Generalised probability distributions for the observed states under Hidden Markov Models to forecast stock prices

This study focuses on developing a Hidden Markov Model (HMM) for the daily closing prices of the National Stock Exchange (NSE) and ICICI Bank. The fluctuating states of the NSE are treated as hidden states, while the corresponding changes observed in ICICI Bank’s closing prices are considered visible (emission) states influenced by the hidden dynamics of the NSE. The parameters of the HMM–namely the Initial Probability Vector (IPV), Transition Probability Matrix (TPM), and Observed Probability Matrix (OPM)–are estimated under the assumption of discrete Markov chains operating (i) within the hidden states (Gain and Loss in NSE) and (ii) between the hidden and observed states (Rise and Fall in ICICI Bank). Separate probability distributions are formulated for sequences of ( n )-day transitions corresponding to the Rise and Fall states. The behaviour of the Rise and Fall states in ICICI Bank’s closing prices is further analysed using explicit mathematical expressions for key statistical measures, along with the associated Pearson’s correlation coefficients. Numerical illustrations are provided to enhance clarity and facilitate understanding of the proposed models. These models are intended to assist short-term investors by offering probabilistic indicators for buying and selling decisions based on the likelihood of emission states. With computer-based automation, the proposed framework can serve as a practical and user-friendly analytical tool for interpreting day-to-day stock market movements.