Pricing Barrier Options with Time-Varying Parameters

Barrier options are path-dependent options whose payoff depends on whether the underlying asset''s price reaches or exceeds a given barrier H. In many numerical methods, the distribution error and the nonlinearity error together make the pricing results converge slowly or even oscillate significantly. This thesis incorporates the time-varying barrier H(t) and volatility sigma(t) into the pricing model to reflect the markets better. However, it would be more difficult to price a barrier option accurately by traditional tree models. First, that the nodes in the tree cannot be made to coincide with the time-varying barrier will magnify the nonlinearity error. Furthermore, working with a time-varying volatility sigma(t) that is consistent with the market would make the tree model uncombined and grow exponentially, unless deliberate efforts are given to modify the traditional tree models.
Amin (1991) proposed a binomial tree model to price vanilla options under time-varying volatility. Its time complexity is O(n^2). Dai and Lyuu (2008, 2010) developed the bino-trinomial tree model (BTT) which reduces the nonlinearity error sharply by adjusting its structure for pricing a wide range of derivatives accurately and efficiently.
In this thesis, we extend Dai and Lyuu’s BTT model (2008, 2010) by combining the method in Amin (1991) to compute accurate estimates of single-barrier option price with a time-varying volatility and an exponential barrier. The proposed pricing model is verified with Monte Carlo simulation, and it achieves accurate results with O(n^2) time. Furthermore, the prices converge smoothly and quickly.