Option Pricing Function Research Articles

Quantile LASSO in arbitrage-free option markets

The option price function and the implied volatility surface are both key tools for the derivative pricing strategies and the financial market analysis. Modern and sophisticated methods are used but their credibility suffered due to the financial crisis in 2007–2010. Instead, a method based on a standard semiparametric smoothing is proposed and the overall complexity and robustness (with respect to various anomalies, such as bid-ask spreads, discrete ticks in price, non-synchronous trading, or even heavy tailed error distributions) is achieved by using the conditional quantile estimation. The overestimation and the sparsity principle are adopted to introduce additional flexibility and the LASSO-type penalty and the set of well-defined linear constraints are employed to produce the final estimate which complies with the arbitrage-free criteria dictated by the financial theory. The theoretical results of the model are discussed, finite sample properties are investigated via a simulation study and a practical application of the proposed method is illustrated for the Apple Inc. (AAPL) call options.

Quantile LASSO with changepoints in panel data models applied to option pricing

Panel data are modern statistical tools which are commonly used in all kinds of econometric problems under various regularity assumptions. The panel data models with changepoints are introduced together with the atomic pursuit idea and they are applied to estimate the underlying option price function. Robust estimates and complex insight into the data are both achieved by adopting the quantile LASSO approach. The final model is produced in a fully data-driven manner in just one single modeling step. In addition, the arbitrage-free scenarios are obtained by introducing a set of well defined linear constraints. The final estimate is, under some reasonable assumptions, consistent with respect to the model estimation and the changepoint detection performance. The finite sample properties are investigated in a simulation study and proposed methodology is applied for the Apple call option pricing problem.

The Second Partial Derivative of Option Price with Respect to the Strike: A Historical Reminiscence

An option’s market price reflects the risk-neutral probability that it will end up in the money. Research has been increasing in recent years that shows how, given a set of market prices for options covering a range of strikes, an estimate of the entire risk-neutral probability distribution can be obtained. The technique is based on the fact that the second partial derivative of the option pricing function with respect to the strike price is the risk-neutral density (discounted from option expiration). This idea is generally attributed to Breeden and Litzenberger’s 1978 paper. In this article, Zimmermann shows that the connection between the second partial derivative of the option price with respect to the exercise price and risk-neutral probabilities has a much longer history, including a little-known 1974 note by Fischer Black, and going all the way back to Bachelier in 1900.

Uniform Confidence Bands for Pricing Kernels

Pricing kernels implicit in option prices play a key role in assessing the risk aversion over equity returns. We deal with non-parametric estimation of the pricing kernel (Empirical Pricing Kernel) given by the ratio of the risk-neutral density estimator and the subjective density estimator. The former density can be represented as the second derivative w.r.t. the European call option price function, which we estimate by non-parametric regression. The subjective density is estimated non-parametrically too. In this framework, we develop the asymptotic distribution theory of the EPK in the L1 sense. Particularly, to evaluate the overall variation of the pricing kernel, we develop a uniform confidence band of the EPK. Furthermore, as an alternative to the asymptotic approach, we propose a bootstrap confidence band. The developed theory is helpful for testing parametric specifications of pricing kernels and has a direct extension to estimating risk aversion patterns. The established results are assessed and compared in a Monte-Carlo study. As a real application, we test risk aversion over time induced by the EPK.

The impact of Kiyoshi Ito´s stochastic calculus of financial economics

We discuss the direct or indirect incorporation into financial economics of Kiyoshi Itô´s work on stochastic calculus, particularly the Itô formula, the relevance of his findings for option pricing theory and the way his work has been used to find a unique option pricing function in a competitive and non-arbitrage market. On that basis, we discuss how the option pricing theory may be linked with the general equilibrium theory and other aspects of conventional economics, and finally, Itô’s role in econophysics.

Call option price function in Bernstein polynomial basis with no-arbitrage inequality constraints

We propose an efficient method for the construction of an arbitrage-free call option price function from observed call price quotes. The no-arbitrage theory of option pricing places various shape constraints on the option price function. For each available maturity on a given trading day, the proposed method estimates an option price function of strike price using a Bernstein polynomial basis. Using the properties of this basis, we transform the constrained functional regression problem to the least-squares problem of finite dimension and derive the sufficiency conditions of no-arbitrage pricing to a set of linear constraints. The resultant linearly constrained least square minimization problem can easily be solved using an efficient quadratic programming algorithm. The proposed method is easy to use and constructs a smooth call price function which is arbitrage-free in the entire domain of the strike price with any finite number of observed call price quotes. We empirically test the proposed method on S&P 500 option price data and compare the results with the cubic spline smoothing method to see the applicability.

Pricing Kernel Modeling

We propose a new method to estimate the empirical pricing kernel based on option data. We estimate the pricing kernel nonparametrically by using the ratio of the risk-neutral density estimator and the subjective density estimator. The risk-neutral density is approximated by a weighted kernel density estimator with varying unknown weights for different observations, and the subjective density is approximated by a kernel density estimator with equal weights. We represent the European call option price function by the second order integration of the risk-neutral density, so that the unknown weights are obtained through one-step penalized least squares estimation with the Kullback-Leibler divergence as the penalty function. Asymptotic results of the resulting estimators are established. The performance of the proposed method is illustrated empirically by simulation and real data application studies.

Pricing kernel modeling

We propose a new method to estimate the empirical pricing kernel based on option data. We estimate the pricing kernel nonparametrically by using the ratio of the risk-neutral density estimator and the subjective density estimator. The risk-neutral density is approximated by a weighted kernel density estimator with varying unknown weights for different observations, and the subjective density is approximated by a kernel density estimator with equal weights. We represent the European call option price function by the second order integration of the risk-neutral density, so that the unknown weights are obtained through one-step penalized least squares estimation with the Kullback-Leibler divergence as the penalty function. Asymptotic results of the resulting estimators are established. The performance of the proposed method is illustrated empirically by simulation and real data application studies.

Asymptotic Analysis of Stock Price Densities and Implied Volatilities in Mixed Stochastic Models

In this paper, we obtain sharp asymptotic formulas with error estimates for the Mellin convolution of functions defined on $(0,\infty)$ and use these formulas to characterize the asymptotic behavior of marginal distribution densities of stock price processes in mixed stochastic models. Special examples of mixed models are jump-diffusion models and stochastic volatility models with jumps. We apply our general results to the Heston model with double exponential jumps and perform a detailed analysis of the asymptotic behavior of the stock price density, the call option pricing function, and the implied volatility in this model. We also obtain similar results for the Heston model with jumps distributed according to the normal inverse Gaussian law.

Uniform Confidence Bands for Pricing Kernels

Pricing kernels implicit in option prices play a key role in assessing the risk aversion over equity returns. We deal with nonparametric estimation of the pricing kernel (Empirical Pricing Kernel) given by the ratio of the risk-neutral density estimator and the subjective density estimator. The former density can be represented as the second derivative w.r.t. the European call option price function, which we estimate by nonparametric regression. The subjective density is estimated nonparametrically too. In this framework, we develop the asymptotic distribution theory of the EPK in the L1 sense. Particularly, to evaluate the overall variation of the pricing kernel, we develop a uniform confidence band of the EPK. Furthermore, as an alternative to the asymptotic approach, we propose a bootstrap confidence band. The developed theory is helpful for testing parametric specifications of pricing kernels and has a direct extension to estimating risk aversion patterns. The established results are assessed and compared in a Monte-Carlo study. As a real application, we test risk aversion over time induced by the EPK.

On implied volatility for options—Some reasons to smile and more to correct

We analyze the properties of the implied volatility, the commonly used volatility estimator by direct option price inversion. It is found that the implied volatility is subject to a systematic bias in the presence of pricing errors, which makes it inconsistent to the underlying volatility. We propose an estimator of the underlying volatility by first estimating nonparametrically the option price function, followed by inverting the nonparametrically estimated price. It is shown that the approach removes the adverse impacts of the pricing errors and produces a consistent volatility estimator for a wide range of option price models. We demonstrate the effectiveness of the proposed approach by numerical simulation and empirical analysis on S&P 500 option data.

Time-Independent Pricing of Options in Range Bound Markets

Assuming that price of the underlying stock is moving in range bound, the Black-Scholes formula for options pricing supports a separation of variables. The resulting time-independent equation is solved employing different behavior of the option price function and three significant results are deduced. The first is the probability of stock price penetration through support or resistance level, called transmission coefficient. The second is the distance that price will go through once stock price penetrates out of the range bound. The last one is a predicted short time dramatic fall in the stock volatility right ahead of price tunneling. All three results are useful tools that give market practitioners valuable insights in choosing the right time to get involved in an option contract, about how far the price will go in case of a breakout, and how to correctly interpret volatility downfalls.

Direct Nonparametric Estimation of State Price Density with Regularized Mixture

We consider the state price densities that are implicit in financial asset prices. In the pricing of an option, the state price density is proportional to the second derivative of the option pricing function and this relationship together with no arbitrage principle imposes restrictions on the pricing function such as monotonicity and convexity. Since the state price density is a proper density function and most of the shape constraints are caused by this, we propose to estimate the state price density directly by specifying candidate densities in a flexible nonparametric way and applying methods of regularization under extra constraints. The problem is easy to solve and the resulting state price density estimates satisfy all the restrictions required by economic theory.

Boundary value methods with the Crank–Nicolson preconditioner for pricing options in the jump-diffusion model

Under a jump-diffusion process, the option pricing function satisfies a partial integro-differential equation. A fourth-order compact scheme is used to discretize the spatial variable of this equation. The boundary value method is then utilized for temporal integration because of its unconditional stability and high-order accuracy. Two approaches, the local mesh refinement and the start-up procedure with refined step size, are raised to avoid the numerical malfunction brought by the nonsmooth payoff function. The GMRES method with a preconditioner which comes from the Crank–Nicolson formula is employed to solve the resulting large-scale linear system. Numerical experiments demonstrate the efficiency of the proposed method when pricing European and double barrier call options in the jump-diffusion model.

Option Pricing With Modular Neural Networks

This paper investigates a nonparametric modular neural network (MNN) model to price the S&P-500 European call options. The modules are based on time to maturity and moneyness of the options. The option price function of interest is homogeneous of degree one with respect to the underlying index price and the strike price. When compared to an array of parametric and nonparametric models, the MNN method consistently exerts superior out-of-sample pricing performance. We conclude that modularity improves the generalization properties of standard feedforward neural network option pricing models (with and without the homogeneity hint).

Shape-Preserving Interpolation and Smoothing for Options Market Implied Volatility

The interpolation of the market implied volatility function from several observations of option prices is often required in financial practice and empirical study. However, the results from existing interpolation methods may not satisfy the property that the European call option price function is monotonically decreasing and convex with respect to the strike price. In this paper, a modified convex interpolation method (with and without smoothing) is developed to approximate the option price function while explicitly incorporating the shape restrictions. The method is optimal for minimizing the distance between the implied risk-neutral density function and a prior density function, which allows us to benefit from nonparametric methodology and empirical experience. Numerical performance shows that the method is accurate and robust. Whether or not the sample satisfies the convexity and decreasing constraints, the method always works.

Higher Order Greeks

In the original Black-Scholes pricing framework, a continuously rebalanced delta hedge of an option against its underlying stock produces a perfectly risk free position. But in the real world, a basic delta hedge is still exposed to quite a lot of risk because rebalancing can only be done at discrete intervals, volatility is stochastically time-varying, and underlying assets do not follow true lognormal diffusions. Discrete rebalancing after a non-marginal price change makes gamma an important extra risk factor and with fluctuating implied volatility, vega becomes another one. This makes delta-gamma-vega hedging the next level of option risk management. But unhedged risk still remains. In this paper, Ederington and Guan investigate whether option risk factors measured by other second and third order derivatives of the option pricing function can capture this residual risk, and if so, which are the most important of these „higher order Greeks.” They find that, at least for S&P 500 index futures options, the additional risk factors can explain a substantial portion of the residual risk in a delta-gamma-vega hedge. A key finding is that an option9s sensitivity to the higher order Greeks depends heavily on its moneyness, with several Greek letter risks that are unimportant for at the money options becoming much more relevant for out of the money contracts. They also show that a hedge strategy that neutralizes these additional factors can lower residual risk by an order of magnitude below that of a delta-gamma-vega hedge.

Option Pricing With Modular Neural Networks

This paper applies a non-parametric modular neural network (MNN) model to price the S&P-500 European call options. The modules are based on time to maturity and moneyness of the options. The option price function of interest is homogenous of degree one with respect to the underlying index price and the strike price. We find that modularity improves the generalization properties of standard feedforward neural network option pricing models (with and without the homogeneity hint), relative to the Black-Scholes model.

Model Risk for European-Style Stock Index Options

In empirical modeling, there have been two strands for pricing in the options literature, namely the parametric and nonparametric models. Often, the support for the nonparametric methods is based on a benchmark such as the Black-Scholes (BS) model with constant volatility. In this paper, we study the stochastic volatility (SV) and stochastic volatility random jump (SVJ) models as parametric benchmarks against feedforward neural network (FNN) models, a class of neural network models. Our choice for FNN models is due to their well-studied universal approximation properties of an unknown function and its partial derivatives. Since the partial derivatives of an option pricing formula are risk pricing tools, an accurate estimation of the unknown option pricing function is essential for pricing and hedging. Our findings indicate that FNN models offer themselves as robust option pricing tools, over their sophisticated parametric counterparts in predictive settings. There are two routes to explain the superiority of FNN models over the parametric models in forecast settings. These are nonnormality of return distributions and adaptive learning.

Forecasting volatility: A reality check based on option pricing, utility function, value-at-risk, and predictive likelihood

We analyze the predictive performance of various volatility models for stock returns. To compare their performance, we choose loss functions for which volatility estimation is of paramount importance. We deal with two economic loss functions (an option pricing function and an utility function) and two statistical loss functions (a goodness-of-fit measure for a value-at-risk (VaR) calculation and a predictive likelihood function). We implement the tests for superior predictive ability of White [Econometrica 68 (5) (2000) 1097] and Hansen [Hansen, P. R. (2001). An unbiased and powerful test for superior predictive ability. Brown University]. We find that, for option pricing, simple models like the Riskmetrics exponentially weighted moving average (EWMA) or a simple moving average, which do not require estimation, perform as well as other more sophisticated specifications. For a utility-based loss function, an asymmetric quadratic GARCH seems to dominate, and this result is robust to different degrees of risk aversion. For a VaR-based loss function, a stochastic volatility model is preferred. Interestingly, the Riskmetrics EWMA model, proposed to calculate VaR, seems to be the worst performer. For the predictive likelihood-based loss function, modeling the conditional standard deviation instead of the variance seems to be a dominant modeling strategy.