Continued from part 2.
If you know a little about options already you will probably be aware that their values depend on something called volatility. Volatility is usually not needed to price derivatives that are not options.
Technically volatility is defined as the annualized standard deviation of the return on an asset (in our case Microsoft stock). They are expressed as percentages.
However, it’s easier to think of it intuitively as the amount that the price will swing around in a given period. Stocks with a high level of uncertainty surrounding them will have high volatilities. An example currently might be the stock of small Russian oil companies. Stocks that are relatively stable (e.g. Microsoft) will have lower volatilities.
Why does volatility affect the price of an option? Again this is because our payoff graph is not symmetrical. A stock that has a high volatility is more likely to swing around, and hence more likely to have a very high value or very low value at maturity. A stock with a low volatility is more likely to be close to its current value at maturity.
Now if the stock price at maturity is below our strike price we don’t care if it’s just slightly below or massively below. In both cases we don’t exercise the option and don’t make any money.
But if the share price at maturity is above our strike we really want it to be as far above the strike as is possible, since we make more money the higher the volume is.
So an option with a high volatility is more likely to make us lots of money if the price goes up, but won’t lose us lots of money even if the price goes down hugely.
As a result options with high volatility are more valuable than options with low volatility.
As we will see below both d1 and d2 in the values discussed above depend on volatility.
Finally, note that if I have bought the call I am paying the cash amount in i) above and receiving the value of the stock ii). So we can say that the value, c, of a European call option on a non-dividend paying stock is:
d1 and d2
As mentioned in the introduction the mathematics behind the calculation of the probabilities in the Black-Scholes formula is fairly complex. It turns out that if N() is the cumulative normal function (a statistical operator) then d1 and d2 can be expressed as below. I’ll just present the results without explanation here. Note that whilst these formulas are complicated, you can just plug in the underlying values and get a result: this is what is known as a ‘closed form’ solution.
Here sigma (σ) is volatility, as discussed above.
Actual Derivation of the Formula
This article has attempted to provide an intuitive interpretation of the Black-Scholes formula, without going into the mathematics behind it. Such an interpretation inevitably glosses over some of the details. I have glossed over risk neutrality considerations above, for instance.
In particular the actual derivation of the Black-Scholes formula was not done directly using the intuitive ideas discussed here. I will discuss this in future articles.
John C. Hull. Options, Futures and Other Derivatives (Sixth Edition)