# 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.

## 2 Answers

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.

• APARCH and NGARCH have asymmetric volatilities for example. – not.so.quanty Nov 26 '14 at 14:49

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.