A General Framework for Pricing Asian Options Under Markov Processes
A paper by Ning Cai (The Hong Kong University of Science and Technology), Steven Kou (National University of Singapore), and Yingda Song (University of Science and Technology of China)
RMI's Director Prof. Steven Kou, who is also the Provost's Chair Professor of Mathematics at NUS, wrote a paper with his coauthors Prof. Ning Cai and Dr. Yingda Song titled, "A General Framework for Pricing Asian Options under Markov Processes." The paper is set to appear in the Operations Research in 2015.
Asian options, whose payoffs are contingent on the arithmetic average of the underlying asset prices over a prespecified period, are among the most popular pathdependent options that are actively traded in the financial markets. The average of the underlying asset prices can be computed either discretely, for which the average is taken over the asset prices at discrete monitoring time points, or continuously, for which the average is calculated via the integration of asset prices over the monitoring time period. The valuation of Asian options is challenging since the arithmetic average usually does not have a simple distribution.
The existing literature mostly focuses on only one type of Asian option (discrete or continuous) and specific Markov processes, e.g. the geometric Brownian motion. However, this research paper provides a unified framework for pricing both discretely and continuously monitored Asian options under general onedimensional Markov process models.
More precisely, the contribution of the paper is threefold. First, under general onedimensional Markov processes, they derive the double transforms of the Asian option prices, either discretely or continuously monitored, in terms of the unique bounded solutions to related functional equations. Second, in the special case of continuoustime Markov chain (CTMC), they demonstrate that the functional equations reduce to linear systems that can be solved analytically via matrix inverses. Lastly, by constructing a CTMC to approximate the targeted Markov process first and then numerically inverting the double transforms related to the constructed CTMC, they show that the Asian option prices can be computed under general one dimensional Markov processes models.
Pricing Asian options under a CTMC is quite different from pricing other pathdependent options. For example, to price barrier options, one needs to first study passage times under a CTMC, which consists of analytical solutions in the form of a matrix, obtained by deleting some rows and columns in the transition rate matrix of the CTMC. Unfortunately, it is not easy to find such simple matrix operations for Asian options. This difficulty is overcome by working on the double transform of the continuously monitored (resp. discretely monitored) Asian option price with respect to the strike price and the maturity (resp. the number of monitoring time points). Then it can be shown that the double transforms are the unique bounded solutions to related functional equations, which can be solved analytically in the case of CTMCs, using the strictly diagonally dominant matrix and the LevyDesplanques theorem.
Numerical results demonstrate that the method proposed in this paper is accurate and fast under popular Markov process models.
