Price Limit、Asymmetric Truncated Levy Distribution And Option Pricing

Since the reform and opening up of China, with the continue innovation of modernfinancial markets, The application of the financial derivatives including options in China getgreat development. Under this background, the ability to reasonably price these options isbecoming particularly important. In the field of option pricing, Black and Scholes got fullyapproval of proposing option pricing formula in1973. However, the Black-Scholes model hasmade certain assumptions that may not hold well in some practical situations, in which case,their pricing options may produce systematic mispricing. Thus, both for their theoretical valueand for their practical significance in managing financial risks, studies the higher consistencyformula with the actual market is particularly important.It is worth to mention that stock return in market of price limit has a scope limitation andasymmetric in the limit time. So we do the work as follow:In the beginning, we use the asymmetric stable truncated distribution to describe thechanges of stock price. Assuming the stock priceS tsatisfy the following formula:bandS0are constants;S0is expressed as the initial stockprice;S tis expressed as the stock price in time t; The probability density function of processwhere γ>0,0<α≤2.Under the hypothesis of risk neutral market, for not trading assets,we deduces the closed solution of the European option pricing formula in two phase asfollow: Secondly, the new option pricing formula is compared with Black-Scholes formula innumerical part and analysised the parameter sensitivity. Finally, we use the new pricingformula to explain the phenomenon of volatility smile in Black-Scholes option pricingformula. The conclusion of this paper is that the reason of implied volatility smile inBlack-Scholes formula which get from empirical analysis is the assumption that the change ofstock price satisfy geometric Brown motion.