Projection of the two-dimensional Black-Scholes equation for options with underlying stock and strike prices in two different currencies

The two-variable Black-Scholes equation is used to study the option exercise price of two different currencies. Due to the complexity of dealing with several variables, reduction methods have been implemented to deal with these problems. This paper proposes an alternative reduction by using the so-called Zwanzig projection method to one-dimension, successfully developed to study the diffusion in confined systems. In this case, the option price depends on the stock price and the exchange rate between currencies. We assume that the exchange rate between currencies will depend on the stock price through some model that bounds such dependence, which somehow influences the final option price.As a result, we find a projected one-dimensional Black-Scholes equation similar to the so-called Fick-Jacobs equation for diffusion on channels. This equation is an effective Black-Scholes equation with two different interest rates, whose solution gives rise to a modified Black-Scholes formula. The properties of this solution are shown and were graphically compared with previously found solutions, showing that the corresponding difference is bounded.

Optimal Pricing and Ordering Strategies with a Flexible Return Strategy under Uncertainty

To coordinate the supply chain risk caused by demand uncertainty, this paper proposed a flexible return strategy under demand uncertainty, in which the retailer can choose return quantity independently by put option after the selling season, while the return quantity is usually determined by the supplier in the classical return strategy. In our novel return strategy, the exercise price is not fixed, and we developed the base model of this strategy, named the selective buyback contracts model. We have solved the optimal pricing and ordering strategies of supply chain members. Numerical studies demonstrated that the contracts can coordinate a supply chain with one retailer and one supplier, and the supplier can adjust the profit distribution of the supply chain by adjusting the option exercise price. Compared with other return strategies, the selective buyback contracts give the retailer more power of choice, and the supplier receives risk compensation from the put options.

Coordination of a fresh agricultural product supply chain with option contract under cost and loss disruptions

This paper analyzes the option coordination problem of a fresh agricultural product supply chain under two supply chain structures, when the production cost and the loss rate are disrupted simultaneously. This paper provides the explicit option coordination conditions for the disrupted supply chain under two supply chain structures, and then explores the effects of the disruptions and supply chain structure on the option coordination conditions. The results suggest that it is unfavorable to apply the original coordinating contracts without disruptions to coordinate the disrupted supply chain. The coordination of the disrupted supply chain can be achieved with knowledge of the distribution of demand. In two coordinating contracts for the disrupted supply chain, the exercise price is still at the original level without disruptions while the option price deviates from the original level without disruptions. Moreover, the relationships of the coordination conditions in two supply chain structures depend on the value of the profit allocation coefficient. When the profit allocation coefficient exceeds (falls behind) a certain threshold, the option price is set at a higher (lower) value in the supplier-led supply chain structure than in the distributor-led supply chain structure, while the exercise price is set at a lower (higher) value in the supplier-led supply chain structure than in the distributor-led supply chain structure. Finally, the disrupted supply chain with any supply chain structure will perform better in the modified coordinating contracts than in the original coordinating contracts without disruptions.

Impacts of put option contract and supply chain structure in a multi-period supply chain with uncertain demand

This paper builds the multi-period optimization models that incorporate put option contract and two supply chain structures to determine the production decision for a supplier and the ordering decision for a manufacturer in a two-stage supply chain. This paper applies the method of dynamic programming to derive the structures of optimal policies and provides an approximate algorithm to evaluate the myopic policies. This paper also conducts numerical examples to illustrate the impacts of put option contract, supply chain structure and demand risk on the members’ decisions and total profits as well as the channel’s total profit. The results indicate that put option contract can motivate to increase the channel’s service level and reduce the manufacturer’s inventory risk under two supply chain structures, when compared to the case without put option contract. In the manufacturer-led structure, the channel always benefits from put option contract, the supplier benefits from put option contract with a high option price and a low exercise price, while the manufacturer benefits from put option contract with a low option price and a high exercise price. In the supplier-led structure, the channel and the manufacturer always benefit from put option contract, while the supplier benefits from put option contract with a high option price and a low exercise price. With put option contract, the supplier is more profitable in the manufacturer-led structure than in the supplier-led structure, while the manufacturer and the channel are more profitable in the supplier-led structure than in the manufacturer-led structure. Without and with put option contract, the optimal total profits of two members and the channel will first decrease and then increase in the demand risk. Finally, this paper identifies the explicit conditions under which the multi-period supply chain can be coordinated via put option contract under two supply chain structures. With a coordinating contract, the supplier and the manufacturer are better off compared to the case without put option contract.

Fair-value analytical valuation of reset executive stock options consistent with IFRS9 requirements

Executive stock options (ESOs) are widely used to reward employees and represent major items of corporate liability. The International Accounting Standards Board IFRS9 financial reporting standard which came into full effect on 1-Jan 2018, along with its Australian implementation AASB9, requires public corporations to report their fair-value cost in financial statements. Reset ESOs are typically issued to re-incentivise employees by allowing the option to be cancelled and re-issued with a lower exercise price or later maturity. We produce a novel analytical Reset ESO valuation consistent with the IFRS9 financial reporting standard incorporating the simultaneous resetting of vesting period, exercise window, reset level and maturity. We allow for voluntary and involuntary exercise. Our analytical result is expressed solely in terms of standardised European binary power option instruments. Using the multi-state mortality model of Hariyanto (2014, Mortality and disability modelling with an application to pricing a reverse mortgage contract, PhD thesis, University of Melbourne), we estimate longitudinal disability and death transition probabilities from cross-sectional data. We determine survival functions for pre-vesting forfeiture or post-vesting involuntary exercise for use with weighted portfolios of our formulae to illustrate the effect of survival on the fair value. We examine the IFRS9 method of valuation using expected time to option exercise and demonstrate a consistent overestimation of fair value of up to 27% for senior executives.

What Does an Executive Stock Sale Tell the Market? Evidence From Sale Transactions Following Stock Option Exercises

We examine the impact of liquidity and diversification, taxes, and loss aversion on the informativeness of insider sales following stock option exercise. We do not find evidence of liquidity and diversification influencing post-sale decisions, suggesting that option-related sales are less impacted by these forces than previously thought. We further find evidence that sales more likely to be influenced by taxes are associated with less negative post-sale returns. This finding extends prior tax research by showing evidence that insiders are selling soon after the tax-advantaged holding period is achieved and also that these sales are less informative about future returns. Finally, we provide a more distinct contribution to the literature in our examination of loss aversion. Specifically, we provide evidence that the exercise price is the most likely reference point for establishing the gain and loss domain for the insider. We further show that as the price approaches this reference point, sales become less informative and insiders are less willing to continue holding. We also examine short-window market reactions to sales among each of these dimensions, finding that the market generally interprets the impact of each of these factors in a manner that is consistent with the post-sale returns.

Efficacy of Nifty Index Options through BSM Model

Options are one of the products in financial derivatives, which gives the rights to buy and sell the product to an option holder in pre-fixed price which known as the strike price or exercise price at certain periods. Options contract was existed in various countries for long time, but it became very popular among the investors when the Fisher Black, Myron Scholes and Robert Merton were introduced the Black-Scholes Model in the year of 1973. This model was formerly developed by these three economists who were also receiving the Nobel prize for finding this innovative model. This model is mainly used to deal with the theoretical pricing challenge in options price determination. In India the trading in Index Options commenced on 4th June 2001 and Options on individual securities commenced on 2nd July 2001. There are many types in options contracts like stock options; Index options, weather options, real options and etc. This study has mainly been focusing on Nifty 50 index options which are effectively trade at NSE. This paper goes to describe about the importance of options pricing and how the BSM model has effectively used to find the optimum price of the theoretical value of call and put options.

Managing the equity risk using Short Put Ladder strategy by barrier options

The main aim of the paper is to measure hedging efficiency using the Short Put Ladder strategy formed by barrier options in the equity market. The researchers hedge full protection against price’s drop, combining the European down and knock-in put options with the lowest exercise price and vanilla or barrier put options with the higher exercise prices. The authors chose the analyzed alternatives according to the requirement of the zero-cost strategy. The aim of the investigated hedging variants is to secure the minimum constant selling price for the underlying asset’s price drop. Theoretical results of this approach were applied in the equity market, i.e., SPDR S&P 500 ETF. The authors analyzed and compared all hedging variants to each other, however, only the selected techniques were presented in the paper. The findings reveal that the barrier options used for managing the equity risk produce significant reductions of that risk. The right combination of options with the strike prices and the barrier levels wisely selected plays a significant role in risk elimination. Finally, according to the findings, the recommendations for potential investors are introduced.

Portfolio Investment with Options Based on Uncertainty Theory

In financial markets, there are situations where investors have the future stock prices according to the experts’ evaluations rather than historical data. Thus, the estimations of the stock prices contain much subjective imprecision instead of randomness. This paper discusses a portfolio investment with options in such a kind of situation. Treating the stock index price as an uncertain variable, we build an uncertain mean-chance portfolio model based on uncertainty theory and provide the equivalent form of the model. Furthermore, we make a comparison of the optimal expected return between portfolio investment with options and without options. An important conclusion is reached: The portfolio investment with options produces a no less expected return than that without options. In addition, we make sensitivity analysis and get two vital corresponding results. As an illustration, a numerical example is presented as well. The numerical results reveal that the options should be considered in portfolio investment. And the call option with maximum exercise price is most valuable per premium cost with the same exercise date.

A NEW STOPPING PROBLEM AND THE CRITICAL EXERCISE PRICE FOR AMERICAN FRACTIONAL LOOKBACK OPTION IN A SPECIAL MIXED JUMP-DIFFUSION MODEL

A new stopping problem and the critical exercise price of American fractional lookback option are developed in the case where the stock price follows a special mixed jump diffusion fractional Brownian motion. By using Itô formula and Wick-Itô-Skorohod integral a new market pricing model is built, and the fundamental solutions of stochastic parabolic partial differential equations are deduced under the condition of Merton assumptions. With an optimal stopping problem and the exercise boundary, the explicit integral representation of early exercise premium and the critical exercise price are also derived. Numerical simulation illustrates the asymptotic behavior of this critical boundary.