| Summary: | 碩士 === 國立臺灣大學 === 數學研究所 === 96 === Barrier options have four types: up-and-in, up-and-out, down-and-in and down-and-out. Barrier options also distinguish between call and put, American and European. In this paper, how to price and discuss the convergence of European barrier options is our goal.
First, we use the reflection principle to discuss the number of paths for the down-and-in and up-and-in options. The price is its discounted expected payoff under the risk-neutral probability measures. Thus, we get the formula for the binomial price of European barrier options. Then we use Uspensky''s method to discuss the convergence of the binomial price to the Black-Scholes price.
The convergence order depends on the strike price. We get the errors are of order $frac{1}{sqrt{n}}$ or $frac{1}{n}$ . But, for some n, the errors are all of order $frac{1}{n}$. Finally, we show the numerical results to verify our conclusions.
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