## Syllabus: M339D

INTRODUCTION TO FINANCIAL MATHEMATICS FOR ACTUARIES

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

**Responsible party**: Milica Cudina March 2015

**Description of the Course**: This couse is intended to provide the mathematical foundations necessary to prepare for a portion of

(1) the joint SoA/CAS exam FM/2, as well as

(2) 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. There is intellectual merit to the course beyong the ability to prepare for a professional exam.

The material exhibited includes: elementary risk management, forward contracts, options, futures, swaps, the simpe random walk, the binomial asset pricing model and its application to option pricing. The remainder of the Exam MFE/3F curriculum is exhibited in course M339W (also offered by the Department of Mathematics).

**Prerequisites**:

(1) Formal: Probability M362K and Theory of Interest M329F with a grade of at least C-.

(2) Actual: A thorough understanding and operational knowledge of (at least) calculus, finite-stage-space probability, and the term structure of interest rates.

Topics Covered

Orientation. Role of financial markets. Bid-ask spread. Commissions.

Standing assumptions. Conventions.

Outright purchase of an asset. Discrete dividends. Simple return.

Continuous-dividend-paying assets. Market Indices.

Short sales.

Static financial portfolios. Initial cost. Payoff.

Profit. Definition of long/short positions. Basic risk management. Forward contracts.

European call options (rationale, definition, implementation).

European call options (payoff/profit).

Hedging using European call options.

Caps, i.e., short intrinsic position hedged with a call.

Covered/naked option writing. Covered calls. European put options (definition).

Hedging using put options. Floors. Covered puts.

Parallels between classical property-insurance policies and put options.

Derivative securities.

Examples of “simplest” derivative securities: All-or-nothing options.

Review of finite probability spaces. Dynamic portfolios. Profit.

Arbitrage portfolio. Arbitrage.

Law of the unique price.

Prepaid forward contracts. Forward and prepaid forward pricing (stocks).

Annualized forward premium. Arbitrage and forwards’ pricing.

Commodity swaps.

Futures.

Put-call parity.

Replicating portfolios. “Synthetic forward contracts”. Chooser options. Straddles.

Gap calls and puts. Gap-option parity.

American options. Options on futures contracts.

Options on currencies.

Exchange options.

Maximum option. Generalized put-call parity.

Option price bounds and monotonicity. Bull spreads.

Option price “slope” bounds. Bear Spreads.

Option price convexity. Butterfly Spreads. Speculating on volatility.

Strangles. Collars. Ratio Spreads. Equity-linked CDs.

Binomial asset-pricing model.

Derivative-pricing by replication. Risk-neutral probability.

The forward tree. Cox-Ross-Rubinstein binomial tree. Jarrow-Rudd binomial tree.

Two-period binomial pricing. Multiple binomial periods.

Early exercise. Bermudan options.

Pricing American options.

Properties of American-option prices.

Asian options and their binomial pricing.

Barrier options and their binomial pricing.

Compound options and their binomial pricing.

Binomial pricing of options on currencies.

Binomial pricing of options on futures contracts.

Interest-rate swaps.