Early Exercise and Dividend Effects in American Call Pricing
Summary
The discussion compares American and European option values computed with an implicit finite-difference method. The questioner reports that the two call values are nearly equal and both sit below the Black–Scholes value, despite expecting the European values to agree and the American call to be worth more. Changing grid size did little to resolve the discrepancy. For puts, the finite-difference results follow the expected ordering, with American puts valued above European puts.
The accepted explanation is that, under the usual assumptions of no dividends, early exercise of an American call is not advantageous: it gives up remaining time value, so American and European calls have the same value. Dividends can change that incentive, particularly around an ex-dividend date, and may make early exercise worthwhile. The post also notes that zero interest rates remove the early-exercise value of American puts. These are conceptual explanations rather than a formal proof or a diagnosis of the reported pricing gap; the document does not specify enough model inputs or implementation details to establish why the numerical call values differ from Black–Scholes.
Key ideas
- Without dividends, early exercise does not increase the value of an American call on a non-dividend-paying stock.
- American and European calls therefore have equal values under the stated assumptions.
- Dividends may make early exercise worthwhile by changing the value of holding the stock through an ex-dividend date.
- American puts can carry early-exercise value, though the discussion says it disappears when interest rates are zero.
- The explanation does not establish the cause of the questioner's finite-difference pricing discrepancy.
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Full text
# Value of American Call vs Value of European Call when using implicit finite differences # Value of American Call vs Value of European Call when using implicit finite differences I calculated values for put options (european and american) using the implicit finite difference method and compared them to black/scholes values. The values for american put options are higher than the values for european put options (BS and finite differences), which is in line with theory. I applied the same code to call options, setting the lower boundary to 0 and the upper boundary to S(max)-K. The values using finite differences for american and european options are roughly the same. The BS Value lies considerably above both values. I expected the values of the european options to be almost the same and the american option value to exceed both. My code is more or less based on the following source (for put options, for call options i simply changed the payoffs to max(S-K,0) and changed the boundary conditions (see above). http://www.quantcode.com/modules/mydownloads/singlefile.php?lid=248 Does anybody have an idea? I changed the grid size, which didn't change much. Thank you in advance! ## Answer by FKaria (score 3, accepted) https://quant.stackexchange.com/a/10173 I guess that, in your model, the stock does not pay dividends. The price of an European Call option written for a stock that does not pay dividends is always higher than its intrinsic value. Therefore, in that case, Prices of European and American Call options are equal. Note that this is not true for Put options, since Put values are short interest rates (Calls are long interest rates). If interest rates are zero, American and European Puts have the same price. Note that you are always better waiting until maturity to exercise the option, if the stock does not pay dividends, since otherwise you will lose the time value. When a stock pays dividends it might be better to exercise the day before the stock goes ex-dividend, because the drop in price of the stock may not compensate for the time value. This is an idea of what is going on, by the way, not a formal proof. Edit: Clarified that, assuming no dividends, an European and American Call options have the same price. Assuming no interest rates ($r=0$) the same happens for Puts. ## Answer by FreshF (score 0) https://quant.stackexchange.com/a/10163 A Log-Transform did not help - I guess it has to do with dividends. When a stock does not pay dividends a risk-neutral investor should have no incentive to exercise a call option before expiration. Correct me if I'm wrong. Including a dividend rate (which diminished the risk free rate) should give incentive to exercise early and therefore make an american call option more worth than an european call option
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