Advanced Mathematics: Financial Tools and Applications

Course Outline

The scope of this course is to revise some of the theories and principles used for calculating the price of financial derivatives. The Black-Scholes model is used as the starting point but gradually the level of complexity is increased to discuss the jump-diffusion models, stochastic volatility models as well as discussions of pricing various exotic pay-offs that will be the new vanilla products tomorrow.

Who The Course is For

Quantitative analysts, risk managers, product controllers, financial engineers, researchers. Past participants have included: Chief investment officers, Asset Managers, Strategists, Private Banks, Relationship Managers.

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Prior Knowledge

Differential calculus, yield curves, duration, convexity, covariance matrix, Riemann integral, ODE (covered in Maths Refresher)

Random variable, expectation and moments, conditional expectation, Central Limit Theorem, Random walk, Markov chain, Wiener process, Geometric Brownian motion, Ito formula, Girsanov transform (covered in Intermediate Mathematics: Understanding Stochastic Calculus)

This program is eligible for 16 Continuing Education credit hours from the CFA Institute. If you are a CFA Institute member, CE credit for your participation in this program will be automatically recorded in your CE Diary.

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Day One

Introduction and Course Outline
Complete-incomplete markets. Non-arbitrage Pricing. Arrow Debreu prices
Black-Scholes Formulas:
- time dependent coefficients;
- jumps;
- dividend yield;
- Margrabe formula;
- convergence of binomial tree to Black-Scholes.

Workshop: Pricing freight derivatives

Options pricing when asset prices are predictable:
- trended O-U process;
- impact of drift estimation on volatility estimation;
- autocorrelation.

Workshop: Pricing real estate derivatives, forward contracts and total return swaps.

Risk-neutral density recovering models:
- Studying the market stability;
- Options pricing with Weibull, generalized gamma, GB2 and g-and-h distribution;

Workshop: Testing for market stability in stock and interest rate markets. Was there any impact of London bombings in July 2005 on the interest markets?

Day Two

Deterministic methods for multi-asset option pricing and other financial calculus:
- Theory based on weak convergence of probability measures and local limit theorems;
- One-dimensional formulas;
- Multi-dimensional formulas.

Application of deterministic algorithms to pricing:
- Asian;
- basket;
- rainbow options.

Workshop: Numerical examples of deterministic algorithms and comparison with Monte Carlo Option Pricing.

Probability distortion operators for option pricing:
- A unified approach to pricing in financial markets and insurance markets;
- Determination of market price of risk;
- No-arbitrage violations.

Workshop: Pricing options with normal and student distortion operators.

Mathematical bounds for options prices; distribution free bounds. A new paradigm in risk management and algorithmic trading.