This paper considers the pricing model of options under illiquidity. According to the model proposed by Frey and Patie (2001), we explore a new numerical approach for solving the nonlinear PDE rather than the Newton method. Furthermore, we employ the Thomas algorithm for solving the tridiagonal system and propose the pattern search algorithm for getting the liquidity parameter, respectively. Therefore, the calculating process is not time-wasting by two algorithms. After a preliminary numerical study of the model, we apply it to stock call option prices for the sample period from January 1, 2000 to December 31, 2004. We demonstrate that the pricing error results from market illiquidity (the bear market) in the first part of empirical study. In the second part of empirical study, we enlarge the sample period and check the practicability of the Frey model for various companies which is listed on CBOE.
The Frey model, for the most of sample, not only exhibits good outcomes regardless the length of the sample period but also presents excellent performance in illiquid market. The Frey model really represents a vital improvement with respect to the BS model in terms of pricing error and it provides a reasonable option pricing model for the pricing of a block order in terms of price impact. We argue that the serious pricing biases of the BS model can be explained by the nonlinear feedback effect and thus if the large trader uses the Frey model rather than the BS model, they could avoid unnecessary loss from the stock option market where illiquidity occurred.
In further research, the singularity separating method (SSM)31 can be applied to the option pricing. Since the SSM is adopted, precise numerical solution can be obtained very quickly. The SSM method is proposed by You-Lan Zhu who improves
31 Some of article says singularity removing transformation (SRT) method but they offer the same concept regarding the numerical method.
the terminal condition of the option32. There is a singularity point in the terminal payoff as the stock price equals to the exercise price and thus numerical solution will have a bad accuracy and reduced convergence rate around the singularity point even though the numerical solution will become smooth finally. However, we do not use the SSM method into our numerical scheme. If we want to get more rigorous solution of the PDE, we should adopt the SSM in our frameworks.
Recently, the field of the computational finance grows up quickly. We believe that the Frey model can be calculated by other numerical schemes but until now we consider that the new approach of mine is the fastest way for solving the Frey model.
Accordingly, the large trader who uses the Frey model obtaining the more accurate theoretical price and the fastest way in a short term period so that they make the right strategy immediately and establish the optimal position in the market.
32 See Zhu, Wu and Chern (2004)
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