Workshop on Computational Finance
(29 - 30 Aug 2005)

Towards the Technical Analysis: Prediction, Estimation of the Changes in Stock Prices, and Price Range Charts
Albert N. Shiryaev and Tatiana B. Tolozova, Steklov Mathematical Institute, Moscow

Albert Nikolaevich Shiryaev obtained his Ph.D in 1961 under the guidance of the legendary Russian mathematician A. N. Kolmogorov and his Doctor of Science in 1967. He is a member of the Steklov Mathematical Institute of the Russian Academy of Sciences, a professor in the Moscow State University, and has supervised 52 Ph.D dissertations. He has published over 10 research monographs/textbooks and more than 160 scientific papers in various fields, such as probability theory, mathematical and applied statistics, optimal stochastic control, financial mathematics and history of mathematics. He is a co-editor of the Springer journal “Finance and Stochastics” and has been on the editorial board of many journals. He was President of the Bernoulli Society for Mathematical Statistics and Probability in 1989-91, and President of the Bachelier Finance Society in 1998-99. His awards include Markov Prize in 1974 and Kolmogorov Prize in 1994. He has been an elected member of the Academia Europea since 1990 and a correspondent member of the Russian Academy of Sciences since 1997.

Abstract - The main aim of the "technical analysis" in the financial engineering is to create some recommendations, some rules for selection of time for buying or selling. Since a long time ago (for example in Japan in XIX century) many different rules, sometimes very sophisticated, were invented and still now are in use on financial markets. Practically, all of these methods have a descriptive character, and a problem of finding explanation of interest in them and a problem of their "mathematization" are very actual. We select for a "mathematization" two well-known Japanese methods, namely the Kagi and Renko charts which give a method of trading based on the analysis of price changes which exceed a certain range H. Very transparent results of our approach will be demonstrated for the case when prices are controlled by a Brownian motion. Also we present our result for problems of prediction and estimation of times of changing (change points) of the trend direction in the stock prices.

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Pricing Mortgage Backed Securities: Continuous Time Stanton's Model
Min Dai, National University of Singapore

The prepayment behavior of mortgage loans plays a critical role in the pricing of mortgage backed securities (MBSs). Stanton (1995) proposed a rational prepayment model where mortgage holders decide whether to prepay their mortgage at random discrete intervals. However, his model essentially is an algorithm or a discrete time model, rather than a continuous time model. In this talk, we present a continuous version of Stanton's model, which is described by a nonlinear partial differential equation. The continuous time model allows us to devise more efficient algorithms. Numerical results are given as well. This is joint work with Yue-Kuen Kwok and Hong You.

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Reclaiming Quasi-Monte Carlo Efficiency in Portfolio Value-at-Risk Simulation
Xing Jin, National University of Singapore

Quasi-Monte Carlo method overcomes the problem of sample clustering in regular Monte Carlo simulation and has been shown to improve simulation efficiency in derivatives pricing literature when price is expressed as a multi-dimensional integration and integrand is suitably smooth. For portfolio Value-at-Risk problems, the distribution of portfolio value change is based on the expectation of an indicator function, hence the integrand is discontinuous. The purpose of this paper is to transform the expectation estimation of an indicator function into that of a smooth function via Fourier transform, so the faster convergence rate of quasi-Monte Carlo method could be reclaimed theoretically. Under fairly mild assumptions simulation of portfolio Value-at-Risk is fast and accurate. Numerical examples demonstrate the advantage of the proposed approach over regular Monte Carlo and straightforward quasi-Monte Carlo methods.

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Feynman Perturbation Expansion for the Price of Coupon Bond Options and Swaptions in a Field Theory of Forward Interest Rates
Belal E. Baaquie, National University of Singapore

European options on coupon bonds are studied in a field theory model of forward interest rates. A approximation scheme for finding the option price is developed based on the fact that the volatility of the forward interest rate is a small quantity. The field theory for the forward interest rates is Gaussian, but when the payoff function for the coupon bonds option is included it makes the field theory nonlinear. Feynman perturbation expansion, using Feynman diagrams, gives a convergent result for the price of coupon bond option. The correlation of two coupon bond options is approximately evaluated. Interest rate swaptions are shown to be a special case of the coupon bond option, and some special features of swaptions are discussed.

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Credit Risk Premia
Kian-Guan Lim, Singapore Management University

Dr Lim Kian-Guan is Professor of Finance at the Lee Kong Chian School of Business of the Singapore Management University. Prior to joining SMU, he was Professor of Finance at the National University of Singapore. He was the founding President of the Association of Financial Engineering (Singapore), founding Director of the NUS Center for Financial Engineering and the Master program in Financial Engineering, and founding secretary of the Asia-Pacific Finance Association that is now merged as the Asian Finance Association. He has been on many journal editorial boards and reviews for journals in finance, mathematical finance, and economics. He has published over 50 refereed journal papers and many other articles. He has also consulted for many banks and corporations. His current research and writing interests are in Quantitative Finance, Asset Pricing, Risk Management, and issues of Estimation and Control.

Abstract - This paper investigates the nature of the credit risk premium in transforming empirical or objective probabilities of credit migration to their equivalent martingale measures (EMMs). We provide a modified version of the risk premium adjustment that allows a linear partition of the credit spread into an empirical default component, a recovery component, and the risk premium component. The log-transform of the default risk premia can be specified as linear regressions on a set of macroeconomic variables testing sensitivities to short-term Treasury rates, forward rates and market returns. This would enable interpretations regarding how default premia would vary with business cycles or with "flight to safety" episodes. By using time-varying empirical default credit migration models, the default premia departs from the trivial role as a credit spread scaling and maps onto the Radon-Nikodym derivative with respect to the EMMs. Various low-dimensional characterizations of the transform used as in Jarrow-Lando-Turnbull (1997), Kijima and Komoribayashi (1998), Lando (1994, 1998), and Lando (2000) may be applied, and which produces different EMMs in the credit migration matrix. In such an incomplete market setting, where the implied risk-neutral default probability is insufficient to uncover the complete set of migration probabilities, the different EMMs lead to model risks since the EMM credit migration matrix is used for pricing and hedging in migration dependent credit instruments. The regression specifications would provide methods of assessing common factors influencing the EMMs and thus the amount of model risk that may be predictable.

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Hedging and Pricing Options with Transaction Costs
Tiong-Wee Lim, National University of Singapore

In the presence of transaction costs, it is no longer possible to perfectly replicate the payoff of a European option by trading in the underlying stock. We present a new option hedging strategy based on minimizing the expected cumulative hedging error and additional cost of rebalancing due to proportional transaction costs. The resulting singular stochastic control problem can be related to an optimal stopping problem, which we solve to show that an optimal hedge consists of selling/buying the underlying stock whenever the number of shares held falls above/below a no-transaction band about the "delta". We also discuss a new approach to pricing using market data on the stock and options, and consider the optimal hedge in the presence of transaction costs. This is joint work with T.L. Lai.

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Weak Interest Rate Parity and Currency Portfolio Diversification
Yonggan Zhao, Nanyang Technological University

Dr. Yonggan Zhao is an Assistant Professor at Nanyang Technological University. He received an M.Sc. in mathematics from Western Kentucky University and a PhD. in finance and management science from the University of British Columbia. He is currently teaching portfolio management, derivative securities, optimization in finance, and theory of investment. His research interests are mainly in the areas of intertemporal equilibrium theory, dynamic portfolio management and option pricing with incomplete market.

Abstract - This paper presents a dynamic model of optimal currency returns with a hidden Markov regime switching process. We postulate a weak form of interest rate parity that the hedged risk premiums on currency investments are identical within each regime across all currencies. Both the in-sample and the out-of-sample data during January 2002 - March 2005 strongly support this hypothesis. Observing past asset returns, investors infer the prevailing regime of the economy and determine the most likely future direction to facilitate portfolio decisions. Using standard mean variance analysis, we find that an optimal portfolio resembles the Federal Exchange Rate Index which characterizes the strength of the U.S. dollar against world major currencies. The similarity provides a strong implication that our three-regime switching model is appropriate for modeling the hedged returns in excess of the U.S. risk free interest rate. To investigate the impact of the equity market performance on changes of exchange rates, we include the S&P500 index return as an exogenous factor for parameter estimation.

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Discrete Credit Barrier Models
Oliver Chen, National University of Singapore

Dr. Oliver Chen is a visiting fellow at the Department of Mathematics and Centre for Financial Engineering of the National University of Singapore. He has previously taught in the Financial Mathematics programmes at the University of Toronto and Imperial College, London. His current research interests are in credit risk and interest rate modeling.

Abstract - A method for generating a stochastic process on a discrete lattice is given. These discrete stochastic processes are based on birth-death processes and have traditional diffusion processes as limits. One can introduce stochastic volatility and jumps with little computational overload. As an application, a credit risk model is proposed in which a credit-worthiness variable reflects the credit rating of each reference entity. The model can be calibrated to match historical credit rating transition and default probabilities. Upon addition of a drift, credit spreads can also be matched.

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The Volatility Risk Premium Embedded in Currency Options
Buen-Sin Low (CPA, CFA), Nanyang Technological University

Buen Sin is Associate Professor of finance and the Director of the MSc (Financial Engineering) Program at the Nanyang Technological University. His publications have appeared in the Journal of Financial and Quantitative Analysis, Journal of Futures Markets, Journal of Business Finance and Accounting, among others. He also has been actively involved in industry consultation and has conducted well-received in-house executive seminars for major organizations in Asia and Europe. He is currently the Technical Consultant for Ernst & Young, Transaction Advisory Services.

Abstract - This study employs a non-parametric approach to investigate the volatility risk premium in the over-the-counter currency option market. Using a large database of daily quotes on delta neutral straddle in four major currencies – the British Pound, the Euro, the Japanese Yen, and the Swiss Franc – we find that volatility risk is priced in all four currencies across different option maturities and the volatility risk premium is negative. The volatility risk premium has a term structure where the premium decreases in maturity. We also find evidence that jump risk may be priced in the currency option market.

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