Investigating Apparent Violations of European Call Price Bounds

Summary

The document raises a practical question about reported European index call prices that appear below a textbook lower bound. The bound compares the call price with the present value of the underlying, adjusted for dividends, less the discounted strike. The author says the apparent violations occur across maturities and moneyness levels, including actively traded contracts after filtering out options with low volume, and uses a Treasury rate as a proxy for the risk-free rate.

The text contains no answer or diagnosis, so it does not establish that arbitrage was available. Its observations motivate checking inputs and conventions: the underlying and option timestamps, dividend assumptions, rate tenor, maturity and settlement details, and whether the bound is being applied to matching prices. The volume filter and Treasury proxy are stated choices, but no transaction costs, bid-ask spreads, data quality checks, or arbitrage execution are assessed. The reported pattern therefore remains an unresolved empirical observation rather than evidence that the theoretical bound fails.

Key ideas

• A European call has a lower bound based on the dividend-adjusted underlying and discounted strike.
• The author reports apparent below-bound prices across maturities and moneyness levels.
• A Treasury yield is used as a risk-free-rate proxy, and low-volume contracts are filtered out.
• The document provides no resolution, and the observations alone do not demonstrate executable arbitrage.

Tags

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


# 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.

Your answer will be appreciated. Thanks.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.