# why many option contract price less than minimum boundary price?

I downloaded data from NSE(National Stock Exchange) website regarding closing price of European Call Option written on Index. From standard textbook, I read that option contract must satisfy $C(t) \geq S(t)e^{-dt} - Ke^{-rt}$ where S(t) is the value of underlying Security, d is the dividend rate, r is risk free rate of interest, k is strike price, and t is time to maturity.

When I calculated minimum price, i found many contract were priced less than their minimum price. Not only these, such contract were evenly distributed across various maturities so not restricted to just near the maturity , say less than one week. Further not only highly deep in the money contract but many contract near the money found below their minimum price.

I read in book, if option traded at below minimum price it would lead to arbitrage opportunity. I want to know why these contracts price below their minimum price?

To calculate Minimum price, I used following methodology : 1) Option contract having volume less than 500 is excluded. So only actively traded option contracts are considered. 2) Risk free rate of interest is proxy by treasury rate.