# 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?

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).

• Why is this only true for volatility derivatives? – D Stanley Mar 10 '17 at 15:27
• Edited. I forgot the most important word "NOT" in that sentence. Thanks @DStanley – amdopt Mar 10 '17 at 15:29
• OK, so why is it NOT true for VIX options? – D Stanley Mar 10 '17 at 15:29
• @amdopt True, you can't short the spot index, but you can short (sell) VIX futures, which should be what the option price is based on. If market expectation is for the index to go down then it's perfectly reasonable for options to be priced less than the "current" intrinsic value (spot - strike), but not less than the "future" intrinsic (future - strike). It's the nearest future that the option should be compared to, not the spot price. – D Stanley Mar 10 '17 at 18:52
• @DStanley Agreed. However, the confusion that I am trying to alert the OP to is that when you look at a VIX option chain it shows you VIX options that settle and pay out based on VIX cash settlement but have little to do with VIX cash until the actual settlement day. If one were unaware, they may think the option (or even the whole chain) is priced incorrectly when it is not at all. – amdopt Mar 10 '17 at 19:01

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.

• The last caveat is very important @A. Oreo. Nice answer D Stanley – Quantuple Mar 10 '17 at 18:35