## Notes on Black-Scholes (From 2003)

The eclectic von Pepe asked me to dig these up… I haven’t reviewed them, but 2003 Bob knew this stuff better than 2016 Bob so let’s hope he didn’t screw up.

Notes on Black-Scholes

Robert P. Murphy

July 2003

A friend from the finance world asked me to review the original Black-Scholes paper on option pricing. I realized that my comments on the journal article might be of interest to others, and so I am preparing these notes for public consumption. Let me confess at the outset that I am an expert in neither finance nor statistics. (Please drop me an email if you think there’s an error below.) Nonetheless, I hope to shed at least some light on what Black and Scholes do in this seminal work.

I will go through the paper, “The Pricing of Options and Corporate Liabilities,” published in the Journal of Political Economy. (I don’t have an exact citation handy at the moment.) These notes will make more sense if you have a copy of the paper yourself.

The Problem

An option is “a security giving the right [but not the obligation] to buy or sell an asset, subject to certain conditions, within a specified period of time” (637). The most straightforward example is a call option, which gives its owner the right (but not the obligation) to buy a share of stock at a certain price (called the “strike price”), in a specified time interval.

The problem is, how much should these options be worth? If a share of IBM stock currently sells for $10, how much should I pay for an option that gives me the right to buy a share of IBM for $15 anytime in the next 12 months? How much should I pay for the option if the strike price is $8?

The Black-Scholes formula gives us a rational procedure for computing the “true value” of options. The problem is not so much in predicting what options will actually sell for on the market; the problem is to define what the “rational” speculator will do on the market. When it comes to regular consumer goods, this problem doesn’t arise: People pay more for steak than hamburger because they enjoy steak more than hamburger. But when it comes to options, it is not clear how much people should value such complicated assets. This is where the Black-Scholes formula comes in.

Abstract

We can first turn to the article abstract, to see how Black and Scholes themselves viewed their contribution:

If options are correctly priced in the market, it should not be possible to make sure profits by creating portfolios of long and short positions in options and their underlying stocks. Using this principle, a theoretical valuation formula for options is derived.

Here Black-Scholes rely on the no-arbitrage condition, which has proved so useful in formal economics. To give some background: In mathematical economics, an equilibrium set of prices is defined as one in which all market participants, given their endowments of resources and the market prices, formulate plans (of buying and selling decisions) that (a) maximize the utility of each agent and (b) are all compatible with each other. For example, if the equilibrium price of apples is $2/pound, then people must collectively desire to sell X apples at this price, and people must collectively desire to buy X apples at this price.

In this framework, a necessary condition for equilibrium is that the price system contains no arbitrage or pure profit opportunities. If this were not the case, then individual agents would attempt to earn an infinite amount of money by exploiting the price differentials; since this is impossible, the prices would have to quickly adjust to eliminate the arbitraging action. But if the prices must change, that means the original set of prices (which contained the pure profit opportunity) must not have been an equilibrium one.

Returning to Black-Scholes, we see that they are employing this technique to determine how options “should” be priced. Black-Scholes are adopting the rather modest position that a necessary condition of option prices is that they do not allow for arbitrage in the securities market.

Before proceeding, consider an analogy with production. Suppose oranges are selling in Florida at $2 per pound, and that the transportation costs to ship them to New York are ten cents per pound. If oranges sell in New York at more than $2.10 per pound, then a merchant can engage in (almost) riskless arbitrage by buying the oranges in Florida and selling them in New York. From an economic point of view, the Florida-oranges-plus-transportation is “equivalent” to New-York-oranges, and so the market price of both “portfolios” must be the same in equilibrium. When the prices of the two portfolios diverge, we would say that the oranges are incorrectly priced.

With this approach, we can say that an option is “underpriced” if its buyer is assured of pure (riskless) profits, and we can say that an option is “overpriced” if its seller is assured of the same. By the process of elimination, if an option is neither underpriced nor overpriced, we say it is correctly priced.

How can someone make sure profits through securities transactions? Well, notice that certain portfolios are “equivalent” from a financial point of view. For example:

Regardless of the movement in the price of the stock, both portfolios will always give the same payoff at expiration. If the price S remains at $x, then the put and call options are worthless at expiration, and the bond pays (at time of expiration) $x while the stock can be sold for $x. If the price of the stock goes above the strike, then at expiration the call is worth S-x, while the put is worthless. Thus the left-hand portfolio above is worth (S-x)+x, while the right-hand portfolio is worth the share price of the stock, S. Finally, if the price of the stock goes below the strike, then at expiration the call is worthless while the put is worth x-S. Thus the left-hand portfolio is worth x while the right-hand portfolio is worth S+(x-S).

Because these two portfolios have the same payoff (at expiration), regardless of what happens to the price of the underlying stock, no-arbitrage requires that the price (at any time prior to expiration) of both portfolios be the same. If the price of one of the portfolios, say the left-hand one, were lower than the other, then a speculator could earn a “riskless” profit by buying the left-hand portfolio and selling the right-hand one.

For example, suppose that a share of stock XYZ sells for $100, and that calls and puts (with strike price $100) sell for $10 and $5, respectively. That means the right-hand portfolio above costs $105. Now if a risk-free bond paying $100 at the time of expiration currently sells for anything less than $95, a speculator can earn a pure profit by buying the left-hand portfolio and selling the right-hand one. (This will happen—i.e. the risk-free bond with face value of $100 will sell for under $95—whenever the relevant interest rate is higher than roughly 5.3%.)

In the example above, all I’ve done is show that the prices of puts, calls, and stocks must obey a certain relationship, given an interest rate. This alone doesn’t tell us how to actually price an option. (Further restrictions are described at this website.) Nonetheless, my example does illustrate the approach of Black-Scholes. A correctly priced option can withstand any type of portfolio selection, not just the specific two that I described above.

Introduction

Black-Scholes specifically state their goal later at the end of their Introduction: “[I]n equilibrium, the expected return on such a hedged position must be equal to the return on a riskless asset. What we show below is that this equilibrium condition can be used to derive a theoretical valuation formula.” (p. 640)

By “hedged position” they are referring to a portfolio that gives a guaranteed return, regardless of the movement of the stock price. For example, suppose an investor buys a share of stock XYZ, buys a (European) put, and sells a (European) call, with strike prices of $100 each, which can only be exercised in twelve months’ time. The resulting portfolio is a hedged position, guaranteeing $100 in twelve months. If the price of XYZ turns out to be $100, then the options expire worthless and the investor can sell the stock. If the price is higher than $100, then the put is worthless, but the sold call will be exercised—the investor will have to surrender his share for only $100. Finally, if the price is lower than $100, then the call will not be exercised, and the investor can use his put to sell his share for $100.

Keep in mind, though, that this guaranteed return can only be enjoyed in one year. During that time, whatever funds the investor originally spent on the portfolio will remain tied up (when they could have been earning interest on a risk-free bond). In the example above, suppose the stock initially sells for $90, while the put sells for $10 and the call sells for $5. In that case, the investor would need to invest $95 today in order to receive $100 (guaranteed) in twelve months. This can only be an equilibrium outcome if the rate of return on risk-free bonds for twelve months is also 5.26% (approximately).

Assumptions

The final thing I will do in these notes is elaborate on one of the Black-Scholes assumptions. (The other assumptions are fairly self-explanatory.) This is the second assumption, which states, “b) The stock price follows a random walk in continuous time…”

A random walk model of stock prices means that, at any time t, the expected value of the stock at any time t+∆ is simply its current price. This doesn’t mean that people “expect” that the price of the stock will actually remain constant. Rather, it means that the “best guess” is the current price, because there is just as much chance for the price to go up as for it to go down.

The motivation for this assumption is that markets are assumed to be efficient. Whatever information individual traders possess about the company or the economy in general, is assumed to have already worked its way into all current share prices.

If this is one’s view of the economy, then it would be silly to argue that, “You should buy shares in Oscar Mayer right before the 4th of July, because hot dog sales will go through the roof.” Presumably, enough people already know this fact, and so the price of Oscar Mayer stock will already take this consideration fully into account. Absent any insider information, then, one should expect the price of Oscar Mayer to be just as likely to go down on July 4th as to go up. (Of course, the price of the stock is higher than it would have been had the government previously announced a cancellation of the holiday, but the point is that the price is at its higher level all along; it doesn’t jump up to the high level on the day of the holiday itself.)

When modeling stock prices, then, Black-Scholes assumes that the change in levels has a mathematical expectation of 0, with a certain variance and other properties that allow for tractable analysis.

Good job. Near the end, you write, quoting them, “The stock price follows a random walk in continuous time.” Assuming you’re quoting correctly, I think they made a mistake. Isn’t it stock price CHANGES that follow a random walk? I’m not saying their article is wrong; I’m saying that they stated their assumption wrong.

Lognormal random walk for stock prices, normal random walk for price changes.

Relevant topic. With low interest rates abound, options gifted to employees at start-up companies under IRS 409a have lower strike prices than otherwise would be. Low interest rates cause the strike price (rf bond) to be discounted less, thus the excess value of the call is diminished. This methodology is often applied to start-ups since their enterprise value is frequently below the “hurdle” value of liquidation preference and debt…just like a stock price being below the strike price. It could result in large tax burdens for the individuals receiving these options in the future if/when interest rates are at a higher level.

“When modeling stock prices, then, Black-Scholes assumes that the change in levels has a mathematical expectation of 0. ”

They don’t actually assume this. This is only true in the risk-neutral framework, and applies to the discounted value of the stock price rather than the actual stock price.

In the real-world framework, the behaviour of the stock is a geometric random [walk] with a constant drift equal to its expected return. However the no-arbitrage argument demonstrates that the value of the option does not in fact depend on the expected return. This is also true for simpler derivatives like futures.

Thanks Zaki I was not sure about this as well, since I knew the stocks would need to be expected to earn more than the risk-free rate.

So were they (slightly) inaccurate when saying they assumed stock prices followed a random walk? Should they have said something like the percentage change in stock prices

~~followed a random walk plus drift~~? (EDIT: I realized this isn’t right, the change itself isn’t a random walk.)I don’t blame you for being unsure. I spent a whole summer studying the Black-Scholes model so I could teach it to postgraduate actuarial science students.

Yes, they were inaccurate, but I didn’t actually look at their paper. Fortunately there are many textbooks that explain the model in a lucid way.

The phrase “Stock prices follow a random walk” is commonly used as a shorthand for the more precise “geometric Brownian motion with constant drift”, but your way of putting it is nice. The drift term is irrelevant to the price of the option, but this has to be proved, not assumed from the start.

BTW, your hedging example is good for explaining what hedging means, but not the dynamic hedging strategy required to work out the price of a call or a put.

“The drift term is irrelevant to the price of the option, but this has to be proved, not assumed from the start.”

The drift term is the risk free interest rate, correct? The “greek” rho measures option value change relative to interest rate change. Is it correct to say that dynamic hedging replaces the expected market return, whatever that is, with the risk free rate?