Winter24

Study plan for Winter Vacations of ‘24.

• Based on https://www.cqf.com/about-cqf/program-structure/cqf-qualification

• Assignment Tracker: https://docs.google.com/spreadsheets/d/12Oytf3DVDPgPQ-wsGGm25sbX30Hj0t0OoBTId9zaknY/edit?usp=sharing

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Topics

Module 1 - Building Blocks of Quantitative Finance

The Random Behavior of Assets

• Different types of financial analysis
• Examining time-series data to model returns
• Random nature of prices
• The need for probabilistic models
• The Wiener process, a mathematical model of randomness
• The lognormal random walk- The most important model for equities, currencies, commodities and indices

Binomial Model

• A simple model for an asset price random walk
• Delta hedging
• No arbitrage
• The basics of the binomial method for valuing options
• Risk neutrality

PDEs and Transition Density Functions

• Taylor series
• A trinomial random walk
• Transition density functions
• Our first stochastic differential equation
• Similarity reduction to solve partial differential equations
• Fokker-Planck and Kolmogorov equations

Applied Stochastic Calculus 1

• Moment Generating Function
• Construction of Brownian Motion/Wiener Process
• Functions of a stochastic variable and Itô’s Lemma
• Applied Itô calculus
• Stochastic Integration
• The Itô Integral
• Examples of popular Stochastic Differential Equations

Applied Stochastic Calculus 2

• Extensions of Itô’s Lemma
• Important Cases - Equities and Interest rates
• Producing standardised Normal random variables
• The steady state distribution

Martingales

• Binomial Model extended
• The Probabilistic System: sample space, filtration, measures
• Conditional and unconditional expectation
• Change of measure and Radon-Nikodym derivative
• Martingales and Itô calculus
• A detour to explore some further Ito calculus
• Exponential martingales, Girsanov and change of measure

Module 2 - Quantitative Risk & Return

Portfolio Management

• Measuring risk and return
• Benefits of diversification
• Modern Portfolio Theory and the Capital Asset Pricing Model
• The efficient frontier
• Optimizing your portfolio
• How to analyze portfolio performance
• Alphas and Betas

Fundamentals of Optimization and Application to Portfolio Selection

• Fundamentals of portfolio optimization
• Formulation of optimization problems
• Solving unconstrained problems using calculus
• Kuhn-Tucker conditions
• Derivation of CAPM

Value at Risk and Expected Shortfall

• Measuring Risk
• VaR and Stressed VaR
• Expected Shortfall and Liquidity Horizons
• Correlation Everywhere
• Frontiers: Extreme Value Theory

Asset Returns: Key, Empirical Stylised Facts

• Volatility clustering: the concept and the evidence
• Properties of daily asset returns
• Properties of high-frequency returns

Volatility Models: The ARCH Framework

• Why ARCH models are popular?
• The original GARCH model 
• What makes a model an ARCH model?
• Asymmetric ARCH models 
• Econometric methods

Risk Regulation and Basel III/IV

• Definition of capital
• Evolution of Basel
• Basel III/IV and market risk
• Key provisions

Collateral and Margins

• Expected Exposure (EE) profiles for various types of instruments
• Types of Collateral
• Calculation Initial and Variation Margins
• Minimum transfer amount (MTA)
• ISDA / CSA documentation

Module 3 - Equities & Currencies

Black-Scholes Model

• The assumptions that go into the Black-Scholes equation
• Foundations of options theory: delta hedging and no arbitrage
• The Black-Scholes partial differential equation

• Modifying the equation for commodity and currency options

• The Black-Scholes formulae for calls, puts and simple digitals
• The meaning and importance of the Greeks, delta, gamma, theta, vega and rho
• American options and early exercise
• Relationship between option values and expectations

Martingale Theory - Applications to Option Pricing

• The Greeks in detail
• Delta, gamma, theta, vega and rho
• Higher-order Greeks
• How traders use the Greeks

Martingales and PDEs: Which, When and Why

• Computing the price of a derivative as an expectation
• Girsanov’s theorem and change of measures
• The fundamental asset pricing formula
• The Black-Scholes Formula
• The Feynman-K_ac formula
• Extensions to Black-Scholes: dividends and time-dependent parameters
• Black’s formula for options on futures

Intro to Numerical Methods

• The justification for pricing by Monte Carlo simulation
• Grids and discretization of derivatives
• The explicit finite-difference method

Exotic Options

• Characterisation of exotic options
• Time dependence (Bermudian options)
• Path dependence and embedded decisions
• Asian options

Understanding Volatility

• The many types of volatility
• The market prices of options tells us about volatility
• The term structure of volatility
• Volatility skews and smiles
• Volatility arbitrage: Should you hedge using implied or actual volatility?

Further Numerical Methods

• Implicit finite-difference methods including Crank-Nicolson schemes
• Douglas schemes
• Richardson extrapolation
• American-style exercise
• Explicit finite-difference method for two-factor models
• ADI and Hopscotch methods

Derivatives Market Practice

• Option traders now and then
• Put-Call Parity in early 1900
• Options Arbitrage Between London and New York (Nelson 1904)
• Delta Hedging
• Arbitrage in early 1900
• Fat-Tails in Price Data
• Some of the Big Ideas in Finance
• Dynamic Delta Hedging
• Bates Jump-Diffusion

Advanced Greeks

• The names and contract details for basic types of exotic options
• How to classify exotic options according to important features
• How to compare and contrast different contracts
• Pricing exotics using Monte Carlo simulation
• Pricing exotics via partial differential equations and then finite difference methods

Advanced Volatility Modeling in Complete Markets

• The relationship between implied volatility and actual volatility in a deterministic world
• The difference between ‘random’ and ‘uncertain’
• How to price contracts when volatility, interest rate and dividend are uncertain
• Non-linear pricing equations
• Optimal static hedging with traded options
• How non-linear equations make a mockery of calibration