December 18, 2018
Robert Almgren Dan Rosen
Leif Andersen William F. Shadwick
Peter Carr Juergen Topper
Matt Davison Stanislav Uryasev
Paul Glasserman Yong Wang
Adam Kolkiewicz Tony Ware
Yuying Li Petter Wiberg
David Pooley  
Philip Protter  

Robert Almgren, University of Toronto
Optimal execution with nonlinear cost functions and trading-enhanced risk

In the trading of large portfolios, the volatility risk eliminated by rapidly completing the trade program must be balanced against the market impact costs incurred when rapid execution is demanded. In previous work by R Almgren and N Chriss, explicit solutions were given for the case in which per-share impact costs are linear in trading rate. In this work we consider two nonlinear extensions. First, we obtain explicit solutions in the case that market impact depends nonlinearly on trading rate, including the popular square-root law. Second, we consider the case in which rapid trading increases not only the expected value of cost but also its uncertainty; although we do not obtain fully explicit solutions, we are able to give a complete asymptotic description and extract conclusions for practical trading.

Leif Andersen, General Re Securities
Primal/dual simulation of high-dimensional American options

This talk describes a practical algorithm based on Monte Carlo simulation for the pricing of multi-dimensional American (i.e., continuously
exercisable) and Bermudan (i.e., discretely-exercisable) options. The method generates both lower and upper bounds for the option price and hence gives valid confidence intervals for the true value. Lower bounds can be generated using any number of primal algorithms. Upper bounds are generated using a new Monte Carlo algorithm based on a duality representation of the Bermudan value function. Computational results for multi-factor equity and interest rate options demonstrate the simplicity and efficiency of the proposed algorithm.

Peter Carr, Courant Institute, NYU
Derivatives, Duality, and Time Reversal (joint with Jesper Andreasen of Bank of America)

We study the probabilistic underpinnings of the Dupire forward partial differential equation for European option values. We provide a link with the literature on time reversal of Markov processes. We derive a new result called Put Call Reversal and provide various applications. In particular, we show how it can be used to simplify semi-static hedging.

Matt Davison, Department of Applied Mathematics, University of Western Ontario
Pricing Swing Options using a Hybrid Spot Electricity Model

We have developed a model for electricity prices which is a hybrid of the bottom-up "stack-based" pricing traditionally used in the power engineering community with the top-down financial time-series driven model customary in quantitative finance. Our model allows engineering data on plant failure and meteorological data on power demand to be used to estimate model parameters - important in the case of electricity markets for which spot price data is often either limited or nonexistent. We simplify and adapt our model to optimize it for the task of pricing a simple swing option to buy electrical power. The resulting pricing model may be written in terms of mixtures of Poisson distributions. The swing option may then be priced using relatively straightforward backward recursion techniques.

Paul Glasserman, Columbia University
Cap and Swaption Approximations in LIBOR Market Models with Jumps

Joint work with Nicolas Merener

This paper develops formulas for pricing caps and swaptions in LIBOR market models with jumps. The arbitrage-free dynamics of this class of models were characterized in Glasserman and Kou (1999) in a framework allowing for very general jump processes. For computational purposes, it is convenient to model jump times as Poisson processes; however, the Poisson property is not preserved under the changes of measure commonly used to derive prices in the LIBOR market model framework. In particular, jumps cannot be Poisson under both a forward measure and the spot measure, and this complicates pricing. To develop pricing formulas, we approximate the dynamics of a forward rate or swap rate using a scalar jump-diffusion process with time-varying parameters. We develop an exact formula for the price of an option on this jump-diffusion through explicit inversion of a Fourier transform. We then use this formula to price caps and swaptions by choosing the parameters of the scalar diffusion to approximate the arbitrage-free dynamics of the underlying forward or swap rate. We apply this method to two classes of models: one in which the jumps in all forward rates are Poisson under the spot measure, and one in which the jumps in each forward rate are Poisson under its associated forward measure. Numerical examples demonstrate the accuracy of the approximations.

Adam Kolkiewicz, University of Waterloo
Pricing American style contracts on multiple assets using simulations

Computational tools in modern finance are often classified as either numerical or simulation methods. While the former provide fast and accurate answers in less complex problem, the latter offer the only viable tools for pricing instruments contingent on several assets. In the talk, we present a framework that allows combining these apparently different approaches. This is possible by introducing a smooth Monte Carlo estimator of transition density functions for stochastic differential equations. The estimator, though nonparametric, is unbiased and exhibits a rate of convergence that is typical to parametric problems. When used to approximate functionals of terminal prices, it reduces variance by a factor that depends on the ``smoothness" of the density estimate. We illustrate some possible applications of the method using European and American style financial instruments. For the latter, we focus on methods based on regression techniques, like the one considered recently by Longstaff and Schwartz, and on the low discrepancy mesh method (Boyle et al.).

Yuying Li, Cornell University
Discrete Hedging Under a Piecewise Linear Risk Minimization

Joint work with Thomas F. Coleman, Yuying Li and Cristina Patron

In an incomplete market, it is impossible to eliminate the intrinsic risk of an option that cannot be replicated. Thus it is unclear what are the
optimal hedging stratey and fair value of an option. Quadratic risk minimization is often used to determine fair value and hedging strategy for an option when the market is incomplete. We investigate hedging strategies and prices from alternative piecewise linear risk minimization. We illustrate that piecewise linear risk minimization often leads to smaller expected total hedging cost and significantly different, possibly more desirable, hedging strategies from that of the quadratic risk minimization. Comparative numerical results are provided for discrete dynamic hedging.

David Pooley, University of Waterloo
Convergence of Numerical Methods for Uncertain Volatility Models

The pricing equations derived from uncertain volatility/transaction cost models in finance are often cast in the form of nonlinear partial differential equations. Implicit timestepping leads to a set of nonlinear algebraic equations which must be solved at each timestep. To solve these equations, an iterative approach is employed. In this talk, we show the convergence of a particular iterative scheme for one factor uncertain volatility models. We also demonstrate how non-monotone discretization schemes (such as standard Crank-Nicolson timestepping) can converge to incorrect solutions, or lead to instability. Numerical examples are provided.

Joint work with Peter Forsyth and Ken Vetzal

Philip Protter, Cornell University
The Longstaff-Schwartz Algorithm for Pricing of American Options Works

We present a precise mathematical description of the Longstaff-Schwartz algorithm, breaking it down into two steps and proving the convergence. The second step is a Monte Carlo step, and we obtain the rate of convergence and the asymptotic normalized error. The talk is based on joint work with E. Clement and D. Lamberton.

Dan Rosen, Algorithmics
Advanced methods for pricing and managing loan portfolios

This paper introduces a general option-valuation framework for loans that provides valuation information at loan origination and supports mark-to-market analysis, portfolio credit risk and asset and liability management for the entire portfolio. We describe, in detail, the main structures found in commercial loans and the practical assumptions required to model the state-contingent cash flows resulting from these structures. We stress the need to account properly for the embedded options such as prepayment, revolvers, and grid pricing. The characteristics of the credit risk model necessary to capture the main features of the problem are described. A case study is used to addressed the data available in practice , calibration methodologies and the impact of various modelling assumptions. Finally, we outline some of the computational challenges of performing portfolio mark-to-market and risk measurement and discuss various solutions. Though we focus primarily on large corporate and middle-market loans, the approach is applicable more generally to bonds and credit derivatives.

William F. Shadwick, The Finance Development Centre Limited
Business and modelling issues in option pricing

Joint work with B.A. Shadwick, The Finance Development Centre Limited

In spite of the growing sophistication of option pricing technology, relatively primitive approaches to the computation of prices and hedging parameters are still in widespread use. This appears to be an inevitable side effect of the growth of derivatives markets. As one practitioner noted, "The problem with the Black Scholes formula is that it makes every idiot think he can price an option." Increasingly, the options he thinks he can price are what used to be known as exotic.

One of the most common examples of this problem is the variety of ad hoc methods which have been devised to deal with volatility smile or skew. There is a straightforward extension of the original Black Scholes Merton 1-factor model which good engineering practice would suggest should be exhausted prior to moving to more complex remedies. However, there is no shortage of examples of trading desks which purport to be using more 'advanced' approaches, many of which prove to be self contradictory under a discouragingly low level of scrutiny.

It is not uncommon to find that traders and quants who espouse these advances are, in their view, routinely making large 'profits' buying or selling long dated options to large sophisticated counterparties. Needless to say, these P&L effects often use mark to model and proprietary risk exposure analysis in an essential way.

We review the maximal 'local volatility' extension of the 1-factor Black Scholes Merton (BSM) model and illustrate one good reason for this situation. We show that the computation of prices and sensitivities in this framework requires numerical expertise sufficient to solve the forced, variable coefficient BSM equation and that as a result, one really does have all or nothing.

The low level of the pde solvers in common use in the finance industry then explains the prevalence and longevity of the ad hoc approaches to variable volatility. We provide some examples which show just how dangerous these approaches can be in the process of pricing or hedging even simple derivative positions.

Juergen Topper, Andersen, Germany
Worst-Case Pricing of Multi-Asset Options

Options on several underlyings are a common exotic product in the equity and FX derivatives market. The value of these kinds of options also depends on the correlation of the underlyings. We will present a model to compute a lower bound for the price of this option. The model, represented by a non-linear parabolic PDE, is implemented with finite elements in order to be able to compute accurate cross-derivatives We will demonstrate the results with several derivatives from the European market.

Stanislav Uryasev, University of Florida
Risk Management Using Conditional Value-at-Risk

Value-at-Risk (VaR), a widely used performance measure, answers the question:
what is the maximum loss with a specified confidence level? Although VaR is a very popular measure of risk, it has undesirable properties such as the lack of sub-additivity, i.e., VaR of a portfolio with two instruments may be greater than the sum of individual VaRs of these two instruments. Also, VaR is difficult to optimize when calculated using scenarios. In this case, VaR is non-convex, non-smooth as a function of positions, and it has multiple local extrema.

An alternative measure of loss, with more attractive properties, is Conditional Value-at-Risk (CVaR), see [6,7,8]. CVaR coincides in many special cases with Upper CVaR, which is the conditional expectation of losses exceeding VaR (also called Mean Excess Loss and Expected Shortfall), see [7]. However, Acerbi et al. [1,2] recently redefined Expected Shortfall in a manner consistent with the CVaR definition. Acerbi et al. [1,2] proved several important mathematical results on properties of CVaR, including asymptotic convergence of sample estimates to CVaR.

CVaR, is a coherent measure of risk [5,7] (sub-additive, convex, and other nice mathematical properties). CVaR can be represented as a weighted average of VaR and Upper CVaR. This seems surprising, in the face of neither VaR nor Upper CVaR being coherent. The weights arise from the particular way that CVaR "splits the atom" of probability at the VaR value, when one exists.

CVaR can be used in conjunction with VaR and is applicable to the estimation of risks with non-symmetric return-loss distributions. Although CVaR has not become a standard in the finance industry, it is likely to play a major role. CVaR is able to quantify dangers beyond value-at-risk, [1,6,7,8,10]. CVaR can be optimized using linear programming, which allows handling portfolios with very large numbers of instruments and scenarios. Numerical experiments indicate that for symmetric distributions the minimization of CVaR also leads to near optimal solutions in VaR terms because CVaR is always greater than or equal to VaR, [6]. Moreover, when the return-loss distribution is normal, these two measures are equivalent [6,7], i.e., they provide the same optimal portfolio. However, for skewed distributions, VaR optimal and CVaR optimal portfolios may be very different, [10]. Similar to the Markowitz mean-variance approach, CVaR can be used in return-risk analyses. For instance, we can calculate a portfolio with a specified return and minimal CVaR. Alternatively, we can constrain CVaR and find a portfolio with maximal return, see [4,7]. Also, we can specify several CVaR constraints simultaneously with various confidence levels (thereby shaping the loss distribution), which provides a flexible and powerful risk management tool.

Several case studies showed that risk optimization with the CVaR performance function and constraints can be done for large portfolios and a large number of scenarios with relatively small computational resources. For instance, a problem with 1,000 instruments and 20,000 scenarios can be optimized on a 700 MHz PC in less than one minute using the CPLEX LP solver. A case study on the hedging of a portfolio of options using CVaR is included in [6]. Also, the CVaR minimization approach was applied to the credit risk management of a portfolio of bonds, [3]. A case study on optimization of a portfolio of stocks with CVaR constraints is included in [4]. The numerical efficiency and stability of CVaR calculations are illustrated with an example of index tracking in [7]. Several related papers on probabilistic constrained optimization are included in [8].

Yong Wang, RBC Financial Group, Toronto
Risk decomposition

Our presentation includes two parts. In the first part, I will discuss some problems, which arise from our trading activities. In particular, I will discuss the exotic type transactions such as variance swap, volatility swaps, and correlation swaps etc. In the second part of the presentation, my colleague, Quan Zhao, will talk about the Risk Decomposition using orthogonal arrays.

Tony Ware, University of Calgary
Partial Differential Equation Techniques for Energy Contracts

We consider the use of partial differential equation-based models for energy contracts. In particular, we describe a semi-Lagrangian finite-element method solving the equations that arise in the context of one- and two-factor models. We demonstrate the application of this method to the pricing of various types of swing options, and make use of the results to explore the properties of swing contracts.

Petter Wiberg, University of Toronto
Dimension reduction in the computation of value-at-risk

The value-at-risk is the maximum loss that a portfolio might suffer over a given holding period with a certain confidence level. In recent years, value-at-risk has become a benchmark for measuring financial risk used by both practitioners and regulators. In this seminar, we discuss value-at-risk from a modeling and simulation perspective. We present a new efficient algorithm for computing value-at-risk and the value-at-risk gradient for portfolios of derivative securities. In particular, we discuss dimensional reduction of the model, and present some recent results on perturbation theory and applications to hedging of derivatives portfolios.


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