Option Pricing Models
Mathematical frameworks used to compute the theoretical fair value of an option contract, with the Black-Scholes-Merton model and the Binomial model being the two most widely referenced approaches in Indian derivatives markets.
The challenge of options pricing is that the value of an option depends on uncertain future events — specifically, where the underlying price will be at expiry. Option pricing models provide a principled way to assign a theoretical value to this uncertain payoff, given a set of observable inputs: the current price of the underlying, the strike price, time to expiry, the risk-free interest rate, and the volatility of the underlying.
The Black-Scholes-Merton (BSM) model, developed in 1973 by Fischer Black, Myron Scholes, and Robert Merton, was the foundation on which modern options markets were built. The model assumed that the underlying price follows a log-normal distribution with constant volatility, that markets are frictionless, that there are no dividends (or dividends are known and certain), and that risk-free borrowing and lending are available at a constant rate. Under these assumptions, BSM derived a closed-form equation for the price of a European call or put option.
In India, NSE published theoretical option prices based on BSM calculations, which traders used as a reference. The model's most critical input — and its most controversial assumption — was volatility. BSM took a single volatility number for the entire term structure, but in practice, different strikes traded at different implied volatilities (the volatility skew). This limitation made BSM a pricing benchmark rather than a perfect predictor.
The Binomial model, introduced by Cox, Ross, and Rubinstein in 1979, worked differently. It modelled the possible paths of the underlying price as a tree of discrete up-and-down moves over multiple time steps. At each node, the probability of an up move and a down move was calibrated to match the assumed volatility. Working backward from the terminal payoff at expiry, the model computed the option value at each prior node through risk-neutral discounting. The binomial model was particularly useful for pricing American-style options and for visualising how option value changed as expiry approached.
For Indian market participants, BSM remained the primary analytical tool. Broking platforms, option chain tools, and implied volatility calculators published by NSE all used BSM inputs and outputs. Understanding the model's mechanics helped traders interpret the Greeks — delta, gamma, theta, vega — which were the partial derivatives of the BSM pricing formula with respect to each input.