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# Calculus

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I have in the past discussed the importance of calculus in option pricing. I have gone into some detail in the following few pages (with the help of an excerpt from "A Mathematical Universe" by William Dunham) for those of you interested in a better understanding of how it works. For those of you who don't care, stop reading now!

There are two types of calculus: integral calculus is concerned with the area under curves (on a graph), while differential calculus deals with the slopes of curves, which defines rate of change. The latter is what traders are concerned with and what I will discuss.

Graphs describe the relationship between one variable and another. The following graph depicts a straight line in a coordinate plane: it shows how the variables x and y vary to each other. It illustrates that if x increases by a certain number of units, y will increase by a certain number of units. The number of units of y for a certain number of units of x depends on the steepness of the line or its slope.

The slope of a line is defined as the change in y per unit change in x (the rate of change of a unit of y per unit of x); a line tells us that this rate of change is constant. If the world were a totally linear place (if planes never changed speed or acceleration), we could graph (describe) all relationships with lines.

But the world is not such a simple place. Markets go up and down at different rates and planes change their speed and acceleration: the real world exhibits non-linear behavior manifest as curves on a graph. Differential calculus was developed as a set of rules that determines the slope of a curve (rate of change) at any coordinate on the curve. Specifically, calculus determines the slope of the tangent line at that coordinate.

Consider the following graph of a parabola. It may represent the speed of a plane (y) over time (x). The speed is changing, defined as acceleration. As the curve declines the plane is slowing down, at the bottom of the curve its speed is constant, and as the curve inclines it is accelerating. The change in acceleration of the plane at any time (x) is measured by the slope of the tangent line at any point on the curve. As the slope of the tangent at any point increases or decreases, so does the rate of acceleration. At the bottom of the parabola where the slope of the tangent is zero, there is no acceleration and the plane is traveling at a constant velocity.

At point (3, 4) where we have drawn the tangent line, imagine the plane splits in two. Plane A is traveling per the parabola, but Plane B stays on the tangent. Plane A begins accelerating at a faster and faster rate (as the slope of the tangent line increases), while Plane B continues to accelerate at a constant rate. So how do we determine the slope of the tangent? Leibniz in the late 17th century probably spent years in trial and error to give us an ingenious short cut. In order to calculate the slope of the tangent, all we need are two points. But at x =3 we only have one point (3, 4). Consider Leibniz' clever solution in the following graph.

We can calculate another point B on the parabola for x =4 (4, 7) and then draw a line called a secant to point A (3, 4) creating line AB. We then calculate the slope of AB as shown in the graph to be 3. But we can get closer to the slope of the tangent at (3, 4) by successively using secants with x coordinates closer and closer to point A. Secant AC has a slope of 2.50 while secant AD has a slope of 2.10. If we kept doing this we would get a slope approaching 2 as the secants become better and better approximations of the actual tangent.

What Leibniz did was streamline this process of approximations that allows us to quickly find the slope of any tangent on the parabola. This requires a more general and algebraic approach as illustrated in the following graph.

If we are trying to determine the slope of the tangent at P, we first consider the point Q with an x coordinate x +h (h is a small unspecified amount). By using x +h as the x coordinate, the y coordinate for Q then becomes:

a(x + h)^2 + b(x + h) + c =
a(x^2 + 2xh + h^2) + b(x + h) + c =
ax^2 + 2axh + ah^2 + bx + bh + c

So Q is the point (x + h, ax^2 + 2axh + ah^2 + bx + bh + c). Point P is (x, ax^2 + bx + c).
To find the slope of this secant we plug these two points into our slope formula:

Slope PQ = y2-y1 = (ax^2 + 2axh + ah^2 + bx + bh + c) - (ax^2 + bx + c)
x2-x1 (x + h) - x

= ax^2 + 2axh + ah^2 + bx +bh + c - ax^2 - bx -c
x + h -x

= 2axh + ah^2 + bh = h(2ax + ah +b)
h h

= 2ax + ah +b

As we move the secants closer to the tangent, h approaches zero; so the ah variable can just be dropped and we get for the slope of the tangent: 2ax + b.

If we apply this formula to the actual parabola y = x^2 - 4x +7 at x =3, where a = 1 and b = -4 (c does not matter), we get 2(1)(3) + (-4) = 2.

The slope of the tangent to a curve is the limit of the slopes of the secants as h goes to zero. This limit is called the derivative and the process of finding it is called differentiation.

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