Quantitative Analysis in Financial Markets

The following sections are included:

• INTRODUCTION

• ACKNOWLEDGEMENTS

• THE CONTRIBUTORS

• CONTENTS

In this article, we suggest an efficient method of approximating a general, multivariate lognormal distribution by a multivariate binomial process. There are two important features of such multivariate distributions. First, the state variables may have volatilities that change over time. Second, the two or more relevant state variables involved may covary with each other in a specified manner, with a time-varying covariance structure. We discuss the asymptotic properties of the resulting processes and show how the methodology can be used to value a complex, multiple exerciseable option whose payoff depends on the prices of two assets.

A systematic time-domain approach is presented to the derivation of closed-form solutions for interest-rate contingent assets. A financial system "asset - interest rate market" is assumed to follow an any-factor system of linear stochastic differential equations and some piece-wise defined algebraic equations for the payoffs. Closed-form solutions are expressed through the first two statistical moments of the state variables that are proven to satisfy a deterministic linear system of ordinary differential equations.

A number of examples are given to illustrate the method's effectiveness. With no restrictions on the number of factors, solutions are derived for randomly amortizing loans and deposits; any European-style swaptions, caps, and floors; conversion options; Asian-style options, etc. A two-factor arbitrage-free Gaussian term structure is introduced and analyzed.

We present a dynamical systems approach for modelling the term structure of interest rates based on a linear differential equation under uncertainty. In contrast to a stochastic process we introduce impulse or point-impulse perturbations on either (a), the spot (shortest-term, risk neutral) interest rate as the unknown function, or (b), its integral, namely the yield function, or both simultaneously. Parameters are estimated by minimizing the maximum absolute value of the measurement errors. Termed the Optimal Observation Problem, OOP, it defines our norm of uncertainty, in contrast to the expectation operator for a stochastic process. Beyond the learning period (the current time), the solved-for spot rate function becomes the forecast of the unobservable function in a future period, while its integral should approximate the yield function well. Non-arbitrage is addressed by providing a necessary condition expressed as constraints in the OOP, under which non-arbitrage is guaranteed. The property of mean-reversion is also preserved, and functional estimates are provided for the market price of risk. Analogous concepts to “drift” and “volatility” are treated in a manner that provides a criterion for the choice of perturbation to employ in a given real situation. Additional constraints, if necessary, guarantee non-negative short and forward rates, a property not automatically fulfilled in the stochastic case. We test the approach empirically with daily Treasury yield curve rates data, mainly for discount bonds having 3- to 6-month maturities over observation periods of up to one year. Computational results are reported for many numerical experiments together with some financial interpretations, see http://kwel.biz.uiowa.edu/

We present a framework for calibrating a pricing model to a prescribed set of option prices quoted in the market. Our algorithm yields an arbitrage-free diffusion process that minimizes the relative entropy distance to a prior diffusion. We solve a constrained (minimax) optimal control problem using a finite-difference scheme for a Bellman parabolic equation combined with a gradient-based optimization routine. The number of unknowns in the optimization step is equal to the number of option prices that need to be matched, and is independent of the mesh-size used for the scheme. This results in an efficient, non-parametric calibration method that can match an arbitrary number of option prices to any desired degree of accuracy. The algorithm can be used to interpolate, both in strike and expiration date, between implied volatilities of traded options and to price exotics. The stability and qualitative properties of the computed volatility surface are discussed, including the effect of the Bayesian prior on the shape of the surface and on the implied volatility smile/skew. The method is illustrated by calibrating to market prices of Dollar-Deutschemark over-the-counter options and computing interpolated implied-volatility curves.

This paper develops static hedges for several exotic options using standard options. The method used relies on a relationship between European puts and calls with different strike prices. The analysis allows for constant volatility or for volatility smiles or frowns.

We first recall the well-known expression of the price of barrier options, and compute double barrier options by the mean of the iterated mirror principle. The formula for double barriers provides an intraday volatility estimator from the information of high-low-close prices. Then we give explicit formulas for the probability distribution function and the expectation of the exit time of single and double barrier options. These formulas allow to price time independent and time dependent rebates. They are also helpful to hedge barrier and double barrier options, when taking into account variations of the term structure of interest rates and of volatility. We also compute the price of rebates of double knock-out options that depend on which barrier is hit first, and of the BOOST, an option which pays the time spent in a corridor. All these formulas are either in closed form or double infinite series which converge like e^−αn2.

Arithmetic Asian options are difficult to price and hedge as they do not have closed-form analytic solutions. The main theoretical reason for this difficulty is that the payoff depends on the finite sum of correlated lognormal variables, which is not lognormal and for which there is no recognizable probability density function. We use elementary techniques to derive the probability density function of the infinite sum of correlated lognormal random variables and show that it is reciprocal gamma distributed, under suitable parameter restrictions. A random variable is reciprocal gamma distributed if its inverse is gamma distributed. We use this result to approximate the finite sum of correlated lognormal variables and then value arithmetic Asian options using the reciprocal gamma distribution as the state-price density function. We thus obtain a closed-form analytic expression for the value of an arithmetic Asian option, where the cumulative density function of the gamma distribution, G(d) in our formula, plays the exact same role as N(d) in the Black-Scholes/Merton formula. In addition to being theoretically justified and exact in the limit, we compare our method against other algorithms in the literature and show our method is quicker, at least as accurate, and, in our opinion, more intuitive and pedagogically appealing than any previously published result, especially when applied to high yielding currency options.

In this article, we present a new method for pricing and hedging American options along with an efficient implementation procedure. The proposed method is efficient and accurate in computing both option values and various option hedge parameters. We demonstrate the computational accuracy and efficiency of this numerical procedure in relation to other competing approaches. We also suggest how the method can be applied to the case of any American option for which a closed-form solution exists for the corresponding European option.

Given noisy data, function estimation is considered when the unknown function is known a priori to consist convex or concave on each of a small number of regions where the function. When the number of regions is unknown, the model selection problem is to determine the number of convexity change points. For kernel estimates in Gaussian noise, the number of false change points is evaluated as a function of the smoothing parameter. To insure that the number of false convexity change points tends to zero, the smoothing level must be larger than is generically optimal for minimizing the mean integrated square error (MISE). A two-stage estimator is proposed and shown to achieve the optimal rate of convergence of the MISE. In the first stage, the number and location of the change points is estimated using strong smoothing. In the second stage, a constrained smoothing spline fit is performed with the smoothing level chosen to minimize the MISE. The imposed constraint is that a single change point occur in a region about each empirical change point from the first-stage estimate. This constraint is equivalent to the requirement that the third derivative of the second-stage estimate has a single sign in a small neighborhood about each first-stage change point. The change points from the second stage are near the first-stage change points, but need not be at the identical locations.

The following sections are included:

• Bias-Versus-Variance Trade-off

• Local Error and Optimal Kernels

• How to Select the Halfwidth

• Plug-in-Derivative Estimates of the Local Halfwidth

• Data-adaptive Smoothing

• Further Reading

• Acknowledgments

• References

We construct a statistical model for term structure of implied volatility of currency options based on daily historical data for 13 currency pairs in a 19-month period. We examine the joint evolution of 1 month, 2 month, 3 month, 6 month and 1 year 50 Δ options in all the currency pairs. We show that from these five observable variables, there exist three uncorrelated state variables (principal components) which account for the parallel movement, slope oscillation, and curvature of the term structure and which explain, on average, the movements of the term structure of volatility to more than 95% in all cases. We test and construct an exponential ARCH, or E-ARCH, model for each state variable. One of the applications of this model is to produce confidence bands for the term structure of volatility.

The conditional volatility of foreign exchange rates can be predicted with GARCH models, and with implied volatility extracted from currency options. This paper investigates whether the difference in these predictions is economically meaningful. In an efficient market, after accounting for transaction costs and risk, no trading strategy should earn abnormal risk-adjusted returns. In the absence of transaction costs, both the delta-neutral and the straddle trading strategies lead to significant positive economic profits against the option market, regardless of which volatility prediction method is used. The agent using the Implied Stochastic Volatility Regression method (ISVR) earns larger profits than the agent using the GARCH method. However, after accounting for transaction costs assumed to equal one percent of market prices, observed profits are not significantly different from zero in most trading strategies; the exception is for an agent using the ISVR method with a 5% price filter.

The following sections are included:

• Introduction

• Portfolio-Based Risk Pricing

• Parameters Estimation
  • Parameters for the GJR-GARCH(l,l)/jump-diffusion model

• Pricing S&P 500 Index Put Options
  • Option parameters for one-year European puts
  • From distributions to prices
  • Volatility smile
  • Comparison with historical market price of puts in 1988–1997
  • Five-year European puts

• Conclusion

• References

This paper proves the existence of a general equilibrium in a financial model with transaction costs. A general equilibrium is shown to exist in a model with convex trading technology, in which the agents include consumers, production firms, brokers or dealers. When the trading technology is non-convex, an individual approximate equilibrium, introduced by Heller and Starr (1976), is proved in the above model. And, moreover, under a further assumption of finite p-convexity on the commodity excess demand correspondence, a general equilibrium for a non-convex exchange economy is obtained for an economy with consumers, brokers or dealers.

A general method is presented for constructing dynamic equity portfolios through the use of mathematical generating functions. The return on these functionally generated portfolios is related to the return on the market portfolio by a stochastic differential equation. Under appropriate conditions, this equation can be used to establish a dominance relationship between a functionally generated portfolio and the market portfolio.