Abstract: Option pricing is one of the core issues in financial engineering research. The value of an option depends on the price of the underlying asset during the period of validity. Therefore, it is the basis and key of option pricing to properly construct the underlying asset price dynamic model. The mainstream option pricing model mainly describes the uncertainty of underlying asset price by stochastic differential equations.However, the financial system is highly complex. Owing to market fluctuations, human error, insufficient information and other reasons, the uncertainty cannot be described only by randomness. For instance, if the stock price is around 50 yuan, it is difficult to model the phrase "around 50" from the perspective of probability theory, obviously it has a fuzzy feature. The uncertainty in financial markets involves at least two aspects, namely, fuzziness and randomness, and these two aspects cannot be substituted. Randomness can be modeled by the stochastic analysis theory, and fuzziness can be modeled by the fuzzy set theory. In addition, the recent behavioral financial studies have shown that decision maker's subjective judgment, such as sentiment, preference and expectation, have a certain impact on decision-making.
Based the above analysis, a fuzzy stochastic process model is constructed for underlying stock price by fuzzy random variables, and the stochastic analysis and fuzzy set theory are used to study the pricing problem of continuous geometric average Asian call options under uncertain environment with fuzziness and randomness taking account of investor's subjective judgment. An Asian option is a popular financial derivative, where the payoff is based on the average underlying asset price over some pre-set period prior to maturity. The payoff structure of Asian options makes them less vulnerable to manipulation.
Firstly, the arbitrary level cut sets of Asian option fuzzy price is deduced according to the definition of expectations of fuzzy random variables, and the problem of determining the confidence degree for any given reference price is converted into solving optimization problem.
Secondly, the Asian option pricing problem is studied considering two kinds of decision-maker's subjective judgment. One is to introduce a fuzzy goal to represent the degree of decision maker's satisfaction for expected option price, and present the permissible range of rational expected Asian option prices such that the degree of reliability of the expected price is greater than the degree of decision maker's satisfaction. The other is to introduce pessimism-optimism index to represent the decision maker's pessimistic degree, and derive the Asian option pricing formula in the sense of possibility mean values defined based on weighing function and possibility evaluation measure.
Finally, some numerical examples are presented to illustrate the feasibility and practicality of the models. The results show that fuzzy stochastic modeling has good reliability and effectiveness. The permissible range of rational expected prices depends on the fuzzy goal selected by the decision-maker. The weighted possibility mean value of Asian option price is an increasing function of fuzzy factor, and is a decreasing function of pessimism-optimism index.
The options pricing model considering decision maker's subjective judgment under uncertain environment with fuzziness and randomness increases the flexibility of investment decision and has more practical significance. This paper makes up for the deficiency of the existing literature, and draws some valuable conclusions. It provides a new idea for the pricing of options and many other financial derivatives, risk management and investment decision under uncertain environment.

Key words: Asian option pricing, uncertainty with fuzziness and randomness, fuzzy goal, pessimism-optimism index, possibility mean value