# Path Integral Methods in Physics & Finance

## Overview

With the gradual acceptance of “path integral” based “string theory” as a strong candidate for unification, knowledge of the nuances of the “path integral” formalism of QFT is indispensable for academic progression in this area. The applications of this versatile concept are also growing by the day, one of the cardinal ones being in pricing of complex financial assets. Scientific risk management by the investor fraternity has become of cardinal necessity for generating competitive returns and surviving in the marketplace. Derivatives have proven to be immensely useful in the management of financial risk. Their vitality can be gauged from the exponential growth in trading volumes as well as the advent of new structured products literally on a day to day basis. Derivatives in petroleum and natural gas industries in the United States are, now, well entrenched, and they are being extensively used in the electricity industry as well. This has called for advent of innovative methods for the pricing of such instruments. With enhanced computing power being accessible, the “path integral formalism” is gradually becoming the method of choice in this context.

(ii) Basics of financial derivatives;

(iii) Senior school mathematics (algebra, calculus & probability).INDUSTRIES SUPPORT :Path integrals form the basis of QFT computations and as such, proficiency in this area will attract immense demand in research and industrial establishments engaged in activities involving applications of field theoretic methods. This course will also attract immense recognition in the entire financial services industry including banks, stock & commodity exchanges, stock & commodity brokers, portfolio managers, investment bankers, market regulators etc as the newer derivative products with sophisticated features hit the market. The pricing of such products will usually involve a role of path integral methods with the payoff becoming complex functions of paths of underlying prices. Needless to add, Academicians will find it a gateway to further work in related areas.

**The audience would comprise of those desirous of getting acquainted with the intricacies of the path integral formalism and its applications in contemporary physics, (quantum field theory, in particular) and finance (pricing of path dependent and exotic options and other derivatives) and also, appreciating the nuances that have led to the origin and extensive development of this field of knowledge.PREREQUISITES : (i) Basics of classical & quantum mechanics;**

INTENDED AUDIENCE :INTENDED AUDIENCE :

(ii) Basics of financial derivatives;

(iii) Senior school mathematics (algebra, calculus & probability).INDUSTRIES SUPPORT :Path integrals form the basis of QFT computations and as such, proficiency in this area will attract immense demand in research and industrial establishments engaged in activities involving applications of field theoretic methods. This course will also attract immense recognition in the entire financial services industry including banks, stock & commodity exchanges, stock & commodity brokers, portfolio managers, investment bankers, market regulators etc as the newer derivative products with sophisticated features hit the market. The pricing of such products will usually involve a role of path integral methods with the payoff becoming complex functions of paths of underlying prices. Needless to add, Academicians will find it a gateway to further work in related areas.

## Syllabus

### COURSE LAYOUT

**Week 1:**Motivation, rationale& concept of path integrals; Elements of probability distributions, Moments &Cumulants& their generating functions; Binomial & Gaussian distributions, Laplace &Stirling formula.

**Week 2:**Gaussian integrals of matrix functions; The Central Limit Theorem; Elementary theory of stochastic processes. Chapman Kolmogorov equation, Master equation, Kramer Moyal equation, Fokker Planck equation; Random Walks & Brownian motion.

**Week 3:**Path integral solution of diffusion equation; Feynman Kac formula; Autocorrelators; Langevin equation; Path integral solution of Schrodinger equation &Langevin equation.

**Week 4:**Fokker Planck vs Langevin equations; Determinism vs Classical & Quantum randomness; Basic theory of QM path integral; Introduction to the QM machinery & notational conventions; Equivalence of Schrodinger & Heisenberg pictures; Single particle non-relativistic QM path integral.

**Week 5:**QM Harmonic Oscillator path integral; Equivalence between Schrodinger & path integral approaches; Correlation functions; Relevenace of functional derivatives.

**Week 6:**Relativistic path integral; Saddle point approach to path integral; Interpretation of QM path integral; Causality violation; Need for QFT.

**Week 7:**0-Dimensional Field Theory, Correlation Functions, Generating Functionals, Field interactions, Perturbative expansions, Convergence issues, Propagators, Feynman Rules, Schwinger-Dyson Equation.

**Week 8:**SDE from Feynman Diagrams, Loop expansions, Limiting behavior, Saddle point approximation, Effective field, Renormalization, 1-D field theory, Feynman Rules & SDE, propagator, SDE in terms of propagators, Explicit expression for propagator, Generalization to D-dimensions, Continuum limits.

**Week 9:**Mode space computations, divergences, loop integrals, Generatingfunctionals, correlators and allied results in D- Dimensional Euclidean space; Field theory in Minkowski space, Scalar field generating functional.

**Week 10:**Scalar field propagator & correlators, Fourier integrals, Causality, Interacting field in Minkowski space, Important identities, Derivation of SDE,Fermionic fields & their path integral quantization.

**Week 11:**Maxwell equations & gauge field quantization, their path integral. Fluctuation properties offinancial assets, Brownian motion & Ito’s lemma; Lognormal distribution & stock price distribution, Formulation of & solution to the Fokker Planck equation for stock price model.

**Week 12:**Basic theory of options; Profit diagrams; Put-Call parity; Option pricing, the Binomial &Black Scholes Models;Risk Neutral valuation; American options; Derivation of and solution to the Black-Scholes PDE; Path integral approach to pricing of path independent options; Path integral valauation of other cash flow structures e.g. zero coupon bonds etc..