Option Strategies

To protect against a fall in a share or trade a view on price moves, you combine options rather than use them one at a time. In covered calls, spreads, straddles, and similar strategies, the mix of purchases and sales sets the caps on profit and loss and the break-even points.

The profit and loss of a combination is the sum of each trade's profit and loss

The profit and loss of a combined strategy at expiry is the sum, at each price at expiry, of the profit and loss at expiry of each trade in the combination. Below, each individual trade in a combination is called a leg. The profit and loss of a single option at expiry, and its shape, is covered in Option Payoffs and Profit/Loss.

The profit and loss at expiry of each leg is as follows. Here $S_T$ is the price of the underlying asset at expiry, and $K$ is the strike price. The symbols $c$ and $p$ are the option premiums of a call option (call) and a put option (put), and $S_0$ is the price at which the underlying was bought. The profit and loss of a sale is that of the purchase with the sign reversed.

If the total premium of the legs you buy is larger than the total premium of the legs you sell, you pay a net premium when you set up the combination. This net payment is called a debit¹. In a combination where you receive more premium than you pay, the premium received is added to the profit and loss whatever the price at expiry.

Consider a hypothetical example in which a share's current price is 100 yen and options on that share with the same expiry have the premiums in the table below.

Combining the underlying with an option

If you hold the underlying and add one option, you can change the shape of the profit and loss you would have from holding the underlying alone. Selling a call brings in a premium but caps the gain from a rise, and buying a put costs a premium but caps the loss from a fall.

Covered call

A covered call is a combination in which you keep holding the underlying and sell a call on that underlying. It is a strategy used, for example, to enhance the yield². Consider an example in which you hold a share bought at 100 yen and sell a call with a strike price of 110 yen for a premium of 2 yen.

If the share price at expiry is 110 yen or lower, the call is not exercised, and the profit and loss is that of the share plus the 2 yen premium received. The break-even point is 100 − 2 = 98 yen. At a share price of 90 yen, the 10 yen loss from holding the share alone becomes an 8 yen loss. If the share price at expiry is above 110 yen, the loss on the call sold offsets the rise in the share, and the profit stops at 110 − 100 + 2 = 12 yen. At a share price of 120 yen, holding the share alone gives a profit of 20 yen, but the covered call gives 12 yen.

Figure description

The profit and loss at expiry of a share held at 100 yen combined with a sold call with a strike price of 110 yen (premium 2 yen). The profit and loss is zero at a share price of 98 yen at expiry, a loss below that, and a profit above it. Up to 110 yen, each 1 yen rise in the share price adds 1 yen to the profit and loss, and from 110 yen upward the profit stays at 12 yen. On the downside, the loss is 2 yen smaller than holding the share alone because of the premium received, but it keeps growing as the share price falls.

The maximum profit is "strike price − purchase price of the share + premium received", and the maximum loss, reached when the share price falls to zero, is "purchase price of the share − premium received"³. If the share price at expiry is above the strike price, the call seller is assigned and has to deliver the share at the strike price⁴.

Protective put

A protective put is a combination in which you keep holding the underlying and buy a put on it. It limits the loss from a fall in the underlying while still letting you gain if it rises⁵. Consider an example in which you hold a share bought at 100 yen and buy a put with a strike price of 90 yen for a premium of 2 yen.

If the share price at expiry is below 90 yen, the profit on the put makes up for the fall in the share, and the loss stops at 100 − 90 + 2 = 12 yen. At a share price of 70 yen, holding the share alone loses 30 yen, but the protective put loses 12 yen. On the other hand, if the share price at expiry is 90 yen or higher, the put is not exercised, and the profit is reduced by the 2 yen premium paid. The break-even point is 100 + 2 = 102 yen. At a share price of 120 yen, the 20 yen profit from holding the share alone becomes 18 yen.

Figure description

The profit and loss at expiry of a share held at 100 yen combined with a bought put with a strike price of 90 yen (premium 2 yen). At a share price of 90 yen or lower at expiry, the loss stays at 12 yen. From 90 yen upward, the profit and loss rises with the share price, reaches zero at 102 yen, and is a profit above that. There is no cap on the profit, but at every share price the profit and loss is 2 yen lower than holding the share alone because of the premium paid.

The maximum loss is "purchase price of the share − strike price + premium paid", and there is no cap on the profit⁶. Seen as a hedge against a fall, the premium is the cost of that protection.

Spreads with different strike prices

If you combine buying and selling calls, or buying and selling puts, with the same expiry and only different strike prices, the profit and loss is sloped only between the two strike prices. You pay less premium than when buying one option alone, but in return the profit is also capped⁷.

Bull call spread

A bull call spread is a combination in which you buy a call with a lower strike price $K_1$ and sell a call with a higher strike price $K_2$. It makes a profit when the share price at expiry rises. If you buy a call with a strike price of 100 yen for a premium of 5 yen and sell a call with a strike price of 110 yen for a premium of 2 yen, you pay a net 3 yen.

Figure description

The profit and loss at expiry of buying a call with a strike price of 100 yen for a premium of 5 yen and selling a call with a strike price of 110 yen for a premium of 2 yen. At a share price of 100 yen or lower at expiry, the net 3 yen paid is the loss. Between 100 yen and 110 yen, the profit and loss rises with the share price and reaches zero at 103 yen. At 110 yen or higher, the loss on the call sold offsets the profit on the call bought, and the profit stays at 7 yen.

With $D$ as the net premium paid, the break-even point, maximum profit, and maximum loss are as follows⁸.

$$\text{Break-even point} = K_1 + D$$ $$\text{Maximum profit} = K_2 - K_1 - D$$ $$\text{Maximum loss} = D$$

In the example, the break-even point is 100 + 3 = 103 yen, the maximum profit is 110 − 100 − 3 = 7 yen, and the maximum loss is 3 yen. Compared with buying one call alone, the loss is smaller by the reduction in premium paid, but the profit is capped.

Bear put spread

A bear put spread is a combination in which you buy a put with a higher strike price $K_2$ and sell a put with a lower strike price $K_1$. It makes a profit when the share price at expiry falls⁹. If you buy a put with a strike price of 100 yen for a premium of 5 yen and sell a put with a strike price of 90 yen for a premium of 2 yen, you pay a net 3 yen.

If the share price at expiry is 100 yen or higher, neither put is exercised, and the 3 yen paid is the loss. As the share price falls from 100 yen to 90 yen, the profit and loss rises, and the break-even point is $K_2 - D$ = 100 − 3 = 97 yen. At 90 yen or lower, the loss on the put sold offsets the profit on the put bought, and the profit stops at $K_2 - K_1 - D$ = 100 − 90 − 3 = 7 yen. The shape of this example is the bull call spread example flipped left to right around a share price of 100 yen.

Straddles and strangles

If you buy both, or sell both, a call and a put with the same expiry, the profit and loss depends on how far the price at expiry moves, not on whether it goes up or down. The combination is called a straddle when the strike prices are the same and a strangle when they differ¹⁰.

Buying and selling a straddle

A long straddle is a combination in which you buy both a call and a put with the same strike price. If you buy a call and a put with a strike price of 100 yen for a premium of 5 yen each, you pay 10 yen in total. If the share price at expiry moves more than 10 yen above or below 100 yen, the profit on the option you exercise exceeds the premiums paid¹¹.

Figure description

The profit and loss at expiry of buying both a call and a put with a strike price of 100 yen for a premium of 5 yen each. The loss is largest, the 10 yen paid, when the share price at expiry is 100 yen. The break-even points are 90 yen and 110 yen. Between them the result is a loss, and below 90 yen or above 110 yen it is a profit. The further the share price moves from the strike price, in either direction, the larger the profit.

The break-even points are "strike price ± total premium paid", and the maximum loss is the total premium paid. There is no cap on the profit when the share price rises, and the profit when it falls is at most 100 − 10 = 90 yen, reached when the share price falls to zero.

A short straddle is a combination in which you sell both a call and a put with the same strike price, and its profit and loss is the long straddle's flipped upside down. In the example, if the share price at expiry is exactly 100 yen, the 10 yen received is all profit, and anywhere between 90 yen and 110 yen some profit remains. If the share price rises above 110 yen, the loss on the call sold grows without limit. Below 90 yen, the loss on the put sold grows, up to a maximum of 90 yen when the share price falls to zero¹². A short straddle is a strategy in which the premium received becomes profit if the price stays within a certain range, but the loss can be large if the expectation turns out wrong¹³.

Buying and selling a strangle

A strangle combines a call and a put with different strike prices. In the examples below, a put with a lower strike price and a call with a higher strike price are combined, and the range of price moves it anticipates is wider than for a straddle¹⁴.

As an example of a short strangle, sell a put with a strike price of 90 yen and a call with a strike price of 110 yen for a premium of 2 yen each. You receive 4 yen in total. This is less than the 10 yen of the short straddle, but in return the prices at which losses begin are further from the current share price of 100 yen (86 yen and 114 yen, against 90 yen and 110 yen for the short straddle)¹⁵.

Figure description

The profit and loss at expiry of selling both a put with a strike price of 90 yen and a call with a strike price of 110 yen for a premium of 2 yen each. If the share price at expiry is between 90 yen and 110 yen, neither option is exercised, and the 4 yen received is the profit. The break-even points are 86 yen and 114 yen, and below or above them the result is a loss. There is no cap on the loss when the share price rises, and the loss when it falls is at most 86 yen, reached when the share price falls to zero.

A long strangle is a combination in which you buy both of the same two options, and its profit and loss is the reverse of the short strangle's. In the example, you pay 4 yen. If the share price at expiry is between 90 yen and 110 yen, the loss is 4 yen, and below 86 yen or above 114 yen the result is a profit. You pay less premium than for a long straddle, but you need a larger price move to make a profit.

Butterflies that concentrate profit near the middle price

A long call butterfly is a combination in which you buy one call with a lower strike price, sell two calls with the middle strike price, and buy one call with a higher strike price. The lower and higher strike prices are the same distance from the middle one¹⁶. The name comes from the shape of the profit and loss, which looks like a butterfly with its wings spread¹⁷.

You buy one call with a strike price of 90 yen for 12 yen, sell two calls with a strike price of 100 yen for 5 yen each, and buy one call with a strike price of 110 yen for 2 yen. The net payment is 12 − 10 + 2 = 4 yen.

Figure description

The profit and loss at expiry of buying one call with a strike price of 90 yen and one with a strike price of 110 yen, and selling two calls with a strike price of 100 yen. The net premium paid is 4 yen. At a share price of 90 yen or lower, or 110 yen or higher, at expiry, the loss stays at 4 yen. The profit is largest, 6 yen, when the share price is 100 yen, and the break-even points are 94 yen and 106 yen.

The maximum profit is "the difference between the middle strike price and an outer strike price − premium paid", which is 10 − 4 = 6 yen in the example. The maximum loss is the 4 yen premium paid, which occurs when the share price at expiry is outside the outer strike prices¹⁸. A short straddle also has its largest profit at the middle price, but because the butterfly buys the outer calls, its profit is smaller and in return its loss is capped. Selling a call and a put with the middle strike price and buying a call with a higher strike price and a put with a lower strike price (a short iron butterfly) also gives a profit and loss at expiry similar in shape to a long call butterfly. In this case, you receive a net premium when you set up the combination¹⁹.

Comparing profit and loss at expiry by strategy

The table lists the figures from the examples above by strategy. In the net premium column, a positive value is a payment (debit) and a negative value is a receipt.

The covered call and protective put rows include the profit and loss from the share price. The net premium of the covered call does not include the cost of the share, and its maximum loss is the value when the share price falls to zero. "No cap" refers to the profit or loss when the share price rises.

The simulator calculates the premiums from volatility, not from the values in the table²⁰. Consider a hypothetical example in which the current price of the underlying is 100, expiry is one year away, and the interest rate is 0. At a volatility of 20%, the premiums of the call and the put with a strike price of 100 are both 7.97. A long straddle costs 15.93 in total, and the break-even points are 84.07 and 115.93. Because you pay more premium than the 10 yen in the table example, the break-even points are also further from the strike price.

Profit and loss before expiry and trading in practice

All the profit and loss above is at expiry. If you end a trade before expiry with an offsetting trade, the profit and loss has a different shape. The price of an option before expiry includes time value, which decreases as expiry approaches. As a result, if days pass while the share price does not move, a long straddle shows an unrealized loss even before expiry²¹. Conversely, for a short straddle or short strangle, the passage of time with no move in the share price works in your favor²².

The effect of a change in volatility also differs by strategy. In a long straddle, which buys both a call and a put, a rise in volatility raises the prices of both options, and a fall lowers both²³. Because a spread holds a bought and a sold option of the same type, the effects of a change in volatility on the bought option and the sold option can largely offset each other²⁴. The sensitivities that measure price moves before expiry (the Greeks) are covered in The Greeks.

When you trade a combination that includes a sold option on an exchange, you post initial margin when you open the position. A combination that only buys options needs no margin²⁵. Also, if you have sold an option that can be exercised before expiry, you may be assigned before expiry. If only the sold leg is exercised, part of the combination breaks apart, and the profit and loss no longer follows the shape at expiry²⁶.

Notes

1. Osaka Exchange, "デビットスプレッド戦略" (Debit spread strategies, in Japanese), the paragraph containing "支払いのことをデビットというので、このポジションはデビットスプレッドと呼ばれます" (a payment is called a debit, so this position is called a debit spread, our translation). ↩

2. Japan Exchange Group, "カバード・コール（用語集）" (Covered call, glossary, in Japanese). ↩

3. The Options Industry Council, "Covered Call (Buy/Write)", "MAXIMUM GAIN" and "MAXIMUM LOSS". The example values were obtained by putting a purchase price of 100 yen, a strike price of 110 yen, and a premium of 2 yen into these formulas. ↩

4. The Options Industry Council, "Covered Call (Buy/Write)", "Description" and "Profit/Loss". ↩

5. Japan Exchange Group, "プロテクティブ・プット（用語集）" (Protective put, glossary, in Japanese), and Central Council for Financial Services Information, "3-5 オプション取引活用の例（やさしいデリバティブ）" (Examples of using option trading, Easy Derivatives, in Japanese), "プロテクティブ・プット" (Protective put). ↩

6. The Options Industry Council, "Protective Put (Married Put)", "MAXIMUM GAIN" and "MAXIMUM LOSS". ↩

7. Osaka Exchange, "デビットスプレッド戦略" (Debit spread strategies, in Japanese), the paragraph between "図表３" (Figure 3) and "図表４" (Figure 4). The maximum loss becomes smaller than when buying a single call, and the profit above the strike price of the call sold is offset. ↩

8. The Options Industry Council, "Bull Call Spread (Debit Call Spread)", "MAXIMUM GAIN", "MAXIMUM LOSS", and "Breakeven". ↩

9. The Options Industry Council, "Bear Put Spread", "Description", "MAXIMUM GAIN", "MAXIMUM LOSS", and "Breakeven". ↩

10. Japan Exchange Group, "ストラドル（用語集）" (Straddle, glossary, in Japanese) and "ストラングル（用語集）" (Strangle, glossary, in Japanese). ↩

11. The Options Industry Council, "Long Straddle", "MAXIMUM GAIN", "MAXIMUM LOSS", and "Breakeven". ↩

12. The Options Industry Council, "Short Straddle", "MAXIMUM GAIN", "MAXIMUM LOSS", "Max Loss", and "Breakeven". ↩

13. Central Council for Financial Services Information, "3-5 オプション取引活用の例（やさしいデリバティブ）" (Examples of using option trading, Easy Derivatives, in Japanese), "停滞を予想するとき" (When you expect the market to stagnate). ↩

14. Japan Exchange Group, "ストラングル（用語集）" (Strangle, glossary, in Japanese). ↩

15. The Options Industry Council, "Short Strangle", the paragraph after "MAXIMUM GAIN" and "MAXIMUM LOSS", and "Breakeven". ↩

16. The Options Industry Council, "Long Call Butterfly", "Description". ↩

17. Osaka Exchange, "日経225先物・オプション基礎講座デルタ編②-1デルタニュートラルとストラドル・バタフライ" (Nikkei 225 futures and options basics, delta part 2-1: delta neutral, straddles and butterflies, in Japanese), the paragraph after "図表５" (Figure 5). ↩

18. The Options Industry Council, "Long Call Butterfly", "Max Loss" and "Max Gain". ↩

19. The Options Industry Council, "Short Iron Butterfly", "Description" and "Variations". ↩

20. The simulator calculates the premiums with the Black-Scholes formula (no dividends). Each premium and the total are calculated from unrounded values and then rounded to two decimal places, so twice 7.97 (15.94) does not match the total (15.93). ↩

21. Osaka Exchange, "コールとプットを両方買う戦略 ～ロングストラドル・ロングストラングル～" (Strategies that buy both calls and puts: long straddles and long strangles, in Japanese), the paragraph after "図表６　ロングストラドルの損益の出方" (Figure 6: How a long straddle makes profit and loss). ↩

22. The Options Industry Council, "Short Straddle" and "Short Strangle", "Time Decay". ↩

23. The Options Industry Council, "Long Straddle", "Volatility". ↩

24. The Options Industry Council, "Bull Call Spread (Debit Call Spread)" and "Bear Put Spread", "Volatility". ↩

25. Central Council for Financial Services Information, "3-5 オプション取引活用の例（やさしいデリバティブ）" (Examples of using option trading, Easy Derivatives, in Japanese), "他にもさまざまな投資戦略がある" (There are many other investment strategies). ↩

26. The Options Industry Council, "Short Straddle" and "Long Call Butterfly", "Assignment Risk". ↩

27. The Options Industry Council, "Short Strangle", "Volatility". ↩