Quantitative, Mathematical and Computational Finance Course

Module 1: Introduction

• Fixed-income securities
  – Valuation
  – Interest rate sensitivity
  – Portfolio management
• Portfolio optimisation
  – Basic mean-variance portfolio optimisation

Module 2: Option Pricing and Binomial Techniques

• Options
  – The no-arbitrage principle
• The binomial model
  – Pricing option by use of binomial methods
• An alternative binomial model

Module 3: Stochastic Differential Equations

• Introduction to the stochastic Ito process
• Introduction to stochastic Ito integral
  – Properties and definition of the Ito integral
• Into Lemma
• Application in the stock market

Module 4: The Black-Scholes Expression

• Black -Scholes expression derivatives
• Black Scholes expression solution
  – Changes to the heat equation
  – Closed-form result of put and call choices
• The Greeks: Hedging portfolio
• Expected volatility

Module 5: Monte Carlo Simulation and Random Numbers

• Pseudorandom numbers
• Transformation of random variables
  – Inverse transformation technique
  – Acceptance-rejection technique
• Generation of normal variates
  – Box-muller method
  – The polar method of Marsa glia
  – Multivariate normal variables
• Monte Carlo integration
• Option costing by Monte Carlo simulation
  – Correlated assets
• Variance reduction techniques
  – Antithetic variates
  – Control variates
• Quasi-monte Carlo simulation
  – Halton sequences

Module 6: Option Costing by Partial Differential Expression

• Allocation of PDEs
• Finite variation techniques for parabolic expression
  – An explicit model
  – An implicit method
  – Crank-Nicolson method
• Option costing by the Heat expression
• Option costing by using black Scholes expression
  – Pricing by an exact method
  – Pricing by an implicit model
  – Costing by the Crank-Nicolson technique
• Pricing options
  – Projected SQR method
• Tree methods and finite differences
  – A trinomial tree
  – A binomial tree

Module 7: The Basics of Financial Reporting

• Introduction to financial reporting concepts
• The financial position statements
• The cash flow statement
• The income statements
• Analytical review and ratio analysis
• Company governance

Module 8: Fixed Income Derivatives and Securities

• Bond market technique
  – Overview and notations
  – Notations
  – Swaps and caps
  – Valuation of primary tools: vanilla choices and zero coupons on zero-coupon
  – Black model
  – Short-rate models
  – Term structure consistent models
  – Inverting the yield curve
  – Affine term structure
  – Problems
• Credit default swap (CDS), securitisation, and exchange-traded funds (ETF)
  – Overview
  – Bond ETFs
  – Commodity ETFs
  – Currency ETFs
  – Inverse ETFs
  – Exchange-traded funds (ETFs)
  – Index ETFs
  – Stock ETFs
  – Leveraged ETFs
  – Credit default swap (CDS)
  – Example of credit default swap
  – Valuation
  – Recovery rate calculation
  – Binary credit basic swaps
  – Basket credit basic swaps
  – Collateralised loan obligations (CLO)=
  – Collateralised bond obligations (CBO)
  – Mortgage-backed securities ( MBS)
  – Collateralised mortgage obligations (CMO)
  – Collateralised debt obligation (CDO)
  – Examples

Module 9: Advanced Methods for Underlying Assets

• Stochastic volatility technique
  – Overview
  – Stochastic volatility
  – Types of progressive-time SV technique
  – Hull-white technique
  – Moment evaluation for CIR-type activity
  – Scott model
  – Stein and stein technique
  – Heston method
  – Formulation of formulae applied: mean-reverting activity
  – The stochastic alpha beta Rho (SABR) method
  – Problems
• Jump diffusion models
  – Introduction
  – The Poisson process (jumps)
  – Suitability of the stochastic method postulated
  – Regime switching jump-diffusion technique
  – The compound Poisson activity
  – The black Scholes technique with jumps
  – Solutions to partial-integral differential structure
  – The option pricing challenge
  – The general PIDE structure
  – Examples
• Generalised levy activity. Long-range correlations and memory impacts
  – Introduction
  – Stable issuing
  – Kurtosis
  – The levy flight technique
  – Self-comparison
  – The H- infinity association for levy flight
  – Sum of stochastic levy variables with various factors
  – Rescaled range analysis and Detrended fluctuation analysis (DFA) applied to economic indices
  – Total of levy random variables with multiple factors
  – Examples and applications
  – Truncated levy models applied to financial indices
  – Total of exponential random variables with other parameters
  – Problems
• Factor and copulas model
  – Introduction
  – Cross-sectional regression
  – Factor technique
  – Expected return
  – Basic factor method
  – Copula technique
  – Families of copulas
  – Macroeconomic factor method
  – problems