Abstract: It is necessary to accurately describe the volatility of asset prices for pricing options. A large number of studies have shown that the traditional assumption that asset price volatility is constant in the market, is gradually not applicable to the development of modern financial market measurement. At present, research on stochastic models mainly includes stochastic volatility models and local volatility models. These two types of models still have some shortcomings: stochastic volatility models are usually difficult to solve, and complete hedging is even more difficult to achieve. Local volatility models usually rely on subjective experience in form, and the volatility functions fitted at different times vary greatly and even present random changes. The random model parameters can only be estimated from the market price of options, which means that option pricing is essentially an approximation of the market price rather than a theoretical price. Strictly speaking, model parameters should be obtained through the statistical properties of stock prices.
To alleviate the shortcomings of the above volatility models, the option pricing system designed in this article seeks a compromise modeling approach between constant volatility and stochastic models, with the model only making predictable assumptions about volatility. For the determination of influencing factors in the process of volatility modeling, existing methods generally rely on subjective manual extraction settings to complete. Manually selecting influencing factors is a very laborious task, and the accuracy of selecting influencing factors largely depends on experience and understanding, lacking theoretical basis. How to automatically, scientifically, and effectively determine the influencing factors is the key to non-parametric volatility modeling, usually with more efficient and effective deep learning algorithms. The backpropagation algorithm is the most well-known deep learning algorithm, but its performance is not very good when learning structures with a large number of hidden layers in practice. Deep Boltzmann Machine (DBM) is an efficient unsupervised learning algorithm in deep learning, which empirically alleviates optimization problems related to deep models. This article first uses DBM to extract the influence factors of the model, and uses the k-step contrast divergence algorithm as the learning algorithm of DBM to establish a volatility DBM-ANN model. Based on this, it uses stochastic differentiation and martingale methods to obtain the closed form solution of European options under risk neutral conditions. The system does not need to assume the distribution form of volatility to determine model parameters through asset prices, which overcomes the shortcomings of traditional manually designed volatility model forms and the parameters can only be estimated using option market prices. In the computational experiment, the numerical results show that the system has ideal accuracy in characterizing the movement of volatility. Through comparison, it is found that the B-S formula often prices 50ETF stock options lower than the option in this paper, and its degree increases with an increase in the remaining time from the expiration date and with S/K→1.
Throughout the outstanding research achievements in option pricing problems, almost all contain a large number of mathematical methods, and complex mathematical financial models occupy a central position. The characteristics and performance of deep learning make it widely applied in various fields of artificial intelligence, and also provide a solid and reliable tool for scientific and systematic quantitative research on financial derivatives. It is an inevitable trend in the future research of mathematical finance. Although some scholars have previously applied deep learning methods to study option problems and achieved many excellent results, the content involved is the prediction of option prices rather than the search for theoretical prices. This article introduces deep learning into the study of option pricing problems, enriching the research methods of option pricing and expanding the application boundaries of deep learning algorithms.

Key words: option pricing system, volatility, DBM-ANN model, risk neutral pricing

摘要： 要进行期权定价就要准确描述资产价格的波动率。大量的研究表明,市场上资产价格波动率为常数的传统假设已渐渐不适用于现代金融市场计量的发展,而随机模型在实际应用中尚存一些难以克服的困难问题。本文设计的期权定价系统在常数波动率和随机模型之间寻求一种折衷建模方式,模型只对波动率做可预测的假设。首先基于深层玻耳兹曼机提取波动率的影响因子建立波动率DBM-ANN模型,基此利用随机微分和鞅方法在风险中性条件下得到期权定价公式。该系统无需假定波动率的分布形式,通过资产价格确定模型参数,一定程度上克服了传统人工设计波动率模型形式,参数只能使用期权市场价格进行估计的不足。计算实验中,数值结果表明系统对波动率运动规律的刻画精度比较理想,并通过对比发现B-S公式对50ETF股票期权的定价往往低于本文期权定价,其程度会随着距离到期日剩余的时间的增加而增加,随着S/K→1时而增加。

关键词: 期权定价系统, 波动率, DBM-ANN模型, 风险中性定价