Arbitrage When a European Call Trades Below Intrinsic Value
Summary
The document explains a no-arbitrage trade when a European call costs less than its intrinsic value, defined here as the underlying price minus the strike. The described position buys the call and shorts the underlying. At expiration, if the underlying is below the strike, the call expires out of the money and the short position can be closed by buying shares more cheaply. If the underlying is at or above the strike, exercising the call covers the short at the strike. The responses show why both expiration cases produce a positive payoff under the stated price inequality.
The argument assumes the call can be purchased and the underlying shorted at the relevant prices, and it highlights dividends and corporate events as potential complications. The answer says the reasoning applies to non-dividend-paying equities and some commodities, but not to every derivative; volatility products are given as an exception. The document does not address practical frictions such as transaction costs, funding, or short-sale constraints.
Key ideas
- When a call is priced below intrinsic value, buying it and shorting the underlying creates the proposed arbitrage position.
- If the underlying finishes below the strike, closing the short at the lower market price produces a gain under the stated inequality.
- If the underlying finishes at or above the strike, exercising the call covers the short at the strike.
- Dividends, corporate events, and market frictions can affect whether the trade is available in practice.
- The reasoning does not apply universally to all derivatives, including volatility products.
Tags
Full text
# How to make the arbitrage if intrinsic value is greater than European call value
# How to make the arbitrage if intrinsic value is greater than European call value
It always says if the intrinsic value is greater than European call value, there will be a arbitrage opportunity，but how to construct the portfolio $(S_t - K)^+$ or how to make this arbitrage.
By the way, is it true for every derivatives?
## Answer by amdopt (score 4, accepted)
https://quant.stackexchange.com/a/32939
> how to construct the portfolio (St−K)+ or how to make this arbitrage
If you have this scenario on your hands then you construct the portfolio by putting as much capital as you can into the trade. It's an all reward and no risk scenario. Max it out! You "make" the arb by buying the call, shorting the equivalent amount the underlying at the current price and wait for expiry to realize the profit. Beware of a pending dividend or other corporate events though!
> By the way, is it true for every derivatives?
No. This will not be true for Volatility derivatives (VIX futures and options).
## Answer by D Stanley (score 4)
https://quant.stackexchange.com/a/32938
The intrinsic value of a call is the price of the underlying minus the strike (`S0-K`), so if you find a european call whose value is less that that you would:
- Sell (or short) the underlying at `S0`
- Use the proceeds to buy the call at `C`
and wait. At maturity, the price of the underlying is `Sm`, and you will make a profit in either case:
If `Sm < K`, the call is out of the money, buy you would buy Sm (to close out your short position)
profit = `S0 - Sm - C`, which is > 0 since
```
S0 - Sm - C > S0 - Sm - (S0 - K) since C < S0 - K
> -Sm + K
> 0 since Sm < K , K - Sm > 0
```
If `Sm >= K`, the call is in the money, and you exercise the call (which closes out your short position.
profit = `S0 - K - C`, which is > 0 since `C < S0 - K`.
> By the way, is it true for every derivatives?
This does not take borrowing costs (since you do not need to borrow money in this scenario) or dividends into account, but it will work on non-dividend paying equities, commodities, etc.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.