Payment Of Dividend In The European Option Pricing With Stochastic Time

In the recent years, with the continuous improvement of people living standards, thepeople’s awareness to investment has been strengthened, which leads the financial market tobecoming more and more active. The pricing theory of option is one of the key problems inthe field of finance research, and many investment strategies avoid risk based on thecombination of option. The financial model derived from the pricing theories of option is alsomore and more, but most of them base on the Black-Scholes pricing formula, just relax theassumptions underlying the Black-Scholes pricing formula. These assumptions are seem toideal in the real financial market, which are not quite match with the actual situation.In the paper we discuss the pricing problem of European option to pay bonuses in thevalid. People buy the stock to require some bonuses regularly in the real life, therefore thedividend policy is particularly important. But the existing dividend policy is not too ideal,either is a fixed proportion of dividends, or is according to the regular dividend. Consideringtwo factors, one is investors, who require more bonuses after buying stock; the other is acompany issuing shares, when the company’s performance begins to rise slowly or decline,the company wants to allocate fewer or no bonuses. Based on this, the paper we propose amore reasonable dividend policy that when the stock price reaches a rated value for the firsttime to pay dividend. The bonus mechanism not only meets investors’ demand of preferringrisk investment, but also makes appropriate adjustments according to the company’sperformance.There has been the literature of the problem of option pricing with payment of dividendat present, we only study to pay dividends in the case of a fixed time. This paper studiesEuropean option pricing problem in the case of a dividend strategy when the underlying assetprice reaches a certain threshold. Because of the randomness of the dynamic characteristics ofthe underlying asset prices, asset prices for the first time reaches a threshold is random, so theproblem of this paper comes down to paying dividends at random time. On the Europeanoption pricing problem we use geometric Brown motion to characterize the dynamic nature ofthe underlying asset price, firstly, we need to derive the Laplace transform of geometricBrown motion by the first passage time. Secondly, we get the probability density function of geometric Brown motion by the first passage time after the inverse Laplace transform. Andthen using the smoothing formula of conditional expectation, we derive the European optionpricing formula of paying bonuses at random time. Finally, the paper respectively discussesthe influence of the parameters on option value by the numerical calculation, where theparameters are price threshold, bonus amount, the riskless interest rate, volatility, time tomaturity and strike price.