Numerical Methods for Mathematical Models on Warrant Pricing

>Magister Scientiae - MScWarrant pricing has become very crucial in the present market scenario. See, for example, M. Hanke and K. Potzelberger, Consistent pricing of warrants and traded options, Review Financial Economics 11(1) (2002) 63-77 where the authors indicate that warrants issuance affects the stock price process of the issuing company. This change in the stock price process leads to subsequent changes in the prices of options written on the issuing company's stocks. Another notable work is W.G. Zhang, W.L. Xiao and C.X. He, Equity warrant pricing model under Fractional Brownian motion and an empirical study, Expert System with Applications 36(2) (2009) 3056-3065 where the authors construct equity warrants pricing model under Fractional Brownian motion and deduce the European options pricing formula with a simple method. We study this paper in details in this mini-thesis. We also study some of the mathematical models on warrant pricing using the Black-Scholes framework. The relationship between the price of the warrants and the price of the call accounts for the dilution effect is also studied mathematically. Finally we do some numerical simulations to derive the value of warrants

Long Memory Options: LM Evidence and Simulations

This paper demonstrates the impact of the observed financial market persistence or long term memory on European option valuation by simple simulation. Many empirical researchers have observed the non-Fickian degrees of persistence or long memory in the financial markets different from the Fickian neutral independence (i.i.d.) of the returns innovations assumption of Black-Scholes' geometric Brownian motion assumption. Moreover, Elliott and van der Hoek (2003) provide a theoretical framework for incorporating these findings into the Black- Scholes risk-neutral valuation framework. This paper provides the first graphical demonstration why and how such long term memory phenomena change European option values and provides thereby a basis for informed long term memory arbitrage. By using a simple mono-fractal Fractional Brownian motion, it is easy to incorporate the various degrees of persistence into the Black-Scholes pricing formula. Long memory options are of considerable importance in corporate remuneration packages, since stock options are written on a company's own shares for long expiration periods. It makes a significant difference in the valuation when an option is 'blue' or when it is 'red.' For a proper valuation of such stock options, the degrees of persistence of the companies' share markets must be precisely measured and properly incorporated in the warrant valuation, otherwise substantial pricing errors may result.Options, Long Memory, Persistence, Hurst Exponent, Identification, Simulation, Executive Remuneration

Pricing equity warrants under the sub-mixed fractional Brownian motion regime with stochastic interest rate

This paper proposes a pricing model for equity warrants under the sub-mixed fractional Brownian motion regime with the interest rate following the Merton short rate model. By using the delta hedging strategy, the corresponding partial differential equations for equity warrants are obtained. Moreover, the explicit pricing formula for equity warrants and some numerical results are given

Actuarial approach in a mixed fractional Brownian motion with jumps environment for pricing currency option

This research aims to investigate the strategy of fair insurance premium actuarial approach for pricing currency option, when the value of foreign currency option follows the mixed fractional Brownian motion with jumps and the European call and put currency option are presented. It has certain reference significance to avoiding foreign exchange ris

Dynamic hybrid pricing formulation for equity warrants

Equity warrants are instruments issued by a company that give the stockholder the privilege of buying a stock at a certain strike price within a particular timeframe. Motivated by empirical studies, the Black-Scholes option pricing model is not suitable to price a warrant since both assumptions of constant volatility and constant interest rates in the model are incompatible. This study proposed the Heston-Cox-Ingersoll- Ross (Heston-CIR) hybrid model to identify the effects of stochastic volatility and stochastic interest rates in pricing equity warrants. The study constructed new analytical pricing formulas for equity warrants by using Cauchy transformation and partial differential equation approaches. The local optimization method is employed to obtain the estimated parameter values by calibrating the Heston-CIR model. The effectiveness of the proposed model is investigated through the empirical study using the data from Bursa Malaysia. The proposed model shows significant improvement on the computation time in estimating nine model parameters, ranging from 38.12 to 62.62 seconds compared to the existing models. Moreover, the empirical study suggested that the proposed model is accurate when compared to the real market over five years period. This model also produced smallest pricing errors among the existing models. The finding also suggested equity warrants in moneyness opportunity, 88.75% of the warrants are profitable. In conclusion, the proposed model performs the best in identifying the effects of stochastic volatility and stochastic interest rates in pricing equity warrants

The valuation of currency options by fractional Brownian motion

This research aims to investigate a model for pricing of currency options in which value governed by the fractional Brownian motion model (FBM). The fractional partial differential equation and some Greeks are also obtained. In addition, some properties of our pricing formula and simulation studies are presented, which demonstrate that the FBM model is easy to use.© 2016 The Author(s). This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.fi=vertaisarvioitu|en=peerReviewed

A comprehensive literature review on pricing equity warrants using stochastic approaches

Prior literature's revealed that most researchers tend to employ the Black Scholes model to price equity warrants. However, the Black Scholes model was found deficient by contributing to large estimation errors and mispricing of equity warrants. Therefore( issues revolving equity warrants are discussed in this paper, by focusing on specific topics and respective stochastic models to provide a basis for improvements in future research. In recent years, stochastic approaches had been used to a great extent among researchers due to the expansive applications in both theoretical and practical sense. Subsequently, this paper provides the results of a comprehensive literature review on various stochastic modelling methods and its applications for pricing financial derivatives in terms of applications, modifications of methods, comparisons with other methods, and general related researches. Focus is given on two types of stochastic mod~ls namely stochastic volatility and stochastic interest rate models; along with the discussions associating these two types of models. This paper acts as a valuable source of information for academic researchers and practitioners not only for pricing financial instruments, but also in various other fields involving stochastic techniques

Hybrid equity warrants pricing formulation under stochastic dynamics

—A warrant is a financial contract that confers the right but not the obligation, to buy or sell a security at a certain price before expiration. The standard procedure to value equity warrants using call option pricing models such as the Black–Scholes model had been proven to contain many flaws, such as the assumption of constant interest rate and constant volatility. In fact, existing alternative models were found focusing more on demonstrating techniques for pricing, rather than empirical testing. Therefore, a mathematical model for pricing and analyzing equity warrants which comprises stochastic interest rate and stochastic volatility is essential to incorporate the dynamic relationships between the identified variables and illustrate the real market. Here, the aim is to develop dynamic pricing formulations for hybrid equity warrants by incorporating stochastic interest rates from the Cox-Ingersoll-Ross (CIR) model, along with stochastic volatility from the Heston model. The development of the model involves the derivations of stochastic differential equations that govern the model dynamics. The resulting equations which involve Cauchy problem and heat equations are then solved using partial differential equation approaches. The analytical pricing formulas obtained in this study comply with the form of analytical expressions embedded in the Black-Scholes model and other existing pricing models for equity warrants. This facilitates the practicality of this proposed formula for comparison purposes and further empirical study

Option pricing in fractional models

This thesis deals with application of the fractional Black-Scholes and mixed fractional Black-Scholes models to evaluate different type of options. These assessments are considered in four individual papers. In the first articles, the problem of geometric Asian and power options pricing is investigated when the stock price follows a time changed mixed fractional model. In this model, an inverse subordinator process in the mixed fractional Black-Scholes model replaces the physical time. The aim of the third paper is to evaluate the European currency option in a fractional Brownian motion environment by the time-changed strategy. Also, the impact of time step and long range dependence are obtained under transaction costs. Conditional mean hedging under fractional Black-Scholes model is the propose of the second article. The conditional mean hedge of the European vanilla type option with convex or concave positive payoff under transaction costs is obtained. In the fourth article, the mixed fractional Brownian motion with jump process are incorporated to analyze European options in discrete time case. By a mean delta hedging strategy, the pricing model is proposed for European option under transaction costs.Väitöskirja tarkastelee fraktionaalisen Black–Scholes -mallin ja sekoitetun fraktionallisen Black–Scholes -mallin käyttöä erityyppisten optioiden arvottamisessa. Tätä tutkitaan neljässä artikkelissa. Ensimmäisessä artikkelissa tarkastellaan geometrisia aasialaisia optioita ja potenssioptioita, kun osakehinta noudattaa aikamuunnettua sekoitettua fraktionaalista mallia. Tässä mallissa sekoitun fraktionaalisen Black–Scholes -mallin käänteinen subordinaattoriprosessi korvaa fysikaalisen ajan. Kolmannen artikkelin tarkoitus on hinnoitella eurooppalainen valuuttaoptio fraktionaalisen Brownin liikkeen mallissa aikamuunnetulla strategialla. Lisäksi aika-askeleen ja pitkän aikavälin riippuvuuden vaikutusta tutkitaan transaktiokulujen alaisuudessa. Ehdollinen keskiarvosuojaaminen fraktionaalisessa Black–Sholes -mallissa on toisen artikkelin aihe. Ehdollinen keskiarvosuojaus eurooppalaiselle vaniljaoptiolle, jolla on konveksi tai konkaavi positiivinen tuottofunktio transaktiokulujen vallitessa, on artikkelin päätulos. Neljännessä artikkelissa tutkitaan eurooppalaisia optioita diskreetissä ajassa mallissa, joka on hypyllinen sekoitettu fraktionaalinen Brownin liike. Käyttäen keskiarvoista deltasuojausstrategiaa artikkelissa johdetaan hinnoittelumalli eurooppalaisille optioille transaktiokulujen vallitessa.fi=vertaisarvioitu|en=peerReviewed