line parallel to the horizontal axis, representing the value of max(0, S-X) and the option premium (PO). When the spot price on the option expiration date moves “to the left,” meaning below the strike price, the buyer of the call option loses the entire premium because there’s no point in exercising the right to buy the shares at a higher (strike price) than the current share price – resulting in a 100% loss of the initial investment (PO). Conversely, if the spot price on the expiration date moves to the right of the strike price, the buyer’s profit or loss depends on the premium paid and will be (S-X-PO).

The following charts display a complete set of call and put options along with long and short positions.

Now, let’s move on to a tedious exercise in numbers.

Long Lcall/LPut position

Example of an LCall (Long Call – purchase of the right to buy shares, buying a call option, holding a long position in a call option):

• The price of one share is $100, and I expect it will rise to $110 within a month. Instead of spending $10,000 to buy 100 shares, I can purchase one option that gives me the right to buy 100 shares at a strike price of $100, with an expiration date in one month. This is called an “at-the-money” option because the strike price of $100 matches the current share price of $100. Assuming the cost of this option is $2 per share, I will spend only $200 to purchase it.
• In the model scenario, if I am correct, the share price will rise to USD 110 in one month. I can then exercise my option to purchase 100 shares at USD 100 per share; however, I am not obligated to do so. If I choose not to act, at the end of the option’s validity period, the price difference will be automatically credited to my account, i.e., USD 1,000 = (USD 110 – USD 100) * 100. Therefore, I spent USD 200 (plus approximately USD 1 in transaction fees, depending on the broker, which we can disregard for now). I have realized a USD 1,000 gain, yielding an approximate 400% rate of return: (USD 1000 – USD 200) / USD 200 (ignoring negligible transaction costs at this scale).
• It’s important to remember that the option premium (in my case, $200) gradually declines to zero as the option expires, and the stock price need not necessarily rise.
• I don’t need to wait until the expiration date and can sell it in the meantime (as long as the instrument is liquid, meaning there are potential buyers, a quote exists, and an order can be successfully placed). Let’s assume that halfway through the expiration period, the stock price is $105. In this case, my option value is $6 = $5 (the difference between the stock price and the $100/share strike price) + $1 (the premium for the remaining time to expiration). I now sell my rights for $6 (or, more precisely, $600 minus a $1 transaction cost). I paid USD 200 for the option, so the rate of return on this transaction is approximately 150% = (600 – 200) / 200 (ignoring transaction costs). It is not 400%, but I already have money in my account and do not risk a change in the trend.
• To make things more complex, I buy a 250 call option—giving me the right to purchase 100 shares of the company’s stock at a strike price of $250 per share. The current stock price is $245, and I expect it to rise to $260 during this period. This option is relatively inexpensive at $2 because it’s an “out-of-the-money” option, meaning the $250 strike price is higher than the current price of $245. So, I spent $200 to buy this option, and within a few days, the stock price rose to $252. As the share price increases, the option price also rises to $7. Consequently, I sell this option before expiration and realize a 250% return: (700-200)/200 = 500/200. Of course, the stock price could fall below the $250 strike price at any moment and stay there until the option expires, resulting in a loss of exactly $200, which is a -100% return: (-200)/200 – I lost all the capital I spent on buying the option.
• I could also buy a call option that is “in the money,” with a strike price of $240, which is below the current share price of $245. In this case, the option price will likely be $7, calculated as $5 (the difference between $245 and $240 per share) plus an approximate $2 time premium. Therefore, I spent $700 to buy this option. Let’s assume the probability of the share price dropping below $240 is very low, so there’s a good chance I won’t lose my entire investment. If I don’t have $24,500 to buy 100 shares and want to use leverage to profit from a price increase to $250 (which I firmly believe will happen), then I’ve only spent $700 but control the entire $24,500.
• If I’m correct and the stock price reaches $250 at any point during the option’s term, the option’s value could exceed $10, calculated as $250– $240 per share plus the time premium.
  • If I bought 100 shares at $245 each and sold them at $250 each, I would spend $24,500 (100 shares at $245 each) and realize a $500 profit, the difference between $25,000 and $24,500. This yields a roughly 2% return, calculated as $500 divided by $24,500.
  • If I buy one option, I would spend only $700 and earn at least $300 = ($10 – $7) * 100, meaning the rate of return would be about 44.5% = 300/700 with less capital involved, but also a higher risk of losing my $700 investment (or even 100%).

Investing in stocks isn’t without risk either. However, the right to own the shares doesn’t expire, and I can sell them later, when and if they “rebound.”

The above reasoning and principles work in reverse when purchasing the right to sell, which is a LPut, or Long Put – that is, buying the right to sell shares (purchasing a put option equals a long position in the put option).

We usually buy a put option when we expect the stock price to fall. In this case, the following situations might also happen when we take a position:

• At the money – when the put option’s strike price equals the current stock price.
• Out of the money – when the put option’s strike price is below the current stock price.
• In the money  – when the put option strike price is higher than the current stock price

General conclusions about long positions (LCall, LPut):

• Immediate payment for the option acquired.
• There is no obligation to exercise the option (i.e., the right to buy or sell shares during the option’s validity period), nor is there an obligation to exercise the option, which means forcing the option writer to act.
• Automatic cash inflow occurs if the option’s strike price is lower (LCall) or higher (LPut) than the share price on the option’s expiration date.
• 100% loss occurs if the option’s strike price is higher (LCall) or lower (Lput) than the share price at expiration, or a slight loss occurs when closing the position early (selling the option before expiration while it still has time value).
• Loss of the entire time premium value at the option’s expiration.

Short SCall/SPut

Example of a SCall (Short Call – selling the right to purchase shares, selling a call option, short position in a call option):

• The price of one share is $100, and I expect it will drop to $90 within a month. Instead of taking a short position in the stock (i.e., selling 100 shares for $10,000 and repurchasing them later for $9,000), I can sell the right to buy 100 shares. For example, I can write (sell) one call option (with a strike price of $100) that expires in a month. This is called an “at-the-money” option because the strike price of $100 per share equals the current share price. Let’s assume the cost of this option is $2, meaning I will receive $200 for writing it. 
• In the modeled scenario, if I am correct, the share price will be USD 90 in one month, and the option value will approach zero. No one will pay for an option that gives the right to buy 100 shares at USD 100 per share (the exercise price) when they can buy a share on the market for USD 90 per share. The option premium (in my case, USD 200) systematically expires to zero as the option expiration date approaches, meaning it remains in my pocket to a greater extent. The rate of return on this investment will be 100%, corresponding to a USD 200 return on USD 200 invested.
• However, I don’t have to wait until the expiration date and can buy back the option in the meantime, provided the instrument is liquid—that is, there are sellers. Assuming the stock price is $97 per share at the midpoint of the expiration period, my option price could be $1. I buy back my right for $1 (or more precisely, $100). Of the $200 I received when writing the option, I returned $100, leaving me with $100 in cash, with no capital outlay. The rate of return is now 50%, calculated as $ 100 divided by $200.
• To make things more complicated, I sell a single 250 call option—giving me the right to buy 100 shares of the company’s stock at a strike price of $250 per share. The current stock price is $245, and I expect it to fall to $240 during this period. This option is relatively “cheap,” costing $2, mainly covering the premium for the time remaining until expiration, since it’s an “out-of-the-money” option, meaning the call strike price of $250 is higher than the current stock price of $245. As a result, I received a $200 premium for writing this option, and within a few days, the stock price fell to $242, reducing the option’s value to $1. I can also buy back this option for $100 before expiration, or wait longer, as time works in my favor, and the buyer of my option will likely not exercise his right to buy at $250 when he can buy on the market for $242.
• Indeed, the stock price could rise above the strike price of $250/share at any moment, for example, to $255/share, and stay there until the expiration date. In such a case, my option value would increase to $6: ($255/share – $250/share) + $1 (time premium), or less if considerable time has passed. I could accept the loss and close this trade by repurchasing the option for $600, resulting in a $400 loss (since I received $200 when I wrote the option). The rate of return would be negative (-200%): -$400/$200. I could, of course, wait to see what happens next and enter trades to modify my position (I plan to discuss this in future articles). 
• The higher the share price rises, the greater the risk that the buyer will exercise my option. He has the right to purchase 100 shares at USD 250 per share, and I am obliged to deliver these shares (i.e., I must maintain a capital reserve of USD 25,000). If the buyer of the option I sold does not exercise it and the share price on the expiration date is USD 255 per share, then USD 500 (computed as USD 255 minus USD 250, multiplied by 100 shares) will be automatically deducted from my account. However, my total loss will be limited to USD 300 because I received a USD 200 premium when I issued this option, and the remaining time premium automatically expired.
• In the previous example, I could also sell a call option with a strike price of $240/share, which is lower than the current share price of $245/share—a so-called “in-the-money” situation. For example, I could sell a 240 call when the share price is $245. Let us assume the option premium will be $7, calculated as $5 (the difference between $245 and $240) plus a time premium of approximately $2. Therefore, I would receive $700 for writing this option. Suppose the probability of the share price dropping below $240 is very high, so there’s a good chance I will keep this premium. If I am right and the stock drops to $230/share at any point during the option’s term, the value of my option will mainly include a premium for the time it remains valid, since the buyer of the option wouldn’t want to force me to sell the stock at $240 when they could buy it in the market for $230.

The above reasoning and principles apply in reverse when selling the right to put = SPut = Short Put – (selling a Put option = short position in the Put option).

We typically sell a put option when we expect a possible increase in the stock price. In this case, when we take a position, the following scenarios may also occur:

• At the money – when the put option’s strike price equals the current stock price.
• Out of the money – when the put option’s strike price is lower than the current stock price.
• In the money – when the put option’s strike price exceeds the current stock price.

General conclusions – short position:

• I sell an option and accept the obligation – I receive cash.
• Obligation to exercise the option on demand by delivering shares or purchasing shares upon request by the option buyer.
• Automatic cash outflow occurs to cover losses when the option strike price is lower (SCall) or higher (Sput) than the share price on the option’s expiration date or on the closing transaction date (when the option is purchased before the expiration date).
• Achieve 100% profit when the option strike price is higher (SCall) or lower (Sput) than the share price at expiration, or a slightly lower profit if closing the position early by buying the option before expiration. At the same time, it still has some time premium.

Don’t read the following article in this series Derivatives Part 2 until you understand the previous one.