统计与数据科学学院知识库

经典Black-Scholes(B-S)模型构建后,期权定价成为学术界研究的热点话题之一. 随着对经典B-S定价模型研究的不断深入,发现原来的部分假设条件难以符合实际金融情况,如连续交易且无交易费用、标的资产价格变化服从几何布朗运动、以及对数收益率服从正态分布.已有的部分文献是单一考虑,鲜有同时考虑这三个假设条件,并且分数布朗运动下的期权定价模型会出现套利机会.

主要研究了混合次分数布朗运动模型建立的欧式期权定价和风险管理问题.研究内容包括四部分.第一部分不考虑交易费用,建立了基于跳环境和混合次分数布朗运动下的欧式期权定价模型.首先,利用Delta对冲原理,获得了欧式期权所满足的随机偏微分方程.其次,使用拟条件期望分别得到欧式看涨、看跌期权定价公式和看涨看跌平价公式.

基于此, 第二部分利用混合次分数布朗运动建立了欧式期权定价模型,同时考虑带交易费用和跳环境来进行资产定价.首先,利用对冲策略,获得了欧式看涨期权所满足的随机偏微分方程.其次,使用自融资策略分别得到欧式看涨、看跌期权定价公式和看涨看跌平价公式.

第三部分通过希腊字母和关于Hurst指数 H 的偏导公式量化了资产风险.最后,数值模拟表明:定价参数中的Hurst指数H和跳跃强度λ对期权价值有显著影响.第四部分中,分别采用"上证指数","市北B股"和"耀皮B股"等的收盘价日线数据,研究表明:在跳环境和混合高斯过程下的欧式期权定价比经典B-S模型更加接近真实值,该研究不仅能够在理论意义中丰富金融统计与风险管理有关期权定价方面的理论,同时在实际意义中也能够为金融市场提供更多的参考依据.

After the classic Black-Scholes (B-S) model is constructed, option pricing has become one of the hot topics in academic research.With the continuous in-depth study of the classic B-S pricing model, it is found that some of the original assumptions are difficult to meet the actual financial situation, such as,continuous trading and no transaction costs,the price of the underlying asset changes obey geometric Brownian motion,the logarithmic rate of return obeys a normal distribution.Part of the existing literature is a single consideration.It rarely considers these three assumptions at the same time.And the option pricing model under fractional Brownian motion will have arbitrage opportunities.

It mainly studies the European option pricing and risk management problems established by the mixed subfractional Brownian motion model.The content includes four parts.In the first part, no consideration of transaction fees,a European option pricing model based on jump environment and mixed subfractional Brownian motion is established.Firstly, using the Delta hedging principle,the stochastic partial differential equations satisfied by European options are obtained.Secondly,use the quasi-conditional expectation to obtain the European-style call and put option pricing formula ,and the call and put parity formula respectively.

Based on this,the second part uses the mixed subfractional Brownian motion to establish a European option pricing model, while considering the transaction costs and jump environment for asset pricing.Firstly,using the hedging strategy, the stochastic partial differential equations satisfied by European call option are obtained.Secondly, the pricing formula of European call, put option and the parity formula of call and put are obtained by using the self-financing strategy.

In the third part,it quantifies the asset risk through the Greek letters and the partial derivative formula on the Hurst index H.Finally, numerical simulations show that the Hurst index H and jump intensity λ in the pricing parameters have a significant impact on the value of options.In the fourth part,the daily closing price data of "Shanghai Stock Index","Shibei B-shares" and "Yaopi B-shares" are used respectively.It shows that European option pricing under the jump environment and mixed Gaussian process is closer to the true value than the classic B-S model.It can not only enrich the theory of financial statistics and risk management related to option pricing in the theoretical sense,but also in the practical sense.It can provide more reference basis for the financial market.