

 Seminar: Winter 2005
Stanford Financial Mathematics Seminar Schedule
The subtle nature of financial random walks


JeanPhilippe Bouchaud (Science & Finance, Capital Fund Management, and Commissariat a l'Energie Atomique, France)
It is known since Bachelier 1900 that price changes are nearly uncorrelated, leading to a randomwalk like behaviour of prices. However, compared to the simplest Brownian motion, price statistics reveal a large number of anomalies, such as fat tails and long memory in the volatility. The detailed study of trade by trade and order book data allows one to provide evidence for a subtle compensation mechanism that underlies the `random' nature of price changes. This compensation drives the market close to a critical point, which may explain the sensitivity of financial markets to small perturbations, and their propensity to enter bubbles and crashes. We argue that the resulting unpredictability of price changes is quite far from the neoclassical view that markets are informationally efficient.
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Portfolio optimization under incomplete information


Simon Brendle (Princeton University)
We study an optimal investment problem under incomplete information for an investor with constant relative risk aversion. We assume that the investor can only observe asset prices, but not the instantaneous returns. Furthermore, we assume that the instantaneous returns follow an OrnsteinUhlenbeck process, and that their initial distribution is Gaussian. We analytically solve the Bellman equation for this problem, and identify the optimal investment strategy under incomplete information. We explore the relationship between the value function under partial observation and the value function under full observation, and derive a formula for the economic value of information. Furthermore, we discuss how the optimal strategy under partial observation can be computed from the optimal strategy for an investor with full observation. Explicit solutions are presented in a model with only one risky asset.
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Strategic Trading, Liquidity, and Information Acquisition 

Tunay I. Tunca (Stanford University Graduate School of Business)
We study endogenous liquidity trading in a market with longlived asymmetric information. We distinguish between public information, tractable information that can be acquired, and intractable information that cannot be acquired. Besides information asymmetry and noise, the adverseselection spread depends on the diffusion of intractable information and on the interest rate. With endogenous liquidity trading, efficiency is lower than that implied by noisetrading models. Liquidity traders benefit from the information released through the insider's trades in spite of their monetary losses. We study factors that affect the insider's information acquisition decision, including the amount of intractable information, observability, and information acquisition costs.
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Risk Architecture and the Bank of the Future


Ron Dembo (Algorithmics, Toronto)
Measuring the risk of a large financial institution is a gargantuan task. There have been major improvements in doing so over the past few years. These have resulted in the ability of institutions to take on more and more complexity, thereby keeping the risk management treadmill alive and well. We discuss how risk is actually measured, some major new accomplishments, such as realtime risk based on simulation and highlight some of the interesting research problems that are being addressed, such as realtime bankwide optimization
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Finding, Measuring and Using Mean Reversion in Financial Data 

George Papanicolaou (Professor of Mathematics, Stanford University)
Mean reversion of volatility for the SP500, or other equity indices, has been observed for a long time but is difficult to quantify by the usual econometric methods, as are its implications for derivative pricing. I will review the inadequate status of understanding of mean reversion in the literature and show that it is not possible to make a simple `stylized' assessment of it. In fact there are at least two distinct rates of mean reversion that can be identified in the SP500 and in the DJ. I will then show how mean reversion can be incorporated in stochastic volatility models for pricing derivatives and the results that come out.
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Exact Simulation of Stochastic Volatility and other Affine Jump Diffusion Processes


Mark Broadie (Graduate School of Business, Columbia University)
The stochastic differential equations for affine jump diffusion models do not yield exact solutions that can be directly simulated. Discretization methods can be used for simulating security prices under these models. However, discretization introduces bias into the simulation results and a large number of time steps may be needed to reduce the discretization bias to an acceptable level. This paper suggests a method for the exact simulation of the stock price and variance under Heston's stochastic volatility model and other affine jump diffusion processes. The sample stock price and variance from the exact distribution can then be used to generate an unbiased estimator of the price of a derivative security. We compare our method with the more conventional Euler discretization method and demonstrate the faster convergence rate of the error in our method. Specifically, our method achieves an O(s^{1/2}) convergence rate, where s is the total computational budget. The convergence rate for the Euler discretization method is O(s^{1/3}) or slower, depending on the model coefficients and option payoff function.
(This is joint work with Ozgur Kaya.)
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Penalty approximation and analytical characterization of the problem of superreplication under portfolio constraints


Alain Bensoussan (University of Texas Dallas)
In this paper, we consider the problem of superreplication under portfolio constraints in a Markov framework. More specifically, we assume that the portfolio is restricted to lie in a convex subset, and we show that the superreplication value is the smallest function which lies above the BlackScholes price function and which is stable for the socalled face lifting operator. A natural approach to this problem is the penalty approximation, which not only provides a constructive smooth approximation, but also a way to proceed analytically.
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Receding Horizon Dynamic Hedging for Derivative Securities under Transaction Costs 

James Primbs (Stanford University Department of Management Science and Engineering)
In this work, we apply receding horizon control to the problem of dynamically hedging a derivative security in the presence of transaction costs. Receding horizon control is a methodology that uses online optimization to determine a control policy, and is well suited to many problems that are not analytically tractable. We demonstrate it on the
problem of hedging a European call option on a single asset, and a basket option on many assets, both under transaction costs. It is shown to significantly outperform BlackScholes and Leland based delta hedging under an expected absolute error criterion. Additionally, we provide some theoretical properties for the performance of receding horizon hedging.
Joint work with Pete Meindl.
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