Modern Pricing Models - Research Paper Example

MODERN PRICING MODELS By Mashael Alanazi This research work explores the different types of modern pricing models used by financial analysts and money experts across the world. These modern pricing models are instrumental in determining the correct price of 
options. The first chapter explored is the Geometric Brownian Motion (GBM), a stochastic processes as applied in the real world. The GBM introduces learners to the stochastic volatility, as an improvement of other models previously used in determination of price changes in stocks and options, such as the Black-Scholes model. It further explores the different features of the GBM such as its characteristics and application in finance. The paper also discusses the Merton model as a financial analysis model, and its application in finance, its uses and properties, alongside its integration with the Black Scholes model. It determines the features of the model that make it a stochastic volatility model. The other pricing model explored in this paper is the Heston model of financial planning and analysis, whereby it explores its basic features, its usage in finance, as well as its application in stochastic volatility. The Bates model is also discussed in the paper, which appears as an improvement of the Merton and Heston models of modern pricing. Finally, the paper explores the jump diffusion model for option pricing, a feature that appears of some of the modern pricing models. Key words: stochastic volatility, jump diffusion. Contents Modern Pricing Models 4 Introduction 4 CHAPTER ONE: THE GEOMETRIC BROWNIAN MOTION (GBM) 5 Definition and the History of the GBM 5 The Stochastic Process of the Brownian motion 6 Analysis of the GBM Equation 7 Monte Carlo Simulation with GBM 7 Specify a model, such as the GBM 8 Basic properties of standard Brownian motion 9 Uses in Finance of GBM 12 Financial application of the GBM model 13 Benefits of GBM 14 CHAPTER TWO: THE MERTON MODEL 15 Introduction 15 Definition and history of the Merton Model 16 The Merton Model (1974) 16 Limitations of the Merton Model 18 Significance 19 Uses of the Merton’s model (1976) in Finance 19 Analysis of the Merton model equations 20 Integration with the Black Scholes model 21 Conclusion 22 CHAPTER THREE: THE HESTON MODEL 23 Definition of the Heston Model 23 History of the Heston Model 25 Analysis of the Heston Model 26 Properties of the Heston Model 26 Financial uses of the Heston Model 27 Applications of the Heston Model in Finance 30 Benefits of Using the Heston Model in Finance 33 CHAPTER FOUR: THE BATES MODEL 34 Definition of the Bates Model 34 History of the Bates Model 35 dc / dt + ½VS2(d2C / dS2) + ½ σ2V(d2C/dv2) + pӨVS(d2C/dSdV) + (r – δ - λk)S(dc / dS) + k(Ө – V)dC / dV) – rC + IC = 0--------------------------------------- EQ 29 36 Introduction to Bates Model 36 Analysis and properties of the Bates Model 36 Financial Applications of the Bates Model 37 CHAPTER FIVE: FURTHER APPLICATIONS 37 Introduction 37 Analysis of the Jump diffusion model 38 Relationship of the Jump Diffusion Model to Other Models of Option Pricing, Such as the Bates Model, and the Heston Model 39 Bates model with jump diffusion 39 Chapter Six: Conclusions 42 Modern Pricing Models Introduction The success of every firm is embedded on its effectiveness to pick a price model that efficiently help attract, serve and retain its customers in an ever increasing competitive environment pricing models keep on changing and become obsolete. Marketing analytic price models (MAPM) is crucial in guiding decisions on options s, key price points, portfolio pricing as well as competitive price matching. In addition, MAPM is also critical in providing insights into correct price of options. In essence, MAPM is the foundation for picking an effective or set of pricing models by a firm. In this essay, the researcher purposes to cover various modern pricing models (MPM) and show case how useful they are for a particular company based on their financial application and compare each model, Such as Geometric Brownian motion (GBM), Merton Model, Heston Model, and Bates Model will form the bedrock of this discussion. CHAPTER ONE: THE GEOMETRIC BROWNIAN MOTION (GBM) Definition and the History of the GBM In every financial modeling for pricing options, the main purpose of applying the financial model is through the determination of correct prices of options. The geometric Brownian motion is an assumption of stock price behavior used by various models through simulations to determine the future prices of stock, especially in the options and stock prices for investors. A geometric Brownian motion (GBM), or an exponential Brownian motion, refers to a stochastic process that runs continuously over time in which case the logarithm of the randomly varying quantity follows a Wiener process, or the Brownian motion with a drift. According to Vose (2008), the Brownian motion is a significant example of the stochastic processes that satisfy a stochastic differential equation (SDE). Most applications of Brownian motion incorporate, in particular, mathematical finance, especially in consideration of the model stock prices, such as is the case with Black-Scholes model. As such, the geometric Brownian motion is a core building block of modern finance. This is particularly in the case of the Black Scholes model whereby the stock price is assumedly in line with the principles and expectations of the GBM dynamics (Vose, 2008). From a technical perspective, the definition of SDE is the stochastic process (St), said to follow a GBM if it satisfies the following stochastic differential equation (SDE), the EQ 1 below. dSt = μStdt + σStdWt ----------- EQ 1 Wt = a Brownian motion or a wiener process. μ= the percentage drift (constant) used to model deterministic trends. σ= the percentage volatility (constant) used to model a set of unpredictable events. occurring during this motion (Vose, 2008). The Stochastic Process of the Brownian motion According to Benth (2004), It is quite fortunate that top economists conducted a great deal of research in the field of Wiener motion, such as Fischer Black and Myron Scholes who came up with the famous Black-Scholes formula instrumental in determining the correct price in the volatility of a stock. The geometric Brownian motion forms an important element of this formula, and as such, serves as an approximate model for stock price fluctuations when traced on a graph. These economists argue that the Wiener motion follows a stochastic process in the manner in which it responds to particular changes in prices of stocks. The Brownian motion associates closely with the Wiener process in such a way that both have a characteristic of a series of random variables that form a consistent time chart of motion. This type of process, known by analysts as the stochastic process, is a very useful for tracking the stock prices over time because it incorporates a certain degree of randomness that is consistent with the fluctuation of option price changes in the market. By taking the Wiener process into account in its formula, and harnessing the stochastic process followed by stock prices over a particular time-period, the Black-Scholes formula accounts for a level of randomness in the market volatility thereby making the formulae more reliable. While it is impossible to predict accurately all the changes that affect a stock on the market, particularly over a very short period of time, the reliance on Brownian motion and the Wiener process makes it easier to make an informed assessment of options and the chances of being in the money. This is important as it enables investors to determine the right prices or the relative changes expected in a given stock price or option. Analysis of the GBM Equation According to Benth (2004), an analysis of the GBM equation requires one to solve the SDE equation, which in turn results to the variety of options available for users to take advantage of while making financial decisions. In solving the SDE, one considers S0 that is an arbitrary initial value of the equation to be solves. As such, the analytic solution of the SDE, given under the Brownian Motions interpretation (Benth 2004) is shown in EQ 2 below: St = S0 exp [(μ-σ2 / 2)t + σWt] ----------- EQ 2 In order to arrive at this formula, one has to undertake a number of mathematical calculations and simulation, such as dividing the SDE by the St, and later write it in the Brownian Motion’s integral (Benth, 2004) form as shown in EQ 3 below: ∫0t(dSt / St) = μt + σWt assuming W0 = 0 ----------- EQ 3 A close analysis of the formula provides that dSt / St appears somewhat related to the other derivative ln St. As such, analysts need to use the Brownian Motion calculus because St is an Brownian Motion process. An application of the Brownian motion’s formulae presented by Glasserman (2004), as presented in EQ 4 below. It provides that d (ln St) = dSt / St – 1 / 2σ2 dt ----------- EQ 4 Analysts obtain the following equation when they plug back the equation derived from the SDE in EQ 5 shown below. ln (St / S0) = (μ – σ2/2)t + σWt----------- EQ 5 Monte Carlo Simulation with GBM The Monte Carlo simulation makes great use of the analysis from GBM by enabling the financiers and economic analysts to determine the level or amount of risk that might accrue from a given investment option. As such, the Monte Carlo simulation makes use of the GBM equation analysis to estimate risks and expected changes in the financial markets. In order to understand the effect of the Monte Carlo simulation (MCS) on the determination of the value of risk, it is best to review the basics of the MCS as applied to a given price of stock. The geometric Brownian motion comes in at this stage of the simulation since this also requires a model to specify the behavior of a price stock, as GBM is one of the most common models used in financial analysis and simulation. As the simulation process begins, it is imperative for investors to note that the Monte Carlo simulation is just an attempt by the financial analysts to predict the future of a stock price many times over. From the simulation, analysts get thousands to millions of random trials that produce a distribution of outcomes usable for analysis and prediction of risk in associated cases. The basic steps to adhere to include the specification of the financial analysis model to apply, and in this case, the Geometric Brownian motion is the best option. The second step is to generate random trials, whilst the third and last step is the process the output (Glasserman, 2004). Specify a model, such as the GBM The best model to use in this simulation is the geometric Brownian motion, technically a Markov process which is known as a stochastic model can be used to model a random system that changes states according to a transition rule that only depends on the current state. Markove peosess enables financiers to estimate the value of risk, as well as the level of exposure to certain risks in the financial market and stock trading. The simulation incorporates beforehand the past price information, whereby the next price movement remains conditionally independent of the past price movements. This then follows the application of the GBM equation whereby “S” reflects the price of stock, “μ” reflects the expected return, “s” reflects the standard deviation on the returns, “t” reflects the time horizon over which the investment process takes place, while “ε” reflects the random variable in the application. ΔS / S = μΔt + σε√ (Δt) ----------- EQ 6 A rearrangement of the EQ 6 above in order to solve for the change experienced in the price of stock provides the stock price “S” as two terms as outlined in the parenthesis in EQ 7 shown below. ΔS = S (μΔt + σε√ (Δt)) ----------- EQ 7 The first term in the equation represents a “drift” while the second term in the equation represents a “shock”. For every time-period under consideration, the GBM model assumes that the stock price will undergo a “drift” upwards in line with the expected value of return. However, in the event of a shock within the drift, such as an addition or a subtraction, the stock price is said to have suffered a random shock. As such, the derivation of the random shock incorporates the product of the standard deviation “s” by a random number “Ԑ”, as a simple way of scaling the standard deviation. This is the key essence of applying the GBM in this simulation. The stock price follows a given series of steps in which every step it undergoes either a drift plus or a drift minus a random shock, which is itself a function of the standard deviation of the stock. Figure 1 The figure below explores a drift and a shock that occurs to an option price. A shock and a drift refer to either a subtraction or an addition in the stock price Figure 1: [online] Available at http://studentblogs.warwick.ac.uk/morse/tag/statistics/ Basic properties of standard Brownian motion Property One and Two The first basic property of standard Brownian motion, denoted as W0indexed with time, is that W at time 0 equals 0. This means that Brownian motion at time =0 takes the value of zero. Brownian motion can take a variety of paths. However, one thing that all realizations of Brownian motion have is that they take a value of 0 at time =0. Therefore, we can denote it W at time 0=0.Clearly, this property is self-explanatory, thus we can proceed to look at the other properties. The second property is that Brownian motion is normally distributed with mean 0 and variance t and Brownian motion increment Wt - Ws is normally distributed with mean 0 and variance (t-s). What does it mean to say that the Brownian motion is normally distributed with mean 0 and variance t? At a point in time, say time =0, I want to know what my Brownian motion is or will be at time, t=2. The Brownian motion can take on a number of single realizations, that is, it can take a range of values. What we do know is the expected value to the distribution of Brownian motion at time =2. Therefore, we know what the center of the distribution is, i.e. what the expected value of the distribution is and this will be the expected value of W2= 0. It will always be zero, regardless of what point in time we view the Brownian motion. The expectation of Brownian motion at all points on a plain at any time is 0 as per property one. Not only will the expected value at any time be 0, but also we are told that it will be normally distributed. The peak of the normal distribution is centered at 0, meaning that the Brownian motion will be distributed as a normal variable with expected value 0 and variance t. Property Three Property three relates to the concept of property number two, i.e. the Brownian motion increment, which is the difference between the two Brownian motions (Wt- Ws). Therefore, the difference between the two Brownian motions is also normally distributed and the variance of the Brownian motion increments (Wt - Ws) is (t-s), where t stands for time and s stands for a point in time which differs from t. (t - s) is the difference in two time periods between measurements of our Brownian motion. Consequently, if we look at our Brownian motion at two different points in time, the expected increment , the expectation of the difference of these two Brownian motions ( E [Wt - Ws])=0 and the variance of this difference ( Var [Wt-Ws]) = t-s. It emerges that the variance is proportional to time. Other properties of Brownian motion state that the process Wt has stationary and independent increments. What does it mean to say that the Brownian motion has stationary increments? The difference between W1 and W0 will be the same as the distribution of this increment. The difference between W (0 + a) and W (1+a) is that they will both be normal with expected value of 0 and variance of (t1-t0) N~ (0, t1- t0). Regardless of where the Brownian motion is, it will always have the same properties. The incremental expectation will always be 0 and the variance will be the difference in time. To sum up, where the Brownian motion does not matter, it will always have the same property in a probabilistic sense. It will always have an expectation of 0 and variance will be proportional to time. Another property is that increments are independent, that is, if W1is less than W0 and W1+a less than W0+a. Independence means that the motion moves in a time interval (between 1 and 0) and will have absolutely no bearing on how it will move in the next time interval (1+a) and (0+a). Property Four Looking at some other properties of Brownian motion, the covariance of two Brownian motion paths is the minimum of time s or t. (cov{W(t), W(s)} = minimum{s, t}. Looking at a Brownian motion at time s and looking at a Brownian motion at time t, and if you try to find covariance of these two terms (Ws and Wt), you will find that the covariance will be the minimum of s and t. If time s comes before time t, then the minimum of the two terms is s. Property Five Property number five states that Brownian motion is a Markov process. This means that the past is irrelevant for where the process will be in the future. It is irrelevant where the process has been in the past for deciding on where it will be in the future. The past has absolutely no bearing on the future evolution of the process. Property Six Property six states that Brownian motion is a martingale. The expected value of Brownian motion, which moved by time s (it is at time t and moved by time s) is conditional on all the information I have at time t and is where the process is now, at Wt. This martingale property can be illustrated. Assuming that the process is at time Wt, it is easier to calculate where the process will be at time Wt+s? According to the martingale property of Brownian motion, the expectation of where the process is going to be time t + s is conditional on all the information known of how the process moved in the past is equal to Wt . What this means is that I do not expect the process to have moved at all. Instead, I expect it stay at the same level i.e. I expect it to be at the same point at time t + s, as when I was at time Wt. Property Seven Brownian motion is continuous everywhere and differential nowhere. What do we mean by saying it is continuous everywhere? There are no jumps or discontinuities in Brownian motion and you cannot differentiate it anywhere. You cannot find a derivative of the Brownian motion, as it is too volatile, it has spikes everywhere. It is impossible to find the derivative and in fact, if you zoom in on a Brownian motion path you would see that it is constantly moving. Uses in Finance of GBM The geometric Brownian motion is an effective for financial analysts to use in determining the various changes that a stock price might undergo through the application of the Black-Scholes model. According to Linghao (2010), the largest usage of the GBM model is in undertaking an analysis of the stock price, such as in the Monte Carlo simulation, the options trading, as well as the changes in the stock prices. Consequently, a number of arguments are fronted by these analysts to support their usage of the GBM model to analyze and evaluate changes in stock prices, as well as estimating the risks associated in the said changes that investors might have to consider before making an investment in a given stock. Some of the arguments given by these analysts in support of them using the GBM model in the analysis of financial risks associated with changes in stock prices are that the expected returns of GBM are always independent of the stock price, which is the value of the processes. This is a concept that is in agreement with what appears as the expected performance in a real market scenario. Another argument fronted by these analysts is that the GBM process only assumes positive values, just as is the case with real stock prices. In addition, the GBM processes exhibit the same kind of roughness in its path. This situation is also similar as observed in real stock prices. Another argument placed in favor of using the GBM model in these analyses is that the calculations relating to the usage of GBM process are relatively easy to apply and undertake as opposed to the application of other financial models in the same spirit of financial risk analysis (Linghao, 2010). On the other hand, it is imperative to note that the GBM model is not entirely a realistic model while making these financial analyses. As such, it is noteworthy to remember at all times when applying the GBM on financial analysis of stock prices that the model may fall short of reality in some points. One point is the fact that GBM assumes volatility to be a constant factor, a concept that is very different in real stock prices whereby the volatility of the stock prices changes over a given period, possible in a stochastic manner. Financial application of the GBM model It so appears, and as mentioned earlier, that the time series generated by the geometric Brownian motion is among the simplest, as well as the most commonly used model in undertaking financial analyses and estimating risks associated with changes in stock prices. Take for instance EQ 8, which provides that the value of the variables change in one unit of time by a given amount that has normal distribution with a variance of 2and a mean of xt + 1 = xt + Normal (μ,σ) ----------- EQ 8 Such a Normal distribution as recorded in the above formula provides a suitable first choice for a number of variables because it is possible for analysts to consider that the model outlines that numerous but independent random variables affect additively the x variable. As such, analysts can proceed to iterate the equation in order to deduce a relationship that exists in EQ 9 and EQ 10 between the following functions xt and xt+2: xt+2 = xt +1 + Normal(μ,σ) xt+2 = [xt+ Normal(μ,σ) + Normal(μ,σ) xt+2 = xt + Normal(2μ, √2σ) ----------- EQ 9 The next step is to generalise to any time interval of T: xt+T = xt + Normal(μT, σ√T) ----------- EQ 10 This equation is rather convenient because of a number of parameters, such as because it continues to use normal distributions, as well as it enables users to determine the correct prise of options between and time intervals that they may select while undertaking the simulation. As such, the EQ 10 above focuses on discrete units. However, it is still possible to write these in a continuous time form whereby the analysts have to consider a small interval of time, and the change incurred within this minor interval provided as t in EQ 11 below. Δx = Normal (μ (Δt), σ√Δt) ----------- EQ 11 Benefits of GBM The geometric Brownian motion is beneficial because it is very effective when applied to different cases of market analysis as discussed above, such as in evaluating the changes in the stock market, especially the way the prices change from time to time. As such, this enables investors to predict future trends or come up with a given pattern that certain selected stock follows. The GBM is also beneficial to financial analysts because it provides optimal dynamic trading that has advantage constraints. In addition, the model also allows investors to undertake an appropriate analysis that allows for investments, uncertainty, as well as creation of price stabilization schemes. Consequently, the results from the analysis of the Brownian motion equation and application of the model gives investors the right information to make important informed decisions concerning their investments, especially those for the stocks . CHAPTER TWO: THE MERTON MODEL Introduction Various models used in mathematics and finance follow a particular process for implementation and the achievement of success. They include the traditional forms of techniques and the more modern types. They are different in the way that the proving opportunities are conducted. As such, varied methods van is employed to distinguish between the two. Among the models is the Brownian model that tends to support a continuous process. In this process, it runs over time with a logarithm that randomly varies in quantity and follows the Wiener process. Alternatively, it follows the Brownian motion with a drift. It is a geometric or an exponential model hence one of the traditional approaches of pricing as is used by most financial institutions such as banks. Besides this form of the model, there is also a more model structure, which is referred to as the Merton model. It is different from the Brownian model in the sense that it satisfies the Stochastic Differential equations. In addition, unlike the Merton motion, the Brownian motion incorporates the mathematical finance that is related to the stock process. The Merton model is thus one of the current pricing models. The difference can be obtained using data from the swap of credit default in the market. In this way, a comparison between the implemen5tation of the Merton’s model and the traditional models that were used in the implementation of pricing options will be done. The Georgian Brownian motion (GBM) acts as the foundation for most of the models used in modern finance. It is the building block of most concepts applied by stock pricing models. This is the case particularly in the Black-Scholes model where the basic price of stock is assumed to follow the basic dynamics of GBM as shown in the equation EQ 12 below: dSt = rSt+ σSdtdWt In this respect, when the I are applied to Lemma (Connor et al, 2010), then the equation becomes St = S0 exp {(r – σ2 / 2) t + σWt} ----------- EQ 12 The principal problem includes the fact that the empirical results are quite different from the assumptions made in these models. The financial returns obtained shows fatter tails that are than those postulated by the Black-Scholes model in the effort to predict the how volatile the stocks in the market are. The main disadvantage of the bond prices is that, most of the time, they serve as indicators, and not the firm quotes. Definition and history of the Merton Model Robert Merton developed this model in 1974. It was proposed and intended to serve as a model that helps in the assessment of the credit risk of the company. It is done by the characterization of the equity in the company as a call option of the assets within it. The model possesses parameters that can be estimated from the implied volatilities among the options of equity within the company (Connor et al, 2010). The Merton Model (1974) A number of companies on a daily basis receive new pieces of information. The publicity of this kind of information affects the stock prices of the company. Such changes occur suddenly. The information is not usually expected, hence, it can leave a huge of an impact on the stock size. It can also be treated as random, in the same way, which the information is obtained randomly. The Merton Model (1974) was named after its founder Robert Merton. He postulated the model to make room for discontinuous trajectories about the prices of the stock. This came after he made some observation regarding the adverse effects of the impact created by publicized private information belonging to a particular company. With the mode, it would be possible to cope with the consequences. Additionally, the model is extended in a way that it adds jumps to the dynamics of the stock price as shown in EQ 13 below: dSt / St = rdt + σdWt + dZt --------- EQ 13 In this equation is a compound process that is mostly related to Poisson regression model. It has a distribution, which is the logarithm of the normal distribution and contains a jump of all sizes. The jump is performed after a homogenous process in the Poisson. The primary assumption that is made by the Merton model is that there is the presence of a debt structure, which is overly simple. It is also constructed on the assumption that the total value At of the assets within a particular firm follows the geometric Brownian motion with reference to physical measure that is represented by EQ 14 below dSt= µStdt+ σ StdWt, S0> 0,. --------- EQ 14 In this case, µ refers to the mean rate of return that is placed on the assets while σ stands for the volatility of the assets. However, there is still a need to make more assumptions. These include the fact that there are no bankruptcy charges. As a result, it indicates that the value of liquidation is equal to the value of the firm. In addition, both equity and the presented debts are tradable assets that lack any form of friction. Firms that tend to depend on large or medium caps usually trade on shares or what is commonly referred to as equity. When the Merton model is applied, it assumes that the debt is made up of a single and outstanding bond that possesses a face value of K and a maturity period of T. At the time of maturity, if the total debt value of the assets possessed by the company is found to be greater than the debts, then the debt is full paid and the remaining value is shared out among the shareholders. However, if the total value of assets is lesser than the outstanding bond with a face value of K, as in STt. In this equation, the capital structure is provided by the relationship of the balance sheet as shown in model M1 St= Et +Dt………………………………… (M1) In the case whereby St>k the debt holders of the company can be paid the full amount of K. In addition, the shareholders still possess a higher value of St-K. On the contrary, if the company does not honour its promise about the debt T and default, then it means that St Read More