Research On The Pricing Method For Warrants Of Long Memory Processes

As the process of nontradable shares reform accelerating and financial supervision continuously improving, warrants have become an important part in China financial derivative market. As a consequence, how to price them reasonly and how to estimate the parameters of these pricing models efficiently have become major focus of attention from corporate and regulatory sectors. Since the 1970s, many scholars have considered the problem of pricing warrants and estimating parameters of pricing model under the geometric Brownian motion environment. However, in recent years, some researches about assets'returns have illustrated that the log-return of asset are not independent and have the properties of self-similar and long memory, and the log-return has a slowly decayed autocorrelation function. These properties indicate that the traditional Brownian motion can not describe the random phenomena well. Fortunately, the random process, namely fractional Brownian motion, which has these properties of both long memory and self-similar, can capture the random phenomena very well. Hence pricing warrants with long memory models and estimating the unknown parameters of these pricing models have important theoretical and practical significance for both determining the fair and reasonable values of the warrants and promoting the strong development of China's financial derivatives market.However, the fractional Brownian motion is not a semi-martingale. Many famous scholars have stated that the application of fractional Brownian motion in finance has some limitations: leading the arbitrage opportunity in finance or making no economic sense. Recently, some famous scholars pointed out that the introduction of risk preferences and mixed fractional Brownian motion ensures absence of arbitrage. Following these ideals, this thesis derives the warrants pricing models in fractional Brownian motion with risk preferences and mixed fractional Brownian motion, respectively. Furthermore, we study three detivative problems arising from issuing warrants based on the characteristic of warrants. In order to appling our pricing models into practice, this thesis uses the maximum likelihood method to estimate the unknown parameters in pricing models effectively. The main results and innovations in this thesis are listed as follows:First, using the stoshastic models with long memory to describe the behavior of underlying asset, this thesis derive the pricing models for warrants.Previous studies dealing with warrants pricing usually assumed that the underlying asset followed the geometric Brownian motion. Using the risk preferences based on the fractional Brownian motion and the mixed fractional Brownian motion to describe the behavior of the underlying asset, this thesis obtains the pricing models by risk-neutral pricing method and the partial differential pricing method. Furthermore, based on the characteristic of warrants, this thesis presents the algorithm for solving the valuation of warrants, i.e., the nonlinear equations solving the valuation of warrants is given. Moreover, using different methods this thesis proves the existence of solutions to these nonlinear equations. The empirical research shows that: The pricing results of the warrants pricing models, which have the long memory property, is closer to the real market value than the results of these pricing models based on traditional geometric Brownian motion. This is mainly because that there exists long-memory in Chinese securities market and hence the introduction of long-memory can improve the warrants pricing.Second, under the risk preferences based on the fractional Brownian motion, this thesis deals with the problem of pricing the butterfly warrants, and the pricing formula of multiple warrants are also proposed.Previous studies dealing with warrants pricing usually assumed that the underlying asset followed the geometric Brownian motion and priced the single warrant. However, in order to reduce risk, some companies issued both put and call warrants, i.e., the butterfly warrants are issued. Taking into account the dilution of cross effect and the features of warrants, this thesis presents the pricing model for butterfly warrants. Further empirical research shows that: the single warrant pricing models, which do not consider the cross dilution effect, will under estimate the valuation of warrants. The pricing results of Chinese warrants using the model proposed in this thesis is more close to the market prices than the results calculated by the simpler single warrant pricing models.Third, under the risk preferences based on the fractional Brownian motion, this thesis presents the distribution processes of both before warrants issuing and after warrants issuing. The pricing model and algorithm for pricing warrants after their issued are also presented. Since warrants are different from stock option, issuing or exercising warrants can increase the number of outstanding shares of the firm and dilutes the stock of the firm, and thus change the behavior of stock. Under the assumption that the underlying asset follows the geometric Brownian motion, this thesis studies the behavior process of stock after warrants issued. The changes of stock price behaviors in both the physical measurement and risk neutral measure are obtained using stochastic analysis method. Moreover, using empirical analysis, this thesis shows that the behavior of stock price has changed after the warrants issued. On the other hand, the warrants historical volatility can be obtained from warrants real market after warrants issued. In order to take the full advantage of historical information and to improve the accuracy of warrant pricing, this thesis proposes the algorithm for pricing warrants after their issued. And some empirical studies have illustated the effectiveness and accuracy of the method.Fourth, using the maximum likelihood approach, this thesis obtains the estimators of unknown parameters of pricing models. The effectiveness of maximum likelihood method and the rate of convergence are also presented.Since the fractional Brownian motion and mixed fractional Brownian motion are used to describe the behavior of the underlying asset, how to estimate the unknown parameters in stochastic differential equation driven by fractional Brownian motion is the critical problem in warrants pricing. In a large sample (observation interval is fixed and observation points are large enough), this thesis estimates the unknown parameters in mixed Brownian motion, geometric fractional Brownian motion, geometric mixed fractional Brownian motion and fractional Ornstein-Uhlenbeck process using maximum likelihood estimation. Furthermore, the convergence and the central limit theorem of these estimators are obtained by using Malliavin stochastic analysis and Laplace transform. Numerical simulation results show that: in the case of small sample size, the means of these estimators are very close to the true values and standard deviations of these estimators are very small. Hence the maximum likelihood approach has the good stability and the fast convergence.