Option Greeks Research Articles

On the optimal wealth process in a log-normal market: Applications to risk management

Using a stochastic representation of the optimal wealth process in the classical Merton problem, we calculate its cumulative distribution and density functions and provide bounds and monotonicity results for these quantities under general risk preferences. We also show that the optimal wealth and portfolio processes for different utility functions are related through a deterministic transformation and appropriately modified initial conditions. We analyze the value at risk (VaR) and expected shortfall (ES) of the optimal wealth process and show how each can be used to infer a constant relative risk aversion (CRRA) investor's risk aversion coefficient. Drawing analogies to the option greeks, we study the sensitivities of the optimal wealth process with respect to the cumulative excess stock return, time, and market parameters. We conclude with a study of how sensitivities of the excess return on the optimal wealth process relate to the portfolio's beta.

On the Optimal Wealth Process in a Log-Normal Market: Applications to Risk Management

Using a stochastic representation of the optimal wealth process in the classical Merton problem, we calculate its cumulative distribution and density functions and provide bounds and monotonicity results for these quantities under general risk preferences. We also show that the optimal wealth and portfolio processes for different utility functions are related through a deterministic transformation and appropriately modified initial conditions. We analyze the value at risk (VaR) and expected shortfall of the optimal wealth process and show how each can be used to infer a CRRA investor's risk aversion coefficient. Drawing analogies to the option greeks, we study the sensitivities of the optimal wealth process with respect to the cumulative excess stock return, time, and market parameters. We conclude with a study of how sensitivities of the excess return on the optimal wealth process relate to the portfolio's beta.

Modeling a Risk-Based Criterion for a Portfolio with Options

The presence of options in a portfolio fundamentally alters the portfolio's risk and return profiles when compared to an all equity portfolio. In this paper, we advocate modeling a risk-based criterion for optioned portfolio selection and rebalancing problems. The criterion is inspired by Chicago Mercantile Exchange's risk-based margining system which sets the collateralization requirements on margin accounts. The margin criterion computes the losses expected at the portfolio level using expected stock price and volatility variations, and is itself an optimization problem. Our contribution is to remodel the criterion as a quadratic programming subproblem of the main portfolio optimization problem using option Greeks. We also extend the margin subproblem to a continuous domain. The quadratic programming problems thus designed can be solved numerically or in closed-form with high efficiency, greatly facilitating the main portfolio selection problem. We present two extended practical examples of the application of our approach to obtain optimal portfolios with options. These examples include a study of liquidity effects (bid/ask spreads and limited order sizes) and sensitivity to changing market conditions. Our analysis shows that the approach advocated here is more stable and more efficient than discrete approaches to portfolio selection.

Option pricing and Greeks via a moving least square meshfree method

We apply a meshfree method using the fast moving least squares approximation to option pricing, particularly for the purpose of obtaining high-order Greeks. The method is shown to be accurate and efficient in obtaining prices and Greeks of European, Asian and Barrier options. We also include a complicated Equity Linked Security (ELS) from the Korean OTC market, as a real-world example.

Calibration of Stochastic Volatility Models on a Multi-Core CPU Cluster

Low-latency real-time option analytics feeds provide tick-by-tick implied volatilities and greeks based on exchange data. In order for the Black-Scholes implied volatility surface to exhibit the empirically observed skew or smile, a stochastic volatility model can be used to compute the option greeks. Because the European price under many stochastic volatility models only exists in semi-analytic form, frequent robust calibration of the model is computationally prohibitive. This paper explores three parallelization approaches for calibrating stochastic volatility models deployed on a multicore CPU cluster. The contribution of this paper is to provide benchmarks demonstrating hybrid shared and distributed memory parallelization techniques using Python packages for robust calibration of stochastic volatility models. The focus here will be on the Heston and Bates models, but the results in this paper generalize to any of the exponential Levy models incorporating stochastic volatility and jumps and whose characteristic function can be expressed in closed form. We evaluated the performance for our implementation on a cluster of 32 dual socket Dell PowerEdge R410 nodes providing 256 cores in total. Using distributed memory parallelization, we obtain a speedup of up to 139x against the sequential version of the calibration error function evaluation and reduce the overall time taken to calibrate a chain of 1024 SPX options by a factor of 37x.

Are we using the wrong letters? An analysis of executive stock option Greeks

Greek letters, in particular delta and vega based on the Black–Scholes model (BS), have been widely used to estimate the sensitivity of CEO wealth to changes in stock price (delta) and stock return volatility (vega) and to evaluate the executive stock options (ESOs) granted on the basis of performance and risk. However, the BS model does not take into account the main features of ESOs and therefore the delta and vega values it produces are not valid. The Cvitanic–Wiener–Zapatero model (CWZ) is an alternative model to Black–Scholes for valuing ESOs. It has a closed formula and considers the main features of ESOs. We carry out a sensitivity analysis to show that research on option-based compensation and its risk-taking effects is not robust in ESO pricing models. The sensitivity analysis consists of comparing the impact of the common parameters of the BS and CWZ models, as well as the effect of the specific parameters of the CWZ model, on the sensitivity of CEO wealth to stock price and stock volatility. Additionally, using panel data methodology, we develop an empirical analysis to illustrate the influence of stock return volatility and different corporate policies on both CEO wealth sensitivities.

Estimating multiple option Greeks simultaneously using random parameter regression

The derivatives of option prices with respect to underlying parameters are commonly referred to as Greeks, and they measure the sensitivities of option prices to these parameters. When the closed-form solutions for option prices do not exist and the discounted payoff functions of the options are not sufficiently smooth, estimating Greeks is computationally challenging and could be a burdensome task for high-dimensional problems in particular. The aim of this paper is to develop a new method for estimating option Greeks by using random parameters and least-squares regression. Our approach has several attractive features. First, just like the finite-difference method it is easy to implement and does not require explicit knowledge of the probability density function and the pathwise derivative of the underlying stochastic model. Second, it can be applied to options with discontinuous discounted payoffs as well as options with continuous discounted payoffs. Third, and most importantly, we can estimate multiple derivatives simultaneously. The performance of our approach is illustrated for a variety of examples with up to fifty Greeks estimated simultaneously. The algorithm is able to produce computationally efficient results with good accuracy.

A new radial basis functions method for pricing American options under Merton's jump-diffusion model

A new radial basis functions (RBFs) algorithm for pricing financial options under Merton's jump-diffusion model is described. The method is based on a differential quadrature approach, that allows the implementation of the boundary conditions in an efficient way. The semi-discrete equations obtained after approximation of the spatial derivatives, using RBFs based on differential quadrature are solved, using an exponential time integration scheme and we provide several numerical tests which show the superiority of this method over the popular Crank–Nicolson method. Various numerical results for the pricing of European, American and barrier options are given to illustrate the efficiency and accuracy of this new algorithm. We also show that the option Greeks such as the Delta and Gamma sensitivity measures are efficiently computed to high accuracy.

Applying option Greeks to directional forecasting of implied volatility in the options market: An intelligent approach

This paper examines movement in implied volatility with the goal of enhancing the methods of options investment in the derivatives market. Indeed, directional movement of implied volatility is forecasted by being modeled into a function of the option Greeks. The function is structured as a locally stationary model that employs a sliding window, which requires proper selection of window width and sliding width. An artificial neural network is employed for implementing and specifying our methodology. Empirical study in the Korean options market not only illustrates how our directional forecasting methodology is constructed but also shows that the methodology could yield a reasonably strong performance. Several interesting technical notes are discussed for directional forecasting.

Valuing Seller‐Defaultable Options

This study analyzes seller‐defaultable options that allow option writers to have a free‐will right to default, along with some prespecified default mechanisms. We analytically and numerically examine the pricing, hedging, defaulting, and profitability of the seller‐defaultable options, considering three possible scenarios for seller default. Analyzing the essential implications of seller‐defaultable options, we show that the option price is positively correlated with the default fine, underlying asset price, and volatility. The seller‐defaultable option's Greeks appear more complicated than those of the plain vanilla options. The likelihood of sellers defaulting increases with the underlying asset price, interest rate, volatility, and maturity time. Subject to the default mechanism, the buyers’ trading involves a trade‐off between profits and costs. © 2012 Wiley Periodicals, Inc. Jrl Fut Mark 33:129–157, 2013

Option Pricing and ARCH Processes

Recent progresses in option pricing using ARCH processes for the underlying are summarized. The stylized facts are multiscale heteroscedasticity, fat-tailed distributions, time reversal asymmetry, and leverage. The process equations are based on a finite time increment, relative returns, fat-tailed innovations, and multiscale ARCH volatility. The European option price is the expected payoff in the physical measure P weighted by the change of measure dQ/dP, and an expansion in the process increment dt allows for numerical evaluations. A cross-product decomposition of the implied volatility surface allows to compute efficiently option prices, Greeks, replication cost, replication risk, and real option prices. The theoretical implied volatility surface and the empirical mean surface for options on the SP500 index are in excellent agreement.

TR-BDF2 for Stable American Option Pricing

The Trapezoidal Rule with Second Order Backward Difference Formula (TR-BDF2) finite difference scheme is applied to the Black-Scholes-Merton PDE on a non uniform grid. American Option Convergence and Greeks stability is studied against studied against popular alternatives, namely Crank-Nicolson and Rannacher time-marching.

Canonical Option Pricing and Greeks with Implications for Market Timing

We introduce canonical representation of a call option premium which solves two open problems in option pricing theory. The first problem was posed by (Kassouf, 1969, pg. 694) who sought “theoretical substantiation” for his robust option pricing power law which eschewed assumptions about risk attitudes, rejected risk neutrality, and made no assumptions about stock price distribution. The second problem was posed by (Scott, 1987, pp. 423-424) who could not find a unique solution to the call option price in his option pricing model with stochastic volatility–without appealing to an equilibrium asset pricing model posed by Hull and White (1987) – and concluded: “[w]e cannot determine the price of a call option without knowing the price of another call on the same stock”. First, we show that under certain conditions derivative assets are superstructures of the underlying. Hence any option pricing or derivative pricing model in a given number field, based on an anticipating variable in an extended field, with coefficients in a subfield containing the underlying, is admissible for option pricing and or market timing. This explains the taxonomy of variegated option pricing formulae presented in Haug and Taleb (2008) critique of the Black and Scholes (1973); Merton (1973) formula. To wit, we claim that investor sentiment is an algebraic number. Accordingly, any polynomial which satisfies those criteria is admissible for price discovery and or market timing. Therefore, at least for empirical purposes, elaborate equations of mathematical physics or otherwise are unnecessary for pricing derivatives because much simpler adaptive polynomials in suitable algebraic numbers are functionally equivalent. Second, we prove, analytically, that Kassouf (1969) power law specification for option pricing is functionally equivalent to Black and Scholes (1973); Merton (1973) in an algebraic number field containing the underlying. In fact, our canonical representation of call option price is (i) convex in time to maturity, and (ii) depends on an algebraic number for the underlying – with coefficients based on observables in a subfield. Thus, paving the way for Wold decomposition of option prices, and providing theoretical contrast with Duan (1995) heuristic GARCH option pricing model. Third, our canonical representation theory has an inherent regenerative multifactor decomposition of call option price that (1) induces a duality theorem for call option premium, and (2) permits estimation of risk factor exposure for Greeks by standard [polynomial] regression procedures. Again, providing a theoretical (a) basis for option pricing of Greeks, and (b) solving Scott’s dual call option problem a fortiori by virtue of Riesz representation theory. Moreover, we show how our regenerative decomposition extends to (Wang and Caflish, 2010, pp. 4-5) Snell‘s envelope representation of a dual [American] call option premium. Fourth, when the Wold decomposition procedure is applied we are able to construct an empirical pricing kernel for call option based on residuals from a model of risk exposure to persistent and transient risk factors. This approach is distinguished from Chernov (2003) two stage procedure.

Meshfree Approximation for Multi-Asset Options

We price multi-asset options by solving their price partial differential equations using a meshfree approach with radial basis functions under jump-diffusion and geometric Brownian motion frameworks. In the geo- metric Brownian motion framework, we propose an effective technique that breaks the multi-dimensional problem to multiple 3D problems. We solve the price PDEs or PIDEs with an implicit meshfree scheme using thin-plate radial basis functions. Meshfree approach is very accurate, has high order of convergence and is easily scalable and adaptable to higher dimensions and different payoff profiles. We also obtain closed form approximations for the option Greeks. We test the model on American crack spread options traded on NYMEX.

Option Pricing with Fourier Series

We propose a simple and fairly general approach for valuing European options in closed-form. This approach leads to a single formula for the price of a European option. This formula is easy to implement, fast to execute, and only involves working with real numbers, in contrast to the Fourier transform based option pricing formulas. It also applies many option pricing models existing in the literature. Furthermore, the formula and the derived expressions of option Greeks are payoff as well as model independent, in the sense that their structure remains unchanged for any payoff and any pricing model, unlike existing pricing formulas.

How do you straddle hogs and pigs? Ask the Greeks!

Evidence of distortions is found in commodity options premiums around informational events. Option Greeks are used to uncover the nature of these distortions in terms of underlying factors. Both changes in underlying futures prices and implied volatility are mispriced.

An Eulerian‐Lagrangian method for option pricing in finance

AbstractThis article is devoted to the development and application of an Eulerian‐Lagrangian method (ELM) for the solution of the Black‐Scholes partial differential equation for the valuation of European option contracts. This method fully utilizes the transient behavior of the governing equations and generates very accurate option's fair values and their derivatives also known as option Greeks, even if coarse spatial grids and large time steps are used. Numerical experiments on two standard option contracts are presented which show that the ELM method (favorably) compares in terms of accuracy and efficiency to many other well‐perceived methods. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 293–329, 2007

Heuristic option pricing with neural networks and the neuro-computer synapse 3

Today’s option and warrant pricing is based on models developed by Black, Scholes and Merton in 1973 and Cox, Ross and Rubinstein in 1979. The price movement of the underlying asset is modeled by continuous-time or discrete-time stochastic processes. Unfortunately these models are based on severely unrealistic assumptions. Permanently an unsatisfactory and quite artificial adaption to the true market conditions is necessary (future volatility of the underlying price). Here, an alternative heuristic approach with a highly accurate neural network approximation is presented. Market prices of options and warrants and the values of the influence variables form the usually very large output/ input data set. Thousands of multi-layer perceptrons with various topologies and with different weight initializations are trained with a fast sequential quadratic programming (SQP) method. The best networks are combined to an expert council network to synthesize market prices accurately. All options and warrants can be compared to single out overpriced and underpriced ones for each trading day. For each option and warrant overpriced and underpriced trading days can be used to ascertain a better buy and sell timing. Furthermore the neural model gains deep insight into the market price sen-sitivities (option Greeks), e.g., ▵, Г, Θ and Ω. As an illustrative example we inves-tigate BASF stock call warrants. Time series from the beginning of 1996 to mid 1997 of 74 BASF call warrant prices at the Frankfurter Wertpapierborse (Frankfurt Stock Exchange) form the data basis. Finally a possible speed up of the training with the neuro-computer SYNAPSE 3 is briefly discussed