logo Tue, 24 Dec 2024 01:15:20 GMT

Mathematical Finance


Synopsis


In recent years the finance industry has mushroomed to become an important part of modern economies, and many science and engineering graduates have joined the industry as quantitative analysts, with mathematical and computational skills that are needed to solve complex problems of asset valuation and risk management. An important parallel story exists of scientific endeavour. Between 1965-1995, insightful ideas in economics about asset valuation were turned into a mathematical 'theory of arbitrage', an enterprise whose first achievement was the famous 1973 Black-Scholes formula, followed by extensive investigations using all the resources of modern analysis and probability. The growth of the finance industry proceeded hand-in-hand with these developments. Now new challenges arise to deal with the fallout from the 2008 financial crisis and to take advantage of new technology, which has revolutionized the practice of trading. This Very Short Introduction introduces readers with no previous background in this area to arbitrage theory and why it works the way it does. Illuminating pricing theory, Mark Davis explains its applications to interest rates, credit trading, fund management and risk management. He concludes with a survey of the most pressing issues in mathematical finance today. ABOUT THE SERIES: The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area. These pocket-sized books are the perfect way to get ahead in a new subject quickly. Our expert authors combine facts, analysis, perspective, new ideas, and enthusiasm to make interesting and challenging topics highly readable.

M. H. A. Davis

Summary

Chapter 1: Introduction to Mathematical Finance

* Summary: Provides an overview of the field of mathematical finance, its applications, and the fundamental concepts it relies on from probability, calculus, and statistics.
* Real Example: A portfolio manager uses mathematical models to optimize the allocation of funds among different asset classes, such as stocks, bonds, and real estate.

Chapter 2: Probability Theory

* Summary: Covers the basics of probability, including concepts like sample spaces, events, random variables, and probability distributions.
* Real Example: A financial analyst uses probability distributions to assess the risk of an investment portfolio, such as the Gaussian distribution to model the returns of a particular stock.

Chapter 3: Random Processes

* Summary: Introduces random processes, including discrete-time and continuous-time processes, as well as important properties like stationarity and ergodicity.
* Real Example: A quantitative trader uses a continuous-time stochastic process, such as the Black-Scholes model, to price options contracts.

Chapter 4: Brownian Motion and Stochastic Calculus

* Summary: Explores the properties of Brownian motion and introduces concepts from stochastic calculus, such as Itô integrals and differential equations.
* Real Example: A financial engineer uses Brownian motion to model the evolution of stock prices and derives hedging strategies for derivatives.

Chapter 5: Theory of Finance

* Summary: Covers the foundational concepts of modern finance, including time value of money, risk aversion, efficient markets, and capital asset pricing models.
* Real Example: A pension fund manager applies the CAPM to estimate the expected return of a portfolio, considering its correlation with the market.

Chapter 6: Fixed-Income Markets

* Summary: Examines the characteristics of fixed-income securities, such as bonds and mortgages, and introduces concepts like yield curves and credit risk.
* Real Example: A bond trader uses a fixed-income model to evaluate the duration and convexity of a corporate bond portfolio.

Chapter 7: Equity Markets

* Summary: Discusses different equity investment strategies, such as fundamental analysis, technical analysis, and risk management.
* Real Example: A hedge fund manager employs a quantitative model based on historical data to identify undervalued stocks.

Chapter 8: Derivatives Markets

* Summary: Introduces the main types of derivatives, including forwards, futures, options, and swaps, and covers their valuation and hedging principles.
* Real Example: A risk manager uses a forward contract to hedge against the risk of future currency exchange rate changes.

Chapter 9: Numerical Methods in Finance

* Summary: Explores numerical methods used in mathematical finance, such as Monte Carlo simulations, finite difference methods, and optimization algorithms.
* Real Example: A portfolio optimizer uses a numerical solver to find the optimal weights of different assets in a portfolio.