June 21, 2021

March 22-24, 2010
Workshop on Computational Methods in Finance

Talk Titles and Abstracts

Liming Feng (Illinois)
Hilbert transform approach to options valuation

A Hilbert transform approach to options valuation will be presented in this talk. For many popular option pricing models with known analytic characteristic functions for the underlying driving stochastic processes, the Hilbert transform approach exhibits remarkable speed and accuracy, with errors decaying exponentially in terms of the computational cost. The pricing of discrete barrier, lookback and Bermudan options will be illustrated. Applications in applied probability will also be discussed.


Peter Forsyth (Waterloo)
Analysis of A Penalty Method for Pricing a Guaranteed Minimum Withdrawal Benefit (GMWB)

The no arbitrage pricing of Guaranteed Minimum Withdrawal Benefits (GMWB) contracts results in a singular stochastic control problem which can be formulated as a Hamilton Jacobi Bellman (HJB) Variational Inequality (VI). Recently, a penalty method has been suggested for solution of this HJB variational inequality (Dai et al, 2008). This method is very simple to implement. In this talk, we present a rigorous proof of convergence of the penalty method to the viscosity solution of the HJB VI. Numerical tests of the penalty method are presented which show the experimental rates of convergence, and a discussion of the choice of the penalty parameter is also included. A comparsion with an impluse control formulation of the same problem, in terms of generality and computational complexity, is also presented.


Jim Gatheral (Merrill Lynch)
Optimal order execution

In this talk, we review the models of Algmren and Chriss, Obizhaeva and Wang, and Alfonsi, Fruth and Schied. We use variational calculus to derive optimal execution strategies in these models, and show that static strategies are dynamically optimal, in some cases by explicitly solving the HJB equation. We present general conditions under which there is no price manipulation in models with linear market impact. Finally, we present some new generalizations of the Obizhaeva and Wang model given in a recent paper by Gatheral, Schied and Slynko, again deriving explicit closed-form optimal execution strategies.

This is partially joint work with Alexander Schied and Alla Slynko.


Kay Giesecke (Stanford)
Asymptotically Optimal Importance Sampling For Dynamic Portfolio Credit Risk

Dynamic intensity-based point process models, in which a firm default is governed by a stochastic intensity process, are widely used to model portfolio credit risk. In the context of these models, this paper develops, analyzes and evaluates an importance sampling scheme for estimating the probability of large portfolio losses, portfolio risk measures including value at risk and expected shortfall, and the sensitivities of these quantities with respect to the portfolio constituent names. The scheme is shown to be asymptotically optimal. Numerical experiments demonstrate the advantages of the algorithm for several standard model specifications.


Mike Giles (Oxford)
Progress with multilevel Monte Carlo methods

The multilevel Monte Carlo path simulation method combines simulations with different levels of resolution to reduce the computational cost for achieving a prescribed Mean Square Error.

Is this talk I will describe the latest progress with this technique,with new applications to jump-diffusion models, multi-dimensional SDEs, the calculation of Greeks, and a stochastic PDE arising from a credit. I will also outline joint work with Kristian Debrabant and Andreas Rossler on the numerical analysis of the multilevel method using the Milstein discretisation.


Garud Iyengar (Columbia)
A behavioral finance based tick-by-tick model for price and volume

We propose a model for jointly predicting stock price and volume at the tick-by-tick level. We model the investor preferences by a random utility model that incorporates several important behavioral biases such as the status quo bias, the disposition effect, and loss-aversion. The resulting model is a logistic regression model with incomplete information; consequently, we are unable to use the maximum likelihood estimation method and have to resort to Markov Chain Monte Carlo (MCMC) to estimate the model parameters. Moreover, the constraint that the volume predicted by the MCMC model exactly match observed volume introduces serial correlation in the stock price; consequently, standard MCMC techniques for calibrating parameters do not work well. We develop new modifications of the Metropolis-within-Gibbs method to estimate the parameters in our model. Our primary goal in developing this model is to predict the market impact function and VWAP (volume weighted average price) of individual stocks.


Petter Kolm (Mathematics in Finance M.S. Program, Courant Institute, New York University)
Algorithmic Trading: A Buy-Side Perspective

The traditional view of portfolio construction, risk analysis, and execution holds that these three functions of money management are separable. Portfolios are constructed without incorporating the costs of execution, and execution is conducted without considering portfolio level risk. With the explosive growth of algorithmic trading, several mathematical and computational methodologies have been proposed for unifying and improving traditional money management functions. This presentation addresses some important developments in this area, including incorporating market impact costs into portfolio optimization, multi-period dynamic portfolio analysis, and high-frequency simulation for dynamic portfolio analysis.


Ralf Korn (TU Kaiserslautern)
Recent advances in option pricing via binomial trees

A survey on some new results obtained in joint work with S. Mueller is given. In particular, we present an optimized 1-D-scheme (the optimal drift model) that is based on overlaying a given binomial scheme with an additional drift process and that obtains a higher than advanced schemes such as the Tian- or the Chang-Palmer approach. Further, we introduce the orthogonal decoupling approach to solve n-D-valuation problems. This approach is based on a non-linear transformation of the state space, always results in well-defined probabilities in the approximating n-D binomial tree, and admits a regular convergence behaviour.


Ciamac Moallemi (Columbia)
A multiclass queueing model of limit order book dynamics

We model the limit order book as system of two, coupled multiclass queues. Specifically, each side of the book is modeled as a single server, multiclass queue operating under a strict priority rule defined by the prices associated with each limit order. We describe the transient dynamics of this system, and formulate and solve the optimal execution problem for a block of shares over a short time horizon.

This is joint work with Costis Maglaras.


Kumar Muthuraman (UT Austin)
Moving boundary approaches for solving free-boundary problems

Free-boundary problems arise when the solution of a PDE and the domain over which the PDE must be solved are to be determined simultaneously. Three classes of stochastic control problems (optimal stopping, singular and impulse control) reduce to such free-boundary problems. Several classical examples including American option pricing and portfolio optimization with transaction costs belong to these classes. This talk describes a computational method that solves free-boundary problems by converting them into a sequence of fixed-boundary problems, that are much easier to solve. We will illustrate application on a set of classical problems, of increasing difficulty and will also see how the method can be adapted to efficiently handle problems in large dimensions.


Phillip Protter (Cornell)
Absolutely Continuous Compensators

Often in applications (for example Survival Analysis and Credit Risk) one begins with a totally inaccessible stopping time, and then one assumes the compensator has absolutely continuous paths. This gives an interpretation in terms of a ``hazard function'' process. Ethier and Kurtz have given sufficient conditions for a given stopping
time to have an absolutely continuous compensator, and this condition was extended by Yan Zeng to a necessary and sufficient condition. We take a different approach and make a simple hypothesis on the filtration under which all totally inaccessible stopping times have absolutely continuous compensators. We show such a property is stable under changes of measure, and under the expansion of filtrations; and we detail its limited stability under filtration shrinkage. The talk is based on research performed with Sokhna M'Baye and Svante Janson.


Chris Rogers (Cambridge)
Convex regression and optimal stopping

There are many examples, particularly in finance, of optimal stopping problems where the state variable is some point in Euclidean space, and the value function is convex in the state variable. This then permits approximation of the value function as the maximum of a sequence of linear functionals, an approach which has various advantages. The purpose of this paper is to present the methodology and explore its consequences.


Birgit Rudloff (Princeton)
Hedging and Risk Measurement under Transaction Costs

We consider a market with proportional transaction costs and want to hedge a claim by trading in the underlying assets. The superhedging problem is to find the set of d-dimendional vectors of initial capital that allow to superhedge the claim. We will show that in analogy to the frictionless case, the superhedging price in a market with proportional transaction costs is a (set-valued) coherent risk measure, where the supremum in the dual representation is taken w.r.t. the set of equivalent martingale measures. To do so, we extend the notion of set-valued risk measure to the case of random solvency cones. Connections to recent results about efficient use of capital when there are multiple eligible assets are drawn. When starting with a vector of initial capital that does not allow to superhedge, a shortfall at maturity is possible. For an investor who finds a hedging error that is 'small enough' still acceptable, good-deal-bounds under transaction costs can be defined.


Georgios Skoulakis (Maryland)
Solving Consumption and Portfolio Choice Problems: The State Variable Decomposition Method

This paper develops a new solution method for a broad class of discrete-time dynamic portfolio choice problems. The method efficiently approximates conditional expectations of the value function by using (i) a decomposition of the state variables into a component observable by the investor and a stochastic deviation; and (ii) a Taylor expansion of the value function. The outcome of this State Variable Decomposition (SVD) is an approximate problem in which conditional expectations can be computed efficiently without sacrificing precision. We illustrate the accuracy of the SVD method in handling several realistic features of portfolio choice problems such as intermediate consumption, multiple risky assets, multiple state variables, portfolio constraints, non-time-separable preferences, and nonredun- dant endogenous state variables. We finally use the SVD method to solve a realistic large-scale life-cycle portfolio choice and consumption problem with predictable expected returns and recursive preferences.


Jeremy Staum (Northwestern)
Déjà Vu All Over Again: Efficiency when Financial Simulations are Repeated

Many computationally intensive financial simulation problems involve running the same simulation model repeatedly with different values of its inputs. Such tasks include pricing exotic options of the same type but of different strikes and maturities, valuation of options given different values of the model’s parameters during calibration, and measuring a portfolio’s risk as the markets move. The basic approach is to run the simulation model using each of the input values in which one is interested. In this talk, we explore generic methods for solving a suite of repeated simulation problems more efficiently, by estimating the answer given one value of the inputs using information generated while running the simulation model with different values of the inputs.


Nizar Touzi (Ecole Polytechnique)
A Probabilistic Numerical Method for Fully Nonlinear Parabolic PDEs

We suggest a probabilistic numerical scheme for fully nonlinear PDEs, and show that it can be introduced naturally as a combination of Monte Carlo and finite differences scheme without appealing to the theory of backward stochastic differential equations. Our first main result provides the convergence of the discrete-time approximation and derives a bound on the discretization error in terms of the time step. An explicit implementable scheme requires to approximate the conditional expectation operators involved in the discretization. This induces a further Monte Carlo error. Our second main result is to prove the convergence of the latter approximation scheme, and to derive an upper bound on the approximation error. Numerical experiments are performed for two and five-dimensional (plus time) fully-nonlinear Hamilton-Jacobi-Bellman equations arising in the theory of portfolio optimization in financial mathematics.



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