Mathematical Modelling in Finance -Puneet Prakash,

Transcript:Mathematical Modelling in Finance -Puneet Prakash,:
Mathematical Modelling in Finance -Puneet Prakash, IME 04-04-2014 1 MMF Overview MPT and Beta MMF Matched funding Emmanuel Derman (2011) “There are no theories in finance... only imperfect models remain” Metaphors, Models & Theories , Quart. J. of Fin., 01, 109 -122(2011) The only “law” in finance? The law of one price If you want to price a security, look for a “similar” one. 04-04-2014 2 The (Financial) Modellers' Hippocratic Oath - Authored by Derman and Wilmott I will remember that I didn't make the world, and it doesn't satisfy my equations. (*) Though I will use models boldly to estimate value, I will not be overly impressed by mathematics. I will never sacrifice reality for elegance without explaining why I have done so. Nor will I give the people who use my model false comfort about its accuracy. Instead, I will make explicit its assumptions and oversights. I understand that my work may have enormous effects on society and the economy, many of them beyond my comprehension. (*) – I have a feeling this was written in response to Llyod Blankfein’s (CEO of Goldman Sachs) claim that “We do God’s work” 04-04-2014 3 Why the brouhaha? Finance does require a certain degree of mathematical sophistication It is NOT rocket science though 04-04-2014 4 Finance: A Typology 04-04-2014 5 Branches of Mathematics Involved Calculus Constrained Optimization Linear Algebra Probability and Statistics Simulations Stochastic Calculus Partial Differential Equations Martingale Theory Econometrics Regression Analysis Time Series Analysis Computational Mathematics Algorithms Discrete Mathematics 04-04-2014 6 The Basic Mathematical Structure A little bit of theory (VERY Informal) If finance is study of cash flows (returns), then The space of cash flows (returns) is a Hilbert space Note: Hilbert space permeates many branches. For example, probability, statistics, differential equations, quantitative finance and even nuclear engineering! E.g.: A probability space (𝜴,𝓕,𝓟) with finite variance is a Hilbert space 04-04-2014 7 Hilbert space A Hilbert space is a Banach space with an additional property that An inner product is defined, and that the norm of an element is defined in terms of the inner product of the element with itself Implications: All properties of Banach spaces hold for Hilbert spaces Projections of one element on another can be taken Banach space? 04-04-2014 8 Banach space It is a complete normed linear space in which the distance between any two points of a sequence can be made arbitrarily