*MATH6127 *Mathematical Finance

## Module Overview

Following an initial discussion of the assessment and measurement of investment risk, mean-variance portfolio theory is introduced and used to determine the risk and return for a portfolio of risky assets, the composition of the optimal such portfolio, and the location of the efficient frontier. Single- and multifactor models of asset returns are then introduced and, in conjunction with concepts from mean variance portfolio theory, lead to the establishment of equilibrium asset pricing models, such as the Capital Asset Pricing Model (CAPM) and Arbitrage Pricing Theory (APT). The various forms of the Efficient Markets Hypothesis are discussed against this background. Attention then turns to stochastic models for security prices, such as geometric Brownian motion, and to the essential mathematical tool required for analysis and solution of the underlying stochastic differential equations, namely the Ito calculus, and, in particular, the Ito integral and the Ito formula. With such a stochastic model for the underlying random variable, it is possible to develop a model for the valuation of a derivative security whose price is contingent on this underlying random variable, and this is a central aspect of the module. The approach to derivative security pricing, and, in particular, option pricing, is built up in stages: first, the discrete-time binomial lattice approach is used; next, the continuous-time Black-Scholes approach is used; and finally, following the introduction of concepts such as martingales and risk-neutral measures, the martingale approach, or, equivalently, the state-price deflator approach is used. Calculations of option prices are extended to the partial derivatives of such prices, the so-called Greeks, and the role of such partial derivatives in the risk management of a portfolio of derivative securities is described. Finally, the risk-neutral and state-price deflator approaches are applied to the pricing of zero-coupon bonds and interest rate derivatives for general single-factor diffusion models of the risk-free rate of interest, such as those of Vasicek, Coss, Ingersoll, and Ross, and Hull and White.

### Aims and Objectives

#### Module Aims

The principal aim of this module is to provide students with an appreciation and understanding of how the application of mathematics, particularly stochastic analysis, to the field of finance may be used to illuminate this field and model its randomness. This results in greater understanding and quantification of investment returns and security prices. In particular, the module describes how to mix risky assets in order to achieve optimal trade-off between investment return and risk, and how to price or hedge a derivative security, that is, one whose value depends on that of an underlying risky asset or random variable.

#### Learning Outcomes

##### Learning Outcomes

Having successfully completed this module you will be able to:

- Demonstrate an understanding of investment risk and how it can be measured using different risk measures such as VaR and CVaR
- Define the Greeks for an option contract and demonstrate an understanding of how these may be used in the risk management of a portfolio of derivatives
- Demonstrate an understanding of the term structure of interest rates and the use of risk-neutral and state-price deflator approaches to the valuation of zero-coupon bonds and interest rate derivatives for single-factor models of the risk-free rate of interest, such as those of Vasicek, Cox, Ingersoll, and Ross, and Hull and White
- Perform calculations using stochastic models for security prices, such as the lognormal model or the Wilkie model
- Describe the main types of single- and multi-factor models of asset returns and perform calculations using such models
- Demonstrate an understanding of simple structural and reduced form credit risk models such as Merton's Model and apply them in pricing defaultable bonds.
- Calculate, using mean-variance portfolio theory, the expected return and risk of a portfolio of many risky assets, the composition of the optimal such portfolio, and the location of the efficient frontier
- Demonstrate an understanding of the relationship between investment risk and return, of the benefits of diversification, and of the distinction between diversifiable and non-diversifiable risk
- Calculate option prices using both the binomial model and the Black-Scholes model
- State and discuss the three forms of the Efficient Markets Hypothesis
- Define and explain the basic properties of Brownian motion, demonstrate an understanding of stochastic differential equations and the Ito integral, and apply the Ito formula
- Define and solve the stochastic differential equations such as geometric Brownian motion and Ornstein-Uhlenbeck processes
- Define the terms arbitrage and complete market, and describe the risk-neutral pricing approach to option valuation, in both binomial lattice and Black-Scholes contexts
- Define the terms martingale, equivalent martingale measure, and state-price deflator, derive the Garman-Kohlhagen form of the Black-Scholes equation, and perform pricing and hedging of simple derivatives contracts using the martingale approach

### Syllabus

• Utility functions and their rule in investment preferences: VNM theorem, first and second order stochastic dominance, indifference curves. • Assessment and measurement of investment risk: distribution of return, variance of return, downside semi-variance of return, Semi-deviation, shortfall probabilities, Value at Risk (VaR), Conditional Value at Risk (CVaR also known as TVaR and AVaR). Role of investor’s utility function. Comparison of investment opportunities using such measures. • Mean-variance portfolio theory: assumptions, opportunity set, efficient frontier, risk-reword diagrams, expected return and risk of a portfolio composed of many risky assets, optimal portfolio. Applying different optimality criteria to obtain the composition of such optimal portfolio with and without short sales. Benefits of diversification. • Single- and multi-factor models of asset returns: macroeconomic models, fundamental factor models, statistical factor models, single index model. Calculations involving such models. Diversifiable and non-diversifiable risk. • Equilibrium asset pricing models: CAPM, APT. Assumptions, principal results and limitations of such models. Calculations involving such models. • The three forms of the Efficient Markets Hypothesis. Evidence for and against each form. Consequences for investment management. • Stochastic models for security prices: lognormal and alternative models. Calculations involving stochastic pricing models. Parameter estimation. • Brownian motion: definition and basic properties. Stochastic differential equations, the Ito integral, and the Ito formula. Diffusion and mean-reverting processes. Solution of the stochastic differential equations such as the geometric Brownian motion and Ornstein-Uhlenbeck processes. • Option valuation. Arbitrage. Factors that affect option prices. Bounds for option prices. Put-call parity. Option valuation using binomial trees and lattices. The risk-neutral pricing measure for a binomial lattice and application to the valuation of equity options. Option valuation using the Black-Scholes model: risk-neutral pricing and equivalent martingale measure, complete markets, derivation of the Black-Scholes equation, including in Garman-Kohlhagen form, assumptions, pricing and hedging of simple derivatives contracts using the martingale approach, option valuation calculations using the Black-Scholes model. Equivalence of state-price deflator approach and risk-neutral pricing approach. The Greeks for an option: definitions, derivation and characteristics for European options, use in risk management of a portfolio of derivatives, including delta-hedging. • Models for the term structure of interest rates. Risk-neutral and state-price deflator approaches to the pricing of zero-coupon bonds and interest rate derivatives for a general single-factor diffusion model for the risk-free rate of interest. Vasicek, Cox-Ingersoll-Ross, and Hull-White single-factor models for the term structure of interest rates. • Credit risk modelling: Structural models such as Merton, Reduced form models such as Jarrow–Turnbull model. Calculating the price of zero coupon bonds with recovery rate.

### Learning and Teaching

#### Teaching and learning methods

Lectures, Work Book, web support materials, assigned problems, private study

Type | Hours |
---|---|

Teaching | 60 |

Independent Study | 90 |

Total study time | 150 |

#### Resources & Reading list

Williams D (1991). Probability with Martingales.

Hull J. An Introduction to Options, Futures and Other Derivative Securities.

Merton R (1992). Continuous-Time Finance.

Wilmott P (1993). Option Pricing: Mathematical Models and Computation.

Elton EJ, Gruber MJ, Brown SJ & Goetzmann WN (2003). Modern Portfolio Theory and Investment Analysis.

Hull J (2005). Options, Futures and Other Derivative Securities.

Wilmott P, Howison SD & Dewynne JN (1995). The Mathematics of Financial Derivatives.

Wilmott P (1998). Derivatives.

### Assessment

#### Assessment Strategy

Purpose of the Assessment: To demonstrate the extent to which students have assimilated material in the syllabus and can apply it to relevant problems and situations. The method of assessment is designed to comply with the requirements of the Institute and Faculty of Actuaries (IFoA) so that students who score sufficiently well in the examination (at a level determined by the IFoA) may gain professional exemption from CT8 of the professional examinations of the IFoA

#### Summative

Method | Percentage contribution |
---|---|

Class Test | 10% |

Examination | 90% |

#### Referral

Method | Percentage contribution |
---|---|

Examination (3 hours) | 100% |

#### Repeat Information

**Repeat type: Internal & External**

### Linked modules

Pre-requisite: MATH6131

### Costs

#### Costs associated with this module

Students are responsible for meeting the cost of essential textbooks, and of producing such essays, assignments, laboratory reports and dissertations as are required to fulfil the academic requirements for each programme of study.

In addition to this, students registered for this module typically also have to pay for:

##### Books and Stationery equipment

Course texts are provided by the library and there are no additional compulsory costs associated with the module

Please also ensure you read the section on additional costs in the University’s Fees, Charges and Expenses Regulations in the University Calendar available at www.calendar.soton.ac.uk.