## 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

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.

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.