The Mathematics of Stock Options

 

You’ll remember that in earlier posts I've discussed a hypothetical widget-making company, XYZ, and said that we could enter various reasonable assumptions about this company’s stock in a calculator available in any of several websites, and receive for our troubles the value of a proposed put or call.

How does the calculator do that?  A skeptic may sneer at the whole idea and say, “only supply and demand determine the value of a financial instrument. The demand for an option on a particular stock may change from day to day – as may the supply, as new writers enter or leave the market – so surely no formula or algorithm can tell us in advance what the value is.”

That’s a plausible response, but it is in error. The value of stock options can be defined mathematically in ways that the value of the underlying stocks cannot. After all, we know intuitively what the value of a stock option is on the expiration date. If it has expired worthless, we know the value is $0. If it hasn’t, we know its value by simple arithmetic.  The value of an instrument that allows me to buy a $45 asset for only $40 is … $5. Valuation of the stock on dates prior to expiration is a bit more complicated, but in its essence no more problematic.

Think of a stock option, if it helps, as a lottery ticket. If the ticket wins, its value on that date is known. If it loses, its value is also known. On dates before the drawing, its value can generally be defined as a fraction of the prize money defined by the number of tickets outstanding. Your one ticket (out of a million) for a $500,000 prize is, on this simple application of arithmetic, worth  $0.50.

It isn’t difficult to make the arithmetic just a little more complicated. Suppose you’ve bought a lottery ticket for a drawing that won’t take place for weeks or months yet. This ticket is worth something less than half a dollar, because neither you nor any of the other contestants will receive that prize until the drawing actually takes place. If you’ve purchased that ticket for $0.50, thinking you’ve made an “even money” bet, you have in fact made an interest-free loan of that amount to the sponsor of the event.  

If we want to determine the true value of the ticket, then, we’ll discount for the rate of interest that you could have received on that money in the interim. If you have perfect confidence that the sponsor of the lottery won’t default between the date of the purchase of the ticket and the date of the drawing, then the discount rate is the risk free rate of interest, which as I've mentioned in earlier posts in this blog is called r [US T bills are often taken as a proxy].

Sometimes much the same function is served by including in the equation the expression B (t,T), which refers to the value of a bond at time t, when that bond will mature at time T.

What about our skeptic’s reference to supply and demand? If there is a secondary market in our lottery tickets, then we can be pretty sure the value of the tickets will stay close to the value determined in the way we’ve described in the last three paragraphs. The more liquid and transparent the market is, the more certain of that we can be, and the smaller will be any deviations from the arithmetically right price. Supply and demand will behave. If they don’t, there will develop a cadre of arbitrageurs willing to assist them. 

But we still don’t have much of a fix on how the algorithms work that tell us the value of stock options. They work on the basis of the Black-Scholes-Merton theorem, a model developed by those three named scholars [Fischer Black, Myron Scholes, and Robert Merton] in 1973, just as the first great options exchange, the CBOE, was getting itself started.

They said that the value of a call for a non-dividend-paying underlying stock, mathematically expressed as C(S,t) equals N(d₁)S – N(d₂)Ke^-r(T-t) .

By this time in our inquiry we know enough about options, and about risk, to understand everything in that equation.

N(d₁)S refers to the extent to which the stock price will be in the money on the day of the expiration of the (European) option, if it finishes in the money at all. To revert to a lottery drawing: N(d₁)S is the amount of money that a particular ticket holder will win if he wins. This is called N because for purposes of this model it is determined by the normal distribution (that is, we assume that the bell curve we’ve discussed illustrates the way in which stock prices move in determining the extent of the prize).

N(d₂) refers to the probability that the ticket will win, that the stock price will finish in the money.

K is the strike price, by which in-the-money is defined.

This leaves us with e raised to a certain power. We discussed e in our first chapter: it is the base of the natural logarithm, and it is here – as often in other contexts – because we are modeling a continuous process. When we had our money invested in stock options, we could have had it in other possible investments, and we could have seen it continuously compounding there.

The power to which e is raised here is –r(T-t). The expression (T-t) just refers to the passage of time between the purchase of an option (or its valuation) and its exercise or expiration. We must calculate the risk free rate of interest r for that span of time.    

The corresponding put option formula is as follows:

P(S,t)=N(-d₂)Ke^-r(T-t) –N(-d₁)S.

There all the expressions have the same meaning we’ve explained above.

It isn’t difficult to have a computer program run through this formula given the necessary data, particularly including volatility (which is critical to the calculation of d₁ and d₂.)