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# Option Strategies

Candidates seem to have a tremendously difficult time with option strategies: how to construct various spreads, how to compute the payoffs on those spreads, how to compute the profits on those spreads, how to compute the breakeven underlying prices on those spreads, and why they would employ a particular spread.

I’m here to tell you that it isn’t nearly as hard as most people make it out to be.

I kid you not.

You can master this stuff, without pharmaceutical assistance.

The principle problem, in my experience, is all the max/min junk that you see in the curriculum: it’s confusing as all get-out.

Rest assured: you won’t see any max/min functions or formulae in these articles.

I’ll show you how to construct the spreads, how to compute the payoffs and profits, and how to determine the break-even points. If you can draw straight lines (technically, line segments and rays), you’re 90% of the way there. (OK, maybe 80%.) The last 10% (20%) is easy.

Finally, I’ll explain the reason for each of the spreads: why you would want to use each one.

In this series of articles I’ll cover these strategies:

- Bear spread
- Box spread
- Bull spread
- Butterfly spread
- Collar
- Covered call
- Inverse (short) straddle
- Protective put
- Seagull spreads (Level III only)
- Short strangle
- Straddle
- Strangle

The nice thing about these articles is that they have lots of pictures.

**Preliminaries: Payoffs on Calls and Puts**

The key to constructing option spreads is a thorough understanding of the payoffs on:

- A long call
- A long put
- A short call
- A short put

If you do not have these payoffs thoroughly memorized – if you cannot draw them in your sleep, after a long night of partying – you need to practice these with intense determination; they have to be innate . . . second nature. First nature’s even better.

Here they are:

The horizontal axis in each of these payoff diagrams represents the price of the **underlying** *at the expiration of the option*. **X** is the strike price of each option. The vertical axis is, naturally, the payoff on the option. The horizontal axis crosses the vertical axis at zero.

There are some general points about these payoff diagrams that warrant cataloguing:

- Long positions (long call, long put)
have**cannot***negative*payoffs: the payoff will be zero or positive. (We’re assuming that you’re not an idiot.) - Short positions (short call, short put)
have**cannot***positive*payoffs: the payoff will be zero or negative. (We’re also assuming that whoever buys your options is not an idiot.) - Call positions (long call, short call) have
*nonzero*payoffs if and only if the price of the underlying at expiration isthan the strike price.**higher** - Put positions (long put, short put) have
*nonzero*payoffs if and only if the price of the underlying at expiration isthan the strike price.**lower** - Call positions have unlimited positive or negative payoffs: there is no upper limit to the price of the underlying at expiration. (Hence, the arrows in the call payoff diagrams, above.)
- Put positions have limited positive or negative payoffs: the price of the underlying cannot fall below zero. (Hence,
arrows in the put payoff diagrams, above.)**no** - For the nonzero portions of these graphs, they rise or fall dollar-for-dollar (or yen-for-yen, or euro-for-euro, or baht-for-baht, or . . . well . . . you get the idea): the slopes are +1 or −1, always.

Finally, note that these are * payoff *diagrams, not

*profit*diagrams. To compute the profit, we need to incorporate the premium paid or received for the option. We’ll handle that at the end of each separate article.

We will use these payoff diagrams in our construction of the payoffs of the various spreads listed above. Whenever we have a kink in the payoff diagram for a particular spread, we will have one or more options with strikes at that point.

Let’s look at how the various option payoffs affect the shape of an option strategy payoff strategy in general; once we know that, it’s an easy matter to decide what sorts of options we need to achieve a given payoff profile.

The effects are perhaps best categorized by considering the slope of a payoff diagram (or a portion of a payoff diagram). The only slopes we’ll see in our payoff diagrams will be 0 (the graph is horizontal), +1 (the graph is going up as we move to the right), and −1 (the graph is going down as we move to the right). Here they are:

Adding a kink to the payoff diagram means changing the slope, which is accomplished by adding an option position. Fortunately, slopes add exactly the way you would expect them to:

Existing Slope |
Slope Added |
Resulting Slope |
||

−1 | + | 0 | = | −1 |

−1 | + | +1 | = | 0 |

0 | + | −1 | = | −1 |

0 | + | 0 | = | 0 |

0 | + | +1 | = | +1 |

+1 | + | −1 | = | 0 |

+1 | + | 0 | = | +1 |

Graphically, here are some examples of how you can add an option position to create a specific kink in a payoff diagram. In these examples, the **yellow** graph represents the payoff on the existing position (without the new option position being added), the (dashed) **blue** graph represents the payoff on the new option position, and the **green** graph represents the payoff on the resulting position (with the new option position added). In each case, you should convince yourself that the slope of the green graph at any *x* (horizontal axis) value is equal to the slope of the yellow graph at that *x* value plus the slope of the blue graph at that *x* value. Chromatically, yellow plus blue gives green, so it should be easy to remember what’s what.

*Example 1*

Here, the existing payoff diagram (**yellow**) is constant: horizontal (slope = 0). We add a long call option with a strike price of X (**blue**): slope = 0 below (to the left of) X, slope = +1 above (to the right of) X. The resulting payoff diagram (**green**) is horizontal (at the original level) to the left of X; to the right of X the payoff has a slope of +1.

If we look at the slopes: to the left of X we have 0 + 0 = 0 (original plus option equals result); to the right of X we have 0 + 1 = +1 (original plus option equals result).

In a table, the payoffs might look like this, where the strike on the option is X = $30:

Underlying Price |
Original |
Option |
New |

$0.00 | $10.00 | $0.00 | $10.00 |

$10.00 | $10.00 | $0.00 | $10.00 |

$20.00 | $10.00 | $0.00 | $10.00 |

$30.00 | $10.00 | $0.00 | $10.00 |

$40.00 | $10.00 | $10.00 | $20.00 |

$50.00 | $10.00 | $20.00 | $30.00 |

$60.00 | $10.00 | $30.00 | $40.00 |

*Example 2*

Here, the existing payoff diagram (**yellow**) has a slope of +1. We add a long put option with a strike price of X (**blue**): slope = −1 to the left of X, slope = 0 to the right of X. The resulting payoff diagram (**green**) is horizontal to the left of X; to the right of X the payoff has a slope of +1.

If we look at the slopes: to the left of X we have (+1) + (−1) = 0 (original plus option equals result); to the right of X we have (+1) + 0 = +1 (original plus option equals result).

In a table, the payoffs might look like this, where the strike on the option is X = $30:

Underlying Price |
Original |
Option |
New |

$0.00 | ($20.00) | $30.00 | $10.00 |

$10.00 | ($10.00) | $20.00 | $10.00 |

$20.00 | $0.00 | $10.00 | $10.00 |

$30.00 | $10.00 | $0.00 | $10.00 |

$40.00 | $20.00 | $0.00 | $20.00 |

$50.00 | $30.00 | $0.00 | $30.00 |

$60.00 | $40.00 | $0.00 | $40.00 |

*Example 3*

Here, the existing payoff diagram (**yellow**) has a slope of −1. We add a short put option with a strike price of X (**blue**): slope = +1 to the left of X, slope = 0 to the right of X. The resulting payoff diagram (**green**) is horizontal to the left of X; to the right of X the payoff has a slope of −1.

If we look at the slopes: to the left of X we have (−1) + (+1) = 0 (original plus option equals result); to the right of X we have (−1) + 0 = −1 (original plus option equals result).

In a table, the payoffs might look like this, where the strike on the option is X = $30:

Underlying Price |
Original |
Option |
New |

$0.00 | $40.00 | ($30.00) | $10.00 |

$10.00 | $30.00 | ($20.00) | $10.00 |

$20.00 | $20.00 | ($10.00) | $10.00 |

$30.00 | $10.00 | $0.00 | $10.00 |

$40.00 | $0.00 | $0.00 | $0.00 |

$50.00 | ($10.00) | $0.00 | ($10.00) |

$60.00 | ($20.00) | $0.00 | ($20.00) |

*Example 4*

For a change of pace, let’s look at an existing payoff diagram that already has a kink in it:

Here, the existing payoff diagram (**yellow**) has a slope of +1 to the left of X, and a slope of 0 to the right of X. We add a short call option with a strike price of X (**blue**): slope = 0 to the left of X, slope = −1 to the right of X. The resulting payoff diagram (**green**) has a slope of +1 to the left of X; to the right of X the payoff has a slope of −1.

If we look at the slopes: to the left of X we have (+1) + (0) = +1 (original plus option equals result); to the right of X we have 0 + (−1) = −1 (original plus option equals result).

In a table, the payoffs might look like this, where the strike on the option is X = $30:

Underlying Price |
Original |
Option |
New |

$0.00 | ($20.00) | $0.00 | ($20.00) |

$10.00 | ($10.00) | $0.00 | ($10.00) |

$20.00 | $0.00 | $0.00 | $0.00 |

$30.00 | $10.00 | $0.00 | $10.00 |

$40.00 | $10.00 | ($10.00) | $0.00 |

$50.00 | $10.00 | ($20.00) | ($10.00) |

$60.00 | $10.00 | ($30.00) | ($20.00) |

At this point you should have the hang of adding kinks to a payoff diagram by adding new option positions. We’ll use this knowledge to build the payoff diagrams for the option strategies listed at the beginning of this article.

Our approach to building the payoff diagrams will be methodical: we’ll start at the left and work our way right, adding options for each kink in the payoff diagram. Then, for an exciting change of pace, we’ll build the same diagram by starting at the right and working our way to the left, adding options for each kink along the way. In this way, you’ll see that there is more than one way to build an option strategy payoff diagram, and you’ll gain more insight into how option strategies work in general. It’ll be fun.

Let me warn you ahead of time that the one bit of work you’re going to have to put into this exercise – apart from having to learn the shapes of the payoff diagrams for individual options, covered above – is to * learn the shapes of the payoff diagrams for each of the option strategies*. This is one area where you’ll benefit from drawing the pictures many, many times. Draw the picture and write the name of the strategy. Do this for each strategy. Then throw away the pictures and do it again. Once you’ve done this a few dozen times, you’ll know these diagrams cold.

The Greeks

The 2020/2021 Level III curriculum changes added discussions of the Greeks to the option readings. Given that I’ve read a lot of comments from candidates that indicate that they really do not understand the option Greeks, I thought that I would add a section on them to each option strategy article.

In brief, the option Greeks are:

- Delta (
*Δ*): The rate of change of the price of the option compared to the change in the price of the underlying: the first (partial) derivative of the option price with respect to the price of the underlying. - Gamma (
*Γ*): The rate of change of the the option’s*delta*compared to the change in the price of the underlying: the first (partial) derivative of the option*delta*with respect to the price of the underlying, or the second (partial) derivative of the optionwith respect to the price of the underlying.*price* - Theta (
*θ*): The rate of change of the price of the option compared to the change in time: the first (partial) derivative of the option price with respect to time. (Note that this is time measured according to the calendar, not time remaining until expiration of the option. As we get nearer to the expiration of the option, time increases.) Why finance people chose theta, the Greek cognate of the English “*th*” instead of tau (*τ*), the Greek cognate of the English “*t*” is a mystery. - Vega (
*ν*): The rate of change of the price of the option compared to the change in the volatility of returns of the underlying: the first (partial) derivative of the option price with respect to volatility. Note a couple of points:- Because we do not (indeed, cannot) know the actual volatility of the underlying’s returns (because it’s the volatility of returns from the current time until expiration; i.e. it’s future volatility, which we cannot observe), the volatility measure used to calculate vega is the implied volatility.
- The symbol used for vega –
*ν*– is, in fact, a lower case Greek letter*nu*; it’s the cognate of the English letter “*n*“, not “*v*“. There is no Greek letter with the sound of a “*v*“.

- Rho (
*ρ*): The rate of change of the price of the option compared to the change in the risk-free interest rate: the first (partial) derivative of the option price with respect to the risk-free rate.

Why delta and gamma are abbreviated with upper case Greek letters while theta, vega, and rho are abbreviated with lower case Greek letters is yet another mystery.

To compute any particular Greek for an option strategy, you simply add the corresponding Greeks for the constituent options. So, for example, if you want to know the delta for a bear put spread, it’s the delta for the put with the higher strike price (the long put) minus the delta for the put with the lower strike price (the short put).

**A Word of Caution about Computing the Profit on Option Strategies**

In the articles on the various option strategies, I explain how to compute the profit on each strategy, which leads to drawing the profit diagram, and forms the basis for determining the breakeven points.

In computing the profit, I add a wrinkle that, as far as I can tell, is not discussed explicitly anywhere else: the (net) cost of the option strategy that you subtract from the payoff to get the profit **must be** the * future value* of the cost (i.e., the future value at the expiration of the options, when the payoff occurs), not the

*present value*of the cost (i.e., when the strategy is undertaken). The reason that I emphasize that it must be the future value of the (net) cost is that for some of the strategies you get a very different cost if you create the strategy with calls than if you create it with puts (and a correspondingly different payoff for the two constructions). If you (incorrectly) calculate the profit using the present value of the (net) cost, then it appears that the profit using calls is different from the profit using puts, suggesting that there is an arbitrage opportunity. However, if you (correctly) calculate the profit using the future value of the (net) cost, the profit is the same in both cases, as it should be.

I make a big point of this because in every textbook on options, in every article you’ll likely ever read on option strategies, and, in particular, **in the CFA curriculum**, the profit is always shown as being calculated using the *present value* of the (net) cost. It’s wrong, but there it is.

**Note well: if you calculate the profit the correct way (using the future value of the (net) cost), you will lose points on the CFA exam; if you calculate the profit the incorrect way (using the present value of the (net) cost), you won’t lose points on the CFA exam. I want you to get the points, so on the exam I want you to do it incorrectly, but I also want you to know that you’re doing it incorrectly. When you get back to work the day (or two) after the exam, do it correctly.**

**As a help, in each article I’ve added a note every time I compute the future value of the cost, letting you know that that’s the step that you need to omit when you’re taking the Level II or Level III CFA exam.**