Option Pricing Equations Research Articles

A Finite Element Like Scheme for Integro-Partial Differential Hamilton–Jacobi–Bellman Equations

We construct a finite element like scheme for fully nonlinear integro-partial differential equations arising in optimal control of jump-processes. Special cases of these equations include optimal portfolio and option pricing equations in finance. The schemes are monotone and robust. We prove that they converge in very general situations, including degenerate equations, multiple dimensions, relatively low regularity of the data, and for most (if not all) types of jump-models used in finance. In all cases we provide (probably optimal) error bounds. These bounds apply when grids are unstructured and integral terms are very singular, two features that are new or highly unusual in this setting.

Dispersion trading: Empirical evidence from U.S. options markets

This paper develops empirical evidence on the viability of a form of volatility trading known as “dispersion trading.” The results shed light on the efficiency with which U.S. options markets price volatility.Using end-of-day implied volatilities extracted from equity option prices for the stocks that comprise the S&P 500, the implied volatility of the S&P 500 is computed using a modification of the Markowitz variance equation. This Markowitz-implied volatility is then compared to the implied volatility of the S&P 500 extracted directly from index options on the S&P 500. These contemporaneous measures of implied volatility are then examined for exploitable discrepancies both with and without transaction costs. The study covers the period October 31, 2005 through November 1, 2007.It is shown that, from a trader's perspective, index option implied volatility tended to be more often “rich” and component volatilities tended to be more often “cheap.” Nevertheless, there were times when the opposite was true; suggesting that potential dispersion trades can run in either direction.

Asymptotic Pricing of Commodity Derivatives using Stochastic Volatility Spot Models

It is well known that stochastic volatility is an essential feature of commodity spot prices. By using methods of singular perturbation theory, we obtain approximate but explicit closed‐form pricing equations for forward contracts and options on single‐ and two‐name forward prices. The expansion methodology is based on a fast mean‐reverting stochastic volatility driving factor and leads to pricing results in terms of constant volatility prices, their Deltas and their Delta‐Gammas. Both the standard single‐factor mean‐reverting spot model and a two‐factor generalization, in which the long‐run mean is itself mean‐reverting, are extended to include stochastic volatility and each is analysed in detail. The stochastic volatility corrections can be used to efficiently calibrate option prices and compute sensitivities.

Systematic Risk and the Price Structure of Individual Equity Options

This study demonstrates the impact of systematic risk on the prices of individual equity options. The option prices are characterized by the level and slope of implied volatility curves, and the systematic risk is measured as the proportion of systematic variance in the total variance. Using daily option quotes on the S, and P 100 index and its 30 largest component stocks, we show that after controlling for the underlying asset's total risk, a higher amount of systematic risk leads to a higher level of implied volatility and a steeper slope of the implied volatility curve. Thus, systematic risk proportion can help differentiate the price structure across individual equity options. The Author 2008. Published by Oxford University Press on behalf of The Society for Financial Studies. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org, Oxford University Press.

Simple Robust Linkages between CDS and Equity Options

We test a theory that provides a simple and robust link between market prices of credit default swaps (CDS) and equity options. The theory links the CDS spread to the market prices of vertical spreads of American put options which are sufficiently out-of-the-money. T he linkage is established under a general class of stock price dynamics for which the default event affects only the state space in which stock prices evolve. Specifically, stock prices are assumed to be bounded below by a barrier B > 0 strictly before the default event, and to drop below an alternative barrier A < B at the default time. We allow random stock price evolution after default, so long as it is bounded above by A. We suppose that investors can take a static position in at least two co-terminal American put options struck within the default corridor [A, B]. We show that a vertical spread of such options scaled by the spread between the two strikes replicates a standardized credit insurance contract that pays one dollar at default whenever the company defaults prior to the option expiry and zero otherwise. Given the above state space behavior, we show that this simple replicating strategy is robust to the details of the pre-default stock price dynamics, the post-d efault stock price dynamics, interest rate dynamics, and default arrival rate fluctuations. We use the value of the American put spread to infer risk-neutral default probabilities and compare them to the default probabilities estimated from CDS spreads on the same reference company. Collecting data from both markets on several reference companies with significant default probabilities, we identify a strong correlation between the defa ult probabilities inferred from the two markets, and find that deviations between the two estimates help predict f uture movements in both markets.

Numerical solution of linear and nonlinear Black–Scholes option pricing equations

This paper deals with the numerical solution of Black–Scholes option pricing partial differential equations by means of semidiscretization technique. For the linear case a fourth-order discretization with respect to the underlying asset variable allows a highly accurate approximation of the solution. For the nonlinear case of interest modeling option pricing with transaction costs, semidiscretization technique provides a competitive numerical solution with respect to others recently given in [B. Düring, M. Fournier, A. Jüngel, Convergence of a high order compact finite difference scheme for a nonlinear Black–Scholes equation, Esaim–Math. Modelling Numer. Anal.–Modelisation Mathematique et Analyse Numerique 38 (2004) 359–369; B. Düring, Black–Scholes type equations: mathematical analysis, parameter identification & numerical solution, Dissertation, University Mainz, July 2005].

Smiling less at LIFFE

AbstractThis study investigates the structure of the implied volatility smile, using the prices of equity options traded on the LIFFE. First, the slope of the implied volatility curve is significantly negative for both individual stocks and index options, and the slope is less negative for longer‐term options. The implied volatility skew can be described by risk‐neutral skewness and kurtosis, with the former having the first‐order effect. Moreover, the implied volatility skew for individual stock options is less severe than for index options. Finally, the relationship between the real and risk‐neutral moments implied in option prices is significant. The results indicate that, for equity options traded on the LIFFE, the slope of the implied volatility skew is flatter than that on the Chicago Board of Exchange (CBOE). © 2008 Wiley Periodicals, Inc. Jrl Fut Mark 28:57–81, 2008

ENERGY SPOT PRICE MODELS AND SPREAD OPTIONS PRICING

In this article, we construct forward price curves and value a class of two asset exchange options for energy commodities. We model the spot prices using an affine two-factor mean-reverting process with and without jumps. Within this modeling framework, we obtain closed form results for the forward prices in terms of elementary functions. Through measure changes induced by the forward price process, we further obtain closed form pricing equations for spread options on the forward prices. For completeness, we address both an Actuarial and a risk-neutral approach to the valuation problem. Finally, we provide a calibration procedure and calibrate our model to the NYMEX Light Sweet Crude Oil spot and futures data, allowing us to extract the implied market prices of risk for this commodity.

Methods for the rapid solution of the pricing PIDEs in exponential and Merton models

The pricing equations for options on assets that follow jump-diffusion processes contain integrals in addition to the usual differential terms. These integrals usually make such equations expensive to solve numerically. Although Fast Fourier Transform methods can be used to to evaluate the integrals at all mesh points simultaneously, they are costly since the computational region must be extended in order to avoid problems with wrap around. Other numerical difficulties arise when the density function for the jump size is not smooth, as in the Kou double exponential model. We present new solution methods which are based on the fact that even when the problems contain time-dependent parameters the integrals often satisfy easily solved ordinary or parabolic partial differential equations. In particular, we show that by using the operator splitting method proposed by Andersen and Andreasen it is possible to reduce the solution of the pricing equation in the Kou and similar models to a sequence of ordinary differential equations at each time step. We discuss the methods and present results of numerical experiments.

Systematic Variance Risk and Firm Characteristics in the Equity Options Market

We construct synthetic variance swap returns from prices of traded options to investigate the pricing of systematic variance risk in the equity options market. Cross sectional tests reveal no evidence of a negative market variance risk premium. Furthermore, we show that a class of linear factor models cannot simultaneously explain index and equity option prices. In particular, equity options appear to be underpriced relative to index options. To exploit the mispricing, we analyze an investment strategy known as dispersion trading, which is implemented by going long a portfolio of equity options, and short a portfolio of index options. After transaction costs, the strategy generates a Sharpe ratio which is more than four times greater than that of the market. We also find that equity option prices are related to underlying firm characteristics: Options on small and value stocks are more expensive than options on large and growth stocks, respectively. We find that these results are not explained by di.erential exposure to risk factors, nor by market microstructure considerations. One interpretation is that investors overestimate risk on small and value stocks. This finding provides a new context for understanding the size and value anomalies in stock returns: It suggests that investors expect higher rates of return on small and value stocks because they perceive them to be riskier than they truly are.

The Firm Under Uncertainty: Theoretical Development and Empirical Tests

Our paper models and empirically tests the production behavior of firms operating under uncertainty. The study compares firms operating in perfect competition with those producing with product market power. Our model predicts that both types of companies produce less under uncertainty from the cost of risk. In addition, our model predicts that companies with market power have a greater market value, face more product market uncertainty, use less leverage but have similar levels of default risk as firms operating under perfect competition. We empirically test our model by comparing manufacturing firms that operate in close to perfect market competition having little product differentiation and operating in low concentrated industries versus those that have product market power using advertising and/or R&D and operating in highly concentrated industries. The empirical evidence supports our hypotheses. We also model and empirically investigate the effect of the option against value on the firm's performance that occurs to companies when the real product market changes greatly reduce or even eliminates firms' product markets. Long, Wang and Zhang (2006) originally presented this idea and tested it with the machine tool industry to investigate dynamic downward changes in real product markets and the resulting affect on individual firms' values. We also find that smaller companies, those with less capital market following are more likely to experience a product market shift going out of business. This is even after we controlled the default probability estimates based on Merton's Equity Option Pricing.

Integro-differential equations for foreign currency option prices in exponential Lévy models

We investigate the rational evaluation of foreign currency option price when the underlying price processes are described by exponential Lévy processes. The main contribution of our study is that we give the integro-differential equations for foreign currency option prices.

Estimating Systemic Risk in the International Financial System

Using a unique and comprehensive dataset, this paper develops and uses three distinct methods to quantify the risk of a systemic failure in the global banking system. We examine a sample of 334 banks (representing 80% of global bank equity) in 28 countries around 6 global financial crises (such as the Asian and Russian crises and September 11, 2001), and show that these crises did not create large probabilities of global financial system failure. First, we show that cumulative negative abnormal returns for the subset of banks not directly exposed to a negative shock (unexposed banks) rarely exceed a few percent. Second, we use structural models to obtain more precise point estimates of the likelihood of systemic failure. These estimates suggest that systemic risk is limited even during major financial crises. For example, maximum likelihood estimation of bank failure probabilities implied by equity prices suggests the Asian crisis induced less than a 1% increase in the probability of systemic failure. Third, we also obtain estimates of systemic risk implied by equity option prices of U.S. and European banks. The largest values are obtained for the Russian crisis and September 11 and these show increases in estimated average default probabilities of only around 1-2%. Taken together our results suggest statistically significant, but economically small, increases in systemic risk around even the worst financial crises of the last 10 years. Although policy responses are endogenous, the low estimated probabilities suggest that the distress of central bankers, regulators and politicians about the events we study may be overstated, and that current policy responses to financial crises and the existing institutional framework may be adequate to handle major macroeconomic events.

Efficient Pricing of Derivatives on Assets with Discrete Dividends

It is argued that due to inconsistencies in existing methods to approximate the prices of equity options on assets which pay out fixed cash dividends at future dates, a new approach to this problem may be useful. Logically consistent methods which are guaranteed to exclude arbitrage exist, but they are not very popular in practice due to their computational complexity. An algorithm is defined which is easy to understand, computationally efficient, and which guarantees to generate prices which exclude arbitrage possibilitites. It is shown that for the method to work a mild uniform convergence condition must be satisfied and this condition is indeed satisfied for standard European and American options. Numerical results testify to the accuracy and flexibility of the method.

Option pricing for the transformed-binomial class

This article generalizes the seminal Cox-Ross-Rubinstein (1979) binomial option pricing model to all members of the class of transformed-binomial pricing processes. The investigation addresses issues related with asset pricing modeling, hedging strategies, and option pricing. Formulas are derived for (a) replicating or hedging portfolios, (b) risk-neutral transformed-binomial probabilities, (c) limiting transformed-normal distributions, and (d) the value of contingent claims, including limiting analytical option pricing equations. The properties of the transformed-binomial class of asset pricing processes are also studied. The results of the article are illustrated with several examples. © 2006 Wiley Periodicals, Inc. Jrl. Fut Mark 26:759–787, 2006

Lie-algebraic approach for pricing moving barrier options with time-dependent parameters

In this paper we apply the Lie-algebraic technique for the valuation of moving barrier options with time-dependent parameters. The value of the underlying asset is assumed to follow the constant elasticity of variance (CEV) process. By exploiting the dynamical symmetry of the pricing partial differential equations, the new approach enables us to derive the analytical kernels of the pricing formulae straightforwardly, and thus provides an efficient way for computing the prices of the moving barrier options. The method is also able to provide tight upper and lower bounds for the exact prices of CEV barrier options with fixed barriers. In view of the CEV model being empirically considered to be a better candidate in equity option pricing than the traditional Black–Scholes model, our new approach could facilitate more efficient comparative pricing and precise risk management in equity derivatives with barriers by incorporating term-structures of interest rates, volatility and dividend into the CEV option valuation model.

Integro-differential equations for option prices in exponential Lévy models

We explore the precise link between option prices in exponential Levy models and the related partial integro-differential equations (PIDEs) in the case of European options and options with single or double barriers. We first discuss the conditions under which options prices are classical solutions of the PIDEs. We show that these conditions may fail in pure jump models and give examples of lack of smoothness of option prices with respect to the underlying. We give sufficient conditions on the Levy triplet for the prices of barrier options to be continuous with respect to the underlying and show that, in a general setting, option prices in exp-Levy models correspond to viscosity solutions of the pricing PIDE.

Does the Firm-Specific Asset Volatility Process Implied by the Equity Market Revert to a Constant Value?

In this paper, two structural models where firms have stationary capital structures and endogenous default barriers are extended to allow the principal value of a firm's debt to grow at a constant rate. This allows firms to have a dynamic capital structure. These two models are then used in conjunction with observable equity data to calculate the implied asset volatilities of a sample of fifty firms. Unit root tests are applied to the implied asset volatility and equity volatility processes to determine whether the processes are mean-reverting. Evidence that asset volatility is mean-reverting is found for forty-six of the fifty firms in the sample, regardless of which structural model is used to calculate the asset volatility, while the number of firms whose equity volatility is mean-reverting is in general lower for the poorer credit classes, consistent with the leverage effect. The mean-reversion of asset volatility has implications for the modelling of both equity and debt, and for the pricing of equity options, corporate bonds and credit derivatives.

Estimating Systemic Risk in the International Financial System

Using a unique and comprehensive dataset, this paper develops and uses three distinct methods to quantify the risk of a systemic failure in the global banking system. We examine a sample of 334 banks (representing 80% of global bank equity) in 28 countries around 6 global financial crises (such as the Asian and Russian crises and September 11, 2001), and show that these crises did not create large probabilities of global financial system failure. First, we show that cumulative negative abnormal returns for the subset of banks not directly exposed to a negative shock (unexposed banks) rarely exceed a few percent. Second, we use structural models to obtain more precise point estimates of the likelihood of systemic failure. These estimates suggest that systemic risk is limited even during major financial crises. For example, maximum likelihood estimation of bank failure probabilities implied by equity prices suggests the Asian crisis induced less than a 1% increase in the probability of systemic failure. Third, we also obtain estimates of systemic risk implied by equity option prices of U.S. and European banks. The largest values are obtained for the Russian crisis and September 11 and these show increases in estimated average default probabilities of only around 1-2%. Taken together our results suggest statistically significant, but economically small, increases in systemic risk around even the worst financial crises of the last 10 years. Although policy responses are endogenous, the low estimated probabilities suggest that the distress of central bankers, regulators and politicians about the events we study may be overstated, and that current policy responses to financial crises and the existing institutional framework may be adequate to handle major macroeconomic events.

Hanke, M.: Credit Risk, Capital Structure, and the Pricing of Equity Options

Hanke, M.: Credit Risk, Capital Structure, and the Pricing of Equity Options