Modern pricing models - Essay Example

THE MODERN PRICING MODEL (GBM, MERTON MODEL, HESTON MODEL, BATES MODEL) By StateDate of SubmissionHeston ModelDefinition of the Heston Model The Heston Model refers to another type of financial pricing model incorporating the stochastic volatility. The person who developed this financial model was Mr. Steven Heston, who was an associate finance professor as at the time of developing this model in 1993. He developed this model in order to undertake a critical analysis of the currency and bond options.

As such, he named the financial model after himself, the Heston Model. As such, he was able to develop a model that provided pricing options with a closed-form solution in an attempt to overcome the challenges arising from the option-pricing model developed by Black-Scholes. Some of the shortcomings experienced with the Black-Scholes model were strike-price bias and return skewness. Consequently, the development of the Heston model came in as the best alternative tool for the purposes of advanced investments (Gilli, Maringer & Schumann 2011, p.257). As any other stochastic volatility model, the Heston model utilizes statistical methods when making calculations or forecasts of the various pricing options in consideration.

As such, it also bases on the assumption that the underlying security or trading option has an arbitrary volatility. Therefore, the Heston model falls among the various different models of stochastic volatility such as the GARCH model, the Chen model, as well as the SABR model. Consequently, the Heston Model also falls under the standard smile model category, with “smile” in this concept referring to the volatility smile. A volatility smile is a graphical representation of various options that have identical expiration date expressing an increasing volatility.

This increase in volatility arises often arises when the options become more out of the money or in the money. The concave shape generated by the graph is what gives rise to the name, the smiles model, as it appears like a smile (Wang 2007, p.3). The Heston Model applies mathematical calculations in describing the process of evolution in volatility that an underlying asset undergoes under the stochastic volatility options. As such, just as other statistical models mentioned above, the Heston Model equally has a number of assumptions, such as the volatility of an asset not being constant, or deterministic, but rather following a random process.

Some the of the basic assumptions of the Heston Model is that the stochastic process determines the asset price, St dSt = μStdt + √VtStdWtsIn this equation, is the instantaneous variancedv = k(Ө - Vt)dt + Ԑ√VtdWtvIn addition, forms part of the Wiener Process as experienced under the GBM (Geometric Brownian Motion) also considered as random walks. As such, the equation has correlation ρ, which is an equivalent of covariance ρ dt. The following represent the parameters used in the above equationsThe asset’s rate of return is μThe long variance is θ, also referred to as the long run variance of the average price.

From the equation, we develop that as t tends towards infinity, the expected value of v tends to θk refers to the rate at which v reverts to θξ refers to the vol of vol, or volatility of the volatility. This determines the variance of v as implied by its nameThe process of remains strictly positive when the parameters in the above equations adhere to the Feller condition as expressed in the following equation (Moodley 2005, p.2)2KӨ >Ԑ2History of the Heston Model The historical era of the Heston model started as early as 1987 when Alan White and John Hull developed and made public in 1987 a one factor stochastic model that they developed.

However, this did not have any significant effects as a game changer in the market. As such, the occurrence of the real paradigm shift was in 1993 when Steven Heston developed a two-factor stochastic volatility model, which was a great improvement from the model developed by Hull and White. At that time, Heston was still an associate professor at Yale University. This model was a pioneer in the field of quantitative finance as it evaluated processes that were more complex as opposed to the GBM.

This was a new way of visualizing volatility and the problems arising from option pricing thereby leading to the most significant break required in quantitative finance, in an era dominated by volatility surface and local volatility. The Heston model proved to be a rigid framework when accounting for all features enshrined in the volatility surface. Therefore, this requires the consideration of additional degrees of freedom to the original model. As such, the first straightforward extension is by incorporating time dependent parameters, thereby transcribing the model dynamics as followsdst = μStdt +√VtStdWtsIn this case,, refers to the instantaneous variance, which is also a time-dependent processdvt = kt(Өt - Vt)dt + Ԑt√VtdWtVIn addition, refers to the Wiener processes or random walks undertaken in the equation with correlation ρ.

As such, one may require a piecewise-constant status in the parameters for him or her to be able to retain tractability of the model. Another process of exploring the volatility surface is to add a second process of variance in the whole equation, which is very independent from the first equation.dSt = μStdt + √vt1StdWts,1 + √vt2StdWts,2dvt1 = k1 (Ө1 – vt1) dt +Ԑ1√vt1dWtv1dvt2 = k2 (Ө2 – vt2) dt +Ԑ2√vt2dWtv2The Chen Model, developed by Lin Chen in 1996 provides a significant extension of the Heston model in its attempt to meet both mean and volatility stochastic.

The following equations specify the instantaneous interests’ rates within the Chen model (Rouah 2013, p.1)drt = (Өt - rt)dt + √rtσtdWt,dαt = (ζt – αt)dt + √αtσtdWtdσt = (βt - σt)dt + √σtŋtdWtAnalysis of the Heston ModelAn analysis of the Heston model provides its risk-neutral measure, which is a fundamental concept in pricing of derivatives. As such, in order to understand the way the Heston model applies to the concept of price derivatives, it is imperative to consider a number of aspects.

The first ones is that one needs to evaluate the value expected from a discounted payoff under a risk neutral measure in order to price a derivative that has a payoff function with one of more underlying assets. A risk neutral measure also refers to an equivalent martingale measure, a measure that is equivalent to a real-world measure and arbitrage free. Consequently, the discounted price in each of the underlying assets becomes a martingale in such a measure as outlined in the Girsanov’s theorem.

It is possible to describe any equivalent measure in the Heston and Black-Scholes frameworks in a very loose through the addition of a drift in each of the Wiener processes, as both frameworks have filtrations generated from linearly independent sets of Wiener processes. In addition, a selection of certain values of these drifts enables one to obtain an equivalent measure that equally fulfills the condition of arbitrage freedom. The best situation to apply this concept is when considering a general situation whereby there are underlying assets, as well as a linearly independent set of Wiener processes.

Such a set of equivalent measures provides a value that is isomorphic to Rm, which refers to the space of possible drifts (Wang 2007, p.12).Properties of the Heston Model Using the Heston Model is a very challenging affair to most financial analysts who still find it difficult to apply its use in determining prices of trading options because of the closed formula that it has, especially when the parameters under consideration are constant or piecewise constant. As such, for any time dependent heston model, it is possible to derive an analytic formula that is accurate for the price of vanilla options, especially after the application of the Malliavin calculus techniques, as well as the small volatility of volatility expansion.

In such a case, the accuracy is usually less than a few bps for various maturities and strikes. Furthermore, this leads to the establishment of error estimates that are tighter. Financial analysts like using this approach because of its advantages over Fourier-based methods because of its rapidity, whereby it can record a gain by a factor 100 or more, yet it still maintains a competitive accuracy. The approximative formula further enables analysts to derive a number of corollaries relating to the calibration procedure as regards to ill-posed problems, as well as to equivalent Heston models.

A number of empirical studies indicate that the long term of an underlying asset does not always have a normal distribution. Similarly, the volatility and the return have negative correlation, thereby making it impossible to reflect these features sufficiently on the Black-Scholes model. The Heston model comes in as a contrast to the Black-Scholes model in a manner that is much more appropriate because it has the capability of presenting a variety of different distributions. The reason why the Heston model has the capability of presenting numerous distributions is the p parameter (the coefficient of correlation between two Brownian motions that are dependent).

The second reason is the representation of the relationship between the volatility and the return of an underlying asset (Wang 2007, p.7).Financial uses of the Heston Model Most financial analysts prefer using the Heston model in determining future prices of stocks and binary options as opposed to other statistical models, such as the GBM because of the efficiency, as well as the high level of accuracy that results from using the Heston Model. As such, the Heston Model stands out as a preferred model of stochastic volatility because it has dual stochastic analysis thereby enabling analysts to obtain fast option pricing, as well as an accurate calibration of their estimates.

Consequently, the Heston model is a market leader in stochastic volatility especially in determining pricing options, volatility and variance swamps within the model. Furthermore, this model also stands out as a standard model in the industry accounting for the volatility smile seen in the market (Gilli, Maringer & Schumann 2011, p.257). The Heston model is gaining fast significance in the financial markets analysis owing to the deficiencies and failures of other major pricing options within the market, such as the Geometric Brownian Motion and the Black-Scholes model.

For instance, despite the high success recorded by the Black-Scholes model as used in option pricing, the model has a variety of known deficiencies that hamper its effective usage in financial analysis and futures price determination. One major deficiency of the Black-Scholes model is the assumption that it makes regarding the volatility of the return on the underlying asset. This model assumes that this volatility is constant, which is not always the case because, in the real world, implied volatilities of trading options vary generally and from time to time.

Most of the variations occur in line with the maturity or with the strike price of the options under consideration. The volatility smile or the volatility skew refers to the variation that occurs with a strike price. Form the above-mentioned deficiencies, an arising question are how best to price the options in a manner consistent to the variations of implied volatility as observed within the market. The stochastic volatility concept comes forward in an attempt to resolve this problem, especially with the wide variety of stochastic models in existent within the market.

The most popular among these stochastic models in the market and preferred for financial modeling and analysis is the Heston model of stochastic volatility. On one hand, the constant volatility derived from the Black-Scholes model analysis corresponds with an assumption that an underlying asset always follows a stochastic process that is lognormal. On the other hand, stochastic models usually have a basic assumption that the variance or the volatility of the underlying asset is always a random variable.

This situation therefore creates two Brownian motions. The first Brownian motion stands for the variance, while the second Brownian motion stands for the underlying. It is imperative to not that these two process are correlated. For instance, in the equity world, the correlation usually stands out as negative whereby a decrease or an increase in the price of an asset ends up with a similar increase or decrease in the volatility of the underlying asset. After obtaining a stochastic variance of the underlying option, closed – form solutions cease to exist, especially for put and call options (Wang 2007, p.12). however, another attractive feature of the Heston model that makes financial analysts apply it more often is the existence of closed-form solutions, or quasi within most markets plain vanilla options, such as the European markets.

This feature thereby enables a feasible calibration of the model.One major area of financial analysis that applies the Heston Model in determining price of options is the European pricing options. In this scenario, there are five independent parameters of the Heston model. The determination of these five parameters can take place through calibration of the European options to market observed prices, with the options have different maturities and strikes. As such, one can easily price other options after he or she manages to determine a set of parameters in such a manner, such as a European option that has a different strike, a more exotic product, of an American option.

Consequently, the underlying asset price follows the prescribed standard lognormal process. On the other hand, the variance, V follows a mean reverting square root process (Haastrecht & Pelsser 2007, p.17).dS = rSdt + √VSdW1dV = -k(V - V∞)dt + w√V dW2In this case, the following are the meaning to the represented functions, whereby r represents the risk-free interest rate ignoring the dividends. dW1 and dW2 represent two correlated standard Brownian motions, while the five parameters of the Heston model evident in the equation are Vo as the initial variance, p as the correlation, k as the speed of the mean reversion, w as the volatility of volatility, and as the long term variance.

Generally, the price of an option at time t, of a given European call option due to mature a time T results from the discounted expectation value equation as outlined belowCT = e-r(T - t)0∫∞ dx(ex - K) + p(x) In this case setting, and refers to the probability density function of the underlying logarithmic asset price. As such, it is always useful and important to work with characteristic function in such cases because it is not always possible to derive the analytic forms for such option prices as outlined in the equation belowΦT (u) = E [eiuxT | xt = x] The above equation represents a Fourier transformation for the density function of the probability from u-space to x-space.

A Fourier transformation refers to an expression of a function of time, amplitude and frequency. This is in accordance with the different simulations expressed in the Bates model of stochastic volatility. Therefore, one can apply the characteristic function in many different ways in order to obtain the price of plain vanilla European options.Applications of the Heston Model in FinanceOne of the major financial applications of the Heston model is its use in practical hedging. A hedging strategy implemented successfully always enables investors to record a significant increase in their expected returns, such as through making an inclusion of a private equity portfolio with the absence of excessive risks.

As such, the practical hedging procedure in the Heston model presents a real-life scenario of risk management and portfolio hedging under the model. The first process in hedging a trading strategy or a derivative that incorporates numerous derivatives is to determine the parameters of the model through a calibration of the Heston model for it so to correspond to the tools and instruments used in evaluation. Secondly, an evaluation of the payoff follows together with the computation of the hedging ratios.

The parametric nature of the Heston model simplifies its usage and application as a hedging ration, as well as for purposes of computation. All one has to do is to make a selection of the suitable instruments of reference to apply, them make a computation of the sensitivities, usually measured in the partial derivatives w.r.t the state variables, of the overall portfolio with respect to the reference instruments. The remainder or the unhedged sensitivity is in reference to the risk exposure of the portfolio under consideration (Bin 2007, p.69). For instance, one financial application of the Heston model assumes that the spot price of an option follows the following diffusion processdSt = St(μdt + √(vtdWt(1)))As such, this process resembles the geometric Brownian motion and has a non-constant instantaneous variance.

In addition, the developer of this model, Mr. Heston, proposed that the following mean reverting stochastic process drives the variance of the optionsdvt = k(Ө - vt)dt + σ√(vtdWt(2))As well as, give room for the two Wiener processes to correlate with one another as outlines in the following equation dWt(1) dWt (2) = ρdt The original users of the variance process were Cox, Ross and Ingersoll who used the variance process to model the short term interest rates of an option, defined equivocally by three parameters which are , , and .

As such, in the same context f the models if stochastic volatility, such an interpretation can be termed as a long-term variance, the rate of mean reversion to the long-term variance, as well as the volatility of variance, usually referred to as the volatility of volatility, in such order respectively (Gilli, Maringer & Schumann 2011, p.257).A surprising fact is that the introduction of a stochastic volatility in the determination of futures prices for options and stock has no effect on the properties of the spot price process, especially in a manner noticeable by a visual inspection of its realizations.

For instance, the following figures try to transcribe the performance of these models across various volatilities. Figure 1 plots the same paths of the spot process and the geometric Brownian motion on a Heston Model. In order to make the comparison more objective, the analysts conducting the simulation had to obtain both trajectories using the same set of random numbers. It is very clear that these are undistinguishable by the normal mere eye. In both cases, the initial spot rate So = 0.84. On the other hand, the foreign and domestic interest rates were 3% and 5% respectively, thereby leading to a yielding drift of μ = 2% .

The volatility of found in the geometric Brownian motion constant was √vt = √4% = 2%In a Heston model, the mean reverting process drives these volatility functions, whereby Vo = 4% the initial variance = 4% the long-term variance = 2 the speed of mean reversion = 30% representing the vol of vol P = -0.05 representing the correlationFIGURE 1:The above figure shows the sample paths taken by the geometric Brownian motion (red dotted line) in comparison to the spot process found in the Heston model (the solid blue line) obtained using the same set of random numbers (transcribed in the left panel).

As such, despite the fact that GBM requires volatility to be constant, the volatility in the Heston model relies on the mean reverting process (transcribed in the right panel). Therefore, it is evident that a mere eye cannot easily distinguish the sample paths.In examining the Heston model closer, it is possible to reveal a number of differences that accrue concerning the GBM. For instance, the probability density functions of the [log-] returns have tails that are heavier, or exponential as compared to Gaussian, considering the illustration in Figure 2.

As such, this scenario compares the two to hyperbolic distributions whereby the log-linear scale resembles hyperbolas rather than resembling parabolas.Figure 2:The figure above represents the marginal probability density function within the Heston Model (the solid blue line) plotted against the Gaussian Probability Density Function (PDF) which is the dotted red line, using the same set of parameter as applied in figure 1 above. The tails of Heston’s marginals have an exponential effect clearly visible through the right panel where the corresponding log-densities are plottedBenefits of Using the Heston Model in FinanceUser of this model are full of praises for the numerous advantages that accrue to them after they apply the Heston model in undertaking financial analysis, as well as the determination of pricing options.

One of the major attractive features of the Heston model is its powerful duality especially in relation to its tractability and the concepts of its correlation to other models of stochastic volatility. The model is very instrumental in the European markets because of its closed form solutions. As such, the model enables investors in this market of European options to undertake a fast calibration of a given market data, thereby leading to efficiency and accuracy in the determination of option prices.

Furthermore, the Heston Model’s price dynamics give room for the application of non-lognormal distributions of probability, which is quite the opposite as with the case of the Black-Scholes model (Rouah 2013, p.1)Financial analysts also enjoy using the Heston model as opposed to other models of stochastic volatility because it fits the implied volatility surface especially for option prices within the market. This gives room for the analysts to conduct a wide range of analyses using this model in determining option prices as opposed to other models.

Furthermore, the volatility of the Heston model is mean reverting. On the other hand, the Heston model also considers the fact that equity returns and implied volatility have negative correlation, usually referred to as the leverage effect under normal circumstance. As such, this gives rooms for changes to take place in the correlation between the volatility and the assets.However, the tide is not always rosy for investors using the Heston model to undertake their financial analysis options because of a number of drawbacks that the model suffers from time to time during its implementation.

For instance, one of the major challenges of using the model, a concept that equally applies to other models of stochastic volatility in general, is the fact that it is difficult to estimate the parameter values of volatility because this is an unobservable concept. As such, it becomes a tough choice for the analysts to come up with the most probable value that relates to the volatility constraint of each performing parameter. In addition, the prices produced by the Heston Model are very sensitive to the correlation coefficient.

This causes the fitness of the model to rely strongly on the calibrations, because these prices change easily with the slightest rise or drop in the ρ parameters used. As such, the price paid for models that are more realistic arises from the increased complexity in the calibration of the model. In most cases, the estimation method for these parameters becomes just as crucial as the model itself. Finally, in some cases, the Heston model fails to provide results that are decent for short maturities because of the failure of the model to create or develop short term skew stronger than the one provided by the market.

Therefore, in order to perform successfully across a large time interval of maturities, it is necessary to undertake further extensions of the model ReferencesBin, C. (2007) Calibration of the Heston Model with Application in Derivative Pricing and Hedging. [Online] Available at http://ta.twi.tudelft.nl/mf/users/oosterlee/oosterlee/chen.pdfFigure 1 and Figure 2. [Online] available at https://www.google.com/search?q=heston+model&client=firefox-a&hs=Z2Q&rls=org.mozilla:en-US:official&channel=fflb&biw=1366&bih=625&source=lnms&tbm=isch&sa=X&ei=_I5sVJ-SOc7haJPogJAB&ved=0CAgQ_AUoAwGilli, M.

, Maringer, D. & Schumann, E. (2011) Numerical Methods and Optimization in Finance. New York City: Academic Press. [Online] Available at http://books.google.co.ke/books?id=FYkgnJeNFrkC&pg=PA257&dq=Heston+model&hl=en&sa=X&ei=GrRrVLjpC-mCsQT1iYLYCw&redir_esc=y#v=onepage&q=Heston%20model&f=falseHaastrecht, A. & Pelsser, A. (2007) Efficient, Almost Exact Simulation of the Heston Stochastic Volatility Model. [Online] Available at http://arno.uvt.nl/show.cgi?fid=92651Moodley, N. (2005) The Heston Model: A Practical Approach With Matlab Code.

[Online] Available at http://math.nyu.edu/~atm262/fall06/compmethods/a1/nimalinmoodley.pdfRouah, F. (2013) The Heston Model And Its Extensions In Matlab And C#. Hoboken, New Jersey: John Wiley & Sons. [Online] Available at http://books.google.co.ke/books?id=SSlwAAAAQBAJ&pg=PA1931&dq=Heston+model&hl=en&sa=X&ei=GrRrVLjpC-mCsQT1iYLYCw&redir_esc=y#v=onepage&q=Heston%20model&f=falseWang, J. (2007) Convexity of Option Prices in the Heston Model. [Online] Available at http://www.diva-portal.org/smash/get/diva2:304668/FULLTEXT01.pdf