Implied Volatility under Alternative Return Distributions
Summary
The answer argues that implied volatility need not be extracted only from the Black–Scholes lognormal model. It describes constructing option valuation formulas under alternative distributions, including log-hyperbolic-secant, log-Cauchy, log-t, and generalized-normal forms. With distribution shape parameters fixed or linked to the risk-free rate, the author says these setups can reduce calibration to one unknown and may be inverted analytically. The resulting implied-volatility shapes can differ substantially from the Black–Scholes smile. The post suggests comparing model-based volatility estimates for possible relative-value or statistical-arbitrage signals. It reports that some approaches appeared tradeable to the author, but provides no data, validation method, or risk analysis supporting those claims. It also acknowledges uncertainty about academic validity. Alternative distributions therefore represent modeling choices whose calibration, arbitrage consistency, and out-of-sample performance require independent scrutiny; the discussion is an anecdotal outline, not evidence that any strategy reliably earns returns.
Key ideas
- Implied volatility can be defined by inverting option prices under distributions other than the lognormal Black–Scholes assumption.
- Fixing distribution shape parameters can reduce calibration to a single unknown in the examples described.
- Alternative distribution assumptions can produce volatility shapes that differ from Black–Scholes.
- Comparisons between models may suggest relative-value signals, but the post supplies no empirical validation.
- Model validity, arbitrage consistency, and out-of-sample performance remain important caveats.
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Full text
# Are there methods of calculating Implied Volatility in the stock market, other than Black-Scholes? # Are there methods of calculating Implied Volatility in the stock market, other than Black-Scholes? (I've gone through many questions/posts on quant StackExchange and only find responses about Black-Scholes) ## Answer by uday (score 1) https://quant.stackexchange.com/a/41663 Yes, there is. Black Scholes is the formula for log-normal distribution. Normal distribution has two parameters - the drift and the volatility. Under options theory (no arbitrage etc.), the drift is the risk free rate, and hence “known” per se. Thus, you pretty much end up with one equation / one variable to solve for. But you can extent it to anything. In a previous firm, we derived IV using options formulas under - log-hyperbolic secant distribution (2-parameter, of which in the option formula one will replace the “equivalent” drift by relating it to risk-free rate) - log-Cauchy distribution(where the Cauchy distribution had a scale and location, i.e. 2-parameters ) - log-t distribution with fixed degree-of-freedom like 3, 5 etc (reduced 3-parameters to a 2-parameters by fixing the “shape” / df parameter) - log generalized normal by fixing the shape of 3 parameters (and replacing the “equivalent” drift by relating it to risk-free rate) E.t.c. Not sure how many of these would be academically valid, but unlike the stochastic volatility or jump diffusion models that have many more parameters, all these models above have just 1 variable to solve for - and can be analytically inverted. Some of these gave very interesting and dramatically different shapes versus the log-normal (Black Scholes) implied IV. If you think outside the box of standard academic arguments , and consider it as a more “statistical arbitrage” problem, it is really one equation / one variable. Do whatever you wish to do in terms of the equation you want to solve. Some of the solutions are very tradeable (high Sharpe ratios ), when BS shows a high IV between two points, while some other models shows a low IV between two points etc.
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