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Black-Scholes Equations
82 MAT4210 Notes by R. Chan i) The asset price follows the geometric Brownian motion discussed in Chapter 6. That is, dS(t) = µS(t)dt+σS(t)dX(t). (1) ii) The risk-free interest rate r and the asset volatility σ are known functions. iii) There are no transaction costs.
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PROOF OF TAYLOR’S THEOREM
www.math.cuhk.edu.hkection on the proof(s) of Taylor’s theorem. First we recall the (derivative form) of the theorem: Theorem 1 (Taylor’s theorem). Suppose f: (a;b) !R is a function on (a;b), where a;b 2R with a < b. Assume that for some positive integer n, f is n-times di erentiable on the open interval
Chapter 3 The Space of Continuous Functions
www.math.cuhk.edu.hkBaire category theorem is proved and, as an application, it is shown that continuous, nowhere di erentiable functions form a set of second category in the space of continuous functions. 3.1 Spaces of Continuous Functions We studied continuous functions on an interval in MATH2050/60 and in a domain bounded
Chapter 3 The Contraction Mapping Principle
www.math.cuhk.edu.hk3.2. THE CONTRACTION MAPPING PRINCIPLE 3 This theorem will be further explained and proved in the appendix. 3.2 The Contraction Mapping Principle Solving an equation f(x) = 0;where f is a function from Rn to itself frequently comes up in application. This problem can be turned into a problem for xed points. Literally,
3.1 Uniform Convergence of Functions - CUHK Mathematics
www.math.cuhk.edu.hkproven. They are reformulated in the context of in nite series of functions in Section 3. The last two important sections demonstrate the power of uniform convergence. In Sections 4 and 5 we introduce the exponential function, sine and cosine functions based on di erential equations. Although various de nitions of
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www.math.cuhk.edu.hk1 Derivatives of Piecewise Defined Functions For piecewise defined functions, we often have to be very careful in com-puting the derivatives. The di↵erentiation rules (product, quotient, chain rules) can only be applied if the function is defined by ONE formula in a neighborhood of the point where we evaluate the derivative. If we want
MATH2060A Solution to Assignment 1
www.math.cuhk.edu.hkexamine the di erentiability of j at 0 using de niton. Indeed, using the fact the cosine function is even, lim h!0 cosjhj cos0 h 0 = lim h!0 cosh 1 h = 0; from which we conclude that jis also di erentiable at x= 0. Hence jis di erentiable everywhere. A shortcut is to realize that the cosine is an even function, so j(x) = cosxis dif-ferentiable ...
Chapter 4 Inverse Function Theorem
www.math.cuhk.edu.hkIndeed, in MATH2060 we learned that if f is continuously di erentiable on (a;b) with non-vanishing f0, it is either strictly increasing or decreasing so that its global inverse exists and is again continuously di erentiable. Example 4.3. Consider the map F: R2!R2 given by F(x;y) = (x2;y). Its Jacobian matrix is singular at (0;0).
Assignment 1 Exercise 1 - Chinese University of Hong Kong
www.math.cuhk.edu.hk1. Carefully derive the equation of a string in a medium in which the resistance is proportional to the velocity. 5. Derive the equation of one-dimensional di usion in a medium that is moving along the xaxis to the right at constant speed V. 6. Consider heat ow in a long circular cylinder where the temperature depends only on tand on the distance
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