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TN 2 – Basic Calculus with Finance [2016-09-03] Page 1 of 16 © G.S. Questa, 2016 TN 2 – Basic Calculus with Finance [2016-09-03] Page 1 of 16 TN 2 - Basic Calculus with Financial Applications 1 Functions and Limits 2 Derivatives 3 Taylor Series 4 Maxima and Minima 5 The Logarithmic and Exponential Functions 6 Linearity and Convexity. Gains from Convexity 7 Integrals 8 Log Yields (Continuously Compounded Yields) Glossary Excel Functions: EXP, LN Notation y fx() d y' f '() x fx () dx f() x f '() x dx Foreword Calculus can be considered as the mathematics of motion and change. It is a BIG topic with applications spanning the natural sciences and also some social sciences such as economics and finance. In this TN we can only review a few basic concepts that are most likely to be useful for some finance-oriented modules of Master courses. The discussion will be conducted with exclusive reference to real-valued univariate calculus (calculus of one variable) to benefit from its analytical simplicity and ease of visualization. §1 Functions and Limits The first use of the word function is credited to Leibniz (1646-1716). Until the mid-1800s the concept of function was that of a relatively straightforward mathematical formula expressing the relationship between the values of a dependent variable ()y and those of one or more independent variables (univariate and multivariate calculus). In the 19th century, the concepts of function and limit were generalized and made a lot more rigorous, thereby providing a solid foundation for the further development of calculus. A real-valued mathematical expression, such as the quadratic function in exhibit 1.1, has no defined numerical value until you assign a value to()x . Thus, we say that ()y is a function of ()x . Functions are also called transformations because they transform the value of()x into a value of()y . A very important element in the definition of a function is the requirement that for every given point on its domain (x-axis) there is one and only one function value. In other words, if you draw a straight line parallel to the y-axis it must cross only once the function’s graph. © G.S. Questa, 2016 TN 2 – Basic Calculus with Finance [2016-09-03] Page 2 of 16 Thus, if there is more than one value of ()y corresponding to one value of ()x we are dealing with two or more functions instead of one (see exhibit 1.2). 8 y yx -2 0.252 6 4 2 0 x -2 -8 -6 -4 -2 0 2 4 6 8 Exhibit 1.1 - Graph of the quadratic function 1 0.5 0 -0.5 -1 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 Exhibit 1.2 - Graph of unit circle (radius = 1). This is composed by two functions: one for the positive values of (y) and one for the negative. Limits The concept of limit is now all-pervasive in calculus and its applications. The rigorous definition of the limit of a function was worked out in the mid-1800s. There are several types of limits. However, we shall concentrate only on two, chosen for their relevance for our studies. FIRST, the limit of a function that tends to zero for()x 0 , where the symbol () stand for “approaches”. This limit is the cornerstone for the definition and calculation of the derivative of a function and will be discussed in §2. limfx ( ) 0 x 0 SECOND, the limit of a function that tends to zero for()x . limfx ( ) 0 x The above limit is true if, given an arbitrarily small number (ε), there is a number (δ) such that: © G.S. Questa, 2016 TN 2 – Basic Calculus with Finance [2016-09-03] Page 3 of 16 x fx() 0 In other words, there is always a value (δ) large enough to obtain the desired result. This is also known as the (, ) approach. Zero Coupon Bond Example Consider the price ()Z of a zero coupon bond due in ()T years, given a constant compounded yield rate(Y 6 %). The time-to-maturity is also known as tenor in finance- speak: 1.1 Limit of the Price of a Zero Coupon Bond ZK()1 YTT $100 1 . 06 lim Z 0 T $100 Z =$100 × 1.06−T $80 $60 $40 $20 T $0 0 10 20 30 40 50 60 70 80 90 100 Exhibit 1.3 – Limit of the zero coupon bond price as a function of tenor. §2 – Derivatives The twin problems of calculating the tangent to a curve and the area delimited by a curve were solved in the late 1600s. It came as a surprise that the tangent and the area problems are interconnected. The tangent is calculated with the derivative and the area with the integral. The central idea of differential calculus is the notion of derivative. The derivative of a real- valued function y fx() in correspondence of a given value of ()x is a number that measures the slope of the function at that point. Thus, a straight line through this point, with a slope equal to the derivative, will be a tangent to the functiony fx(). More generally, the derivative of a function (often indicated as the primitive) is another function that gives the slope of fx() for each value of ()x on the domain of the function. Derivatives can be denoted in many different ways. 2.1 Notation for Derivatives dy d fx() f '() x y ' dx dx Exhibit 2.1 visualizes the derivative and tangent of the price of a zero coupon bond as a function of yield (for K = $100, T = 30, and Y = 4%). The derivative of bond prices as a function of yield is widely used in fixed-income and is the foundation of duration and convexity © G.S. Questa, 2016 TN 2 – Basic Calculus with Finance [2016-09-03] Page 4 of 16 analysis. We have chosen a long-maturity zero coupon bond (30 years) to make convexity clearly visible. $100 Z =$100 × 1.06−30 $80 $60 $40 $20 $0 0% 2% 4% 6% 8% 10% 12% Exhibit 2.1 – Derivative and tangent to the price of a zero coupon bond as a function of spot yield (at Y = 4%) Derivative and Limit The formula for calculating a derivative relies on the concept of limit and is both rigorous and intuitively obvious. 2.2 Derivative and Limit d fx( h ) fx () y' fx ( ) lim dx h0 h There are two properties to keep in mind because they play a role in understanding a number of economic and financial applications: Adding a constant (a) to a function will just shift the function upwards or downwards, while leaving its slope (derivative) unaltered Multiplying the independent variable by a constant (b) multiplies its derivative by (b) We can show how this works by using the quadratic function: 2.3 Derivative of a Quadratic Function d b() x h22 bx bx 2 lim dx h0 h bx2 b2 xh bh 22 bx lim h0 h b2 xh bh lim lim b22 x bh bx hh00h Differentiable Functions We should add that, to be differentiable for a given value of()x , a function must be both continuous and “smooth” (well-behaved) at that point. Two intuitive examples are provided by exhibits 2.2 and 2.3, which show that a differentiable function may well be non- differentiable in one point. It is interesting to note that there are functions which are both everywhere continuous and nowhere differentiable. One of these function is the Brownian motion, which plays a © G.S. Questa, 2016 TN 2 – Basic Calculus with Finance [2016-09-03] Page 5 of 16 central role in the theory of option pricing in finance. However, this is an advanced topic in stochastic calculus that cannot be covered in these induction lectures. 6 yx=1/ 4 2 0 -2 -4 -6 -10 -8 -6 -4 -2 0 2 4 6 8 10 Exhibit 2.2 – This is a rectangular hyperbola. It is everywhere continuous and differentiable, apart from a discontinuity (singularity) for x = 0. 10 yx=|| 8 6 4 2 0 -10 -8 -6 -4 -2 0 2 4 6 8 10 Exhibit 2.3 – This function is everywhere continuous. However it is not differentiable for x = 0 where the slopes on the left and on the right are not the same. In fact the derivative jumps from -1 to +1. Higher Order Derivatives If we take the derivative of a derivative we get what is called the second order derivative, which is usually written as: dy2 fy''( ) dx 2 The second derivative plays a relevant role in a number of applications, such as: Taylor series approximation Measuring convexity Determining maxima and minima of a function §3 – Taylor Series If we examine again exhibit 2.1 we can’t fail noticing that, over a small interval, the tangent is a good approximation to the original function. Therefore, in a small interval around a value © G.S. Questa, 2016 TN 2 – Basic Calculus with Finance [2016-09-03] Page 6 of 16 ()x0 the function can be expressed with the following equation which is a one-term Taylor series. (This underlies the duration metric in bond mathematics). 3.1 First Order Taylor Expansion fx() fx ()0 f '()( x 00 x x ) In bond mathematics we also use a two-term Taylor expansion, which uses the first and the second derivative of the function. 3.2 Second Order Taylor Expansion fx''( ) fx() fx () f '()( x x x )0 ( x x )2 0 002 0 §4 – Maxima and Minima of Functions The problem of finding the maximum or the minimum value of a function is one of the most pervasive in the sciences and in economics.
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