Call Option Intrinsic Value, Arbitrage Bounds, and Dividends
Summary
The document explains why an in-the-money call generally cannot trade below the underlying stock price minus the strike, using a no-arbitrage argument. If it did, an investor could sell the stock, buy the underpriced call, and invest the strike amount. The resulting position would retain a floor while producing an initial cash gain, creating an arbitrage opportunity.
The explanation has important qualifications. The simple bound applies when dividends are absent; expected dividends reduce the call’s value and enter put-call parity. It also distinguishes American calls, which can be exercised before expiration and therefore allow the underlying to be acquired indirectly, from European calls, which can only be exercised at maturity. A stock with a large dividend before expiration can make a European call worth less than the current stock price minus strike. The argument is a pricing principle, not a claim that every call always exceeds that difference under all contract and dividend conditions.
Key ideas
- A call’s price is supported by no-arbitrage bounds relative to the underlying price and strike.
- If an in-the-money call is priced below intrinsic value, a stock-sale and option-purchase strategy can create an arbitrage.
- Expected dividends reduce call value and affect the relevant put-call parity relationship.
- American exercise rights support an immediate-exercise bound, while European calls may fall below the simple bound when dividends are substantial.
Tags
Full text
# "The potential gain of a Call Option is always incorporated in the Option's price" - Why is that? # "The potential gain of a Call Option is always incorporated in the Option's price" - Why is that? I've heard this but I don't understand why. The demonstration of this is that the Ask Price of a Call Option is always higher than the difference between the Strike Price and the price of underlying stock. Why is this the case? And why should it be the case? ## Answer by Bob Jansen (score 3, accepted) https://quant.stackexchange.com/a/55459 When the call is at or out of the money, the result is obvious: the call will have some value but the difference between stock price and strike is nonpositive. Consider the case that the option is in the money and its current price $C$ is lower than the difference between stock price $S$ and the strike $K$, in symbols: $C < S-K$. If I own the stock, I could sell it to buy the option and lent out $K$ until expiration, at the end of the period I either get $K$ if the option is worth nothing or $S$ is the option is in the money. So by selling the stock for the option and lending out the difference: I've put a floor on the value of my position and pocketed $S - K - C$. This would be a free lunch. As @BabaYaga points out, the statement and my answer above only hold in the absence of dividends. The statement can be fixed by including the dividend in the difference. Put-call parity before expiration is $$C - P = S - K - D$$ where $P$ is the price of a put with the same strike and $D$ the dividend expected between now and maturity. The value of the call is reduced by the dividends. If the dividends are sufficiently large, the original statement would be false. ## Answer by BabaYaga (score 4) https://quant.stackexchange.com/a/55465 > the Ask Price of a Call Option is always higher than the difference between the Strike Price and the price of underlying stock [more precisely, the price of the underlying minus the strike]. This is definitely true for American call options, that can be exercised immediately (as well as all the way to maturity): roughly speaking, you can "indirectly" purchase the underlying asset by purchasing an american call and paying the strike price to exercise the option-- thus the price of the asset can't exceed the price of the option plus its strike price, or there would be significant arbitrage opportunities. I'd say it is not necessarily true for European calls, that can be exercised only at maturity. For example, consider a very, very stable utility stock, that is currently priced at 1000 EUR, and will pay a 50 EUR dividend in 4 days from now; and a European call option on the stock with a strike of 1 EUR maturing in a week from now. The call option price is then very close to 950 EUR, so it's still quite a bit below the current stock price (1000 EUR) minus the strike (1 EUR), violating the statement above.
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