Practice Problems 7

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Published on March 6, 2014

Author: anhtuantran509


Mathematical Methods for Quantitative Finance Practice Problems 7 1. Consider using Lagrange’s method to solve the following optimization problem. minimize: 3x1 − 4x2 + x3 − 2x4 subject to: −x2 + x2 + x2 = 1 2 3 4 3x2 + x2 + 2x2 = 6 1 3 4 Write down the Lagrangian and its gradient and explain why it is difficult to find a critical point. 2. Compute the gradient of the Lagrangian for the maximum expected return portfolio subject to risk = (20%)2 . 3. Consider the Black-Scholes formula for the price of a European call option as a function of a single variable S (i.e., treat all the other inputs as constant). Write down a second order Taylor polynomial around S0 for C(S) in terms of ∆ and Γ. 4. Compute the Taylor series expansion T (x) of the function f (x) = 1 1+x around the point a = 0 and find its radius of convergence. Does T (x) = f (x) on the domain of convergence? 5. Let P0 , P1 , . . . be a sequence of market prices for the same asset. The arithmetic return on the asset during period t is defined to be Rt = Pt − Pt−1 Pt−1 and the log return is defined to be rt = log Pt Pt−1 . Show that rt ≈ Rt for small values of Rt (say on the order of 1%). 6. Use the integral form of the Taylor approximation error to find the order of the Taylor polynomial needed to compute e0.25 to six digits of accuracy.

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