On the Construction and Complexity of Bivariate Lattices

• Main Authors: Chuan-Ju Wang, 王釧茹
• Other Authors: 呂育道
• Format: Others
• Language: en_US
• Published: 2011
• Online Access: http://ndltd.ncl.edu.tw/handle/66706022118681442778

博士 === 國立臺灣大學 === 資訊工程學研究所 === 99 === Derivatives are popular financial instruments whose values depend on other more fundamental financial assets (called the underlying assets). As they play essential roles in financial markets, evaluating them efficiently and accurately is critical. In 1973, Black and Scholes arrived at their ground-breaking analytical pricing formula, which assumes that the underlying asset follows a lognormal diffusion process with a constant risk-free interest rate and a constant volatility of the underlying asset. However, the lognormal diffusion process, which has been widely used to model the underlying asset''s price dynamics, does not capture the empirical findings satisfactorily. Therefore, many alternative processes have been proposed, and a very popular one is the jump-diffusion process. Additionally, since interest rates do not stay constant in the real world, many stochastic interest rate models are put forward. Most derivatives have no analytical formulas once one goes beyond the most basic setup; therefore, they must be priced by numerical methods like lattices. A lattice divides the time interval between the derivative''s initial date and the maturity date into $n$ equal time steps. The pricing results converge to the theoretical values when the number of time steps increases. Unfortunately, the nonlinearity error introduced by the nonlinearity of the value function may cause the pricing results to converge slowly or even oscillate significantly. This dissertation first proposes an accurate and efficient lattice for the jump-diffusion process. The proposed lattice is accurate because its structure can suit the derivatives’ specifications so that the pricing results converge smoothly. To our knowledge, no other lattices for the jump-diffusion process have successfully solved the oscillation problem. In addition, the time complexity of our lattice is lower than those of existing lattice methods by at least half an order. Numerous numerical calculations confirm the superior performance of our lattice to existing methods in terms of accuracy, speed, and generality. As for the stochastic interest models, the second part of this dissertation shows that, when the interest rate models allow rates to grow without bounds in magnitude, previous work on the lattice methods shares a fundamental flaw: invalid transition probabilities. As the overwhelming majority of stochastic interest rate models share this property, a solution to the problem becomes important. This thesis presents the first bivariate lattice that guarantees valid probabilities even when interest rates can go without bounds. Also, we prove that the proposed bivariate lattice grows (super)polynomially in size if the interest rate model allows rates to grow (super)polynomially. Finally, we show the optimality of our bivariate lattice.