Program on Financial Mathematics, University of Chicago
The Geometry of Financial Times.
Multiscale nature of Financial Markets is illustrated by using
European options and term structure of the US Treasures.
- Gamma Saddle & Euler's Numbers
- Magic Distributions of Changes
- Treasury Signature
- Galton's Quincunx
Professor, Courant Institute of Mathematical Sciences, New York University
Weighted Monte Carlo: new techniques for calibrating multifactor asset-pricing models
A general approach for calibrating Monte Carlo models to the market prices of benchmark securities is presented. Starting from a given valuation model (such as HJM), the algorithm corrects for price-misspecifications and finite-sample effects by computing adjusted "probability weights" on the simulated paths. The resulting ensemble prices the set of market instruments exactly. Concrete applications to the calibration of multi-factor interest rate and stochastic volatility models are discussed. Hedging and sensitivity analysis are discussed as well.
Firmwide Risk Management Goldman Sachs & Co.
Risk Constrained Dynamic Active Portfolio Management
Active portfolio management is concerned with objectives related to outperfroming the return of a target benchmark portfolio. In this talk, after reviewing some previous related results, we consider the dynamic active portfolio managment problem with a new objectives for the risk/return tradeoff, where risk is defined directly in terms of probability of shortfall relative to the benchmark and return is defined in terms of the expected time to reach investment goals relative to the benchmark. As a special case, our analysis includes the problem where the investor wants to minimize the expected time until a given performance goal is reached subject to a constraint on the shortfall probability.
Jennifer N. Carpenter
New York University
The Optimal Dynamic Investment Policy for a Fund Manager with an Incentive Fee
This paper solves the dynamic investment problem of a risk averse fund manager compensated with an incentive fee, a call option on the assets he controls.
Under the optimal policy, the manager either significantly outperforms the benchmark or else incurs severe losses. As the fund value goes to zero, volatility goes to infinity. However, the option compensation does not strictly lead to greater risk-seeking. Sometimes, the manager's optimal volatility is less with the incentive fee than it would be if he were trading his own account. Furthermore, increasing the incentive fee percentage causes the manager to decrease volatility.
Eidgen Technische Hochschule
Centre Solutions, Zurich Financial Services Group
Probabilities and Pricing
Attempts to control financial future of a contract and approximate it with a single number (the price) have been frequently considered as the most important task of financial mathematics. When the forced prophesy doesn't work and market values of "risk-neutral" portfolios go wild the analysts and their tools are subject to curse. We argue that the risk management and VaR (inverted PDF) provide the missing division of labor. Now financial math can focus on modeling the probabilities of different outcomes while pricing remains partially subjective. We illustrate this by using the conventional example of dynamic hedging strategy, and indicate the dynamic efficient frontier.
Benoit B. Mandelbrot
Yale University, Mathematics Dept, New Haven, CT
THE MULTIFRACTAL TIME MODEL FOR FINANCIAL PRICES
Many financial prices are shown to be very well represented by a fractional Brownian motion, but one that does not proceed in clock time, rather in a trading time that is a multifractal function of clock time. The presentation will sketch a generalized multifractal formalism developed for this purpose and end with an account of experimental tests.
The Risk Engineering Company
Clearing Systems and the The Transmission of Systemic Risk
We consider default by firms by that are part of a single clearing mechanism in which the repayment obligations of all firms within the system are determined simultaneously in fashion consistent with the priority of debt claims and the limited liability of equity. We will show, via a fixed-point argument, that there always exists a ``clearing payment vector,'' specifying the total payment made by each node in the system, which clears the obligations of the members of the clearing system. Next, we perform comparative statics on this clearing payment vector, determining the nature of its dependence on operating income and the architecture of financial liabilities. These comparative statics imply that unsystematic shocks to the network will lower the value of all firms in the network. Moreover, in contrast with the standard single-firm results, these risk increases may also lower the value of equity. To further analyze financial architecture, we consider in detail properties of the clearing system architecture focusing on the degree to which ability to pay of a given firm depends on inflows from other firms. Using these results we characterize upper and lower bounds on the performance of the financial system and consider the effect of bilateral netting.
We show that, in some cases, bilateral netting of claims can reduce the resilience of the financial system.
University of Illinois
Tutorial: Risk Neutral Valuation and Martingale Measures:
In an effort to understand Black and Scholes' famous result more carefully, researchers discovered that for a stochastic process model of securities prices to be sensible from the economic point of view, it must be a martingale (or close to it) under some change of probability measure (the so-called risk neutral probability measure). They also discovered that stochastic integrals of predictable trading strategies with respect to price processes are natural models for the change in a portfolio's value. These ideas have led to the risk neutral valuation approach for the pricing and heddging of European options, American options, and interest rate derivatives. In addition, a modern approach for optimal consumption/investment problems has emerged.
Rutgers University, New Brunswick
A New Model for Stock Price Fluctuations Based on Delayed Information
In the stock market information is hoarded and is of value to the source only until it becomes known to others. Sometimes disinformation is used, e.g., to mislead others into supporting a price that should fall.
A new model sharpens the usual Black-Scholes-Merton-Samuelson model which assumes that information plays no role in price fluctuations because those who know the truth will exploit it and so the rest of us are allegedly protected by those who have already removed the drift in the price. But there is a delay until the prices readjust and this delay seems to have been overlooked.
The self-evident reality that information in the market is only available with delay gives rise to a new model for price fluctuations presented here which is expected to give more accurate predictions of future prices and more accurate formulas for hedge option valuations. These new prices have been calculated for the various standard options but using the new model in a recent PhD thesis by Xin Guo which will be reviewed here.
Robert J. Vanderbei
A martingale system theorem for stock investments
We show that the dollar-cost-averaging investment strategy yields no advantage over any other non-clairvoyant strategy by showing that the difference between any two such strategies is a mean zero martingale. An interesting corollary of this result is that if X(t) is a positive martingale then the process Y(t) = X(t) int_0^t 1/X(s) ds - t is also a martingale.
Advanced Analytics, Inc.
Lessons from Financial Disasters and A Dynamic Asset Pricing Model
We shall discuss lessons from recent global financial crises and LTCM; their potential impact on financial modeling and risk management, on the debate on Random Walk Hypothesis, Efficient Market Hypothesis, etc.
The global economy and the Capital Market are complex adaptive systems. Adaptive models using simulation are proposed as the key to study such systems. Our methodology provides a unified framework for multi-disciplinary approaches.
Financial mechanics is introduced as the foundation for a new dynamic asset pricing theory which also models the competing market forces. It can be used to describe asset price processes not just in market equilibrium, but also in disequilibrium, chaos, and catastrophic events. It may be used for predication of market trends by either fundamental analysis or technical analysis; modeling short term dynamics associated to event shocks; market/credit risk management; and accurate pricing and hedging of derivatives.
Our model explains naturally the anomalies such as the fat tails and the volatility smile. Only in the equilibrium limit and degenerated case our model reduces to the Black-Schole's model and its variations.