Option Greeks Research Articles

RISK ANALYSIS OF GOOGL & AMZN STOCK CALL OPTIONS USING DELTA GAMMA THETA NORMAL APPROACH

Stocks, as investment products, tend to carry risks due to fluctuations. The tendency of stock prices to rise over time leads investors to opt for call options, which are one of the derivative investment products. However, call options are influenced by several factors that can pose risks and have nonlinear dependence on market risk factors. Therefore, methods are needed to measure the risk of call options, such as Delta Normal Value at Risk and Delta Gamma Normal Value at Risk. Delta and Gamma are part of Option Greeks, parameters that measure the sensitivity of options to various factors used in determining option prices with the Black-Scholes model. This study uses an approach with the addition of Theta, which can measure the sensitivity of options to time. This study aims to analyze Value at Risk with the Delta Gamma Theta Normal approach for call options on Google (GOOGL) and Amazon (AMZN) stocks. The analysis uses closing stock price data from September 7, 2022, to September 7, 2023, and three in-the-money and out-of-the-money call option prices. The study begins by collecting closing stock prices and call option contract components, testing the normality of stock returns, calculating volatility, , Delta, Gamma, and Theta, then calculating the Value at Risk. Based on the analysis, it is found that GOOGL and AMZN call options have a Value at Risk of $0.89588 and $0.92760, respectively, at a 99% confidence level with a strike price of $120. Furthermore, based on the comparison of Value at Risk between in-the-money and out-of-the-money call options, it can be concluded that out-of-the-money call options tend to have larger estimated losses.

React Financial Analysis with Interactive Pay-Off Graph and Options Data

Abstract: We propose a System which represents a web-based financial application designed for managing options trading strategies. Developed using React, it offers an intuitive user interface for strategy selection, parameter configuration, and visualization of profit and loss via payoff graphs. The application provides real-time market data, P&L grid views, and insights into option Greeks. It integrates with AWS services for backend processing, handles order placement and status monitoring, and efficiently manages external services and a database. This system architecture is structured for scalability and high-performance in options trading, serving as a comprehensive tool for traders and investors.

Approximating Option Greeks in a Classical and Multi-Curve Framework Using Artificial Neural Networks

In this paper, the use of artificial neural networks (ANNs) is proposed to approximate the option price sensitivities of Johannesburg Stock Exchange (JSE) Top 40 European call options in a classical and a modern multi-curve framework. The ANNs were trained on artificially generated option price data given the illiquid nature of the South African market, and the out-of-sample performance of the optimized ANNs was evaluated using an implied volatility surface constructed from published volatility skews. The results from this paper show that ANNs trained on artificially generated input data are able to accurately approximate the explicit solutions to the respective option price sensitivities of both a classical and a modern multi-curve framework in a real-world out-of-sample application to the South African market.

QUASI-MONTE CARLO METHODS FOR CALCULATING DERIVATIVES SENSITIVITIES ON THE GPU

The calculation of option Greeks is vital for risk management. Traditional pathwise and finite-difference methods work poorly for higher-order Greeks and options with discontinuous payoff functions. The Quasi-Monte Carlo-based conditional pathwise method (QMC-CPW) for options Greeks allows the payoff function of options to be effectively smoothed, allowing for increased efficiency when calculating sensitivities. Another way to increase efficiency is to utilize the increased computational speed by applying GPUs to highly parallelisable finance problems such as calculating Greeks. We pair QMC-CPW with simulation on the GPU using the CUDA platform. We estimate the delta, vega and gamma Greeks of three exotic options: arithmetic Asian, binary Asian, and look-back. The increased computational speed through usage of the GPU is also shown with achieved speedups over sequential CPU implementations of more than 200x for our most accurate method.

A reduced-order model based on integrated radial basis functions with partition of unity method for option pricing under jump–diffusion models

The current research aims to develop a fast, stable and efficient numerical procedure for solving option pricing problems in jump–diffusion models. A backward partial integro-differential equation (PIDE) with diffusion and advection terms was investigated. Up to the best knowledge of the authors, some special numerical methods and strategies must be selected to solve advection–diffusion problems with reliable stability and accuracy. For the mentioned aims, the first- and second-order derivatives are approximated by integrated radial basis function based on partition of unity method. The IRBF-PU method is local mesh-free method that provides high order accurate result and is flexible for PDEs problems with sufficiently smooth initial conditions and also has a moderate condition number. In particular, we highlight European and American style put options, whose underlying asset follows a jump–diffusion model. For the distribution of the jumps, the Merton and Kou models are studied. Furthermore, the main model is classified in advection-x-diffusion category. As a result, we must increase the number of collocation points as well as the time steps to arrive at the final time. This procedure lengthens the execution time. To address this issue, we use the proper orthogonal decomposition (POD) method to reduce the size of the final algebraic system of equations. This numerical procedure is known as the proper orthogonal decomposition-IRBF-PU method (POD-IRBF-PU). The presented computational results (including the computation of option Greeks) and comparisons with other competing approaches suggest that the IRBF-PU and POD-IRBF-PU methods are efficient and reliable numerical methods to solve elliptic and parabolic PIDEs arising from applied areas such as financial engineering.

Barrier Options and Greeks: Modeling with Neural Networks

This paper proposes a non-parametric technique of option valuation and hedging. Here, we replicate the extended Black–Scholes pricing model for the exotic barrier options and their corresponding Greeks using the fully connected feed-forward neural network. Our methodology involves some benchmarking experiments, which result in an optimal neural network hyperparameter that effectively prices the barrier options and facilitates their option Greeks extraction. We compare the results from the optimal NN model to those produced by other machine learning models, such as the random forest and the polynomial regression; the output highlights the accuracy and the efficiency of our proposed methodology in this option pricing problem. The results equally show that the artificial neural network can effectively and accurately learn the extended Black–Scholes model from a given simulated dataset, and this concept can similarly be applied in the valuation of complex financial derivatives without analytical solutions.

Incorporating Indirect Channels into Option Greeks

Incorporating Indirect Channels into Option Greeks

Quasi-Monte Carlo-Based Conditional Malliavin Method for Continuous-Time Asian Option Greeks

Although many methods for computing the Greeks of discrete-time Asian options are proposed, few methods to calculate the Greeks of continuous-time Asian options are known. In this paper, we develop an integration by parts formula in the multi-dimensional Malliavin calculus, and apply it to obtain the Greeks formulae for continuous-time Asian options in the multi-asset situation. We combine the Malliavin method with the quasi-Monte Carlo method to calculate the Greeks in simulation. We discuss the asymptotic convergence of simulation estimates for the continuous-time Asian option Greeks obtained by Malliavin derivatives. We propose to use the conditional quasi-Monte Carlo method to smooth Malliavin Greeks, and show that the calculation of conditional expectations analytically is viable for many types of Asian options. We prove that the new estimates for Greeks have good smoothness. For binary Asian options, Asian call options and up-and-out Asian call options, for instance, our estimates are infinitely times differentiable. We take the gradient principal component analysis method as a dimension reduction technique in simulation. Numerical experiments demonstrate the large efficiency improvement of the proposed method, especially for Asian options with discontinuous payoff functions.

PRICING ASIAN OPTIONS WITH CORRELATORS

We derive a series expansion by Hermite polynomials for the price of an arithmetic Asian option. This requires the computation of moments and correlators of the underlying asset price which for a polynomial jump–diffusion process are given analytically; hence, no numerical simulation is required to evaluate the series. This allows to derive analytical expressions for the option Greeks. The weight function defining the Hermite polynomials is a Gaussian density with scale [Formula: see text]. We find that the rate of convergence of the series depends on [Formula: see text], for which we prove a lower bound to guarantee convergence. Numerical examples show that the series expansion is accurate but unstable for initial values of the underlying process far from zero, mainly due to rounding errors.

Application of Quasi Monte Carlo and Global Sensitivity Analysis to Option Pricing and Greeks: Finite Differences vs. AAD

Application of Quasi Monte Carlo and Global Sensitivity Analysis to Option Pricing and Greeks: Finite Differences vs. AAD

An adaptive and explicit fourth order Runge–Kutta–Fehlberg method coupled with compact finite differencing for pricing American put options

We propose an adaptive and explicit Runge–Kutta–Fehlberg method coupled with a fourth-order compact scheme to solve the American put options problem. First, the free boundary problem is converted into a system of partial differential equations with a fixed domain by using logarithm transformation and taking additional derivatives. With the addition of an intermediate function with a fixed free boundary, a quadratic formula is derived to compute the velocity of the optimal exercise boundary analytically. Furthermore, we implement an extrapolation method to ensure that at least, a third-order accuracy in space is maintained at the boundary point when computing the optimal exercise boundary from its derivative. As such, it enables us to employ fourth-order spatial and temporal discretization with Dirichlet boundary conditions for obtaining the numerical solution of the asset option, option Greeks, and the optimal exercise boundary. The advantage of the Runge–Kutta–Fehlberg method is based on error control and the adjustment of the time step to maintain the error at a certain threshold. By comparing with some existing methods in the numerical experiment, it shows that the present method has a better performance in terms of computational speed and provides a more accurate solution.

Bias Correction for Bond Option Greeks via Jackknife

The underlying models for bond options are often based on some linear drift functions such that the option Greeks depend crucially on the mean reversion parameters. Substantial estimation bias may arise when these parameters are estimated using standard methods such as maximum likelihood estimation, leading to a bias in estimating the Greeks. To address this issue, following Phillips and Yu (2005), a jackknife method is adopted in this article. In particular, we apply the method directly to the estimation of option Greeks, rather than the estimation of parameters. This approach is general and computationally inexpensive; hence, it is convenient in practice. The finite-sample performance is investigated in several Monte Carlo studies. At last, in dynamic Delta hedging, we show that the bias reduction in the estimation of option Greeks using the proposed method can achieve some economic value. <b>TOPICS:</b>Derivatives, options, fixed income and structured finance, quantitative methods, simulations <b>Key Findings</b> ▪ Like option pricing, the bias problem is still serious in estimating the Greeks of the options. ▪ The jackknife method is a good method to reduce the estimation bias in estimating the Greeks. ▪ The bias reduction in the estimation of option Greeks using the proposed method can achieve some economic values in option markets around the world.

An unsupervised deep learning approach to solving partial integro-differential equations

We investigate solving partial integro-differential equations (PIDEs) using unsupervised deep learning. The PIDE is employed for option pricing, when the underlying process is a Lévy process with jumps. The unsupervised deep learning approach employs a neural network as the candidate solution and trains the neural network to satisfy the PIDE. By matching the PIDE and the boundary conditions, the neural network would yield an accurate solution to the PIDE. Unlike supervised learning, this approach does not require any labels for training, where labels are typically option prices as well as Greeks.

A Closed-form Pricing Solution for Options on Assets with Pricing Errors

This paper examines the effects that pricing errors in the underlying asset have on options prices, their Greeks, and their implied risk neutral densities. Pricing errors can be viewed as a random proportional transaction cost. When pricing errors are information-unrelated, options prices are unambiguously higher than the Black-Scholes case and increasing in the pricing error variance. Hedging volatility is higher and the optimal exercise price for American put options is decreased. The option implied risk-neutral density and option Greeks are materially affected, which leads to suboptimal risk management and hedging when pricing errors are not accounted for.

Option Greeks, Insider Trading, and the Heinz Acquisition

Option Greeks, Insider Trading, and the Heinz Acquisition

Two‐dimensional Haar wavelet based approximation technique to study the sensitivities of the price of an option

AbstractIn the present work, a two‐dimensional Haar wavelet method is proposed to study the sensitivities of the price of an option. The method is appropriate to study these sensitivities as it explicitly gives the values of all the derivatives of the solution. A Black–Scholes model for European style options is considered to analyze the physical and numerical aspects of the put and the call option Greeks. We use the concept of coordinate transformation to make the Black–Scholes equation dimensionless and to resolve the obstacle in approximating the Greeks having non‐smoothness at the strike price. The infinite spatial domain is truncated into the finite domain to avoid large truncation errors. Through rigorous analysis, the method is shown first‐order accurate in the L2−norm. The numerical computations performed to approximate the option price and various Greeks, like delta, theta, gamma, and so forth, confirm the theoretical results in L2−norm. The relative errors and the maximum absolute errors are also presented. The motivational work of option Greeks analysis may leave a significant impact on financial institutes; it helps them to manage the risk by setting the portfolio's new value and to estimate the probability of losing money.

On Option Greeks and Corporate Finance

This paper has proposed new option Greeks and new upper and lower bounds for European and American options. It shows that because of the put-call parity, the Greeks of put and call options are interconnected and should be shown simultaneously. In terms of the theory of the firm, it is found that both the Black-Scholes-Merton and the binomial option pricing models implicitly assume that maximizing the market value of the firm is not equivalent to maximizing the equityholders’ wealth. The binomial option pricing model implicitly assumes that further increasing (decreasing) the promised payment to debtholders affects neither the speed of decreasing (increasing) in the equity nor the speed of increasing (decreasing) in the insurance for the promised payment. The Black-Scholes-Merton option pricing model implicitly assumes that further increasing (decreasing) in the promised payment to debtholders will: (1) decrease (increase) the speed of decreasing (increasing) in the equity though bounded by upper and lower bounds, and (2) increase (decrease) the speed of increasing (decreasing) in the insurance though bounded by upper and lower bounds. The paper also extends the put-call parity to include senior debt and convertible bond. It specifies the lower bound for risky debt and the conditions under which American put option will not be early exercised.

Option valuation under no-arbitrage constraints with neural networks

In this paper, we start from the no-arbitrage constraints in option pricing and develop a novel hybrid gated neural network (hGNN) based option valuation model. We adopt a multiplicative structure of hidden layers to ensure model differentiability. We also select the slope and weights of input layers to satisfy the no-arbitrage constraints. Meanwhile, a separate neural network is constructed for predicting option-implied volatilities. Using S&P 500 options, our empirical analyses show that the hGNN model substantially outperforms well-established alternative models in the out-of-sample forecasting and hedging exercises. The superior prediction performance stems from our model’s ability in describing options on the boundary, and in offering analytical expressions for option Greeks which generate better hedging results.

On Option Greeks and Corporate Finance

This paper has proposed new option Greeks and new upper and lower bounds for European and American options. It also shows that because of the put-call parity, the Greeks of put and call options are interconnected and should be shown simultaneously. In terms of the theory of the firm, it is found that both the Black-Scholes-Merton and the binomial option pricing models implicitly assume that maximizing the market value of the firm is not equivalent to maximizing the equity-holders’ wealth. The binomial option pricing model implicitly assumes that further increasing (decreasing) the promised payment to debt-holders affects neither the speed of decreasing (increasing) in the equity nor the speed of increasing (decreasing) in the insurance for the promised payment. The Black-Scholes-Merton option pricing model, on the other hand, implicitly assumes that further increasing (decreasing) in the promised payment to debt-holders will: (1) decrease (increase) the speed of decreasing (increasing) in the equity though bounded by upper and lower bounds, and (2) increase (decrease) the speed of increasing (decreasing) in the insurance though bounded by upper and lower bounds. The paper also extends the put-call parity to include senior debt and convertible bond. It is found that when the promised payment to debt-holders is approaching the market value of the firm and the risk-free interest rate is small, both the owner of the equity and the owner of the insurance will be more reluctant to liquidate the firm. The lower bound for the risky debt is: the promised payment to debt-holders is greater or equal to the market value of the firm times one plus the risk-free interest rate.

Optimal uniform error estimates for moving least‐squares collocation with application to option pricing under jump‐diffusion processes

AbstractIn this study, we derive optimal uniform error bounds for moving least‐squares (MLS) mesh‐free point collocation (also called finite point method) when applied to solve second‐order elliptic partial integro‐differential equations (PIDEs). In the special case of elliptic partial differential equations (PDEs), we show that our estimate improves the results of Cheng and Cheng (Appl. Numer. Math. 58 (2008), no. 6, 884–898) both in terms of the used error norm (here the uniform norm and there the discrete vector norm) and the obtained order of convergence. We then present optimal convergence rate estimates for second‐order elliptic PIDEs. We proceed by some numerical experiments dealing with elliptic PDEs that confirm the obtained theoretical results. The article concludes with numerical approximation of the linear parabolic PIDE arising from European option pricing problem under Merton's and Kou's jump‐diffusion models. The presented computational results (including the computation of option Greeks) and comparisons with other competing approaches suggest that the MLS collocation scheme is an efficient and reliable numerical method to solve elliptic and parabolic PIDEs arising from applied areas such as financial engineering.