Mathematical Finance

Learning Outcomes

Learning Outcomes

Having successfully completed this module you will be able to:

• 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
• Demonstrate an understanding of simple structural and reduced form credit risk models such as Merton's Model and apply them in pricing defaultable bonds.
• 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
• Use basic programing and simulation techniques, for instance, in derivative pricing and risk management.
• Define and solve the stochastic differential equations such as geometric Brownian motion and Ornstein-Uhlenbeck processes
• Describe the main types of single- and multi-factor models of asset returns and perform calculations using such models
• 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
• 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
• Perform calculations using stochastic models for security prices, such as the lognormal model or the Wilkie model
• Calculate option prices using both the binomial model and the Black-Scholes model
• 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 investment risk and how it can be measured using different risk measures such as VaR and CVaR
• State and discuss the three forms of the Efficient Markets Hypothesis
• 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
• 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

• 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.

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

Summative

This is how we’ll formally assess what you have learned in this module.

Referral

This is how we’ll assess you if you don’t meet the criteria to pass this module.

Repeat

An internal repeat is where you take all of your modules again, including any you passed. An external repeat is where you only re-take the modules you failed.