Quantitative Analysis in Financial Markets

The following sections are included:

• INTRODUCTION

• ACKNOWLEDGEMENTS

• THE CONTRIBUTORS

• CONTENTS

This paper applies to interest rate models the theoretical method developed in Aït-Sahalia (1998) to generate accurate closed form approximations to the transition function of an arbitrary diffusion. While the main focus of this paper is on the maximum-likelihood estimation of interest rate models with otherwise unknown transition functions, applications to the valuation of derivative securities are also briefly discussed.

Most approaches in forecasting merely try to predict the next value of the time series. In contrast, this paper presents a framework to predict the full probability distribution. It is expressed as a mixture model: the dynamics of the individual states is modeled with so-called "experts" (potentially non-linear neural networks), and the dynamics between the states is modeled using a hidden Markov approach. The full density predictions are obtained by a weighted superposition of the individual densities of each expert. This model class is called "hidden Markov experts".

Results are presented for daily S&P500 data. While the predictive accuracy of the mean does not improve over simpler models, evaluating the prediction of the full density shows a clear out-of-sample improvement both over a simple GARCH(1,1) model (which assumes Gaussian distributed returns) and over a "gated experts" model (which expresses the weighting for each state non-recursively as a function of external inputs). Several interpretations are given: the blending of supervised and unsupervised learning, the discovery of hidden states, the combination of forecasts, the specialization of experts, the removal of outliers, and the persistence of volatility.

Continuous-time stochastic processes have become central to many disciplines, yet the fact that they are approximations to physically realizable phenomena is often overlooked. We quantify one aspect of the approximation errors of continuous-time models by investigating the replication errors that arise from delta hedging derivative securities in discrete time. We characterize the asymptotic distribution of these replication errors and their joint distribution with other assets as the number of discrete time periods increases. We introduce the notion of temporal granularity for continuous-time stochastic processes, which allows us to quantify the extent to which discrete-time implementations of continuous-time models can track the payoff of a derivative security. We show that granularity is a function of the contract specifications of the derivative security, and of the degree of market completeness. We derive closed form expressions for the granularity of geometric Brownian motion and of an Ornstein–Uhlenbeck process for call and put options, and perform Monte Carlo simulations that illustrate the practical relevance of granularity.

This paper argues that asset price processes arising from market clearing conditions should be modeled as pure jump processes, with no continuous martingale component. However, we show that continuity and normality can always be obtained after a time change. We study various examples of time changes and show that in all cases they are related to measures of economic activity. For the most general class of processes, the time change is a size-weighted sum of order arrivals. The paper provides a number of new processes for modeling prices. Characteristic functions for these processes are also given in closed form.

We present a family of hedging strategies for a European derivative security in a stochastic volatility environment. The strategies are robust to specification of the volatility process and do not need a parametric description of it or estimation of the volatility risk premium. They allow the hedger to control the probability of hedging success according to risk aversion. The formula exploits the separation between the time-scale of asset price fluctuation (ticks) and the longer time-scale over which volatility fluctuates, that is, the observed "persistence" of volatility. We run simulations that demonstrate the effectiveness of the strategies over the classical Black-Scholes strategy.

Using only the implied volatility smile of a single maturity T and an assumption of path-independence, we analytically determine the risk-neutral stock price process and the local volatility surface up to an arbitrary horizon T′ ≥ T. Our path-independence assumption requires that each positive future stock price S_t is a function of only time t and the level W_t of the driving standard Brownian motion (SBM) for all t ∈ (0, T′). Using the T-maturity option prices, we identify this stock pricing function and thereby analytically determine the risk-neutral process for stock prices. Our path-independence assumption also implies that local volatility is a function of the stock price and time which can be explicitly represented in terms of the known stock pricing function. Finally, we derive analytic valuation formulae for standard and exotic options which are consistent with the observed T-maturity smile.

Using market European option prices, a method for computing a smooth local volatility function in a 1-factor continuous diffusion model is proposed. Smoothness is introduced to facilitate accurate approximation of the local volatility function from a finite set of observation data. Assuming that the underlying indeed follows a 1-factor model, it is emphasized that accurately approximating the local volatility function prescribing the 1-factor model is crucial in hedging even simple European options, and pricing exotic options. A spline functional approach is used: the local volatility function is represented by a spline whose values at chosen knots are determined by solving a constrained nonlinear optimization problem. The optimization formulation is amenable to various option evaluation methods; a partial differential equation implementation is discussed. Using a synthetic European call option example, we illustrate the capability of the proposed method in reconstructing the unknown local volatility function. Accuracy of pricing and hedging is also illustrated. Moreover, it is demonstrated that, using different implied volatilities for options with different strikes/maturities can produce erroneous hedge factors if the underlying follows a 1-factor model. In addition, real market European call option data on the S&P 500 stock index is used to compute the local volatility function; stability of the approach is demonstrated.

This paper constructs a model for the evolution of a risky security that is consistent with a set of observed call option prices. It explicitly treats the fact that only a discrete data set can be observed in practice. The framework is general and allows for state dependent volatility and jumps. The theoretical properties are studied. An easy procedure to check for arbitrage opportunities in market data is proven and then used to ensure the feasibility of our approach. The implementation is discussed: testing on market data reveals a U-shaped form for the "local volatility" depending on the state and surprisingly, a large probability for strong price movements.

A general approach for calibrating Monte Carlo models to the market prices of benchmark securities is presented. Starting from a given model for market dynamics (price diffusion, rate diffusion, etc.), the algorithm corrects for price-misspecifications and finite-sample effects in the simulation by assigning "probability weights" to the simulated paths. The choice of weights is done by minimizing the Kullback–Leibler relative entropy distance of the posterior measure to the empirical measure. The resulting ensemble prices the given set of benchmark instruments exactly or in the sense of least-squares. We discuss pricing and hedging in the context of these weighted Monte Carlo models. A significant reduction of variance is demonstrated theoretically as well as numerically. Concrete applications to the calibration of stochastic volatility models and term-structure models with up to 40 benchmark instruments are presented. The construction of implied volatility surfaces and forward-rate curves and the pricing and hedging of exotic options are investigated through several examples.

A new valuation method is proposed that can be mathematically viewed as a numerical shortcut to approximately solve the partial differential equation written for Expected Instantaneous Return. The method derives OAS as "static spread plus cost of convexity" and is based on some simplified parametric assumption about the static spread's time behavior. The modeling and numerical procedure details are disclaimed with proven accuracy and time efficiency in option-adjusted valuation.

The method is especially effective for multi-scenario pricing, portfolio pricing, risk management and reporting, and for quantifying the impact of "non-traded" factors on reward and risk of holding mortgages. A systematic methodology for comprehensive multi-factor analysis is covered.

We use simulation to develop a Markov chain approximation for the value of caplets and Bermudan interest rate derivatives in the Market Model developed by Brace, Gatarek, and Musiela (1995) and Jamshidian (1996a, b). One and two factor versions of the Market Model were numerically studied. Our approach yields numerical values for caplets which are in close agreement with analytical solutions. We also provide numerical solutions for several Bermudan swaptions.

It is well-known that self-similarity is a powerful concept which reveals the fundamental laws of physic. This concept has important financial applications as well. It allows one to study the properties of derivatives from a unified prospective and significantly simplifies their mathematical modelling and hedging. In the present paper, I show how the concept of self-similarity can be used in order to find similarities between various types of path-dependent options and price them in the unified framework. Specifically, I consider lookback, passport, Asian, and imperfectly hedged European options. I present some new important valuation formulas and rederive a few known ones by elementary means.

This article presents an innovative approach to measuring intra-day VaR that combines the use of a robust parametric technique, Gamma VaR,^a with Monte Carlo simulation to capitalize on the respective strengths of these models. The simulation is optimized by using parametric VaR results to limit the required number of portfolio revaluations to those random scenarios that are statistically relevant given the greek-estimated profit and loss distribution, and as a variance reduction tool to minimize the standard Monte Carlo error term. As the results presented here will show, these techniques, combined with portfolio and market risk factor compression, significantly enhance the performance and precision of the Monte Carlo engine. Although VaR alone, no matter how sophisticated the model, is not sufficient to effectively capture all possible market moves, it is an invaluable intra-day tool for risk managers.

The following sections are included:

• Introduction

• Background

• Model Formulation
  • The absolute deviation trade-off model for fixed-income portfolio selection
  • Step-shaped programs
  • Decomposition for step-shaped programs
  • Variable decomposition for linear step-shaped programs
  • Decomposition for disjunctive step-shaped programs
    • Disjunctive linear programs
    • Combined depth-first search and decomposition

• References