Ivan BercovichSenior Lecture Series

Friday, April 17th, 2009

Asymptotic analysis , Calculus, Differential equation, Ergodic theory, Numerical

analysis, Real analysis, Probability, (Probability distribution, Binomial distribution, Log-normal distribution, Expected value), Value at risk, Risk-neutral measure, Stochastic calculus (Brownian motion, Lévy process), Itô's lemma,

Fourier transform, Girsanov's theorem, Radon-Nikodym derivative, Monte

Carlo method, Quantile function, Partial differential equations (Heat equation), Martingale representation theorem, Feynman Kac Formula, Stochastic differential equations, Volatility (ARCH model, GARCH model), Stochastic volatility,

Mathematical modeling, Numerical method (Numerical partial differential

equations, Crank-Nicolson method, Finite difference method), Information Theory

Financial Mathematics

Merton and Scholes received the 1997 The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel for this and related work.

Financial Mathematics: Assumptions

Efficient market hypothesis: at any given time, the price of a financial instrument reflects all the known information ("informationally efficient“)

Rational Pricing: the price of a financial asset reflects its arbitrage free value, as any deviation will be “arbitraged away” by the market.

Risk Neutral Measure: certain assets such as T-Bonds are considered risk-free and are used as a reference to price risk-carrying assets.

Stocks follow a log-normal model: Central Limit Theorem (CLT)

R = H ( x) − H y ( x)

where R is the amount of information one can receive, H is the amount of information a source sent and Hy(x), the conditional entropy, is called equivocation.

In Finance, Entropy is the value of information

The value of information is inversely related to the number of people who understand it. (note that it does not matter if the information is known, but rather if it is understood)

Issues: price of a financial instrument at time (t) is not independent from price at (t + ε)

Issues: information is not instantaneous

Information is expensive to obtain.

Information is hard to understand

New information can only be understood gradually by human beings.

Issues: random events are more common than we think

Looking at the Dow Jones index with signal processing:

Running Mean

Noise (Volatility)

Autocorrelation

Consequences: Fat Tails

Q: How do financial institutions price risk? Courtesy of A: Value at risk

Mean of both distributions is zeroStandard Deviation 100 million

For both distributions:84% VaR 100 million

P(x<-100 million) = 0.15895% VaR 200 million

P(x<-200 million) = 0.046

P(x ≤ -100 billion) = ~0P(x ≤ -636.13 million) = ~0.0000000001

P(x≤-100 billion) = 0.0005P(x ≤ -636.13 million) = ~0.0005000001

Consequences: Fat Tails

One solution (for long term investors): Dollar Cost Averaging

The metrics

Investing Strategies: Dollar-Cost Averaging vs Lump

k

Dollar-Cost : What is the right time interval?

Loosely Correlated Mildly Correlated Highly Correlated

Lump SumDollar-Cost Average

Dollar-Cost Averaging vs Lump Sum investing strategies: Discrepancy on a good year vs bad year

Conclusion and Final Comments:

The financial markets are extremely complicated highly mathematical environments.

It’s okay to make assumptions in order to simplify mathematical models, but it is important to understand this assumptions when making decisions.

Most models work well under optimal conditions. They should be tested for deficiencies under poor conditions as well.

Interdisciplinary Research is cool (it allows you to see things experts in a field might overlook)

ECE314, ECE563, ECE564, ECE603,Stat515, Stat526, Math 441, Math497G,FINOPMGT 422