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- 1. A Brief History of Mathematical Finance Mathematics Used in Mathematical Finance Application of Financial Mathematics Research Areas in Mathematical Finance Current Issues Conclusion Mathematical Finance (POW) Kenneth K. Mwangi NORTHERN ARIZONA UNIVERSITY DEPARTMENT OF MATHEMATICS AND STATISTICS Some materials have been extracted from a presentation by Peter Carr,PhD ’A Practitioners Guide to Mathematical Finance’ with permission. Kenneth K. Mwangi Mathematical Finance (POW)
- 2. A Brief History of Mathematical Finance Mathematics Used in Mathematical Finance Application of Financial Mathematics Research Areas in Mathematical Finance Current Issues Conclusion Table of contents 1 A Brief History of Mathematical Finance Before the 1900 2 Mathematics Used in Mathematical Finance Math in MF continued 3 Application of Financial Mathematics Hedging and Risk Management Algorithmic Trading Asset Liability Modelling Portfolio Optimization 4 Research Areas in Mathematical Finance 5 Current Issues Open Problems in Mathematical Finance 6 Conclusion Kenneth K. Mwangi Mathematical Finance (POW)
- 3. A Brief History of Mathematical Finance Mathematics Used in Mathematical Finance Application of Financial Mathematics Before the 1900 Research Areas in Mathematical Finance Current Issues Conclusion Brief History of Mathematical Finance 1900: Bacheliers dissertation Theorie de la Speculation taking Brownian motion as a market model. Kolmogorov/Levy/Doeblin/Ito 1973: Geometric Brownian motion - Fischer Black and Myron Scholes partial diﬀerential equation approach using a risk-free portfolio of bond,option and stock. 1981: Harrison-Pliska: Martingale Theory. 1990s: Mathematical Finance/Computational Finance established as a discipline. Many educational programs start in the US and in the UK. Kenneth K. Mwangi Mathematical Finance (POW)
- 4. A Brief History of Mathematical Finance Mathematics Used in Mathematical Finance Application of Financial Mathematics Before the 1900 Research Areas in Mathematical Finance Current Issues Conclusion Brief History of Mathematical Finance: Before the 1900 Thale’s call option: 624 B.C. 17th century - The Netherlands: put options on tulip bulbs. 1637: tulip bulb crash: put option sellers cannot meet their buying obligations. This leads to a bad reputation of options in continental Europe. 18th century - London: problem - no legal frame for default of counterparties. Kenneth K. Mwangi Mathematical Finance (POW)
- 5. A Brief History of Mathematical Finance Mathematics Used in Mathematical Finance Application of Financial Mathematics Math in MF continued Research Areas in Mathematical Finance Current Issues Conclusion Math in MF Stochastic Calculus: Markov Processes, Itos Lemma, Girsanovs Thm Linear Nonlinear PDEs - primarily 2nd order linear, esp. parabolic Monte Carlo Simulation Complex Analysis - for inverting transforms Finite Diﬀerences, Finite Elements, and Spectral Methods Functional Analysis - semi-groups Integral Transforms Fourier/Gaussian/Hilbert/Laplace/Radon Game Theory Inverse Problems - Calibration Statistics, Econometrics, esp. time series. Kenneth K. Mwangi Mathematical Finance (POW)
- 6. A Brief History of Mathematical Finance Mathematics Used in Mathematical Finance Application of Financial Mathematics Math in MF continued Research Areas in Mathematical Finance Current Issues Conclusion Math in MF continued Pseudo Diﬀerential Operators Maximum Principle Fundamental Theorem of Linear Algebra Hahn Banach Theorem Lie Groups Regular and Singular Perturbations Optimal Control Variational Inequalities Diﬀerential Geometry String Theory Kenneth K. Mwangi Mathematical Finance (POW)
- 7. A Brief History of Mathematical Finance Mathematics Used in Mathematical Finance Application of Financial Mathematics Math in MF continued Research Areas in Mathematical Finance Current Issues Conclusion Why does Financial Mathematics Exist? Because ﬁnancial institutions are selling extremely complex ﬁnancial derivatives to clients to hedge their risk exposure and to speculate on the direction of the markets. These ﬁnancial institutions have to make sure they price these derivatives correctly and manage them eﬀectively. This has created a booming area of research in applied probability and other ﬁelds to try to answer very complicated mathematical questions. Kenneth K. Mwangi Mathematical Finance (POW)
- 8. A Brief History of Mathematical Finance Mathematics Used in Mathematical Finance Hedging and Risk Management Application of Financial Mathematics Algorithmic Trading Research Areas in Mathematical Finance Asset Liability Modelling Current Issues Portfolio Optimization Conclusion Risk Measurement and Management 1938 Bond duration 1952 Markowitz mean-variance framework 1963 Sharpe’s single-factor beta model 1966 Multiple-factor models 1973 Black-Scholes option-pricing model, ”understanding the greeks” 1983 Risk-adjusted return 1986 Limits on exposure by duration bucket 1988 Limits on ”greeks”, Basel I 1992 Stress testing 1993 Value-at-Risk (VAR) (Banks are exposed to market risk) 1994 RiskMetrics 1997 CreditMetrics 1998 Integration of credit and market risk 2000 Enterprisewide risk management 2000-2008 Basel II Kenneth K. Mwangi Mathematical Finance (POW)
- 9. A Brief History of Mathematical Finance Mathematics Used in Mathematical Finance Hedging and Risk Management Application of Financial Mathematics Algorithmic Trading Research Areas in Mathematical Finance Asset Liability Modelling Current Issues Portfolio Optimization Conclusion Algorithmic Trading If market prices are martingales, then one can neither gain nor lose on average from non-anticipating trading strategies. However sometimes you can partially anticipate market movements, eg. NAVs of mutual funds with cross country positions. Finance academics used to believe that markets are too eﬃcient in order to systematically proﬁt from predictable market movements. The rise in CPU power lead many academics to abandon that view and start hedge funds to exploit anomalies. Simultaneous advances in automated trading and in the theory of market micro-structure have lead to the formation of companies such as Automated Trading Desk (ATD) which place orders hundreds of times per second. Kenneth K. Mwangi Mathematical Finance (POW)
- 10. A Brief History of Mathematical Finance Mathematics Used in Mathematical Finance Hedging and Risk Management Application of Financial Mathematics Algorithmic Trading Research Areas in Mathematical Finance Asset Liability Modelling Current Issues Portfolio Optimization Conclusion Asset Liability Modelling Pension plans have long term ﬁxed liabilities whose magnitude is based on actuarial estimates of retirement age, future salaries, mortality rates, etc. Similarly, insurance companies promise ﬁxed annuity payments whose length extends from the beneﬁciarys retirement until death. In both cases, it is common practice to invest some fraction of the companys assets in equities to partake of their higher average growth over the long term. This leads to Asset and Liability Management (ALM), a ﬁeld which has long used quantitative methods, but is just beginning to succumb to market-oriented quantitative techniques. Kenneth K. Mwangi Mathematical Finance (POW)
- 11. A Brief History of Mathematical Finance Mathematics Used in Mathematical Finance Hedging and Risk Management Application of Financial Mathematics Algorithmic Trading Research Areas in Mathematical Finance Asset Liability Modelling Current Issues Portfolio Optimization Conclusion Portfolio Optimization Main Concern: How do we minimize the trading cost associated with portfolio management; How do we design a trading system; The tools used for Optimization of the main concern include; Time series analysis of very high-frequency data Advanced statistics Dynamic optimization Kenneth K. Mwangi Mathematical Finance (POW)
- 12. A Brief History of Mathematical Finance Mathematics Used in Mathematical Finance Application of Financial Mathematics Research Areas in Mathematical Finance Current Issues Conclusion Research Areas in Mathematical Finance 1 Pricing of Derivatives 2 Hedging Strategies for Derivatives 3 Risk Management of Portfolios 4 Portfolio Optimization 5 Model Choice and Calibration 6 Default Risk and Credit Derivatives Kenneth K. Mwangi Mathematical Finance (POW)
- 13. A Brief History of Mathematical Finance Mathematics Used in Mathematical Finance Application of Financial Mathematics Open Problems in Mathematical Finance Research Areas in Mathematical Finance Current Issues Conclusion Current Issues Pricing in Incomplete Markets. Correlation risk modeling in large portfolios. Finding the right model, lots about Levy processes,fractional Brownian motion. Forward PDE for American Options in Local Volatility model. Closed Form Formula for American Put in Black Scholes (Simple). Kenneth K. Mwangi Mathematical Finance (POW)
- 14. A Brief History of Mathematical Finance Mathematics Used in Mathematical Finance Application of Financial Mathematics Research Areas in Mathematical Finance Current Issues Conclusion Conclusion As Casey Stengel once said: In theory, theory and practice are the same. But in practice, they are very diﬀerent. MF is too important to be left solely to academics; there is also a thriving subculture of quants, happily doing research,trading, and risk management in banking, insurance, software, money management, and even the public sector. Continuing demand from industry for quantitatively-oriented students bodes well for Masters and PhD level programs in MF. Kenneth K. Mwangi Mathematical Finance (POW)
- 15. A Brief History of Mathematical Finance Mathematics Used in Mathematical Finance Application of Financial Mathematics Research Areas in Mathematical Finance Current Issues Conclusion References Robert Jarrow and Philip Protter A Short History of Stochastic Integration and Mathematical Finance; The early years, 1880 to 1970. COX, J.C., INGERSOLL, J.E. and ROSS, S.A. (1985). A Theory of the Term Structure of Interest Rates. GULKO, L. (1999). The Entropy Theory of Stock Option Pricing. International Journal of Theoretical and Applied Finance, Vol. 2, No. 3. pp 331-355. Chavez-Demoulin, V., Embrechts, P., Neslehova, J.: Quantitative models for operational risk: extremes, dependence and aggregation, Journal of Banking and Finance 30(10). SHREVE, S.E. (1996). Stochastic Calculus and Finance. Lecture notes,Carnegie Mellon University Kenneth K. Mwangi Mathematical Finance (POW)

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