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The widely known Brownian motion was first introduced in 1828 by a Scottish botanist named Robert Brown. Brown used this concept in order to illustrate an irregularity in patterns of movement by pollen grains suspended in liquid. In 1900, Louis Bachelier considered the Brownian motion as a possible solution to modeling stock market prices. As a result, in 1923, Norbert Wiener was the first to give the Brownian Motion a rigorous definition, reconstructing the model. This use of the Brownian theory is often referred to as the Wiener process. Finally, after a series of efforts from a number of scientists, Black and Scholes (1973) introduced the famous Black-Scholes option pricing formula in a paper that remained unpublished. However, later that year, Robert Merton published a follow-up paper that integrated the no-arbitrage condition, generalizing in this way Black-Scholes formula.
After Wiener's work, Brownian motion was considered, until very recently, the most appropriate process for describing asset returns when operating within a continuous time framework. Nevertheless, a large number of relatively recent studies conclude that Brownian motion may not be the most appropriate process. There are three principal objectives that illustrate this opinion in real life data. First, the volatility of returns is not constant. It changes stochastically, according to time variation. Secondly, asset prices for real data are not continuous, but instead they demonstrate jumps. This, in turn leads to the non-normality of returns. Finally, the returns and their volatilities are not actually independent, rather they often illustrate signs of correlation and sometimes the correlation might even be negative. In particular, Fama (1963) stated that returns are actually more leptokurtic than the normal ones, especially when the holding period is small.
In addition, prices on options demonstrate the famous volatility smile and are actually higher than what the Black-Scholes formula predicts. It is also important to note that the Black-Scholes formula assumes normality in log-increments. However, empirical evidence has proved that this is actually false, leading researchers to initiate studies in other ranges of processes. One of the most famous of such family models is the Lévy processes. Their greatest advantage is that they allow the flexibility of not requiring returns to follow a normal distribution.
Finally, the growth of the market has led to the demand of new and more complex financial derivatives for which the Black-Scholes model cannot be used for pricing. As an example, exotic options illustarte this difficulty in pricing very clear, since most times it is impossible to derive a closed-form solution for pricing them. Nonetheless, there are other factors that amplify the complication of pricing an option, as stated in Anyaoku (2005). One such factor, is the early exercise, seen in American options or credit derivatives constructed from a mixture of correlated underlying assets. The path dependent derivatives such as Lookback or Asian options should also be included since their payoff depends on past values of the asset. This kind of options have actually increased in the Over-The-Counter (OTC) market and in order to be traded and hedged, they first need to be priced.
The aim of this paper is to exhibit the use of Lévy processes. In particular, we will examine the use of a Variance-Gamma (VG) process (Madan et al. (1998)) in overcoming all the issues of the Brownian motion, mentioned above, when pricing options and in particular American and Asian options. It will turn out that the empirically examined properties of real world data fit a lot better with Lévy processes than the Black-Scholes model. Jump-diffusion processes and Lévy models have been widely related with option pricing, since they take into account the implied volatility smiles and generally capture all the flaws mentioned above in the Black-Scholes pricing model. The pricing will be employed using Monte Carlo simulation method since it has so far proved an excellent tool for solving problems of higher dimensions.
2. Literature Review
2.1. Construction of Lévy processes
There is an enormous amount of studies illustrating different option pricing models, as also different processes describing asset prices that were produced as an alternative to the flawed Black-Scholes model. Lévy process was one of the most widespread alternatives and researchers up until now are studying the different versions of it. In particular, the first studies about the Lévy process go back to the late 20's but its final structure was gradually discovered by a number of researchers such as De Finetti, Kolmogorov, Lévy, Khintchine and Itô.
Broadly speaking, a Lévy process is a stochastic process in continuous time with independent and stationary increments, similar to the independent and identically distributed improvement of discrete monitoring. The two most famous Lévy processes are, the Brownian motion with drift, used in the Black-Scholes model (1973) and the compound Poisson process which underlies Merton's (1976) jump diffusion model.
Carr and Wu (2004) mentioned in their paper about time-changed Lévy processes that a pure jump Lévy process generates non-normal innovations and in order to capture the stochastic volatility, they applied a stochastic time change to the process. In addition, they mentioned that in order for the correlation between returns and their volatilities to appear, they had to allow improvements in the process to be correlated with improvements in the random clock on which it is run. In the case where the later correlation becomes negative, it means that when the Lévy process falls the clock has a faster running trend. This is what detains the leverage effect described in Black (1976).
There is a huge variety of different types of Lévy processes that are used in pricing options and researchers share separate opinions about which one is actually more suitable. This variety of opinions was clearly illustrated by Carr and Wu (2004). One sort consists of those who believe that compound Poisson processes are appropriate for fitting jumps as it was first mentioned by Merton (1976). Similar to this was Heston's (1993) idea in using a mean reverting square root process when modeling stochastic volatility. Among those who undertook the same research path were Andersen et al. (2002), Bates (2000) and Pan (2002).
The second sort considers general jump structures which allow an infinite number of jumps to arise within a finite time interval. One of the most commonly used models in this strand is the inverse Gaussian model, introduced by Barndorff-Nielsen (1998), as well as the Variance-Gamma (VG) model, introduced by Madan et al. (1998). This strand basically supports the use of infinite-activity process when modeling returns, which leads to the recognition of stochastic volatility.
The best elements of the two strands mentioned above are suitably combined in time-changed Lévy processes, creating an even better fit. In particular, the time-changed framework relaxes the affine requirement and allows greater generality for the jump structure. In addition, it allows volatility changes to be correlated arbitrarily with asset returns. This leads to the privilege of attaining both the leverage effect missing from the Black-Scholes model and the high jump activity as seen in Carr and Wu (2004).
2.2. Recent work on pricing using Lévy Processes
Option pricing is an issue that concerned researchers for decades and it will likely continue to be an issue for years to come. The methodology used for observing pricing behavior varies largely, depending on the type of options and process you are dealing with. Most of the times, pricing of options requires numerical integration or solving a partial differential equation (PDE). However, this is not always feasible. When the dimension of the problem is large, both numerical integrals and solving PDEs become hard to implement since formulas become intractable and a large amount of accuracy is lost (Jia (2009)). This is where estimation methods are introduced, in order to offer more accurate results. Several groups of numerical methods, implemented in recent studies, exist for analyzing option pricing. The most commonly used are the Monte Carlo simulations and Fast Fourier Transforms (FFT). Monte Carlo simulation, in particular, can be applied in evaluating options that contain multiple sources of ambiguity or obscured characteristics. However, a number of other differing analytical approximations are used, as well as derivations of integral equations.
One approach was that of Madan et al. (1998) who examined a stochastic process, in which financial information arrived via jumps. In their study, they applied a high-activity process with infinite small jumps, combining them with the lower-frequency large jumps. Following their work, Carr and Wu (2003) examined the necessity of a diffusion component, when using high-activity pure jump processes. However, they did not manage to reach an exact conclusion since often jump processes in the limit imitate the performance of a diffusion process. Nonetheless, they realized that when pricing short-term index options at-the-money, a diffusion component offers some contribution.
Benhamou (2000), in his effort to model the smile effect, implemented both Fourier and Laplace transforms in a semi-parametric method. The only assumption required was that Lévy processes are appropriate in modeling underlying price processes. However, no other constraints were employed on the price process. In this way, he managed to broaden the Black-Scholes model to a variety of Lévy processes, due to the fact that the latter includes both continuous time diffusion and jump process.
Another approach was that of Lord et al. (2007). They examined the pricing of early-exercise options by introducing a novel quadrature-based method which mainly relies on Fast-Fourier transformations called the Convolution method. Their work was a combination of the recent quadrature pricing methods of Andricopoulos et al. (2003) and O'Sullivan (2005) and the Fourier transformation methods initiated by Carr and Madan (1999), Raible (2000) and Lewis (2001). The main idea of the procedure applied was to reformulate the commonly used risk-neutral valuation formula on the basis that it forms a convolution. The Convolution method is then used for pricing American and Bermudan options. In order to implement this method, they had to impose only one restriction, basically that of a known conditional characteristic function for the underlying asset. However, since the method was applied within an exponential Lévy framework, which includes the exponential affine jump-diffusion models, the constraint for a known conditional characteristic function was fulfilled. In the report, the flexibility of choosing the asset price process within a range of jump processes was illustrated in numerical examples by examining three different processes; namely, the Geometric Brownian Motion (GBM), the VG and the CGMY (Carr et al.(2002)).
Finally, an apparent example on the use of Monte Carlo simulation in pricing exotic options by making use of general pricing techniques for vanilla options is given by Schoutens and Symens (2002). A thorough observation should be made of their work, since this report will use the same procedure in pricing American and Asian options. The processes exploited when pricing the options were the VG process, the Normal Inverse Gaussian (NIG) process and the Meixner process, which were then used to implement a Lévy Stochastic Volatility (Lévy-SV) model. Broadly speaking, in a stochastic volatility model, the time of the process becomes stochastic. This means that time is running slow when we are at times of low volatility and time is running fast when we are at times of high volatility. As a rate of time change, they used the Cox-Ingersoll-Ross (CIR) model and they calibrated the Lévy-SV model to fit their data set, which was completed with mid-prices of a set of European call options on the S&P 500 index. The calibration produced the risk-neutral parameters for each model, on which they carried out simulations and obtained the option prices for all proposed models. Finally, the use of the technique of control variates was made in order to decrease the standard error of the simulations to a minimum. As a result, they realized that although in the Black-Scholes framework, prices of exotic options depend highly on the volatility parameter chosen, which is not apparent; in the Lévy-SV model, prices are almost equal. This is what led them to the conclusion that pricing exotic options on the Lévy-SV model is more reliable than the Black-Scholes model.
3. Data Requirements and Methodologies to be used
Following the procedure above by Schoutens and Symens (2002), we will price an American option and an exotic option, specifically an Asian fixed strike call option with arithmetic average, using Monte Carlo simulations.
Generally, there are two types of research approaches: quantitative and qualitative. The main difference of the two is that the first is objective in nature, i.e. it is not influenced, whilst the second approach is subjective in nature, i.e. it is influenced. The data we would use to implement our model will be FTSE 100 index prices for a 5 year period. The payoff of an American and an Asian option are given in the following way.
Where is the stopping time representing the optimal stopping time and is the payoff of the American contract to be priced.
Asian Fixed Strike Call Option:
Let = average of the asset price over Assume stock prices are recorded every periods from time to time , then there are periods over .
The payoff of the Asian fixed strike call option is given by , where is the strike price.
By using a VG model, we will calibrate model prices in order to try and match them to market prices by minimizing the least squared error of their differences. In particular, for computing the average absolute error we would use the following formula
The procedure begins by simulating different paths for our stock prices process (in our case the VG process) and then for each path evaluate the payoff function for , where is the number of options in our data set. From that, we calculate the Monte Carlo estimate of the expected payoff as
The above estimate is used to discount the final option price, i.e. . The standard error of this estimate is then required, which is given by:
Generally, it is observed that in order to decrease the standard error of the estimate, you need to increase the number of simulations. However, one should note that when increasing the number of simulations, the speed of the procedure decreases. We continue by simulating our VG process up to time
The final step includes the rescaling of our path based on the path of the stochastic business time of our choice and then inserting it in the formula for the behavior of the stock price. Finally, we should note that the stochastic time we chose to use is the Cox-Ingersoll-Ross (CIR (1985a/b)) model.
In this report, our aim is to illustrate a way of pricing American and Asian options that detain all key aspects on real world financial securities; mainly, jumps, stochastic volatility and the leverage effect. By using a VG process, we obtain a framework that allows jumps and with the help of the CIR model, as the stochastic time, we allow for stochastic volatility to be introduced. Finally, the leverage effect is captured through the correlation between the VG innovation and the time change.
Generally, the Black-Scholes model will always remain as a point of reference against all other models or extensions pioneered and it will continue to be seen as a paradigm of option pricing regardless of the amount of research that has occurred, proving the opposite. However, recent studies have been evident of the flaws of the Black-Scholes model when applied to the real world framework. In addition, it has been proven that Lévy processes can be considered as the most recent effective fit to the actual market. Nevertheless, in future research everything is possible to revolutionize.