Pricing Barrier Option Under A Log Student’s T-distribution With Jumps

With the vigorous development of the global financial markets, in order to satisfy thespecial requirements of the investors, a lot of exotic options have be born, such as Barrieroptions, Lookback options, Asian options and so on. These exotic options are more complexand profitable than vanilla options, and theirs trading numbers and transaction amounts arevery large, many financial institutions continue to innovate new exotic options. So how topricing these options has become a key topic in modern financial engineering. In this paperwe mainly study the pricing problem of barrier options. We know that the classicalBlack-Scholes model assumes that the underlying asset price obeys geometric Brown motion,however, a large number of empirical studies have shown that: in the actual market, thedistribution with a peak occurred fat tail phenomenon, and on the impact of some unexpectedevents the price generated jumps. So our article is using Student’s t-distribution instead of thenormal distribution, to pricing barrier option under a log Student’s t-distribution with jumps.In this paper, we first review the research on option pricing systematically, thenintroduce the behavior of financial knowledge briefly. Followed by the introduction of theclassical Black-Scholes model, we introduce the pricing of barrier option under the situationof continuous and discontinuous. Secondly, we introduce the pricing of European option andbarrier option under a log Student’s t-distribution. The last part is the main results of thispaper: Combinating with behavioral finance knowledge,we obtain the analytical solution ofthe barrier option pricing formula under a log Student’s t-distribution with jumps byconditional delta avoid strategy; We use the minimal mean-square-error avoid strategy to getthe market price of the barrier option under incomplete information., and we discover that theoption prices are mean reversion, which is in accord with the point of behavioral finance; Wealso propose a procedure to estimate the volatility parameterσ—using value at risk VAR toestimate volatility parameters; In addition, we compare the implied volatility between B-Smodel and Student’s t-distribution model, and find the implied volatility curve underStudent’s t-distribution model is more smooth.