Option pricing by finite difference methods 1 Table of Contents 1.0 Introduction ........................................................................................................................... 4 1.1 Finite difference methods of option pricing .......................................................................... 4 1.2 Price differential equation (PDE) .......................................................................................... 5 1.3 Purpose and problem formulation ......................................................................................... 5 1.4 Research questions ................................................................................................................ 5 1.5 Objective of the research ....................................................................................................... 6 1.6 Limitations of the research .................................................................................................... 6 1.7 Methodology ......................................................................................................................... 6 1.8 Overview of the thesis ........................................................................................................... 6 Chapter two ..................................................................................................................................... 8 2.1 Introduction ........................................................................................................................... 8 2.2 Option pricing and its history ................................................................................................ 8 2.3 Historical approach towards options ..................................................................................... 8 2.4 Hedging with options ............................................................................................................ 9 2.5 Put options ............................................................................................................................. 9 2.5.1 Sensitivities of option ................................................................................................... 10 2.5.2 Delta.............................................................................................................................. 11 2.5.3 The gamma ................................................................................................................... 12 2.5.4 The Vega....................................................................................................................... 12 2.6 American option .................................................................................................................. 12 2.7 Arbitrage.............................................................................................................................. 13 2.8 Efficient market Hypothesis ................................................................................................ 13 Chapter three ................................................................................................................................. 15 3.1 Introduction ......................................................................................................................... 15 2 3.2 Option pricing _Black Sholes equation ............................................................................... 15 3.2.1 Assumptions with regard to stock economic environment. .......................................... 15 3.3 Finite difference method ..................................................................................................... 19 3.4 Implementation of the methods ........................................................................................... 19 3.4.1 Forward approximation ................................................................................................ 20 3.5 Application of finite method in option pricing ................................................................... 22 3.6 Backward approximation .................................................................................................... 26 3.7 Central approximation method ............................................................................................ 26 3.8 Implicit method ................................................................................................................... 26 3.9 Using Crank-Nicholson Method ......................................................................................... 28 Chapter Four ................................................................................................................................. 31 4.0 Introduction ......................................................................................................................... 31 4.1 Choice of the best alternative to be used in option pricing ................................................. 31 3 Chapter one 1.0 Introduction An option is a contract which offers the owner also known as the holder of the option rights to purchase (buy) or dispose (sell) the asset at a fixed price also known as the exercise or a strike price. It is disposed or purchased at a specified underlying asset for example a stock index or at a foreign curacy at any period using a foreign currency before maturity or expiry of the underlying asset at a stated time. Option at its up initial nature possesses some specific features that makes it unique from other financial and tradable instruments in the financial market. Option gives the holder right to hold the asset but with no obligation and the holder is not obliged to exercise contract if not at a convenient with him or her to dos so with the option .On the other hand, options gives seller the right to write when the holder has made a decision to close or dispose the option. It is rationale at any point before even making a step to determine the various methods of pricing the different kinds of options to be traded in the financial market. to make such a distinction , it helps one to differentiate those options that gives one right to buy also known as call options and those that gives the holder the rights to sell the option commonly known as the put options .It can also be distinguished based on the time that the option is deemed to be exercised . That is it can be classified as either on the expiry or maturity date or it can be excursed only on the maturity or expiry date. Options to be exercised on days before expiration is known as American option and those to be excursed on the day of maturity is called European option. There are also many other ways of classifying options , that is on the basis of security kind, rate of payment on the owners , the strike price it pays to the holders of the option and the role of the strike price in that option . All this classification is based on the three factors that revolves around the obligations, rights to the holder and the buyer and at the same time its impact on the strike pricing. 1.1 Finite difference methods of option pricing This is a method that entails numerical valuation of options using a mathematical approach. It is a valuation method that was first employed in option valuation by the proponent Eduardo Schwartz in the year 1977. 4 1.2 Price differential equation (PDE) In this finite difference methods, options can be valued on a continuous time differentiation equation which gives a description on how an option is valued based on its evolution over a set of time. This differentiation method can also be used to solve option prices using a calculation approach by use of model partial differentiation equation (PDE) that will act as a function of time and at the same time differentiated prices of the option. Under this approach, Black Sholes PDE .When this method is applied, then it means that a finite differentiation model can be obtained and then the best valuation is obtained. Those pricing problems that are considered to possess the same level of complexity can be solved with the use of tree approach. At the end of this research, application of different methods of option pricing will be applied with a practical computation and valuation to reach at the conclusion on the best approach to be employed. Step by step methods will also be analyzed in different methods of option computation. 1.3 Purpose and problem formulation The main aim of this thesis is to find the important theories and the application of financial mathematics and other methods of valuation of stock that is the finite method of option pricing at different financial market. From the valuation of the options, then an optimal analysis if prices will be performed as shown by the step to step valuation methods to be adopted in this system of computation. 1.4 Research questions The thesis will also strive to answer the following research questions with an aim of arriving at the conclusion of the finite option pricing models: How does a change in the volatility that is dividends yield when given a free risk interest rate asset in an optimal price exercised option? What is the optimal pricing option that needs to be used with the mathematical approach in arriving at the optimal pricing? How does a change in volatility that is in terms of dividends rate of yield, risk free interest rate collateral affect the exit pricing in an option valuation? 5 How do we find the optimal valuation pricing of an option given the methods? 1.5 Objective of the research The main objective of the research is to assess different methods of finite option pricing To assess the steps of pricing options To help in assessing the best method with its implication in the option pricing. Implicit finite difference methods for option pricing 1.6 Limitations of the research The main limitations towards the attainment of the project objectives is that valuing options under different methods needs an equation and for example with Black Scholes, it is a complex method to derive and use in the Europe option pricing. 1.7 Methodology For us to fulfill the objectives of this research, there are two approaches that is deemed as important. The two thesis methods includes: A review of various literatures Finite (numerical computations) The main aim of reviewing the previous literature is lay a solid foundation to the discussion, give a sold theoretical valuation of options and help in building on stock valuation methods to be discussed and assessed. The main role of the computation is to get the results of the methods to be used in option valuation and demonstrates clearly and step wise how the methods can be used practically to arrive at the computations valuation. 1.8 Overview of the thesis The theism will be organized in the following format: Chapter two will give an overview of various literature laying the foundation of the study and building framework that will integrate the steps to be used in option valuation Chapter three will give an overview of mathematical computations, that is the finite stock valuation models with its derivative formulas 6 Chapter four will offer a presentation on how we apply finite difference method in solving his problems that were presented in chapter three and evaluation of different methods of arriving at the valuation models to be employed in the computation. Chapter five will present the American option problems and solutions to such problems as a demonstration towards stock valuation Chapter six will offer the comments from the valuation models Chapter seven will offer the conclusion and the recommendation towards a sustainable model. 7 Chapter two 2.1 Introduction Since there are different finite model of options valuation, a review of the past literature will be done on this chapter so as to help in identifying the gaps that needs to be filled by the research. In this chapter, journals, publications and books and other materials will be reviewed so as to give a strong background of the thesis statement that will be utilized in solving the research problem as indicated in the research. 2.2 Option pricing and its history An option is a contract between two parties that agrees upon either selling or buying an asset at a determined strike price in the future. The buyer of an option will pay a premium to get the right to hold the contract. The price of the option depends on the underlying asset, which most commonly is either a stock, commodity, currency or an index. From a game theory point of view, options are a zero-sum game because the sum of each party’s gain or loss is exactly equal Options main aim is set to eliminate the risk that exist in the pricing of financial assets . Options are also deemed to have a negative correlation with its portfolio hedge and it is set with an aim of hedging the risk. Decrease in the value of option lowers its affinity towards the risk analysis and this lowers the ability of the firm to hedge against the risk of such an asset. European and American options are some of the fundamentals options whose formula and valuation will be used in different methods of stock valuation .The main difference between the two is that one has the right prior to date expiration, that is American option and the other lacks the right to be exercised at the point of stock expiration. 2.3 Historical approach towards options Options computation is traced back in 350 BC when one of the philosopher Aristotle made Miletus fortune from option rights to use olive presses. In 1970s the first attempt when Brownian motion was introduced in determining the future pay-off. Black Sholes and Merton created a finding and formula to compute the European pricing option and made an introduction to the theory of financing portfolios. From there henceforth, it indicates that there have been a tremendous popularity growth in option pricing and it is now deemed to be the most common financial derivatives. 8 2.4 Hedging with options In finding a hedging option, it is rationale that a measure needs to be made so as to make the right decision of pricing the option. In this case, there are various options that will be discussed with its aim being to find the right valuation process and the methods that is deemed important in valuing the options. 2.5 Put options They are referred to as financial contracts or derivatives that are placed in order to regulate the selling of assets. The main bearer of the contract is to have a short term underlying assets that possesses the right but with no obligation of liability to dispose the asset to the strike price on a future determined value .The one who rights the option as compared to the holder will long the underlying asset. The case of a put option is to ensure that the rise of the portfolio is reduced to an acceptable level where the sell cannot be executed. For example, if one has a portfolio that has a value of $ 200 and holds 20 stocks, for example Common wealth bank having a stock price of $10, the owner will have aim to minimize the risk of the portfolio and speculates a reduction in the price of the stock .The owner would therefore purchase the 20 stock at $1 for each strike price. Under the case above, it will lead to a profit of (20-15)-1 where the $ 15 is deemed as the time to expiration pricing. This will lead to a profit of $ 4 from the sale execution. If from the computation, the underlying portfolio lead to a loss, then the hedging factor could have been considered so as to mitigate the option against the loss. If the underlying asset is deemed to have never had a strike price, then the owner of the stock portfolio could have registered a loss. The graph below shows the relationship between strike price and the premium with the gains on a call options Strike price Profit or loss 9 Premium Share price The above indicates how the strike price and premium have an impact on the gains or loss in determining the call option of a share. In this case, the following determinants and mathematical computations needs to be used in approaching the above options: C(S, t) = max(S − K, 0) The determination of gains is equal to pay-off less the premium Profit=C(S, t) which is the pay-off –Premium In this case, we can conclude that in option determination, a payoff represents the expected amount of price that is backed up by the seller of the option and written in the option and his p premium is the price that is deemed higher than the stated price in the option 2.5.1 Sensitivities of option This are also known as Greek letters of options and its main rationale is to determine the different risks that are associated with the measure of an option that is to be traded in the market. Each risk measure is deemed to be a derivative of the value of an option with respect to the parameters that underlies the option. The three common sensitive used in the pricing of options are as follows: The delta The gamma The Vega 10 2.5.2 Delta It is represented as ∆ and it is a measure that represents a change in the value of an option with respect to the changes in the underlying derivative or assets. The change in the value of an option is a representation of the change in the value of the underlying asset or the derivative. The importance of this measure is that it can help the holder of the asset to determine the sensitivity of the asset and gives one the room to hedge the asset against any possible risk that might affect the asset. In the case of neutrality, that is the point where the change is zero, then the asset is not sensitive and to the holder, it means that holding or disposing the asset has no implication to the call option. Changes in the underlying price of an asset over time leads to either a positive static gain or negative static loss. In this case, it indicates that a static loss requires the bearer of the asset to undertake a hedging process so as to prevent one from losing the value of the asset and increasing the risk speculation. Hedging needs to be performed on a periodic basis so as to help one obtain a rebalance and obtain a successful hedging process. From the graph above, it is an indicator of different delta values of an asset. Assume that the asset has K=100, T=1, α=0.2 and r=0.1 Then, the delta is deemed to be affected by the changes in the price of the option. In this case, it is a dynamic option and it requires periodical rebalancing so as to help hedge the option against any possible risk that might occur in the value determination of the asset. For the European option, the following delta equation will be computed as follows: 11 Call option: e−qtr. φ (d1) Put option: e−qT [1 − φ (d1)] ln(S/K)+(r+δ2/2(T−t) Where d1= 𝛿 √𝑇−𝑡 2.5.3 The gamma This is the second sensitivity representation that is used in finance in determining the price of an asset and its sensitivity value. In this case, a gamma measure the delta change in respect to the value of an underlying asset portfolio. The gamma is used as a hedge protector more so when a postion hedging is to be performed in different level of measures. It helps in performing a successful protection of a hedge and in the long run, it helps in delta rebalancing ad bettering the delta value .a gamma that is perceived to be neural miss the one that has a full protection against changes in the price of an asset. In most cases, when hedging is done, gamma and delta normally assumes a constant volatility in the asset valuation. It is normally a limitation as it varies due to various factors that affects the price of an underlying asset. In most cases, volatility of an asset that is a change in the value of an asset tends to affect its hedging and in the long run, it affects the prices of an asset. At a point where volatility is deemed to be zero for an underling asset, then the bearer of the asset is called to perform a rebalancing and in this instance, it will broaden the range of an asset pricing. Hedging should normally be performed in assets that have a lower volatility as this will held in developing an option strategy that will lead to a Vega hedging. 2.5.4 The Vega It is a hedging strategy that helps to protect asset value against the fluctuation or changes in volatility. In the case where the V is highly negative or positive, then the sensitivity position is deemed to be very high to the volatility ratio. In the case where the value is nearly zero, it indicates that the hedging limit for the option is close. 2.6 American option This is an option which gives the holder an additional rights to hold the contract. It is an option that is deemed to be exercised at any duration prior to or before the lapse of time or its expiration. It is worth more than the European option since it possesses an additional right to the contract. If not exercised prior to the set time of lapse, then its value is deemed the same with 12 that of the European option. Time of excursed is important as it also determines the risk that is subject to the underlying asset and in the long run, it will also have an impact to its overall value in the asset determination. Early excursive that is performed on the American option is an indicator that it has a higher value because in this case , dividends would not have been distributed and even if they possess the same option value , they tend to have a variation in the pricing of an option . 2.7 Arbitrage In most cases , when one has an asset , that is acquired then resale the asset within the shortest time when the value have increased , than one gains a profit known as an arbitrage profit . In many cases, it is referred to as an imbalance between the market difference and the profit that is deemed as risk free with no any underlying risk of holding the asset. If a trade is performed, then an arbitrage opportunity is realized as its market value is not the same as the priced value. It can be presented as (n+1) where the trade is performed when the future prices of an asset is an asset is known but have not been traded at a discounted price where the rate of interest is deemed to be free from risk . In this case, the dimension of θ (t) = 10b (t), θn (T) needs to satisfy the following conditions when T, that is time of all expiration is greater than zero. T>0 V θ (0) = 0 V θ (T)) ≥ 0 P (V θ (T) > 0) > 0 , where V θ is the value of the portfolio and P denotes the probability. 2.8 Efficient market Hypothesis It is represented as EMH. In this case, it states that three are certain required conditions that needs to be satisfied by a trade for one to have an efficiency in the market. the proponent states that stocks that are traded at their par value at any time given in a market that is deemed to be efficient which implies that it is impossible for the buy to disvalue the assets to which the investor has a right to obtain in excess of the return . Then the following needs to be satisfied as it will also help in stock pricing models: 13 In the long run , there is no arbitrage There should be no information asymmetry or in congruency and all the information should be made available freely and can be accessed easily. existing information should not in all cases leads to creation of excess returns Market value assets rationally, even with a single trade needs to act irrationally. The main implication here is that with the investor, the net value effect should create rationality in the investors trade and if the investor is to make profit , then this will create irrationality in the trade Efficiency in the market should take the three forms that is each efficient market, strong and semi strong efficient market. Here, the main rationality in differentiating the three markets is their lack of predictability and their earning power with respect to the successful dispensation of the information that is needed in the market. In this case, a weak efficient market is subject to a lot of speculation but lacks predictability and in this case, returns cannot be determined with certainty as needed in an efficient market. In a semi strong efficient market, information is publicly offered and in many instances, factors that affect the prices of stock, that is macro factors are put into consideration here. Data that exist can be utilized for one to actively determine the future of an asset. Information of the asset that is new in the market can be reflected easily and in many cases, it is helpful when determining the value of an asset. Information that is hidden is the one that is deemed important and it can be used in many cases to get the excess returns that is needed in the market. In a string efficient market, all the information about the trade have been availed and offered for use. Internal and external users can rely on the information has a gap have not been created or information asymmetry have not been created. Stock is traded at their true value and possibility of excess return as a strategy is impossible in this market. 14 Chapter three 3.1 Introduction This chapter gives different methodologies and approach that is used in determining the price of options with various mathematical approach in determining the actual option and trade execution needs. In the long run, derivative and derivations of various finite mathematical equations will be done with the main aim being to offer the best solution to the option pricing methods. 3.2 Option pricing _Black Sholes equation In this part of our computation, the rationality is to try to derive the equation of Black-Scholes. This model is mostly used in pricing options on European stocks. To find the prices of options under this model, assumptions regarding distribution of prices of stock are distributed as per the economic environment of the stock market .This assumptions to be considered and held constant in this computations includes the following: The volatility of returns that are deemed to be continuous and are regarded as constant over time Continuous returns of stock are compounded ad distributed normally and the distribution is also deemed to be independent over time. The amount of returns on investment that is the future projected dividends are known with certainty in dollar value and are deemed to be fixed. 3.2.1 Assumptions with regard to stock economic environment. The rate of risk is deemed to be free and it is considered as constant Taxes with regard to the option pricing are deemed to be tax free and no cost at all There is also a higher possibility of short selling with zero cost and to make a borrowing under free risk rate over time of short borrowing. The above assumptions are deemed to be held constant and it can only be subject to changes over time and it can be adjusted to fit different options, stocks, currencies and futures in their valuation. In the determination of options valuation, that is its prices, it is also rational to put into consideration the following variables that will be used in price determination. 15 Price of underlying asset: it is represented by(S). This represents the current value (price) of the asset (option) in the market. Volatility value (α): It gives a measure on how much the value of the asset, that is the security price moves as it is being traded over subsequent consecutive periods .It is one of the main challenge that affects the pricing of options as it keeps on changing and it is greatly affected by time and environment .It is also deemed as non-reliable model to be included in the computation Strike price. This is the price that is considered as the option exercise price. It is represented by (K) and it’s deemed as the price which the buyer gives to the seller on the execution of an option transaction. Rate of interest. It is represented by (r). It shows the risk free interest rate that is charged on the asset. Time to option expiration (T): it represents the time gap between the interest exercise date and the time of computation. Yield (δ): this represents the yield of the option in terms of dividends. This gives a representation of stock yield that pays dividends as its returns on any executed sales. In deriving the formula of option valuation using this model, European calls and puts formula will be used as the beginning point that will support our computation: C=Se^-δTN (d1)-Ke^-RT N (D2) P= Ke^-rTN (-d2) - Se^-δTN (-d1) From the above formula, it uses the risk adjusted chances that is the probability in determining the value of an option. N (d1) is the risk adjusted probability that is deemed to be used upon the time lapse or expiration of time of the u option upon complete the contingent amount that had been stated on the option. It is also the risk adjusted rate of option exercising through probity determination. D1 and D2 values are the normal distribution cumulative factors and in the computation it can be derived as follows: 𝑆 𝐾 𝑙𝑛( )+(𝑟−𝛿+0.5𝜕2 )𝑇 d1= 𝛿 √𝑇 16 𝑆 𝐾 𝑙𝑛( )+(𝑟−𝛿−0.5𝜕2 )𝑇 D2= 𝛿 √𝑇 From the above computation , it indicates that the main use of this model, that is black Sholes model is to compute the theoretical value of European options and its valuation in the market .It cannot be used in valuing other stocks for example with the American ones, one cannot use this model in making a valuation . To prove the above method, let us take an example of an option with the value of Ω. To find the computation of the value of this option given that a constant unit of the underlying asset is constant, then a change is asset value will follow the following steps in finding its valuation: Ω = ∆S − f(S, t) This is the first step in asset valuation, it is a representation of the total value of the option to be traded in the market. The second step is determination of the rate of change in the value of the option with respect to do Ω This will be: d Ω = ∆dS – df if we assume that this asset follows the ;eg –normal , that is the application of the geometric Brownian motion , then it will satisfy the following computation : dS = µSdt + σSdW, where µ is the drift of the motion and σ is the volatility, so by Itô’s lemma we have: 𝑑𝑓 𝑑𝑓 1 𝑑2𝑓 d Ω=(∆-𝑑𝑆 )dS-( 𝑑𝑡 + 2δ2S2 𝑑𝑆2)𝑑𝑡 from the above derivative, it means that if there is change that has been imposed on the value of the option, then it creates randomness .For s to attain a perfect value of a portfolio and less risk in the valuation of the asset, then we can employ the second derivative at least for us to mitigate the level of risk that is attached to the level of the asset. Here, the process that is termed as risk hedging is what needs to be done so as to help mitigate the risk and attain the least risk valuation of the option. 17 To satisfy the above condition; it shows that a portfolio is deemed to be risk free or non-risk when: 𝜕𝑓 ∆=𝜕𝑆 So when we factor out the above riskless factor, we shall remain with the following risk variable: 𝑑𝑓 1 𝑑2𝑓 d Ω=-( 𝑑𝑡 + 2δ2S2 𝑑𝑆2)𝑑𝑡 From the above computation, it shows that the rationale of the valuer here is to find out the value of the asset and make an estimation of the changes in the portfolio. The level where the formula for valuation has no profit , the at this point , we call the valuation model non –arbitrage level of valuation as the computation possess a no gain and it leads to a loss in the short run of its operations . This also shows that the main rationale of the investors here needs to place their investment cash in a less risk banks as it will help in reducing the risk that is attached to the asset and it will increase the value of the firm. It will also reduce the risk that is associated with the option investment as the gains from the valuation above might lead to arbitrage losses. If we satisfy this: dΩ = Ωrat Where: r=this is the risk less rate of interest The we can obtain our first valuation model that is Black-Scholes equation 𝑑𝑓 rf= 𝑑𝑆 𝑟𝑆-( 𝑑𝑓 𝑑𝑆 1 𝑑2𝑓 2 𝑑𝑆2 + δ2S2 This is what we therefore call the black-Scholes equation. It is used in option valuation at different level of asset operator and it is one of the rational method that can be used in the European option value determination. 18 3.3 Finite difference method It is a partial differential equation that is used in finding the value of options and various financial derivatives. In financial mathematics, this method is deemed to be an easy method to implement than the explicit method. It is also one of the stable method in finding the valuation of stock and it introduces three models that will be latter explained. That is a model by CrankNicholson and Courtadon method. In larger scale, using this method gives a true representation of stock pricing and in the long run, it helps in creating a stock environment where all the models are stable and can be used in derivative value determination. 3.4 Implementation of the methods In this method, the most rationale thing to put into consideration is the fact that approximation is nearly to be replaced by differential equations through an algebraic equation. According to Taylor polynomial the following equation is the start point to journeying the option pricing methods: F(x0+ h) = f(x0) + f 0(x0) h + f 00(x0)2! h2 + ... + f nn (x! 0) hn + Rn(x) From the Taylor polynomial equation, the derivative equation comes from: F(x0 + h) = f(x0) + f 0(x0) h + R1(x) Through rearranging the equation by making f(x0) the subject; then it will take the following format F(x0 ) = 𝑓(𝑥0) − 𝑓 0(𝑥0) + 𝑅1(𝑥) ℎ Where the value of R1 is the difference between the value of approximation and the value of actual figure that is placed on the asset to be traded. The value will also move towards the zero figure when h approaches that zero figure. In the first derivative where the value of h is deemed to be zero, it will portray the following formulae: F(x0) = 𝑓(𝑥+h)−f(𝑥) ℎ + 0ℎ In this case, when one is computing the value of option that is nearly having a zero value, then it means that the value of H is zero and in the computation, it can be narrowed down to the following when h is deemed to be zero 19 F(x0) = 𝑓(𝑥0) − 𝑓 0(𝑥0) + 𝑅1(𝑥) ℎ Therefore, this represents the finite value of an asset with x valuation features. Under the finite price valuation model, there are three approaches that will be discussed, that is Forward approximation Central approximation Backward approximation 3.4.1 Forward approximation In determining the various methods of approximation, then the following figure is used as a representation From the figure that s shown above, it indicates that there are three points that is f1, f2 and f3. The difference between the three points is h and this represents height and the difference between the points of actual and market value of the securities that are traded between the shares. With the three approximations that are stated in the text, it shows that the three f value have the following implementation: FL= refers to the forward approximation and the following formula is used in determining the price of the asset. f3 0: f(x + h) = (f(x+h) −f(x))/h→ f1 0 = (f2− f1)/h Central approximation is represented by f2= which in this case it is computed as follows F(x) = (f(x+h) −f(x))/h2→ f2 = (f3− f1)/h2 In the third derivative, the following formula is used in the computation of the backward approximation f3= f(x − h) = (f(x+h) −f(x))/h→ f3 = (f3− f1)/h Using the black Scholes partial differential method the equation to be used needs to be combined with the above so as to find a perfect approximation of Taylor polynomial. The derivative that is obtained from the approximated sum of the forward approximating and the backward approximation should take the format of a polynomial equation as shown below. 20 f(x + h) =f(x) +h2’(x) +1/h2”(x) +….. In the backward approximation, the following needs to be done, f(x - h) =f(x)-h2’(x) +1/h2”(x-….. With backward approximation, it shows that the determination of backward approximation is negative and this shows that the second derivative needs to be obtained so as to find the value of f F(x + h) + f(x − h) = { f(x) +h2’(x) +1/h2”(x) +…..}+ { f(x)-h2’(x) +1/h2”(x)-…} Then this equates to F(x) + f(x) + h2 0(x) − h2 0(x) + 1/2h2f(x) + f(x) + 1/2h2f” (x) hf 0(x) This equates to F(x+h) +f(x-h) =f2(x) +h2”(x) +R2(x) Make f”(x) the subject of the formula F”(x) =(f(x+h)-2f(x)+f(x-h))/h2-(R2(x))/h2 When we make a rearrangement of the equation by making f”(x) the formula subject, then we shall obtain the following F”(x) =(f(x+h)-2f(x)+f(x-h))/h2-(R2(x))/h2 R2 is the difference between the approximation and the value of the actual figure in the asset valuation .When the value reaches the zero, that is when the h value is equal to zero, then it indicates that we need to obtain the second derivative so as to help in determining the h F”(x) =(f(x+h)-2f(x)+f(x-h))/h2-o(h2) In the above, when we do an approximation and the second derivative, then we shall have a negative value an in this verge, we can conclude that a numerical algorithmic value greater than the truncation error as we compare it with the first order derivative. Therefore in conclusion with the three methods of approximation and combining it with the second derivative, then it shows that the approximate value will be obtained and the end rationale 21 is that when the value of h is negative, then backward approximation have been obtained and this shows that when a sale is executed at this point of asset valuation, then it will lead to a loss. When the value of h is positive, it shows that the asset valuation is positive and when a sale is executed, it indicates that an approximation level is obtained and the end rationale is that it will lead to a gain when it is executed. Therefore in this two cases , that is combining the Black Scholes approximation level and Finite difference method , that is backward ,forward and central approximation method , it will lead to a definite option value in determining the price of an option . 3.5 Application of finite method in option pricing In determining the prices of options using the finite method, there price and time mesh needs to be explained and introduced .This is the model that shows that prices of various stocks and derivatives that are being traded in the stock market. In this case, it indicates that using the price time mesh, one will be in a postion of determining the prices of financial instruments that are being traded in the stock market at a particular time. It shows a combination of various prices in different market and it is up to the investor to make a speculation on the prices and choose the best square within the mesh where and when a trade can be executed either in terms of a sale or a purchase in the financial market. The derivative from mesh is divided in two axis that is vertical and horizontal shows significance in terms of changes and the importance of each division within the mesh. By combining the two, that is price and time, one is in a postion of making the right decision through observing the angles within the square and this will help in defining the right position where the sale will be executed in the industry. The mesh is drawn down from top up from low. In each and every point that one moves, there is a high possibility that below the points that are stated within the trade, then one is moving to a point known as squaring off the price. That is below the angle 45% in the mesh, this s the region that needs to be considered with utmost importance and in the end, it will help in determining the days fashion and the best point where a sale can be executed in the industry. If the prices of one falls below the level and then it bounces back to the touch line , that is the level above angle 45, then in such an occasion , the trade is deemed to be good and can be 22 executed based on the terms that govern the price execution within the industry . It also depends on the option that is being traded and its features as stated earlier upon the time of trade expiration and execution. If the price comes short below the 0.5 level of probability, then it indicates that time and prices are moving at a higher rate and the probability of trade execution is lowered and it can be executed easily within the stated timeline. There are levels within the chart that are considered dangerous. This are the points where they are risky and when a trade is executed, then it may either lead to a loss on the amount invested or it increase the speculation of the failure in the trade. Each point in the mesh have certain time and certain price. at any given point, it is deemed that jδS is equal to the stock price and iδt is equal to the time of stock execution .At the points in between the mesh, that is boundaries in the mesh then any computation can be made as it contains all the needed data based on the prices and time to execute the sale. The upper boundary of the mesh indicates the maxim and is deemed to be large enough. It is always represented as Smax .At this point, it is deemed to be large enough and when chosen, the put option is deemed to be equal to zero as per the expected pay-off returns with a function of max (K-S, 0). For all the boundary condition , it shows that the call option at any given point is equal to zero and the lower condition that needs to be satisfied by the computation needs to show that S=o. That is s is always equate to zero .The same way, the prices of an option within the boundaries of the mesh is always equated as. In the boundary condition, it shows that the value of the option needs to be discounted as the time of strike price in all times within the mesh. From then minimum or the lower side of the mesh where the prices is equal to zero, the righter boundary condition, that is pay off function at the time of stock expiration is where we compute the expected payoff. The time t=T and the option at different stock price varies based on the boundary at which it is deemed to lay on from the mesh. Prices at the point where time t=0 can be calculated using the back ward approximation method as stated earlier. In this case from t=T Price –Time mesh diagram 23 S JδS S T Iδt In the figure above , it gives an explanation on the different methods of price-time mesh and it is major important to the financial analysis and financial mathematicians together with investors in implementing the method of finite methods of differences as used in stock pricing . The various as that is the y and axis have a representation in form of time and price of stock at different points. In between the mesh is also located with boundaries that determines the maximum and the minimum value of prices at a given time and the stock prices within the meh can be executed at any given time. The y axis is represented with a j and the x axis is represented with an i. The distance between the two has a δ symbol that is embedded with either the price of stock or the time of stock. The δ represents the sensitivity of trade with regard to time and price of stock at any given point in time of trade and pricing of a trade. Using the earlier computations that were made, that is using the black Scholes method of partial method of differentiation, we can differentiate the equations within the boundaries of the mesh so as we can obtain the actual prices they ought to be computed within the mesh. As stated earlier in our previous step of forward approximation, FL= refers to the forward approximation and the following formula is used in determining the price of the asset. f3 0: f(x + h) = (f(x+h) −f(x))/h→ f1 0 = (f2− f1)/h Central approximation is represented by f2= which in this case it is computed as follows F(x) = (f(x+h) −f(x))/h2→ f2 = (f3− f1)/h2 In the third derivative, the following formula is used in the computation of the backward approximation 24 f3= f(x − h) = (f(x+h) −f(x))/h→ f3 = (f3− f1)/h Using the black Scholes partial differential method the equation to be used needs to be combined with the above so as to find a perfect approximation of Taylor polynomial. The derivative that is obtained from the approximated sum of the forward approximating and the backward approximation should take the format of a polynomial equation as shown below. f(x + h) =f(x) +h2’(x) +1/h2”(x) +….. In the backward approximation, the following needs to be done, f(x - h) =f(x)-h2’(x) +1/h2”(x-….. With backward approximation, it shows that the determination of backward approximation is negative and this shows that the second derivative needs to be obtained so as to find the value of f F(x + h) + f(x − h) = { f(x) +h2’(x) +1/h2”(x) +…..}+ { f(x)-h2’(x) +1/h2”(x)-…} Then this equates to F(x) + f(x) + h2 0(x) − h2 0(x) + 1/2h2f(x) + f(x) + 1/2h2f” (x) hf 0(x) This equates to F(x+h) +f(x-h) =f2(x) +h2”(x) +R2(x) Make f”(x) the subject of the formula F”(x) = (f(x+h)-f2(x)+f(x-h))/h2-(R2(x))/h2 When we make a rearrangement of the equation by making f”(x) the formula subject, then we shall obtain the following F”(x) = (f(x+h)-2f(x) +f(x-h))/h2-(R2(x))/h2 With 𝜹𝒇 𝒇𝒊+𝟏.𝒋−𝒇𝒊,𝒋 𝜹𝒕 = 𝛅𝐭 25 3.6 Backward approximation 𝜹𝒇 𝒇𝒊,𝒋.𝒋−𝒇𝒊−𝟏,𝒋 𝜹𝒕 = 𝛅𝐭 3.7 Central approximation method 𝜹𝒇 𝒇𝒊+𝟏.𝒋−𝒇𝒊−𝟏𝒋 𝜹𝒕 = 𝟐𝛅𝐭 this is the first derivative, then we need to find the second derivative so as we can get the actual differential equation as portrayed by the central approximation method 𝜹𝟐𝒇 𝒇𝒊𝒋+𝟏−𝟐𝒇𝒊,𝒋+𝒇𝒊,𝒋−𝟏 𝜹𝑺𝟐 = (𝛅𝐒)^𝟐 From the above, the approximation of the derivatives will be used in computing the prices of options and obtaining a differential black Scholes equation partial differential equation. In each method, that is after integrating the differential equation is when we can obtain the value of stock. The two methods to be used in the Explicit, and the Crank-Nicholson and implicit method. 3.8 Implicit method It is more stable than the explicit method but is more complex as it requires more computational needs. The method cannot be used on its own, it requires other sublimit methods so as to add the quantum of the existing methodologies that were discussed. Both states, that is in its price determination needs the last state equations and the current consolidated state equation so as to help in obtaining the actual pricing system of options Points in the Time price mesh that is used in the computation From the above, the implicit method shows different points in the mesh are used in determining the implicit methods of price values using different methods. 26 The green color indicates the values that are to be determinable and known with certainty and on the other hand, the red ones represent values that are unknown. When using the implicit approach in computing the price of the option, forward approximation will be used in finding the value of 𝜹𝒇 . On the other hand, combining the approximation with 𝜹𝒕 Black Scholes model, then we have the following differential equation as per the problem stated: 𝒇𝒊+𝟏.𝒋−𝒇𝒊,𝒋 𝒇𝒊+𝟏.𝒋−𝒇𝒊−𝟏 𝒇𝒊𝒋+𝟏−𝟐𝒇𝒊,𝒋+𝒇𝒊,𝒋−𝟏 𝛅𝐭 𝟐𝛅𝐒 (𝛅𝐒)^𝟐 +jδS +1/δ2j2 (δS) 2 -rfij=0 This can therefore be rewritten as follows: Fij=[1/2δt(rj-δ2j2)]fi-1j-1+[1+δt(δ2j2+r)]fi-1j+[-1/2δt(rj+δ2j2)]fi-1j+1 When we consider the equation and replace a2, bj and c j as the coefficient in the equation, then we obtain the following as the results and this is known as the implicit equation fig=αjfi-1, j-1+bjfi-1,j+cjfi-1,j+1 In the numerical computation, we can conclude that suing the equation, it can also be converted into a matrix form so as we can integrate it with a numerical computation. In this case, the results that will be obtained when we factor is the above equation as a matrix will be Fi=BFi-1 Where Fi= Fi,1 Fi,2 … … Fi,M-1 27 Then the value of B= B= b1 c1 0 … a2 b2 c2 : : : ….. … 0 0 …. am-1 bm-1 … 0 ; Therefore using the matrix above, then it is comfortable for one who needs to find the value of option to calculate as this gives us the option of computing the value together with its market sensitivity. Combining the approximation model with the Black Scholes differential equation has helped in attaining the rationale of value determination and the end results will be to obtain the total value of the asset that is to be traded in the financial market. 3.9 Using Crank-Nicholson Method This is the implicit differential method that is used in pricing options in a stock market through a consideration of weights between the two explicit and implicit difference method. The method also takes into consideration the price time mesh in its determination of asset values within the boundaries as shown earlier by the method of computation that was done using the Finite differential method with the use of forward , central and backward approximation methods . This indicates the following as its main components: 28 From the above the model, all the points that are indicated in the mesh with a green color are deemed to be the known value and on the other hand, all the values that are indicated with the red are deemed to be the unknown values. In this the most significance of this method is that it is deemed to be one of the stable method as it shows that convergence and shows the most accurate value when it comes to the computation of the value of a derivative . In this method compared to the finite method, it uses the central approximation method as the start point in finding the value of derivative. In this case, the value 𝜹𝒇 𝜹𝒕 Is the derivative to be used in the approximation as it is inserted into the figures and the Black Scholes differential equation so as to help attain the rationale in the long run? Together with the implementation of the jδS, then we can follow the following equation from the approximation to value options in the stock market [−δt 4(σ2j2 − rj)]fi−1,j−1 + (1 − [−δt2 (σ2j2 + r)])fi−1,j − [δt 4(σ2j2 + rj)]fi−1,j+1 =δtδtδt== [δt4(σ2j2 − rj)]fi,j−1 + (1 + [−δt2 (σ2j2 + r)])fi,j + [δt4 (σ2j2 + rj)]fi,j+1 From the above, it indicates that we can replace the expressions in the inside of the equation that is replacing the coefficient with aj, bj and c1 Then when we replace, we obtain the following: −ajfi−1, j−1 + (1 − bj) fi−1, j − cjfi−1, j+1 = ajfi, j−1 + (1 + bj) fi, j + cjfi, j+1 The next step is to formulate it in the form of a matrix so as to help in finding the actual value of an asset when the product is multiplied by the factors that affects the prices of option. This will produce a differential matrix equation as follows: 1-b1 -a2 : 1- -c1 0 … 0 b2 -c2 … ; : : ….. … 29 0 0 …. -am-1 --bm -1 F1= Fi, 1 Fi,2 … … Fi,M-1 C= 1+b1 c1 0 … 0 a2 b2 c2 … ; 0 a3 1-b3 ….. … 0 0 …. am-1 1+bm 1+ In this case, the above matrix shows that sensitivity values together with how to compute various component that affect the prices of options. The C value is the matrix inverse in computing the reversal resell of an option whereas the F1 value is the actual matrix determinant that is used in finding the value of an asset. 30 Chapter Four 4.0 Introduction This chapter will give the best alternative options that needs to be chosen when pricing an option in the financial market . The end rationale here is to determine the best method out of the alternative given so as to offer guide to the investors on when at what level of accuracy should they place when making an investment decision . It is also important when one needs to execute a trade within the financial institution. 4.1 Choice of the best alternative to be used in option pricing In choosing the best method of valuing options, it shall also be used in finding the value of American options and stock loans so as to make a conclusion as from the computations to be performed .According to this theory, there is a tendency that the following needs to be applied in making the best decisions on which option is deemed to be necessary in finding the actual value of an option. Explicit method is the easiest method and due to values that are known with certainty, there is a high tendency that the method can be easily employed. The greatest limitation of using this method is that it is unstable and can lead to a problem carry forward when used in the analysis. In finding the new values in the pricing of options, there is a high tendency that the method is stable, involve a consolidation of complex equations to lead to a rational matrix that is used in final determination of values. This method is known as implicit finite method of asset valuation. It also covers changes in time and at the same time a change in price and with using the mesh to find the boundaries, it can be utilized in determining prices at a particular period of time when making certain computations in the asset valuation. The final method that uses the implicit finite difference method is the application of CrankNicholson method in determining a price of an option. In this method, it combines the features that are offered by the explicit method and the implicit method. Using this formula, it helps in attaining stability nd a faster method in assessing the prices of various issues and a determination of the final prices of an asset. From my opinion and the computations that are done, the best and a rational method that could be used in pricing options is the Crank-Nicholson method as it offered a combined stable implicit and explicit valuation method. 31 The table below indicates the prices and time for various securities .It shows their actual values and values that are obtained using the methods that is the three implicit methods of valuation. This is used to determine the accuracy of each method as shows: Methods Price Explicit S True Implicit Value Durati Accur True Tim Accur on acy value e acy 1.859 0.024 99.984 1.8599 0.20 100.03 1 1 % 63 3.212 0.024 99.975 8 2 % 5.09 0.024 99.976 3 % 7.509 0.024 99.980 4 1 % 10.450 10.44 0.024 99.983 6 88 1 % 13.857 13.85 0.025 99.988 9 62 6 % 17.663 17.66 0.026 99.992 16 5 % 21.790 21.78 0.027 99.994 5 92 7 % 26.169 26.16 0.029 99.997 81 6 % Value 80 85 90 95 100 105 110 115 120 Crank-Nicholson 1.8594 3.2136 5.0912 7.5109 3.2135 5.0906 7.5098 Accura e cy 1.859 0.03 100.00 % 5 02 5% 0.20 100.00 3.213 0.02 100.00 93 % 6 96 0% 0.21 99.99 5.091 0.02 100.00 08 % 3 87 2% 0.20 99.99 7.511 0.02 100.00 91 % 2 66 4% 99.99 10.45 0.02 100.00 % 15 47 9% 99.99 13.85 0.02 100.00 % 9 99 8% 99.99 17.66 0.03 100.00 % 42 41 7% 99.99 21.79 0.03 100.00 % 19 55 6% 100.00 26.17 0.04 100.00 % 05 02 6% 10.4491 0.19 5 13.8564 0.21 23 17.6616 0.22 95 21.7894 0.24 18 26.1682 0.28 16 32 Value Tim From the above, the greater the accuracy percentage shows that it is the best preferred method in valuing the firm’s options. In this case, I would choose the Crank –Nicholson as the best model over the other models since it possesses more 100% level of accuracy in the pricing of stocks at different times and at different stock prices. It is also a good method as it portrays a short performance and high level of accuracy over a price time mesh. This shows that utilizing the model, one needs to utilize less information from the eternal market and this will determine the actual pricing in the model. Implicit finite model accuracy 100,0300% 100,0200% 100,0100% 100,0000% 99,9900% 99,9800% 99,9700% 0 20 40 60 80 100 120 Explicit Implicit Crank-Nicholson Линейная (Explicit) Линейная (Implicit) Линейная (Crank-Nicholson) 140 The model above also shows that with the use of Crank-Nicholson model, there is a high level of accuracy that have been obtained and at the end , it will lead to a positive high output results in the long run analysis of the pricing option . In deciding on this model, I would recommend investors to use this model when they need accurate decision model in pricing the values of stock in a financial market. 33 34