David P. Newton

Universal option valuation using quadrature methods

Abstract This paper proposes and develops a novel, simple, widely applicable numerical approach for option pricing based on quadrature methods. Though in some ways similar to lattice or finite-difference schemes, it possesses exceptional accuracy and speed. Discretely monitored options are valued with only one timestep between observations, and nodes can be perfectly placed in relation to discontinuities. Convergence is improved greatly; in the extrapolated scheme, a doubling of points can reduce error by a factor of 256. Complex problems (e.g., fixed-strike lookback discrete barrier options) can be evaluated accurately and orders of magnitude faster than by existing methods.

UK Fixed Rate Repayment Mortgage and Mortgage Indemnity Valuation

We use a mean-reverting interest rate model and a lognormal house price diffusion model to evaluate British fixed rate repayment mortgage contracts with (embedded) default and prepayment options. The model also provides values for capped mortgage indemnity guarantees and the corresponding (residual) lenders coinsurance. Since the partial differential equation incorporating the general features of these mortgage contracts does not have a closed-form solution, an explicit finite difference method is used for the valuation (and sensitivity) results, with solution improvements to deal with error bounds. Then we provide graphical representations of each mortgage component as a function of house prices and interest rate levels, along with interpretations of the analysis. We calculate precisely the lenders (residual) exposure to house price risk, given the borrowers options, house and interest rate uncertainty, and customary mortgage indemnity insurance for high loan/collateral ratio mortgages. Copyright 2002 by the American Real Estate and Urban Economics Association.

Enhancing the Accuracy of Pricing American and Bermudan Options

A technique recently developed by Longstaff and Schwartz (LS) significantly enhances the Monte Carlo technology for pricing American options, by using regression to streamline the analysis of the subsidiary future paths in modeling the early exercise decisions. But the procedure still requires a large amount of computation and the degree of difficulty explodes as the number of stochastic factors in the problem increases. In this article, the authors show several alterations to the LS approach that can increase its efficiency substantially. The most important involves setting up the problem as a kind of Monte Carlo analysis of Monte Carlo models. For each run, they construct three estimates of the option value using different numbers of simulated paths, for example 1,000, 2,000, and 4,000 paths. This process is then repeated for many runs. Averaging across these multiple Monte Carlo estimates produces a set of three average values that are then used in an extrapolation procedure to estimate the option value that would be produced with an infinite number of paths. The improvement in performance from this two-part strategy is striking, as the authors demonstrate by pricing a Bermudan put option on five underlyings.

Curtailing the range for lattice and grid methods

Numerical option pricing methods discretize a continuous-time continuous-state state space and approximate the solution to the continuous problem with the results of local computations on all of the nodes of the discrete lattice. This procedure converges to the correct solution as the time and price intervals go to zero, but at the cost of larger and larger numbers of calculations. Many of the calculations on a fine lattice are irrelevant to valuing the derivative. They may occur at stock prices that have virtually no probability of being reached. Or the stock price may be reachable from the initial price, but be so far away from the strike price that the option is either (almost) certain to be out of the money at maturity, meaning it is worthless now, or (almost) certain to be exercised at maturity, meaning it can be priced as a forward now, with no need for any further calculations. Eliminating the superfluous calculations at these nodes can greatly improve the performance of a lattice valuation technique, and the possible increase in computation speed becomes much greater as the number of stochastic variables in the lattice increases. Here, Andricopoulos et al., show how to curtail the price range for a numerical valuation technique and demonstrate the substantial improvement in speed that it can produce.

Singular Perturbation Techniques Applied to Multiasset Option Pricing

It is well known that option valuation problems with multiple-state variables are often problematic to solve. When valuing options using lattice-type techniques such as finite-difference methods, the curse of dimensionality ensures that additional-state variables lead to exponential increases in computational effort. Monte Carlo methods are immune from this curse but, despite advances, require a great deal of adaptation to treat early exercise features. Here the multiunderlying asset Black–Scholes problem, including early exercise, is studied using the tools of singular perturbation analysis. This considerably simplifies the pricing problem by decomposing the multi-dimensional problem into a series of lower-dimensional problems that are far simpler and faster to solve than the full, high-dimensional problem. This paper explains how to apply these singular perturbation techniques and explores the significant efficiency improvement from such an approach.

Involuntary excess reserves, the reserve requirements and credit rationing in China

Using a sample of 95 banks that covers the period 2000–2011, this article examines Chinese banks’ credit lending behaviour in response to the changes in the reserve requirement ratio in the presence of involuntary excess reserves (IERs) in the banking system. The study finds that Chinese banks with positive IERs one period after a reserve requirement shock experience a significantly increased credit supply in response to an increase in reserve requirement ratio. However, the reserve requirements have no significant impact on the credit supply in Chinese banks that have negative IERs one period after a reserve requirement shock. This article sheds lights on the effectiveness of Chinese monetary policy, which uses reserve requirements as the primary tool to sterilize excess liquidity and restrain credit expansion.

A Bridge between American and European Options: The “Ameripean” Delayed-Exercise Model

A new class of option-pricing model is discussed, motivated initially by the practical observation that contracts with embedded options are not always exercised immediately when an implicit barrier is breached; this may occur for a number of reasons, for example linked to behavior of the investor. Rather than using a conventional barrier, this option class model takes the first touching of the payoff function by the option value as the start of a waiting period before exercise. This presents itself as a free-boundary problem, similar to, but somewhat more complicated than, that found with the usual American option. It turns out that this gives insight into the dynamics of the American option itself, as the “Ameripean” delayed-exercise option model provides a fluid link between a European and an American option. It also prompts the development of an improved numerical technique, based on boundary-fitted coordinates, together with some useful asymptotic analyses (which give further insights into valuations).

Stock Returns and Profitability Forecasting by Quantile Regression

We show that quantile regression is better than ordinary-least-squares (OLS) regression in forecasting profitability for a range of profitability measures following the conventional setup of the accounting literature, including the mean absolute forecast error (MAFE) evaluation criterion. Additionally, we perform both a simulated-data and an archival-data analysis to examine how the forecasting performance of quantile regression against OLS changes with the shape of the profitability distribution. Considering the MAFE and mean squared forecast error (MSFE) criteria together, quantile regression is more accurate relative to OLS when the profitability to be forecast has a heavier-tailed distribution. An application of the distributional shape analysis framework to cash flows forecasting demonstrates the usefulness of the framework beyond profitability forecasting, supports the notion of an inverted-U-shape effect of distribution asymmetry on the incremental forecasting accuracy of quantile regression, and provides additional empirical evidence on the positive effect of tail-heaviness.

Option Pricing via QUAD: From Black–Scholes–Merton to Heston with Jumps

The Black–Scholes model is the rare closed-form formula for pricing options, but its shortcomings are well known. Adding stochastic volatility, American or Bermudan early exercise, non-diffusive jumps in the returns process, and/or other “exotic” payoff features quickly takes one into the realm where approximate answers must by computed by numerical methods. Binomial and other lattice models, Monte Carlo simulation, and numerical solution of the valuation partial differential equation are the standard techniques. They are well known to eventually arrive at as close an approximation to the exact model value as one wants but can require a large amount of calculation and execution time to get there. In this article, the authors present what amounts to the culmination of a series of papers on using quadrature methods to solve option problems of increasing complexity. Quadrature entails modeling and approximating the transition densities between critical points in the option’s lifetime. Depending on the specifics of the option’s payoff, looking at only a few points in time can greatly reduce the amount of calculation needed to price it. For example, a Bermudan option that can be exercised at three possible early dates needs quadrature calculations for only those three dates plus expiration, while other numerical methods require calculations for every date and possible asset price. This article summarizes how to apply quadrature to standard option problems, from plain-vanilla Black–Scholes up to Bermudan and American options under a stochastic volatility jump-diffusion returns process. The procedure can be speeded up even more by use of Richardson extrapolation and caching techniques that allow reuse of results calculated in intermediate steps. The range of derivatives models that can be valued extremely efficiently using quadrature is very broad.