Syllabus - Computational Finance & Modeling (CB802-(B))

Computer Science and Business Systems (CSBS)

Computational Finance & Modeling (CB802-(B))

VIII

UNIT – I

Numerical Methods and Models

Numerical methods relevant to integration, differentiation and solving the partial differential equations of mathematical finance: examples of exact solutions including Black Scholes and its relatives, finite difference methods including algorithms and question of stability and convergence, treatment of near and far boundary conditions, the connection with binomial models, interest rate models, early exercise, and the corresponding free boundary problems, and a brief introduction to numerical methods for solving multi-factor models.

UNIT – II

Black-Scholes framework

Black-Scholes PDE: simple European calls and puts; put-call parity. The PDE for pricing commodity and currency options. Discontinuous payoffs - Binary and Digital options. The Greeks: theta, delta, gamma, vega&rho and their role in hedging. The mathematics of early exercise - American options: perpetual calls and puts; optimal exercise strategy and the smooth pasting condition. Volatility considerations - actual, historical, and implied volatility; local vol and volatility surfaces. Simulation including random variable generation, variance reduction methods and statistical analysis of simulation output. Pseudo random numbers, Linear congruential generator, Mersenne twister RNG. The use of Monte Carlo simulation in solving applied problems on derivative pricing discussed in the current finance literature. The technical topics addressed include importance sampling, Monte Carlo integration, Simulation of Random walk and approximations to diffusion processes, martingale control variables, stratification, and the estimation of the “Greeks.”

UNIT – III

Application areas

Pricing of American options, pricing interest rate dependent claims, and credit risk. The use of importance sampling for Monte Carlo simulation of VaR for portfolios of options.

UNIT – IV

UNIT –V

Statistical Analysis of Financial Returns

Fat-tailed and skewed distributions, outliers, stylized facts of volatility, implied volatility surface, and volatility estimation using high frequency data. Copulas, Hedging in incomplete markets, American Options, Exotic options, Electronic trading, Jump Diffusion Processes, High-dimensional covariance matrices, Extreme value theory, Statistical Arbitrage.

Course Objective

Understand existing financial models in a quantitative and mathematical way. Apply these quantitative tools to solve complex problems in the areas of portfolio management, risk management and financial engineering. Explain the approaches required to calculate the price of options. Identify the methods required to analyse information from financial data and trading systems

Course Outcome

The student will be able to: Understand existing financial models in a quantitative and mathematical way. Apply these quantitative tools to solve complex problems in the areas of portfolio management, risk management and financial engineering. Explain the approaches required to calculate the price of options. Identify the methods required to analyse information from financial data and trading systems

Practicals

Reference Books

  • R. Seydel: Tools for Computational Finance, 2nd edition, Springer-Verlag, New York, 2004.

  • P. Glasserman: Monte Carlo Methods in Financial Engineering, Springer-Verlag, New York, 2004.

  • W. Press, S. Teukolsky, W. Vetterling and B. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 1997. Cambridge University Press, Cambridge, UK. Available on-line at: http://www.nr.com/.

  • Lewis: Option Valuation under Stochastic Volatility, Finance Press, Newport Beach, California, 2000.

  • Pelsser: Efficient Methods for Valuing Interest Rate Derivatives, Springer-Verlag, New York, 2000.

  • M. Capinski and T. Zastawniak, Mathematics of Finance: An Introduction to Financial Engineering, Springer, 2010

  • S. M. Ross, An Elementary Introduction to Mathematical Finance, Cambridge University Press, 2011.

  • D. Ruppert, Statistics and Data Analysis for Financial Engineering

  • R. Carmona: Statistical Analysis of Financial Data in S-Plus

  • N. H. Chan, Time Series: Applications to Finance

  • R. S. Tsay, Analysis of Financial Time Series

  • J. Franke, W. K. Härdle and C. M. Hafner, Statistics of Financial Markets: An Introduction