What is Quantitative Finance? Why quantitative? Increasingly, mathematical and statistical methods are being applied by hedge funds and asset managers to generate superior returns while minimizing their risk exposures. Notable examples include the Renaissance Technologies’ Medallion Fund in the U.S, and Quantedge Capital in Singapore. Strong quantitative skills are the foundation for these hedge funds. They are extremely good at applying Quantitative Finance models to extract critical investment and trading signals from big data. For day-to-day risk management in any bank these days, quantitative skills are also indispensable to quantify market risks, credit risks, liquidity risks, interest rate fluctuations, funding costs, capital adequacy, and the list goes on.
This 101 course introduces you to the essentials of Quantitative Finance models, starting with three basic principles to look at risk and return. It's going to be cool and fun to see how the math you have learnt so far (Pre-U Math) can be applied to solve problems faced by quantitative strategists and risk analysts.
Week 1: Introduction to Quantitative Finance
— Pre-U math: pre-Calculus (algebraic gymnastics) for all lectures
— QF: installation of Python on your notebook
Week 2: Four Major Asset Classes
— Pre-U math: exponential function, logarithmic function, integration (definite integrals), geometric series, arithmetic mean, geometric mean, standard normal random variable
— QF: simulation of geometric Brownian motion, compactification of Treasury's coupon bond by geometric series
Week 3: Principles of Quantitative Finance
— QF: philosophical overview of QF, three principles, model risk, risk neutral measure and pricing
Week 4: Interest Rates
— Pre-U Math: summation of series, limit, Maclaurin's series, differentiation
— U Math: l'Hôpital’s rule, partial differentiation, single-variable Taylor's expansion
— QF: continuous compounding and discounting, duration, convexity, yeild curve modeling
Week 5: Net Present Value
— QF: self-financing pricing strategy, pricing of equity forward, FX forward, forward rate agreement, interest rate swap, and cross-currency interest rate swap, discount factors, OIS, multi-curve approach
Week 6: Options
— U Math: discontinuous function f(x)= (x-a)+
— QF: put-call parity, implied forward price, Modigliani-Miller's Proposition I, monotonicity, gradient boundedness, and convexity of an option's price curve function of the strike price, box spread strategy
Week 7: VIX Indices
— Pre-U Math: integration by parts, differentiation
— U Math: Dirac's delta function, step function, Riemannian integral
— QF: static replication of any twice differntiable function, variance swap, VIX, model-free approach
Week 8: Mid-Term Test
Week 9: Binomial Models
— Pre-U Math: probability, mean, variance, conditional probability, binomial random variaable, central limit theorem
— QF: one-dimensional random walk, recombinant binomial tree, risk-neutral probability, delta hedging, binomial option pricing
Week 10: Ito's Calculus
— Pre-U Math: integration by parts, differential equation
— U Math: bivariate Taylor's expansion, Itô process and Itô formula
— QF: Einstein's scaling law, Bachelier's scaling law, Brownian motion, symbolic rules of Itô calculus, geometric Brownian motion, the Black-Scholes equation, heat equation
Week 11: American Options
— U Math: Jensen's inequality
— QF: early exercise premium, put-call inequalities, binomial tree for American options, futures options, American option exercise strategies
Week 12: Portfolios
— Pre-U Math: scalars, vectors, inner product
— U Math: matrix, matrix arithmetic, transpose, linear independence, matrix multiplication, trace, rank of matrix, inverse matrix, determinant, characteristic equation, eigenvalues, eigenvectors, vector differentiation
— QF: Google's PageRank algorithm, constrained portfolio optimization, Markowitz's efficient frontier
Week 13: Summary and Revision