Markov Chain Monte Carlo Method for Estimating Implied Volatility in Option Pricing

Abstract

Using market covered European call option prices, the Independence Metropolis-Hastings Sampler algorithm for estimating Implied volatility in option pricing was proposed. This algorithm has an acceptance criteria which facilitate accurate approximation of this volatility from an independent path in the Black Scholes Model, from a set of finite data observation from the stock market. Assuming the underlying asset indeed follow the geometric brownian motion, inverted version of the Black Scholes model was used to approximate this Implied Volatility which was not directly seen in the real market: for which the BS model assumes the volatility to be a constant. Moreover, it is demonstrated that, the Implied Volatility from the options market tends to overstate or understate the actual expectation of the market. In addition, a 3-month market Covered European call option data, from 30 different stock companies was acquired from Optionistic.Com, which was used to estimate the Implied volatility. This accurately approximate the actual expectation of the market with low standard errors ranging between 0.0035 to 0.0275.

Quantum Monte Carlo simulations for estimating FOREX markets: a speculative attacks experience

“…where f(X)P(Y|X) represents the likelihood of finding the system in the vicinity of state X, denoted by f(X), multiplied by the conditional probability of the system transitioning from state X to state Y, denoted by P(Y|X) (Ayekple et al, 2018). The detailed balance condition plays a crucial role in maintaining the correct distribution in the algorithm and is employed to determine the acceptance probability of proposed moves in the Metropolis-Hastings algorithm.…”

“…The balance condition ensures that the desired distribution π(•) is the stationary distribution of the Markov chain. In other words, as the Markov chain converges, the distribution of states should approach the desired distribution π(•) (Ayekple et al, 2018). The balance condition is satisfied when the acceptance probability α(X,Y) is defined as α(X,Y) = min{1, π(Y)/π(X)}.…”