پیش بینی توزیع نوسانات ضمنی برای قیمت گذاری گزینه ای

In this paper, we propose a method that predicts a distribution of the implied volatility functions and that provides confidence intervals for the option prices from it. The proposed method, based on a Bayesian approach, employs a Bayesian kernel machine, so-called Gaussian process regression. To verify the performance of the proposed method, we conducted simulations on some model-generated option prices data and real option market data. The simulation results show that the proposed method performs well with practically meaningful option ranges as well as overcomes the problem of containing negative prices in their predicted confidence intervals by the previous works.

Since the appearance of the Black and Scholes model in 1973, many option pricing formulas have been developed to overcome the restrictive assumptions of Black and Scholes models and to give more accurate prices (Lajbcygier, 1999). Most of the methods are focused on a point prediction of option price. Since there is always discrepancies between the option prices predicted by the models and the real option market prices due to some market frictions such as bid-ask spreads, noisy information, and etc., it is desirable to have a predictive distribution of option prices. To give a distribution of option prices, the use of a neural network kernel-based Bayesian method is addressed in Jung, Kim, and Lee (2006). Han and Lee (2008) suggested the uses of mixed kernels to give more accurate ranges of option prices than other neural networks models in Choi et al., 2004, Gencay and Qi, 2001 and Hutchinson et al., 1994 and applied it to pricing one-sided equity-linked warrants (ELWs, only the buy-side is allowed but the sell-side is limited for the investors). However, these previous approaches have serious problems in their predicted confidence intervals (CIs) of option prices. First, some option prices in their ranges can take negative values in the out of the money (OTM) regions. Second, when the moneyness goes to OTM or in-the-money (ITM) regions, the predicted CIs becomes too broad. Finally, the predicted option ranges around at-the-money (ATM) regions becomes too narrow to encompass the high volume of traded options. To overcome such problems, in this paper, we propose a method that estimates option prices with more practically useful range of options. The proposed method consists of two phases: first phase for constructing a predictive distribution of implied volatility functions and second phase for estimating a confidence intervals of option prices using the predicted volatility distributions. The proposed method will, through simulation results, be shown to contain only positive option prices in their predicted confidence intervals as well as provides more tight option ranges for ITM and OTM regions and more broad option ranges near ATM. The paper is structured as follows. In Section 2, we give some preliminary notions and terminologies that are needed for the subsequent sections. In Section 3, we give a short review of Bayesian linear basis function models and present a method to constructing a distribution of implied volatility functions using a recently developed Gaussian processes regression models. In Section 4, we then provide a way to compute a confidence interval (CI) for option prices using the predicted distribution of implied volatilities. We show some simulation results in Section 5 and conclude the paper in Section 6.

In this paper, we proposed a method that gives confidence intervals (CIs) for the option prices through a predicted distribution of the implied volatility functions. The method predicts a distribution of implied volatilities by applying a Gaussian process regression, a class of Bayesian kernel machines, to the data of implied volatilities and construct confidence intervals for the option prices through inverted Black–Sholes formula. To verify the performance of the proposed method, we used call option prices data obtained from the CGMY model and the ELW real market data traded at KRX. Through simulation, it is shown that the proposed method can overcome the drawbacks of the previous works such as negative option prices in the predicted CIs, too broad option CIs for deep OTM and ITM, and too narrow option CIs near ATM. Applications of the methods to other complex derivatives securities as well as extensions of the methods that provides more accurate option ranges to large option data remain to be further investigated.