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Subject Syllabus

CB-802 (B) - Computational Finance & Modeling

CSBS · Semester 8

Course Modules

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Numerical Methods and Models: Numerical methods relevant to integrationdifferentiation and solving the partial differential equations of mathematical finance: examples of exact solutions including Black Scholes and its relativesfinite difference methods including algorithms and question of stability and convergencetreatment of near and far boundary conditionsthe connection with binomial modelsinterest rate modelsearly exerciseand the corresponding free boundary problemsand a brief introduction to numerical methods for solving multi-factor models.
Black-Scholes framework: Black-Scholes PDE: simple European calls and putsput-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 putsoptimal exercise strategy and the smooth pasting condition.Volatility considerations - actual, historical, and implied volatilitylocal vol and volatility surfaces.Simulation including random variable generation, variance reduction methods and statistical analysis of simulation output.Pseudo random numbersLinear congruential generatorMersenne 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 samplingMonte Carlo integrationSimulation of Random walk and approximations to diffusion processes, martingale control variables, stratification, and the estimation of the "Greeks. "
Financial Products and Markets: Introduction to the financial markets and the products which are traded in them: Equities, indices, foreign exchange, and commodities.Options contracts and strategies for speculation and hedging.
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.
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.CopulasHedging in incomplete marketsAmerican OptionsExotic optionsElectronic tradingJump Diffusion ProcessesHigh-dimensional covariance matricesExtreme value theoryStatistical Arbitrage.