Stop-Loss Start-Gain Strategies, Self-Financing, and Quadratic Variation
Summary
The note examines why a binary stop-loss/start-gain trading strategy can appear to conflict with standard option replication and pricing. The explanation distinguishes continuous underlying paths with bounded variation, where the strategy is self-financing, from continuous paths with unbounded variation, where self-financing fails around the strike. This qualification resolves part of the apparent paradox: continuity alone does not ensure that the trading argument works in every path setting.
The strategy is also interpreted as converting an option’s hockey-stick payoff into local time at expiry, roughly associated with accumulated squared price changes near the strike. Integrating across a range of strikes links a portfolio of coterminal options to quadratic variation over the corresponding price interval, providing a theoretical route to synthesizing variance swaps and, ultimately, the VIX. A second explanation describes buying stock and borrowing at the strike on upward crossings, then liquidating and repaying on downward crossings. The note gives a conceptual account, not a full proof or practical trading specification.
Key ideas
- A continuous underlying with bounded variation supports the binary strategy’s self-financing property.
- With unbounded variation, self-financing fails around the option strike.
- The binary strategy relates an option payoff to the underlying’s local time at expiry.
- Integrating option payoffs across strikes connects the strategy to quadratic variation and variance-swap synthesis.
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Full text
# Stop-loss start-gain paradox: Why is it a 'paradox'? # Stop-loss start-gain paradox: Why is it a 'paradox'? 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 http://engineering.nyu.edu/files/slsg.pdf 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? ## Answer by peter carr (score 15) https://quant.stackexchange.com/a/38586 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. ## Answer by dm63 (score 4) https://quant.stackexchange.com/a/38584 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.
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