Math Refresher - Calculus

Transcription

1 Math Refresher - Calculus David Sovich Washington University in St. Louis

2 TODAY S AGENDA Today we will refresh Calculus for FIN 451 I will focus less on rigor and more on application We will cover both differentiation and integration

3 UNIVARIATE DIFFERENTIATION The derivative of a function f : R R at c R, when it exists, is given by f f (x) f (c) (c) = lim x c x c Other notations for the derivative include: f (c) = df dx (c) = f x(c) The n th derivative of f is often denoted by: f (n) = dn f dx n

4 GEOMETRY OF THE DERIVATIVE

5 PROPERTIES OF THE DERIVATIVE Let f and g be real-valued functions that are both differentiable at c R and let a R. Then: 1. (a f ) (c) = a f (c) 2. (f ± g) (c) = f (c) ± g (c) 3. (f g) (c) = f (c)g (c) + f (c)g(c) 4. (f /g) (c) = g(c)f (c) f (c)g (c) g(c) 2 Some handy rules are: 1. d dx x a = ax a 1 for a 0 2. d dx e x = e x 3. d dx log(x) = 1 x where log denotes the natural logarithm. 4. d dx a = 0

6 PRACTICE PROOF: PRODUCT RULE Claim: (f g) (x) = f (x)g (x) + f (x)g(x) Proof: The definition of the limit is (fg) (x) = lim h 0 f (x + h)g(x + h) f (x)g(x) h Adding and subtracting 0 = f (x + h)g(x) f (x + h)g(x) yields f (x + h)g(x + h) f (x + h)g(x) + f (x + h)g(x) f (x)g(x) lim h 0 h And by properties of limits, this equals ( )( ) ( ) g(x + h) g(x) f (x + h) f (x) lim lim f (x + h) +g(x) lim h 0 h h 0 h 0 h Which is clearly g (x)f (x) + f (x)g(x).

7 COMPUTATIONAL PRACTICE PROBLEMS Practice: Differentiate the quantity 8x x2 Solution: 56x 6 + x Practice: Differentiate the quantity xe x Solution: e x + xe x Practice: Differentiate the quantity 5x 2 log(x) Solution: 10xlog(x) + 5x

8 THE CHAIN RULE Theorem (Chain Rule): Let f : R R and g : R R and consider the composite function g f defined by (g f )(x) = g(f (x)). If f is differentiable at c R and g is differentiable at f (c), then g f is differentiable at c and we have (g f ) (c) = g (f (c)) f (c) Practice: Differentiate the quantity e 5x Solution: d dx e 5x = 5e 5x Practice: Differentiate the quantity (x 2 5x) 20 Solution: d dx (x 2 5x) 20 = 20(x 2 5x) 19 (2x 5)

9 PRACTICE PROOF: CHAIN RULE Proof: The definition of the limit is (g f ) (x) = lim h 0 g(f (x + h)) g(f (x)) h Multiplying by 1 = (f (x + h) f (x))/(f (x + h) f (x)) yields ( ) ( ) g(f (x + h)) g(f (x)) f (x + h) f (x) lim lim h 0 f (x + h) f (x) h 0 h Define f (x + h) f (x) = h. Also, since f (x) R we define f (x) = y. Then we have ( g(y + h ) ( ) ) g(y) f (x + h) f (x) lim h 0 h lim h 0 h Which is clearly g (y)f (x) = g (f (x))f (x).

10 FUNCTIONS OF SEVERAL VARIABLES We have so far dealt only with univariate real-valued functions f : R R We will often deal with multivariate real-valued functions f : R N R Example: The Black-Scholes formula for a European call option has six arguments c(s,k,t t,r,q,σ) = Se q(t t) Φ(d1) Ke r(t t) Φ(d2)

11 PARTIAL DERIVATIVES Partial derivatives help us understand how multivariate real-valued functions change when their inputs are perturbed The partial derivative of a multivariate function f (x 1,...,x N ) with respect to x i at a point x 0 R N is defined by f f (x 0,1,...,x 0,i + h,...,x 0,N ) f (x 0 ) (x o ) lim x i h 0 h Suppose we have the bivariate function f (x,y). We will sometimes use the notation f y f y

12 GEOMETRY OF f (x,y)

13 TANGENT PLANE INTERPRETATION Consider a function f (x,y). The two partial derivatives f x and f y, combined with an initial point (x 0,y 0 ) trace out a tangent plane z = f (x 0,y 0 ) + f x (x 0,y 0 )(x x 0 ) + f y (x 0,y 0 )(y y 0 )

14 PARTIAL DERIVATIVES To compute partial derivatives, we use the same rules for ordinary derivatives, treating all other variables as constants Practice: Differentiate f (x,y) = 2xy + x 2 + 3y 2 with respect to x and y Solution: f f x = 2y + 2x and y = 2x + 6y Practice: Diffferentiate g(x,y) = x 2 y + e xy3 with respect to y y g(x,y) = [ x 2 y ] + [e xy3] y y = x e [ xy3 xy 3 ] y = x 2 3xy 2 e xy3

15 CHAIN RULE FOR MULTIVARIATE FUNCTIONS Consider the bivariate function f (x, y) = f (g(t), h(t)) where g, h are differentiable functions Then the multivariate Chain Rule says f t = f x x t + f y y t In general, we have that f = t j N i=1 f x i x i t j

16 PRACTICE PROBLEM: BOND DURATION Consider a risk-free bond with annual coupons X, time to maturity T, and yield-to-maturity of y. The bond price is P(y,X,T) = T Xe yt t=1 Practice: What is the duration of this bond? In other words, what is D(y,X,T) = P y (y,x,t)/p(y,x,t) Solution: Distributing the partial differentiation operator yields: D(y,X,T) = X ( T i=1 ye yt) X ( T i=1 e yt)

17 CONVEXITY As Jason will explain, options are all about the interaction of convexity and volatility A function f : R R is said to be convex* if for any x 1 x 2 and any α (0,1) we have f (αx 1 + (1 α)x 2 ) αf (x 1 ) + (1 α)f (x 2 ) Some well known functions are convex x a for any a > 1. For example, x 2 is convex e x is convex

18 A QUICK AND EASY CHECK FOR CONVEXITY Proposition: A continuously differentiable function f is convex if and only if f > 0 Proof: We need to prove that if f is convex, then f > 0, and if f > 0, then f is convex For the direction, we use the definition of the limit (on board) For the direction, we use Taylor s Theorem (on board) Practice: Show that e 2x and x 4 are convex

19 WHAT DOES CONVEXITY MEAN? In contrast to a linear function, a convex function exhibits curvature For example, consider our previous bond price function P(y) The duration of the bond provides the first derivative approximation of P(y) But P(y) is a convex function, and hence its second derivative plays a role (P > 0) Bond convexity C is valuable. If C 1 > C 2 and P(y;C 1 ) = P(y;C 2 ), then for any R: P(y + ;C 1 ) > P(y + ;C 2 )

20 WHY IS BOND CONVEXITY VALUABLE? Duration overestimates (underestimates) the price losses (gains) associated with increases (decreases) in yield Convexity corrects for this curvature. More curvature = decreases (increases) in yields associated with larger gains (smaller losses) in price

21 RIEMANN INTEGRATION The integral b a f (x)dx describes the area under the curve The integral sign is simply a version of the sum Σ Let Γ = {a = x 1,x 2,...x N = b} be a partition of [a,b] and t i [x i 1,x i ] i. The Riemann sum R Γ (x) is R Γ (x) = N i=1 f (t i )(x i x i 1 )

22 PROPERTIES OF THE RIEMANN INTEGRAL Sufficient conditions for existence: If f is continuous or monotonic on [a,b], then b a f (x)dx exists Let f and g be Riemann integrable on [a,b] and let c R, then 1. b a cf (x)dx = c b a f (x)dx 2. b a (f (x) + g(x))dx = b a f (x)dx + b a g(x)dx 3. b a f (x)dx = b 1 a f (x)dx + b b 1 f (x)dx for b 1 (a,b)

23 FUNDAMENTAL THEOREM OF CALCULUS The Fundamental Theoreom of Calculus links differentiation and integration Fundamental Theorem(s) of Calculus: Let f be Riemann integrable on [a,b] and define the function F by x F(x) = f (t)dt a ;x [a,b] then F (x) exists x (a,b) with F (x) = f (x) and b a f (x)dx = F(x) b a = F(b) F(a)

24 WHAT IS THE FTC SAYING? The FTC says that differentiation and integration are inverse operations The derivative of the area under the curve f (x) (F (x)) is the curve f (x) Intuition: The area under the curve for f (x) between x + h and h is given by F(x + h) F(x)

25 WHAT IS THE FTC SAYING? As h becomes small, the linear approximation f (x) h becomes a better approximation of this stip x+h x+h F(x + h) F(x) = f (t)dt f (x) dt = f (x) h x x Thus, f (x) (F(x + h) F(x))/h As h approachs 0, we have that (F(x + h) F(x))/h F (x)

26 PRACTICE PROBLEMS Practice: Compute b a xn dx Solution: 1 n+1 xn+1 b a Practice: Compute 5 ( 2 3x 2 + 2x + 1 ) dx Solution:x x x 5 2 = 141 Practice: Compute 4 ( 1 e x + 1 ) x dx Solution: e 4 e 1 + log(4) log(1) = e 4 e 1 + log(4)

27 USEFUL ITEMS FOR PROBLEM SET 0 We will cover probability and statistics next time - but for Problem Set 1 you should know: The expected value of a continuous random variable with density f is E[X] = R xf (x)dx where f 0 The expected value of a function of a continuous random variable is E[g(X)] = R g(x)f (x)dx Densities must integrate to one: P[X ] = F(X ) = R f (x)dx = 1

28 WHAT WE LEARNED So far we have covered the basics of logic and calculus When we come back we will cover probability and statistics