Directional Volatility, Drift, and Black–Scholes Option Pricing
Summary
The document considers whether option pricing should distinguish upward from downward volatility. One response frames asymmetric volatility as a choice of a different underlying model, with risk-neutral pricing still used to value options. It suggests exploring GARCH variants and Monte Carlo simulation if a closed-form solution is unavailable.
A second response explains that Black–Scholes can accommodate a nonzero return drift, but the drift does not affect its option value under the model’s assumptions. The key diagnostic is therefore the distribution of log returns: departures from normality may make Black–Scholes less suitable. The discussion gives conceptual guidance rather than a tested pricing comparison, and it does not derive or validate a specific asymmetric-volatility model.
Key ideas
- Asymmetric upward and downward volatility would represent a different price model, not merely a small adjustment to Black–Scholes.
- Risk-neutral pricing can be applied under alternative models, including models explored through simulation.
- Black–Scholes option values do not depend on the return drift when its assumptions hold.
- The distribution of log returns, especially departures from normality, matters when assessing Black–Scholes suitability.
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Full text
# Does it make sense to use upward and downward volatility in option pricing?
# Does it make sense to use upward and downward volatility in option pricing?
Historically stocks have a higher likelihood to increase in price than to fall in price. As such would it make sense to split a stocks volatility measurement into upward and downward components?
For example if the above strategy was to be used in options pricing one would need to re-work the Black-Scholes equation to accommodate the two volatility measurements which could possibly result in more accurate options pricing.
## Answer by SRKX (score 4, accepted)
https://quant.stackexchange.com/a/15638
Well if you think that this model represents reality more accurately than the Black-Scholes assumptions. A lot of people do indeed think so.
But I wouldn't say you're "tweaking" Black-Scholes... you're just assuming another model altogether and you will use risk-neutral pricing to compute the fair value of the option at time $t$, just like BS.
Frankly, I'm not sure you'll get very far trying to get a closed-form solution for asymmetric volatility, but you might have a look at variations of the GARCH model, which you can then simulate using a Monte-Carlo simulation for example.
## Answer by airguru (score 3)
https://quant.stackexchange.com/a/15640
Actually BS model is still applicable in the market where the upwards/downwards move is much more probable than move in the opposite direction. The Black-Scholes price process model has the form:
$\frac{dS}{S} = \mu dt + \sigma dW$
And with significantly non-zero $\mu$ (called drift) it will capture just what you are talking about. Quite surprisingly, the option value does not depend on the drift, as you can see explained in (Question: Why Drifts are not in the Black Scholes Formula). I prefer my explanation using the put/call parity (which I can probably elaborate further in the comments).
You would seek another pricing formula only if the price behavior would be significantly different than the formula above can capture.
Compute log-returns of your prices. The drift $\mu$ is estimated by their mean, and volatility $\sigma$ by their standard deviation. If the returns are predominantly positive, your $\mu$ would turn out positive and it is not a problem for a BS model and not a factor in the option price in case other BS assumptions are valid. The real concern here is the statistical distribution of these log-returns. The more it differs from a normal distribution, the less usable BS model is.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.