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.
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.
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.
