## Syllabus: M339W

FINANCIAL MATHEMATICS FOR ACTUARIAL APPLICATIONS

**Text: Robert L. McDonald, Derivative Markets, 2nd Edition (2015) Prentice Hall, ISBN 9780321280305**

**Responsible party**: Milica Cudina March 2015

**Description of the Course**: This course is intended to provide the **mathematical foundations** necessary to prepare for a portin of the SoA Exam MFE and the "financial economics" portion of the CAS Exam 3.

Additionally, the course is aimed at building up the vocabulary and the techniques indispensable in the workplace at current financial and insurance institutions. **This is not an exam-prep seminar**.

The materal exhibited includes: an in depth study of the normal and log-normal distributions, the simple random walk, basics of stochastic calculus, the Samuelson (geometric Brownian motion) stock-price model and the Black-Scholes formula, analysis of option Greeks, market making, non-deteministic interest rate models (both discrete, and continuous-time), bond pricing, Monte-Carlo simulations. The remainder of the Exam MFE/3F curriculum is exhibited in course M339d (also offered by the Department of Mathematics).

**Prerequisites**:

(1) **Formal**: Introduction to Financial Mathematics for Actuaries M339D with a grade of at least C-.

(2) **Actual**: A thorough understanding and operational knowledge of (at least) classical calculus, calculus-based probability (with exphasis on the normal distribution), the term structure of interest rates, and the principles of risk-neutral pricing in the binomial asset-pricing model

Topics Covered

Orientation. Standing assumptions. Conventions.

Binomial interest rate models.

Black-Derman-Toy.

Review of uniform distribution. Random number generation.

Probability on the cointoss space. Simulation of random walk.

Law of Large Numbers. Risk-neutral pricing by similuation (the binomial case).

Scaled random walk. 11.3: Proceeding to continuous time.

Normal and log-normal distributions.

Log-normal stock-price model.

Brownian motion.

Introduction to formal stochastic calculus for financial mathematics.

Stochastic integral ("definition", obstacles). Itô-Doeblin Lemma. Itô processes.

Samuelson's model for stock prices. Portfolio representation.

Sharpe ratio. The risk-neutral probability measure.

Black-Scholes PDE. Risk-neutral pricing.

Black-Scholes pricing formula. Price curve prior to expiration.

Black-Scholes pricing for options of futures, currencies, discrete-dividend-paying stocks.

Correlated assets. Exchange options. Black-Scholes pricing and exotic options.

Forward prices for powers of the underlying.

Implied volatility.

Greeks in the Black-Scholes pricing. A detailed look on the ∆. Option elasticity.

“Greeks” in the binomial tree. Market making and ∆−hedging.

Self-ﬁnancing portfolios. Overnight proﬁt/loss. Γ−hedging.

Market-making and bond-pricing. Duration-hedging.

The Ornstein-Uhlenbeck process. Continuous-time interest rate models.

Black formula.

Monte Carlo valuation.

Variance reduction methods. Control variate method.