Martin Widdicks

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.

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.

Bankruptcy Probabilities Inferred from Option Prices

In times of financial crisis, solvency concerns are reflected in market prices for financial instruments. Credit default swap (CDS) spreads provide a direct measure of the market’s (risk-neutral) expectation regarding a firm’s probability of bankruptcy. The options market is another place where investors’ projections of future credit conditions can be seen, in high prices for deep out-of-the-money puts, for example. Although CDS spreads are available for horizons of 1 to 10 years and more, option maturities are much shorter, offering a window onto short-term expectations. Moreover, unlike CDS, the availability of market prices for multiple options with the same expiration but different strike prices allows an entire risk-neutral density (RND) for the future stock price to be extracted. In this article, the authors extend the strategy of fitting the RND as a mixture of two lognormals by adding a discrete probability of bankruptcy. Applying the approach to analyze the RNDs from equity options on Bear Stearns, Lehman, Merrill Lynch, and AIG in the run-up to their 2008 credit events (two of which ended in forced mergers and the other two in insolvency), the authors find that the mixture of lognormals plus bankruptcy RND fits the options market data better than alternative models. They also show that the implied short-run bankruptcy probabilities from equity options aligned perfectly with the one-year CDS spreads and also that the RND default probabilities for J.P. Morgan and Bank of America, which took over the failing Bear Stearns and Merrill Lynch, respectively, were much lower than for the four firms that experienced credit events.

A Model of Equity Based Compensation with Tax

In this paper, we develop a two-stage continuous time model of employee stock option (ESO) valuation under different tax regimes. We show that tax rules can have significant effects on ESO exercise behavior. In addition, we find that incentive stock options (ISO) are the optimal form of compensation for all levels of employees in the UK. In the US, restricted stock plans are preferred, and tax breaks offered by incentive schemes are only beneficial to employees with high liquid wealth (or small option holdings relative to wealth) or low risk aversion. We also analyze 83b elections for restricted stock plans in the US and find that making an election is a sub-optimal decision for both the employee and the firm.

Predictive Power of Option-Implied Densities from High-Frequency Data

Duffie, Pan, and Singletons (2000) model is used to estimate implied densities using daily and high-frequency FTSE 100 index option contracts from 2005 to 2009. The empirical results suggest the following phenomena during the financial crisis: (1) more negative relationships between variance jumps and price jumps; (2) a larger magnitude of the negative mean of price jumps; (3) a larger variance of price jumps and a larger mean of variance jumps; and (4) a higher jump intensity. Further findings are as follows: (1) high-frequency data provide superior predictive power; and (2) RNDs exhibit satisfactory predictive power for option expiration dates.