Lecture Notes 1 Part B: Functions and Graphs of Functions In Part A of Lecture Notes #1 we saw man examples of functions as well as their associated graphs. These functions were the equations that gave us either the value or the profit of a call or a put at expiration. We did not consider how to create the graphs, but the skill to do so will serve us well when we consider more complex securities, such as combinations of the underling asset with various derivatives. The best wa to learn how to do graph the more complex strategies is to learn a sstematic method to both generate the equation and its associated graph. We start first with simple strategies, such as bu a call, or write a put, and then we can easil appl the same method to complex strategies such as a covered call. In fact, for the complex strategies we will learn two methods, so one can serve as a check for the other! Functions allow us to express contracts, ideas, and relations in a more concise manner than through the use of words alone. Although mathematical functions can take on man complicated forms and involve numerous input variables, we can restrict our present stud to univariate linear functions. Univariate means there is onl a single input variable, which will be represented b the letter x. The output variable is represented b the letter : x functionff The box represents the function, which assigns to each input number exactl one output number. Although more than one x can lead to a particular, the value must be unique for each x. A general notation for a univariate function is = f(x). Linear functions take the form = f(x) = ax + b, where a and b are constants. The are called linear because when graphed using coordinate axes, the form a straight line. This feature makes such functions eas to plot. It is often desirable to plot functions so we can visualize the relation. We will use this tool for two ver important concepts in our stud of derivatives. We look at plots of profits from derivatives to visualize when we are winning or losing with a particular position. The input variable will be the underling asset s price at expiration or deliver. We can also use a function to obtain the value of the derivative toda. The use of functions for derivative values will help us understand how investors value derivatives. As in profit diagrams, the value function can provide specific numeric results and the graph of the function can help us visualize a whole range of output values as we var one specific input variable. The rest of this document is a general discussion of linear functions and their graphs with no particular focus on derivatives. Financial Derivatives Steven Freund 1
An example of coordinate axes is shown on this page. Roman numbers indicate the four quadrants in the conventional manner. The gridlines numbers and axes labels are suppressed in the following graphs to make them more readable, but ou should generall label axes in our graphs, particularl when the represent a particular variable. 4 II I 2-4 -2 0 2 4 x III -2 IV -4 We will work on some general examples of plotting linear functions using all four quadrants, but when we assign stock price to the x axis, we onl need to consider quadrant I and IV, because stock price is never negative. Financial Derivatives Steven Freund 2
Linear functions are eas to graph if we identif the x and variables, and simplif to the form = ax + b. We know that it will plot as a straight line, with the value of a as the slope, and b as the -axis intercept. The intercept is the point on the -axis where the function intercepts (passes through) the axis. It is also the value of when x = 0. The value of the slope indicates the number of units the function increases or decreases for each unit increase in the input variable. Some examples: = 2x + 1 = ax + b = 2x + 1 What I have done is to write the general form of the linear function right above m specific equation, and I can easil see that it is a linear function as well as identif both the slope a and the intercept b just b inspection. For this example, the slope is clearl two and the intercept is equal to one. x Sometimes we represent points on the coordinate plane as (x,). For example: (-3, -5). Notice that a positive slope means sloping upward from left to right. In this example, increases two units for ever unit of x. If the intercept is equal to zero, the graph of the function goes through the origin (0,0). Financial Derivatives Steven Freund 3
If the slope is equal to zero, the linear function is known as a constant function. For example: = 5 = ax + b = (0)x + 5 = 5 The intercept is equal to five and the slope is equal to zero. x = 5 No matter what value the x variable takes on, the function value is five. A constant function is alwas a horizontal line. Also, it is not possible for a function to plot as a vertical line since this implies that more than one unique output value is assigned to single input value. Is the following function linear? = -2 + 2x 3x + 3 Combine terms and ou see that it is: = -x +1, with intercept equal to one and slope equal to negative one. x Financial Derivatives Steven Freund 4
Notice that this function slopes downward. Also, a slope of plus one or negative one is at an angle of 45 degrees. Functions with a non-zero slope are sometimes called ramp functions. Also notice that for this function, the x intercept has the same value as the intercept. With a positive slope of positive one, the would have also have an equal value but with an opposite sign. Check it out! Sums of Linear Functions In this section, we will consider the sum of two functions and the graphs of these functions. Although this document is about functions in general, I want to give an example of a situation using derivatives where we ma want to add two functions. Suppose a linear function tells us the profit from one derivative based on a particular expiration stock price. If we combine this derivative with another derivative, also based on the same expiration stock price, we could consider the sum of the two as representing profit from a portfolio where we hold both derivatives. We would be interested in the portfolio profit as a function of the expiration stock price. To differentiate between the two component functions which make up the sum function, I will call the first function f(x), and the second function g(x). We can call the sum function h(x). Therefore, we are saing h(x) = f(x) + g(x). In this example, all of these functions represent profit. Now back to functions in general (with no connection to derivatives): Suppose our two equations are constant functions f(x) = 1 and g(x) = 2: f(x) = 1 + g(x) = 2 h(x) = 3 What ou are doing essentiall is adding up the right hand side of each equation! Financial Derivatives Steven Freund 5
You can also see that in this particular case ou are summing the vertical distance of each graph from the x-axis. Tr adding two constant functions when one is below the x-axis! A more complicated sum would be a ramp function and a constant function. Suppose that f(x) = -2 and g(x) = x: f(x) = -2 + g(x) = x h(x) = x 2 The sum function has a slope of one and a intercept of negative two: g(x) = x f(x) = -2 h(x) = x 2, the sum function. Check if each individual function has the correct plot, and see if the distance from the x- axis rule (negative sign if below the x-axis, rather than absolute value, as distance is often measured) also works? For example, consider the points where x = 2: sum of f(x) and g(x) where x = 2: 2 + (-2) = 0 Financial Derivatives Steven Freund 6
In Assignment #1, ou will be asked to create our own graphs of functions and sums of functions. You will also be asked to show onl the positive segment of the x axis. This is because later the x axis will represent stock prices, which can onl take on positive quantities. Suppose the previous problem is asked as homework problem: Plot each pair of equations separatel and together as a sum. Plot each pair and their sum on the same set of axes, and restrict the domain to x 0. You should answer as follows: f(x) = 2 and g(x) = x Equation 1: f(x) = 2 slope = 0 and intercept = 2 Equation 2: g(x) = x slope = 1 and intercept = 0 Sum equation: h(x) = x 2 slope = 1 and intercept = 2 Notice that after writing down each equation, the slope and intercept is identified before plotting the equation below! g(x) = x h(x) = x 2 0 x axis f(x) = 2 Financial Derivatives Steven Freund 7
Sometimes, for option values, we need to consider an area under or above a linear function. This is described b an inequalit such as > 2x + 1. First ou graph the line as equalit. If it is a strict inequalit (>), ou do not include points on the line. If it is, ou do. For this example, the function is valid for points above the line. If the function would be < 2x + 1, it would be valid for points below the line. An application for this would come up when we would discuss boundaries for rational option values. > 2x + 1 Financial Derivatives Steven Freund 8