What is the Black Scholes pricing model for?

What is the Black Scholes pricing model for? This is a very simple question that deserves a simple answer.

The Black Scholes pricing model is an equation that expresses the price of a European call or put option in terms of the price of the underlying stock, the time to expiration, the risk-free rate of return, and the volatility of the underlying stock.

I remember hearing about Back Scholes years ago and it always had this mystique about it. This was probably because I was reading a lot about finance and options around the late 90s and Black and Scholes were awarded the Nobel Prize for Economics for their work on this model in 1997. So there was a lot of media buzz.

But I think we would be forgiven for thinking that Black Scholes represents some kind of holy grail of financial derivatives pricing that changed the world of finance forever. Or maybe they did and I’m just not seeing it.

Yes, there is some clever mathematics behind it all, but as many in finance know, theories and clever maths aside, it is still human emotions and crowd psychology that drive prices, and prices are still set in a market by the interplay between supply and demand.

The Black Scholes model provided a theory-based pricing model for financial derivatives that gave rational calculations for the fair prices of options. Whereas, prior to the existence of the model and other models like it, traders and brokerage houses only had the market to go by.

Other pricing models

There are other pricing models and for very good reason. Firstly, Black Scholes only works easily for European-style options. European style options can only be exercised at the expiration of the contract, whereas the other common type, American style options can be exercised at any time before expiration if the holder or buyer of the options so decides. There are other types of options, like the Bermudan style option which can be exercised at distinct dates leading up to expiration.

The binomial pricing model is another major derivatives pricing model that is used to price, American-style options. The binomial pricing model uses a statistical decision tree and requires many more calculations and hence processing time than does the Black Scholes model, which uses a single equation.

Mathematical finance

The Black Scholes, the binomial, and a host of other pricing models are used to determine the fair price of financial derivatives, not only by investors, traders, fund managers, and brokers but also for example to determine the fair value of employee stock options.

Many of us will remember the bad days of market turmoil that started towards the end of 2007 and the financial meltdown that happened. Mathematical finance, financial engineering, and other clever ways of using computers to trade were in large part blamed for what happened.

I remember there was a fair bit of deep thinking that went on afterward trying to understand the root causes of what happened. One conclusion that came out of it all was that many fund managers had fooled themselves that they could hedge away their risk by effectively selling or trading it to other people through financial derivatives.

These fund managers created computer-driven trading systems to trade in options, futures, and the underlying securities based on these hedging theories that risk had been traded away.

But the key point was that the risk was all still there, and in fact, all of this trading away risk resulted in a situation where the overall level of risk in the market was greater. So when the big unwind came, it was that much more catastrophic in impact.

It is also worth remembering that one of the key assumptions of the Black Scholes model is precisely this, that you can eliminate risk in an option position by taking long and short positions in the underlying stock.

It is true to say that the financial mess of 2007, 2008 had many complex elements. A litany of buzz phrases comes to mind: sub-prime mortgages, mortgage-backed securities, credit-default swaps, predatory lending tactics, and automated trading systems.

But analysts agree that one of the root causes was the unregulated use of derivatives. And it would be fair to say that one of the key assumptions built into that unregulated use, was this notion that the risk of holding a particular option position could be eliminated by taking long and short positions in the underlying stock.

In other words, a misuse of one of the key assumptions behind the Black Scholes pricing model contributed to the 2007, 2008 financial crisis.

The Black Scholes equation

Since we are on the subject of Black Scholes, at the risk that anyone suffering from a condition of acute mathemaphobia running for the hills, here is the Black Scholes equation.

Where:

V = the price of the option

S = the price of the underlying stock

t = time to expiration

r = the risk-free interest rate

σ = volatility expressed as the standard deviation of a log-normal distribution of the underlying stock price.

We should note that the option strike price doesn’t figure in this equation. That is because this equation is only expressing the extrinsic value of an at-the-money option. In other words, it ignores the intrinsic value component.

Demystifying Black Scholes

If we remind ourselves of some basic facts that we also intuitively know about what governs the price of for example a call option, here are those facts.

The price of a call option …

• decreases as the expiration date approaches
• increases as the volatility of the underlying stock price increases
• decreases as the risk-free interest rate increases
• increases as the price of the underlying stock increases

We can also recognize some of the Greeks that relate to option pricing. Here is an article that gives a brief summary of the Greeks in options pricing.

The interesting part is that we can recognize both the Greeks and the basic facts of options prices in the parts of the Black Scholes equation.

This is the first derivative of the price of the option with respect to time to expiration. And let’s remember that the Greek theta, is the rate of decline of the option price with respect to small changes in the time to expiration. What do we see in the next part of the equation?

There is more going on in this second part. We can express this as, half of the square of the volatility, times the square of the price of the underlying, times the second derivative of the option price in relation to the price of the underlying.

Here it is relevant to remember three options price Greeks. Vega is a measure of the change in the option price in regard to changes in the volatility of the underlying price. Delta is the change in the option price in relation to incremental changes in the price of the underlying. Gamma is the rate of change of delta for small increments in the price of the underlying, which is the same as the second derivative of changes in the option price in regard to the price of the underlying.

Now let’s look at the third part of the equation.

This says the risk-free interest rate, times the price of the underlying, times the first derivative of the price of the option with respect to the price of the underlying. The first derivative is just the rate of change of the option price with respect to the underlying price. The relevant option pricing Greeks here are, rho, how the option price varies with the risk-free interest rate, and again delta the rate of change of the option price with respect to the underlying price.

And now to the last part of the Black Scholes equation.

This is just minus the risk-free interest rate, times the price of the option.

Does any of this mean anything?

I am quite sure that seasoned mathematicians would be groaning in agony at the lack of mathematical rigor here. But we are not trying to understand the complex math of the equation. What I am trying to illustrate is that we can find links on the one hand between all the options pricing Greeks, and on the other hand in what we intuitively know about what should drive option prices, and the terms of the Black Scholes option pricing equation.

So next time you are in conversation about financial derivative pricing and Black Scholes comes up, as it inevitably will, you could say something like.

Well, of course, the Greeks can all be found in the Black Scholes equation as you would expect. Or words to that effect.

If you are interested in a simple way to price an asset or a portfolio of assets, as opposed to derivatives, this article explains the capital asset, pricing model.

Here is a general article that explains options pricing, and here is an article that explains how to work with options.

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Questions and answers

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Q. Why does Black Scholes underestimate put prices?

A. The Black Scholes pricing model assumes that the prices of options and of the underlying are adjusting to market conditions on a continuous basis. However, the prices of the underlying and of options sometimes gap up or gap down when closed markets reopen. This tends to undervalue call and put options that are deep-out-the-money.

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Q. Does Black Scholes work for futures?

A. The Black Scholes model doesn’t work for futures. However, Black modified the model to create the Black-76 model, in 1976, – who would have thought. – The Black-76 model incorporates forward prices whereas the Black Scholes pricing model only used spot prices and added other modifications to allow the model to be used for futures and a variety of other financial derivatives.

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Q. How accurate is the Black Scholes model?

A. In practice the Black Scholes model is not very accurate at predicting options prices. You could argue that just goes to show that there are still significant elements of emotion and less rationality in the real market prices of options.

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Single-page summary

Here is a single-page PDF summary of the Black Scholes pricing model.

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