A Review of the Recent Advances Made in the Black-Scholes Models and Respective Solutions Methods

Corresponding Author: Gurudeo Anand Tularam Department of Mathematics and Statistics, Griffith Sciences (ENV), Environmental Futures Research Institute (EFRI), Griffith University, Brisbane, Australia Email: a.tularam20@gmail.com Abstract: The Black-Scholes model has been a major advance in finance over a period of time; this paper examines this model in some detail, in terms of the latest developments in both analytical and numerical solutions. The paper initially briefly examines historical aspects but quickly moves to a critical analysis of the works in the applications, solutions and implications for the efficiency and use of these. More specifically, this paper critically reviews the existing literature on the proposed exact as well as the numerical solutions to the Black-Scholes model. For this purpose, the exact solution literature on the Black-Scholes model includes speed based solutions and techniques based on valuation issues, time varying instruments and stochastic volatility. Similarly, the key numerical solutions include finite difference methods, the semidiscretization technique, Crank-Nicolson and the R3C scheme, the cubic spline wavelets and multi-wavelet bases method, the two-step backward differentiation formula in the temporal discretization and a High-Order Difference approximation with Identity Expansion (HODIE) scheme and fractional Black-Scholes model (TFBSM) along with Fourier analysis. This analysis reveals that transaction costs, high volatility, illiquid markets and large investor preferences are the key issues of today’s financial derivatives markets, especially after the Global Financial Crisis (GFC). These issues require non-linear solutions to the Black-Scholes models; therefore, Crank-Nicolson and the R3C scheme should be focused upon more by incorporating more and more real-life assumptions of current day trading.


Introduction
Black-Scholes (a pricing options equation) has been used over many years in financial markets particularly for various types of options including 'exotic' options. Financial markets are becoming complex but have some common issues, such as transaction costs, illiquid markets, large investors and risks from an unprotected portfolio. Such issues need to be incorporated whilst providing either an exact or a numerical solution to different types of options. Comparatively, recent studies have attempted to incorporate all of the relevant issues in attempts to provide a solution to the Black-Scholes model. This study briefly explains all of these developments in the Black-Scholes equation along with a historical perspective of the model. It also analyses the implications of the Black-Scholes model in different fields ranging from business (Corrado and Su, 1996) to construction projects (Barton and Lawryshyn, 2011). This is followed by a detailed examination of exact solutions, with given additional conditions involved in the Black-Scholes questions. The introduction of nonlinearity in the Black-Scholes means that the solutions will require the use of numerical methods. Some key studies regarding the exact solutions and numerical methods are critically examined. Consequently, the first aim of this study is to explore briefly the historical perspective, implication and advancement of Black-Scholes in terms of exact and numerical solutions. The other aim is to provide a categorical analysis and a critical review of the Black-Scholes in terms of both exact and numerical solutions. A brief discussion and a summary followed by the main findings will conclude this paper.

Background of the Black-Scholes Model
It is noted that the Black-Scholes equations are still used a lot in the financial and investment world. This is because the derivative markets have become significantly important and continuing to grow. Therefore, the key concern for all finance professionals is to understand the mechanism and functioning of these markets. The proper understanding of fair prices of these derivatives can help mitigate the risks for financial professionals. The second key motivation to properly understand this mechanism is to avoid a crisis similar to the recent global financial crises (2007)(2008)(2009) since it were the derivatives created for the residential mortgage in the USA that played a significant role in these crises (Hull et al., 2013).
The history of options markets goes back to the middle ages, when the futures were created in order to meet the need of merchants and farmers. Consider the position of a farmer in March who will harvest in June. Now, he is uncertain about the price of grains. The option market was developed to manage these kinds of risks. Later on, these trades were formalized through the trading boards, when the Chicago Board of Trade was established in 1848 in order to bring the merchants and farmers onto a single platform. It was not long after that this board opened the Chicago Board Option Exchange in 1973 (Hull et al., 2013). It was then that academicians and financial researchers started to focus on the valuation methods, such as the Black-Scholes equation to model the prices of the options.
Despite some estimation issues, as pointed out in Duan (1999;Yang, 2006;Harun and Hafizah, 2015), the Black-Scholes model has received considerable attention over past two decades-especially in underlying probability attributes of a European call option when written on a non-dividend stock. The Black-Scholes model estimates the probability of a European call option, which is frequently used in the investment decisions (see the work of Fischer Black, Myron Scholes and Robert Merton; they started publishing work related to the pricing of mainly European stock options). The focus was not on American options since the European one was much easier to deal with at the time.
More specifically, Fischer was working on the stock warrant valuation models when Scholes became interested in his work and joined his research. Based on this, they developed their initial versions of the Black-Scholes equation. The initial version of Black and Scholes (Black and Scholes, 1973) was rejected twice, by: (i) Review of Economics and Statistics and (ii) Journal of Political Economy. Later on, this work was further developed by Merton (1973) and interestingly, the Merton paper was accepted earlier than the Black and Scholes (Black and Scholes, 1973). It seems that, on grounds of fairness, Robert Merton asked the journal editor to hold this publication until the acceptance and publication of Black and Scholes (Black and Scholes, 1973).
An acknowledgement of the importance of the Black-Scholes model came in 1997, when Myron Scholes and Robert Merton were awarded the Nobel Prize in 1997. Fischer Black passed away on 30th August 1995; otherwise, he would undoubtedly also have been one of the recipients of the Nobel Prize. This model has now become one of the most important applications of Ito calculus in financial engineering. In this context, Wilmott et al. (1995) pointed out that the Black-Scholes or Black-Scholes-Merton model is the basic building block of the financial derivatives theory. Further, this model played a vital role in the growth and success of financial engineering. The above discussion raises an important question as to how Black, Scholes and Merton made their breakthrough.

Theoretical Justification of the Breakthrough
In terms of options discussed earlier, researchers have used various assumptions in order to calculate the expected payoff of the European options (Feltham, 1968). But in most instances, it was difficult to arrive at the correct discount rate, which is the key element in the calculation of the expected payoff of the European option, as elaborated in section 12.2 of Hull et al. (2013). In order to resolve this lingering issue, Black and Scholes used the capital asset pricing model to determine the association between the markets' required return on the option and the required return on the stock.
This was a rather complicated issue since the underlying relationship was dependent upon: (i) The stock price and (ii) time. In this nexus, Merton's approach was different, involving a riskless portfolio of the option and underlying stock. This riskless portfolio was based on the argument that the return on the portfolio over a short period of time must be equal to the risk-free return. This approach is more general when compared to Black and Scholes' approach since it does not rely on the assumption of the capital asset pricing model in valuing stock options, namely, the Black-Scholes-Merton Model. This model helped, in the end, to turn around the guessing game of forward pricing of options into a mathematical science, which in turn further aided in the development of pricing more exotic options and many other different tools in financial engineering.
In addition to the assumptions mentioned, MacBeth and Merville (1979) stated some other key assumptions, namely, (i) the underlying stock pays no dividend, (ii) the risk-free rate remains constant over the period of an option, (iii) the market players can lend and borrow at the risk-free rate, (iv) the price of the stock one period ahead has a log-normal distribution with constant mean and variance and finally, (v) the number of shares in the stock outstanding remains the same.

Implications of the Black-Scholes Model
During the 1970s, after the publication of Black and Scholes (Black and Scholes, 1973), researchers focused on extracting some empirical evidences from the key financial markets, including insurance sector giants. For instance, MacBeth and Merville (1979) conducted a comparative analysis of the real market prices of call options with the prices predicted by Black and Scholes (Black and Scholes, 1973). These studies attracted researchers from other fields to use these models in the related price predictions. Years later, the Black-Scholes model has been widely used in different fields ranging from business (Corrado and Su, 1996) to construction projects (Barton and Lawryshyn, 2011).
During the 1980s, MacBeth and Merville (1980) applied the model to test the Cox call option valuation for the constant elasticity of variance diffusion processes against the Black-Scholes model. This study observed the movement of common stock prices in line with the constant elasticity of variance diffusion processes. These results explored a new horizon for the capital market investors to predict the prices of common stock along with some other financial instruments. After a couple of years, Chesney and Scott (1989) applied the Black-Scholes model to hedging the risks against underlying securities. During the same period, the model was used in the developing economies in order to establish an optimal educational policy. More specifically, here the model was used to quantify the trade-off between providing the generalizing training as well as the specific skill training, as can be observed from Miller (1990).
From 1990 to 2017 this model has been applied to many cases and in different fields ranging from construction projects (Barton and Lawryshyn, 2011) to evaluation of information technology projects (Benaroch, 2002). In the construction industry projects, Barton and Lawryshyn (2011) priced the real options under the risk-neutral measure with a closed-form solution in order to observe the association between cash flow and value of the project. This study suggested that some of the risk can be mitigated by a delta-hedging strategy, where the project is related to the market.
The Black-Scholes model was also important in information technology. Benaroch (2002) presented an approach for managing information technology investment risk through the use of options in order to optimally control the balance between reward and risk. This study extended the application of model to the establishment of an internet sales channel and their related investment. Del Giudice et al. (2016) conducted a review of Black-Scholes. An overview of this qualitative approachbased study reveals that most of the application of the Black-Scholes model lies in the business studies sector focusing on the financial markets, including investment in research and development-especially in pharmaceutical companies (McGrath, 1997;, customer relationship management (Maklan et al., 2005), assessment of bonds and derivatives (Singh, 2014) and management and evaluation of intangible assets (Park et al., 2012).
Along similar lines, Cutland et al. (1995) explored new horizons of implication of the Black-Scholes model, including ascertaining the value for the deposit insurance, loans for students, farm prices support, patents, pollution rights, policies for governments and drilling rights. Despite all this, the key relevant literature focuses on the stock markets in both equity and fixed income instruments (Geske et al., 1983;Rubinstein, 1983;Scott, 1987).
Comparatively recently, McKenzie, Gerace and Subedar (2007) tested the underlying probabilities attributed within the model in the Australian Stock Market. Applying qualitative regression and a maximum likelihood approach, the results of this study are in line with the Black-Scholes model. This study also includes alternative approaches such as jump-diffusion and implied volatility. Hong (2004) used the data from 1994 to 2003 from Malaysian stock markets and reported that the overall Black-Scholes model prices were significantly below the market prices; both market prices and the Black-Scholes model deviated in a certain systematic pattern. The Black-Scholes model can be applied as an investment strategy in the Malaysian stock market once the systematic pattern of deviation is clear for the specific investment. On similar lines, Mohanti and Priyan (2014) applied the Black-Scholes model and dynamic hedging strategy using daily closing prices of S&P CNX Nifty index options contracts from 1 April 2008 to 31 March 2012 in the Indian stock market. This study reported that the Indian index option market is efficient.
This brief review shows that Black-Scholes model has a number of different applications. In some cases, the results are in line with the model predictions, while in other cases; it seems there are some discrepancies. Nevertheless, the model has clearly the ability to examine various types of valuation of derivatives in different markets. Second, we note that financial derivatives are relatively less used in the electricity market and indeed in the Australian power market. This paper aims to provide a critical analysis of the advancement in the exact and numerical solutions to the Black-Scholes. Such an analysis of the latest advancements will allow researchers to choose more advanced analytical and technical methods of solutions for their research studies and model solutions.

The Black-Scholes Analytical Solution
During the last couple of years, different studies have provided solutions covering different quantitative aspects of the model. For instance, Shin and Kim (2016) focused on the Black-Scholes terminal value problem and provided its solution through the Laplace transform. This study claimed that the proposed method is simpler than the existing methods. In the early 1990s, Harper (1994) applied the generalization technique in order to provide an exact solution for the Black-Scholes equation by reducing parabolic partial differential equations to canonical form. In this generalized form, the variables corresponding to the time appear to run backwards. These fluid mechanics can be applied in other financial derivative products where there are similar informationtheoretic reasons behind the products.
Later, Forsyth et al. (1999) applied the finite element method under stochastic volatility to provide the exact solution of the Black-Scholes equation using the moneyvanilla European option. In this approach, the outgoing waves are also correctly modelled whilst the boundary equations are discretised. For future research, this proposed technique can be applied on American options since the European option was used for the same time remaining until the expiry time.
In the early 2000s, the advanced financial engineering techniques become a part of financial products issued for the housing market. During this time, Jódar et al. (2005) considered the final-value problem and applied the Mellin transform to provide a new method of direct solution of Black-Scholes equation. This proposed method can be applied to the relevant option pricing models.
A year later, Rodrigo and Mamon (2006) incorporated the time varying factor into different aspects of options in order to provide the exact solution of an explicit formula for the price of an option on a dividend-paying equity. In this study, Rodrigo and Mamon (2006) used options of (i) dividend-paying equity and (ii) non-dividend paying equity. As output, this provides a simple derivation of the explicit formula of an option in time dependent parameters of the Black-Scholes partial differential equation. Thus, the pricing of other return-based equity instruments is possible through this mechanism.
Bohner and Zheng (2009) went further and applied the Adomian approximate decomposition technique to provide an exact solution to the equation. The authors suggested that the proposed technique be applied when studying some other problems in finance theory.
Comparatively recently, Edeki et al. (2015) modified the classical Differential Transformation Method (DTM) and used its modified version, Projected Differential Transformation Method (PDTM), to provide a faster exact solution to the Black-Scholes equation for European option valuation. A critical analysis reveals that this fast solution is efficient, reliable and better than the classical DTM. Additionally, it is recommended that both linear and nonlinear stochastic differential equations be encountered in the field of financial mathematics. For future research, this algorithm can be applied on the European put options. Table 1 summarizes the above discussion of the analytical techniques.
Numerical Solutions to Black-Scholes Forsyth et al. (1999) used the finite element approach to the pricing of discrete lookbacks with stochastic volatility. Along similar lines, Tangman et al. (2008) considered High-Order Compact (HOC) schemes for quasilinear parabolic partial differential equations to discretise the Black-Scholes PDE for the numerical pricing of European and American options. Dremkova and Ehrhardt (2011) then presented compact finite difference schemes to solve nonlinear Black-Scholes equations for American options with a non-linear volatility function. Since a compact scheme cannot be applied on the American type options, the study used a fixed-domain transformation. Around the same time, Song and Wang (2013) applied symbolic calculation software to provide a numerical solution using the implicit scheme of the finite difference method. This study combined the timefractional Black-Scholes equation with the conditions satisfied by the standard put options. Two years later, Uddin et al. (2015) presented the numerical result of semi-discrete and full-discrete schemes for European call option and put option by Finite Difference Method and Finite Element Method. In a recent study, Zhang et al. (2016)  The nonlinear Black-Scholes models are now gaining popularity because most of the realistic assumptions including transaction costs, high volatility, illiquid markets and large investor's preferences can also be included. Ankudinova and Ehrhardt (2008) analysed that the Crank-Nicolson and the R3C scheme are the most accurate techniques to price the European call option. This study incorporated different volatility problems in stock price, option price and its derivatives.
Recently, the two-dimensional Black-Scholes equation was explored by Černá et al. (2016) using cubic spline wavelets and multi-wavelet bases. The proposed method suggests some key advantages, including (i) high-order accuracy, (ii) a small number of degrees of freedom and (iii) a relatively small number of iterations. Rao (2016) applied the two-step backward differentiation formula in the temporal discretization and a High-Order Difference approximation with Identity Expansion (HODIE) scheme-concerned with the generalized Black-Scholes models for European call option. This study presents the case of the European call option. The solution noted second-order accuracy in time and third-order accuracy in space.
In a recent study, Zhang et al. (2016;Yang, 2006) applied fractional Black-Scholes model (TFBSM) along with Fourier analysis; this study extends different numerical solution literature on the price change of the underlying fractal transmission system. First, this study derives the fractional Black-Scholes model with an α-order time fractional derivative by applying numerical simulation. Second, the proposed techniques and method are used to price different European options. The review of the above discussion on the numerical solution is categorized and presented chronologically in Table 2.

Discussion of Analytical Solutions
The exact solution literature provided in Table 1 addresses a couple of real-word phenomena of the financial markets. Considering these phenomena, this literature can be categorized into the following: i. A speed-based solution in which Edeki et al. (2015) applied Projected Differential Transformation Method (PDTM), a modification of classical Differential Transformation Method (DTM) for a faster solution to Black-Scholes equation: ii. Valuation issues-based techniques using which Shin and Kim (2016) addressed the terminal value problem of the Black-Scholes model by using the Laplace transformation: Jódar et al. (2005) applied the Mellin transformation to solve the final-value problem: iii. A time-varying instruments-based technique that Rodrigo and Mamon (2006) used time-varying parameters for the options of dividend paying equity as well as the non-dividend paying equity: All of these proposed techniques provide analytical and/or quasi-exact solutions by incorporating different issues of financial derivatives. Therefore, these techniques have their importance for the applicability purpose.

Discussion on Numeric Solutions
The nature of the conditions used in the valuation methods make models such as Black-Scholes a nonlinear one and they then require numerical approaches to reach satisfactory solutions. Some of the aspects that influence the nature of the model are: Transaction costs, illiquid markets, large investors, risks from an unprotected portfolio, weights allocation methods along with the long movements and jumps over the small time steps are the common problem in the financial derivatives markets. All of these issues are incorporated into different studies (Dremkova and Ehrhardt, 2011;Song and Wang, 2013;Tangman et al., 2008;Uddin et al., 2015;Zhang et al., 2016), all of which attempted to provide the numerical solution to the Black-Scholes model using finite difference methods. Other proposed techniques in numerical solution approaches are as follows: i. Semi-discretization technique (Company et al., 2008): ii. Crank-Nicolson and the R3C scheme-the most accurate techniques to provide a solution to the nonlinear Black-Scholes models (Ankudinova and Ehrhardt, 2008): iii. The cubic spline wavelets and multi-wavelet bases method to provide a solution for the two-dimensional Black-Scholes equation (Černá et al., 2016): It has been noted that the transaction costs, high volatility, illiquid markets and large investor's preferences are some key issues in today's financial derivatives markets, especially after the event of GFC. These inclusions require the solving of a nonlinear solution to the Black-Scholes models. It is recommended therefore that the Crank-Nicolson and the R3C scheme be focused on more because they can accommodate more and more real life assumptions of current-day trading (Ankudinova and Ehrhardt, 2008).

Conclusion
This paper critically analyse the recent advances made in the Black-Scholes model and solution methods. The authors examine the historical aspects, various applications, analytical and numerical solutions and efficiency and use of the same. The historical perspective and implication section of this study reveals that the Black-Scholes model is able to examine various types of valuations of derivatives in different markets. Many researchers have attempted to obtain the solution of the Black-Scholes model analytically or numerically, thereby adopting and using various direct and iterative methods, respectively. In the literature on the Black-Scholes model, exact solutions come in the form of speed-based solutions and techniques based on valuation issues, time-varying instruments and stochastic volatility. All of these proposed techniques provide analytical and/or quasi-exact solutions by incorporating different issues of financial derivatives. Therefore, these techniques have advanced their importance in the applicability as well as purpose. The numerical solutions include finite-difference methods, semi-discretization technique, Crank-Nicolson method, the R3C scheme, cubic spline wavelets and multiwavelet bases method, the two-step backward differentiation formula in the temporal discretization and a High-Order Difference approximation with Identity Expansion (HODIE) scheme, fractional Black-Scholes model (TFBSM) using a Fourier analysis. The main findings show that transaction costs, high volatility, illiquid markets and large investor preferences are key issues of today's financial derivatives markets-especially after the Global Financial Crisis (GFC). Complexity issues require non-linear solutions to the Black-Scholes model; therefore, the Crank-Nicolson method and the R3C ought to should be focused on more by incorporating more real-life assumptions of current day trading. Moreover, the Finite difference methods are by far the simplest, except when mesh adaptively is required; in which case it is rather difficult to control the numerical error. It is noted that the FEM performs better with respect to the convergence than the explicit FDM particularly when small number of time points is used. The FEM converges to the solution when the explicit scheme of FDM diverges.