This paper conducts a comprehensive study of the optimal exercise policy for an executive stock option and its implications for option cost, average life, and alternative valuation concepts. The ...paper is the first to provide analytical results for an executive with general concave utility. Wealthier or less risk-averse executives exercise later and create greater option cost. However, option cost can decline with volatility. We show when there exists a single exercise boundary, yet demonstrate the possibility of a split continuation region. We also show that, for constant relative risk averse utility, the option value does not converge to the Black and Scholes value as the correlation between the stock and the market portfolio converges to one. We compare our model's option cost with the modified Black and Scholes approximation typically used in practice and show that the approximation error can be large or small, positive or negative, depending on firm characteristics.
Multitouch Options Guillaume, Tristan
Journal of risk and financial management,
06/2023, Volume:
16, Issue:
6
Journal Article
Peer reviewed
Open access
In this article, the multitouch option, also called the n-touch option (or the “baseball” option when n=3) is analyzed and valued in closed form. This is a kind of barrier option that has been traded ...for a long time on the markets, but that does not yet admit a known valuation formula. The multitouch option sets a gradual knock-out/knock-in mechanism based on the number of times the underlying asset has crossed a predefined barrier in various time intervals before expiry. The higher the number of predefined time intervals during which the barrier has been touched, the lower the value of a knock-out contract at expiry, and conversely for a knock-in one. Multitouch options can be viewed as an extension of step barrier options, preserving the ability of the latter to adjust the exposure to risk over time, while eliminating the notorious danger of “sudden death” that holders of step barrier options are faced with. They are thus less risky and more flexible than step barrier options, and all the more so when compared to standard barrier options. This article also provides closed-form valuation of multitouch options with nonstandard features such as an outside barrier or a barrier defined as a continuous function of time.
•This paper investigates Asian-barrier option in uncertain financial market.•Some Asian-barrier option pricing formulas are derived.•Several numerical examples are given to illustrate the ...effectiveness of the proposed model.
Barrier option is an exotic option on an underlying asset whose payoff depends on whether or not the underlying asset’s price has reached predetermined barrier level. This paper mainly investigates Asian-barrier option pricing problem under uncertain financial market. Assume that stock price follows an uncertain differential equation, some Asian-barrier option pricing formulas are derived. Moreover, several numerical examples are given to illustrate the effectiveness of the proposed model.
The determination of optimal exercise boundaries is a critical aspect of pricing American options, which often requires costly numerical methods. This paper proposes a new approach that employs ...multi-step rebate options to approximate American option prices. Since the rebate options offer payoffs when the multi-step boundaries are touched, the prices of American options are estimated by maximizing the multi-step rebate option prices, and the optimal multi-step barriers replace the true optimal exercise boundaries. To this end, the closed-form pricing formulas for multi-step rebate options are derived and utilized to approximate several American option prices. Through extensive numerical experiments, we demonstrate the validity and performance of our approach.
•Derive distribution of the first-hitting time of a Brownian motion for given multi-step boundaries.•Derive multi-step-first-touch digital option prices to engineer multi-step rebate options.•Utilize derived pricing formulas to approximate the prices of the American-type derivatives.•Numerical examples for various instruments validate our approach.
In this paper, we study the pricing problems of basket options and spread options under the Normal Tempered Stable Lévy model, which is a general model for financial assets and covers many well-known ...models as special cases such as the Variance Gamma model, Normal Inverse Gaussian model etc. Our approach draws inspiration from the lower bound approximation strategy used in Gaussian models in Bjerksund and Stensland (2014). The approximation formula we derived involves some one-dimensional integrations. We calculate these integrals using the generalized Gauss–Laguerre quadrature rule and Taylor expansion methods. In particular, we derive an analytical approximation formula under the Variance Gamma model for some exchange options. Moreover, we extend the approximation formulas proposed by Kirk (1995) and Carmona and Durrleman (2003b) to the Normal Tempered Stable Lévy model. Numerical tests show that our approximation formulas are highly accurate. Furthermore, we show that our approximation formulas outperform the Fourier inversion method introduced by Caldana et al. (2016) in accuracy, especially for low prices cases.
•We study basket option pricing under the Normal Tempered Stable Lévy model.•Closed-form approximate pricing formulas are derived.•Two classic pricing methods are extended to the Normal Tempered Stable Lévy model.•Numerical tests show our pricing formulas are highly accurate.
We contribute to the behavioral theory of the firm and the behavioral agency model by developing a theoretical framework that predicts the differential interaction effects of performance feedback and ...values of stock option grants of multiple agents on firm risk taking. We explain how chief executive officers (CEOs) versus outside directors awarded with stock option grants perceive negative or positive deviations from prior performance. We argue that in a negative attainment discrepancy context, high values of option grants will increase the risk aversion of CEOs who already bear excessive employment and compensation risks, resulting in less risk taking; however, it will enhance the risk-taking propensity of influential outside directors who increase monitoring and support for risky projects because their risk preferences are better aligned with those of shareholders. In a positive attainment discrepancy context, high values of option grants will amplify risk aversion in both CEOs and outside directors who perceive risky strategies as potential threats to anticipated incentive values associated with a gain domain, thereby reducing risk-taking activities. Analysis of panel data from 1992 to 2006 on the research and development spending of U.S. manufacturing firms based on Arellano–Bond dynamic panel regression reveals findings largely consistent with our predictions.
•Efficient numerical solution to high-dimensional parametric PDEs by neural networks.•Unsupervised approach without the need for sample solutions or simulations.•Real-time evaluation of solution and ...sensitivities for a range of parameter values.•Accurate results for option pricing problems in the multivariate Black–Scholes model.•Numerical results of up to 25 dimensions confirm effectiveness of the approach.
We propose, formalise and analyse the deep parametric PDE method to solve high-dimensional parametric partial differential equations with a focus on financial applications. A single neural network approximates the solution of a whole family of PDEs after being trained without the need of sample solutions. As a practical application, we compute option prices and Greeks in the multivariate Black–Scholes model as there is an urgent need for highly efficient methods. After a single training phase, the prices and sensitivities for different times, states and model parameters are available in milliseconds. Exploiting the PDE framework and incorporating a-priori knowledge of no-arbitrage bounds improves the performance significantly. We evaluate the accuracy in the price, the Greeks and the implied volatility with examples of up to 25 dimensions. A comparison with alternative machine learning methods confirms the effectiveness of the new approach and reveals advantages of the underlying PDE formulation.
This paper intends to investigate an interesting class of barrier options, called step barrier options, whose barrier levels are a piecewise constant function of time. These options, while having ...transparent, simple, and flexible payoff structures, allow for explicit pricing formulas under the Black-Scholes model, and thus can be easily embedded into equity-linked products to enhance the yield or reduce the downside risk. Moreover, the class can be further generalized by attaching vertical branches of barriers to the horizontal one as in Lee and Ko (2018). Using the actuarial method of Esscher transform and the factorization formula, we derive the option pricing formulas under a more general framework with vertical branches attached to horizontal barriers. We explore the formulas through numerical examples, demonstrating their applicability to equity-linked investment with the step barrier option embedded.
The coupled nonlinear volatility and option pricing model presented recently by Ivancevic is investigated, which generates a leverage effect, i.e., stock volatility is (negatively) correlated to ...stock returns, and can be regarded as a coupled nonlinear wave alternative of the Black–Scholes option pricing model. In this Letter, we analytically propose vector financial rogue waves of the coupled nonlinear volatility and option pricing model without an embedded w-learning. Moreover, we exhibit their dynamical behaviors for chosen different parameters. The vector financial rogue wave (rogon) solutions may be used to describe the possible physical mechanisms for the rogue wave phenomena and to further excite the possibility of relative researches and potential applications of vector rogue waves in the financial markets and other related fields.
► We investigate the coupled nonlinear volatility and option pricing model. ► We analytically present vector financial rogue waves. ► The vector financial rogue waves may be used to describe the extreme events in financial markets. ► This results may excite the relative researches and potential applications of vector rogue waves.
It has been observed that the stock price process can be modeled with driving force as a mixed fractional Brownian motion with Hurst index H>34 whenever long-range dependence is possibly present. We ...obtain a closed form expression for the price of a geometric Asian option under the mixed fractional Brownian motion environment. We consider also Asian power options when the payoff function is a power function.
•Stock price is modeled with driving force as mixed fractional Brownian motion.•Closed form for the price of the geometric Asian option is derived.•Closed form for the price of the geometric Asian power option is derived.
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