The Stop-Loss Start-Gain Paradox and Option Valuation: A New Decomposition into Intrinsic and Time Value, by Peter P. Carr and Robert A. Jarrow, in The Review of Financial Studies, Volume 3, Issue 3, 1 July 1990, Pages 469–492


In this article a paradox is explained, and I don't quite understand why this is a paradox. I do understand the trading strategy concept, but in order for the strategy to be a 'paradox' at some point we need to assume that the strategy is self-financing. But what indicates that?

In other words: Why does the strategy seem to contradict the usual Black-Scholes replication and pricing?


2 Answers 2


I am one of the two authors of the paper. The continuity in time of the path of the underlying suggests that at every trading time, the strategy is self-financing. In fact, if the underlying random process had continuous sample paths of bounded variation, then the binary trading strategy is actually self-financing. In contrast, when these continuous sample paths have unbounded variation, then the strategy is not self financing, but the failure occurs only around the strike price.

You can think of the binary strategy as a way to convert the hockey stick payoff of an option into the local time of its underlying evaluated at expiry. Roughly speaking this local time is the accumulation of squared underlying price changes experienced around the strike. By integrating across strike from strike A to strike B, the payoff from a portfolio of coterminal options struck within (A,B) can be converted into the quadratic variation of the underlying generated while the underlying is in the stock price interval (A,B). From here, it is fairly easy to synthesize variance swaps. This synthesis was the theoretical basis for the construction of VIX.


The strategy seems to be self financing because the investor's only actions are to buy stock in the market when the stock price increases to the exercise price K, simultaneously borrowing K dollars, or to liquidate the stock in the market when it decreases to K, simultaneously repaying the K dollar loan.

  • $\begingroup$ In plain english (words) and not formulas; Why is this strategy not self financing? $\endgroup$
    – user25295
    Mar 3, 2018 at 14:29
  • 2
    $\begingroup$ Because when the stock price increases to K, you can't buy it quick enough, it is already higher than K. Likewise when it decreases to K, you can't sell it fast enough, it is already lower than K. $\endgroup$
    – dm63
    Mar 10, 2018 at 13:26

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