**522—Financial Mathematics I. (3)**{=MATH 514} (Prereq: MATH 241 or 250 with a grade of C or better) Probability spaces.
Random variables. Mean and variance. Geometric Brownian Motion and stock price dynamics.
Interest rates and present value analysis. Pricing via arbitrage arguments. Options
pricing and the Black-Scholes formula.

**Usually Offered:** Fall even years.

**Purpose:** Complex mathematical techniques are now widely used in finance. This course is an
elementary introduction to these techniques and to the fundamental concepts of financial
mathematics. We examine some mathematical models that are used in finance, e.g. to
model risk and return of financial assets and to model the random nature of stock
prices. After developing the required mathematical tools, we derive the celebrated
Black-Scholes formula for the price of a call option, which formed part of the work
for which Merton and Scholes were awarded the Nobel Prize in Economics in 1997.

The first part of the course is an introduction to probability theory. Random variables -- numerical quantities whose values are determined by the random outcomes of an experiment -- are introduced. Motivated by the example of normal random variables, whose probabilities are determined by a `bell-shaped curve', we consider the concepts of probability density, mean, and variance of a random variable. We also discuss the Central Limit Theorem, perhaps the most important theoretical result in probability, which states that the sum of a large number of random variables is approximately a normal random variable. Then we introduce the random process of `geometric Brownian motion' as a limit of simpler processes called random walks. Geometric Brownian motion is the standard model used in finance to describe the random fluctuations of stock prices.

In the second part of the course we use mathematical techniques, especially the probability theory described above, to solve some problems in finance. First we consider interest rates and the problem of determining the present value of a series of future payments, e.g. mortgage payments or payments to and from a retirement account. We also discuss coupon bonds, continuously varying interest rates, and the `yield curve'.

Next we introduce the fundamental financial concept of `arbitrage'. Simple mathematical arguments based on the `No Arbitrage Principle' determine prices in a variety of situations. Examples of this method include the pricing of forward and futures contracts and a mathematical result known as the put-call option parity formula. To obtain more sophisticated results on options pricing we use geometric Brownian motion as our model of stock price `dynamics'. Approximating this process by a geometric random walk leads to the `multiperiod binomial model', from which we derive the famous Black-Scholes formula for the price of a call option. We examine the dependence of this formula on its various parameters and discuss the `delta-hedging' strategy whereby the option is `replicated' by a continuously readjusted portfolio.

**Current Textbook:** *An Elementary Introduction to Mathematical Finance* (second ed.) by Sheldon M. Ross. Cambridge University Press, 2003.

Topics Covered |
Time |

Probability: Probability spaces. Outcomes and Events. Conditional probability. Random variables. Bernoulli and binomial random variables. Expected value. Variance and standard deviation. Independence. Algebras and subalgebras. Conditional expectation with respect to a subalgebra (finite case only). | 3 weeks |

Continuous Random Variables: Probability density functions. Cumulative distribution functions. The normal distribution. Sums of independent normal random variables. Discussion of the Central Limit Theorem. Normal approximation to the binomial distribution. The lognormal distribution. | 2 weeks |

Geometric Brownian Motion: Random walks and geometric random walks. The Gambler's Ruin Problem. Brownian Motion and Geometric Brownian Motion viewed as limits of random walks. The drift and volatility parameters. The standard model of stock price dynamics. | 2 weeks |

Present Value Analysis: Interest Rates. Present value of an income stream. Abel summation and its application to present value analysis. Coupon and zero-coupon bonds. Yield to maturity and duration. Continuously varying interest rates and the yield curve. | 2 weeks |

Arbitrage: The No Arbitrage Principle. The Law of One Price. Pricing via arbitrage arguments. Forward contracts. Futures contracts. Options. Simple bounds for options prices. Payoff diagrams. The Put-Call Option Parity Formula. | 2 weeks |

Options Pricing Theory: Generalized options. The single period model. The replicating portfolio. Risk-neutral valuation. The multiperiod binomial model. Self-financing trading strategies. Discounted processes and martingales. The equivalent martingale measure. The Black-Scholes Formula. Partial derivatives and the `Greek parameters': delta, gamma, vega, etc. The delta-hedging arbitrage strategy. | 2 weeks |

*The above textbook and course outline should correspond to the most recent offering
of the course by the Statistics Department. Please check with the instructor for the
course regulations, expectations, and operating procedures.*

**Contact Faculty:** Stephen Dilworth (Mathematics)