Research On Uncertainty Option Pricing Models And Related Issues

The content of this dissertation is research on option pricing and its related issues. We ex-tend our study of Black-Merton-Scholes model to many other aspects to improve the deficiency that exists in its previous assumptions. Being one of derivatives, option provides a very useful tool to hedge downside (upside) risk for investors. And option market has developed into an important part of global financial market in recent decades. In our opinion, one of the main rea-sons of such rapid development of this market is that pricing of the option, or estimation of its changes can be made through model and modern methods. The empirical analysis shows that there exist some shortages in Black-Merton-Scholes’s method. Our research is mainly concen-trated in two aspects, that is, constant volatility and geometrical brownian motion. Generally speaking, there are some factors that may influence the option price, such as price of underling asset, interest rate, maturity, strike price and volatility. We can observe or estimate most of these factors from market except volatility. Therefore, how to measure volatility becomes the key point for option pricing. In this thesis, we will use volatility uncertainty model to illustrate the change of stock price. The original idea of this model is that we may be able to get the range of volatility’s variation, and as result we can obtain the optimal span of option price. But there are still some problems when we use this kind of models. A major difficulty is that, one is faced with a family of measures which are mutually singular. To solve this problem, I start to consider this model under G-Expectation framework. G-Expectation was firstly introduced by professor Peng in recent years. It’s a kind of sublinear expectation. It provides us with a new perspective about classic probability theory. Many results become interesting again under this new structure. At the same time, it’s a theory rooted in Knightian uncertainty, which may be not susceptible to measurement.G-Expectation framework is a powerful instrument to study Knightian uncertainty. We can use it to study volatility uncertainty model in financial market.To compare with the above results, we will construct two classes of models which concen-trate in the behavior pattern’s improvement of asset price. The first one is Levy model which is driven by Levy process. Levy process is based on infinitely divisible distributions which is a more general distribution than the Normal. Such distributions take into account skewness and excess kurtosis. Levy process is Markovian and semimartingale as well. It could be continuous or be allowed jump. That makes Levy model more flexible than B-M-S model. The other one is fractal model which is driven by fractional brownion motion. Fractional brownian motion is not a semimartingale in the condition of Hurst index H≠1/2. That could make us think about op-tion pricing problem outside the box of semimartingale framework. It is worth being notied that fractional brownian motion has’long memory’property. So we can use it to illustrate fractal structure of financial market.Moreover, since predictions about financial variables’evolution take strongly into account the knowledge of their past, we will also consider delay effect in above models.Main contents:1. We will construct a class of stochastic delay model under G-Expectation framework, and consider a stock whose price at time t is given by a stochastic process S(t) satisfying the following stochastic delay differential equation In the above equation, the initial process is a C([-τ,0];R) value stochastic process.{B(t),t≥0} is G-Brownian motion, which means {<B>(t), t≥0} is the relevant quadratic variation process. As a matter of fact, it’s a volatility uncertainty model. We will follow Arriojas et al(2007)’s work and take previous stock price into account. Based on Peng(2007), Bai-Lin(2010) and Ren(2013,2011)’s study, we will show that this model is reasonable and we can price European option with it. 2. To compare with the above result, we construct two kinds of stochastic delay models which are driven by Levy process and fractional brownian motion separately. Based on risk-neutral pricing method, we will price European call option under some regular conditions. Note that Levy model corresponds to incomplete market, which inspires us to find Follmer-Schweizer minimal measure to obtain our pricing formula.3. As an example of above models, we will study a class of fractional brownian motion and Levy process environment model without delay. It’s a special financial market where the jump of Levy process is power-jump. We will show that our model is complete and there is no arbitrage in the condition of Hurst index3/4<H<1.At the end of this part, we will use this model to price European option.4. As the theoretical preparations for the future research, we extend Yamada, Yor and Yan ’s work and get the generalized ito formula for G-Brownian motion(namely Yamada formula).