Financial Mathematics

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)

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

Before the 1900
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.


Before the 1900
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.


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.


Current Issues
Conclusion


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


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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.


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

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.


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.


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


Current Issues
Conclusion


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


Open Problems 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).


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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.


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


Financial Mathematics

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