In which Sherlock Holmes explains forward pricing, options theory and other financial arcana.

By John Price
With apologies to Sir Arthur Conan Doyle.

I've been hoodwinked,” I said, as soon as I entered the sitting room. "And by a gentleman.”
Sherlock Holmes glanced at me for a moment from above his magnifying glass and then continued examining the documents he had on the small table beside his chair. I knew by the look on his face that it was no use trying to talk to him when his faculties of deduction were so fully engaged. So I sat in the armchair by the fire and waited as patiently as my state of agitation would permit.
After a few minutes he looked up and said, "I see you have been to the London Club, where you have been talking with Admiral Smithies.” Before I could reply, he added, "And you have shifted your shaving mirror to the east wall of your bathroom.”

"How on earth could you know that?” I asked. Even after all these years, it still astonished me the way Holmes seemed to pluck these facts out of the air.

"It's really quite obvious, my dear Watson,” he answered. "First of all, for weeks you have been talking about nothing but stocks, forwards and options, so I deduced that your disturbed state of mind must have something to do with them. Second, the only gentleman who could possibly pull the wool over your eyes in this area is Admiral Smithies. And finally, today is Tuesday, and the Admiral always spends Tuesdays at the London Club. So there you have it.”

"Simple enough, I suppose, when you explain it like that. But what about the shaving mirror? I moved it only this morning.”

"My good fellow, I know you well. For years your shaving mirror was on the west wall of your bathroom, and since the window is in the south, you were always more closely shaven on the left side of your face than on the right. Now when you appear with a closer shave on the right side, what else can I deduce? But this is a trifling matter. Tell me what has upset you so regarding your financial transactions with Admiral Smithies.”

"It all started when I told the admiral that I was interested in selling my stocks in the Greater London Gas Works. They are currently worth 100 pounds and don't pay any dividends. He said that in a year he was coming into some money and that when he did he would be willing to purchase them. I said that this would please me provided we could come to some arrangement as to what he would pay me for them in a year's time. They call this a forward contract.”

"I am well aware of what a forward contract is,” Holmes interjected. "So the problem is to decide on a fair price for this contract.”

"Exactly,” I replied. "We decided that unless we were both completely happy with the agreed price, we wouldn't do it. After some discussion we resolved to go and see George Oracle. He is Greek, you know. I think his family comes from Delphi, or somewhere near there. They had quite a reputation for being able to predict the future. George is probably not up to their standard, but he is often close to the mark.”

"He said that in a year he was coming into some money and that when he did he would be willing to purchase my stocks. I said that this would please me provided we could come to some arrangement as to what he would pay me for them in a year's time. They call this a forward contract.”

"I have no need for such unscientific methods,” Holmes said sententiously. "But do go on, for I am curious as to what he had to say. He runs a fruit barrow in the West End, does he not?”

"Yes, that's him. When we saw him yesterday he said that he couldn't tell us exactly what the price of the stock would be in 12 months, but he could tell us that the expected value would be 120 pounds. In other words, the expected return from owning the stock for a year is 20 percent. ‘That's good enough for me,' I said to Smithies. ‘Let's agree on 120 pounds as the forward price.' But the admiral said that he would think about it and would let me know the next day provided I would be good enough to meet him in the London Club. He also mentioned something about treasury bonds and arbitrage.”

"So you had your meeting with him today and he offered you a smaller price.”

"Yes, exactly. He said that treasury bonds were paying 10 percent and that the forward price for my stock should be based on this. All he would agree to was that in 12 months he would pay me 110 pounds. So that's the story. I feel that Smithies is being most unreasonable. But I put it to you Holmes, as a dear friend and as someone with remarkable deductive abilities. What is the fair price, 110 pounds, 120 pounds or something else?”

Holmes stared vacantly out the window for a few moments and then started to make some calculations on sheets of paper that he pulled from a small drawer in his table. I knew that Holmes had a bent for mathematics. After all, in his attempt to understand the diabolical Professor Moriarty, he had studied all that Moriarty had written in mathematics and other areas. Moriarty had authored a treatise on the binomial theorem which was of such quality that he was awarded a chair at one of our smaller universities. But I had the feeling that what he was about to explain to me went far beyond anything that he had attempted before. Finally he looked at me and said that it all depended on what assumptions you are making about the marketplace.

"Assumptions? What do you mean assumptions?” I said with some surprise. "This is real money we are dealing with here—not some academic flight of fancy.”

"But that's just it,” Holmes replied. "As soon as you start to talk about fair prices and expectations, you are making assumptions about the market place. Let me explain.”

"I wish you would.”

"Let us suppose that George Oracle is correct when he says that the expected value of the stock in one year is 120 pounds. Then it would seem that this would be a reasonable figure for the forward price.”

"That's exactly what I said to Admiral Smithies,” I said quickly. "I am pleased that you support me on this matter.”

"But now,” Holmes continued, putting his fingertips together, "imagine that you sell your stock for the current price of 100 pounds and invest the money in treasury bonds. At the end of 12 months, you will have no stock and 110 pounds. So it seems only fair that the forward agreement with the admiral should be at least that amount.

"I'll say so. It should be 120 pounds,” I said with enthusiasm.

"Let us see. Suppose that the admiral borrows 100 pounds at the treasury rate of 10 percent and uses the money to buy Gas Works stock. In one year, he will have to pay back 110 pounds but will still own the stock. In effect, he will have bought stock in one year with a forward price of 110 pounds. It is not reasonable, therefore, for him to agree to any forward price that is greater than 110 pounds.”

"Are you saying that it is always possible to own the stock in one year's time and still only be out of pocket by 110 pounds at that time,” I queried, with little enthusiasm.

"Precisely.”

"So that there is no need for the admiral to agree to a forward price of 120 pounds?”

"There can be but one conclusion—the only forward price that is fair for both of you is 110 pounds. This is called the arbitrage price of the forward. If the forward price is greater than the arbitrage price, you can make a guaranteed profit no matter what the market does. On the other hand, if the forward price is less than the arbitrage price, the admiral can make a guaranteed profit.”

"Even though I can follow your logic, the conclusion is very strange. And, I hasten to add, most unfortunate, for it will cost me 10 pounds. Can it be that George Oracle is also correct? In any case, everyone agrees that the likely price of the stock will be more than 110 pounds in one year.”

"But I put it to you Holmes, as a dear friend and as someone with remarkable deductive abilities. What is the fair price, 110 pounds, 120 pounds or something else?”

"Let's go back a step. Do you agree that 100 pounds is a fair current price for Greater London Gas Works stock?

"Certainly I do, since this is its price at the London Stock Exchange. Moreover, since trading volume is at its usual level, I think that everyone else considers that it is currently at a fair price. In this regard, I would like to say one more thing. I believe that knowing past prices of any stock is no more help than knowing its current price when it comes to investing in it.”

"You have just stated the weak form of the efficient market hypothesis.”

"Really?” I asked, feeling rather pleased with myself. "Also I believe that all information about the past activities of the company and opinions regarding its future growth in terms of earnings and dividends is already reflected in the price of the company's stock.”

"This is the strong form of the efficient market hypothesis. Let us see where it leads us. Associated with the various forms of the efficient market hypothesis is the view that stock prices can be described as random walks. Roughly speaking, a random walk is formed by tossing a coin at regular intervals to see if you should step left, right, up or down in the case of a stock price.”

"You mean those traders I see behind the Exchange tossing coins are really giving directions at whether individual stocks should go up or down? All the time I thought that they were involved in some sort of gambling in which …”

"Of course not,” Holmes interjected. "No one is tossing coins to decide if a particular stock should go up or down. It is just that, viewed over an extended period of time, the statistics of a random walk and the statistics of a stock price are essentially the same. The claim is that the stock market adjusts so quickly and efficiently to new information that no one can consistently buy or sell fast enough to benefit. Hence, for all practical purposes, it is a random walk. When the sizes of the time intervals and the step sizes shrink, the limiting case of such a random walk is called a Brownian motion. It is named after the Scottish botanist Robert Brown. He did a first-rate job of expounding on one of my scientific trifles. Pollen in a test tube, I recall. The microscopic grains of pollen in the fluid could be seen to move erratically, indicating that they were being bombarded by particles.”

I was amazed at hearing of this discovery by Holmes, but before I could say anything he continued.

"You can think of stock prices as similar to one of the grains of pollen. A purchase gives it a bump in the upward direction, and a sale gives it a bump in the opposite direction.”

"Does this really give a true picture of the movements of stock prices?” I asked.

"Let us just say that it is a starting point.”

"If stock prices are just random walks, how can investors make money?”

"Random walks can have different characteristics. You could imagine that the jump sizes could be smaller or larger. Or some could be biased more in the upward direction than in the downward direction. It is always hoped that the larger the jump sizes, the more the bias is in the upward direction. In this way, investors anticipate that they will be rewarded for taking on greater risk.”

"What other views do people have about the stock market?”

"First of all, there are technical analysts. These people carefully study the graphs, or charts, of past prices of stocks, often along with their trading volumes, looking for cycles or patterns that will help them determine future movements.”

I recalled some of the conversations I had overheard where people talked about momentum and moving averages.

"At the opposite end of the spectrum are fundamental analysts,” Holmes continued. "Their goal is to determine a stock's proper value, called its intrinsic value, by estimating the company's future growth and earnings and discounting these back to the present. They care little about past patterns. If the intrinsic value is above the actual market price, they argue that the market will eventually realize this and the price of the stock will rise accordingly. Similarly, if the intrinsic value is lower than the actual price, the stock price will eventually fall.”

"I think I am beginning to see. But what about the claim by George Oracle that the expected value of the price for Greater London Gas Works stock in 12 months is 120 pounds?”

"Well, I hope that you didn't pay him anything for this, because there is no way of verifying whether his statement is correct or not.”

Immediately I thought of our promise to buy our fruit from him for the next year at his exorbitant prices. I tried not to show any outward sign of my discomfort, but nothing escapes the sharp eyes of Sherlock Holmes.

"Never mind,” he said gently, much to my relief. "Let's make sure you don't get caught next time. The point is that there is no way of testing the validity of a probabilistic statement by a single observation. It's like trying to verify with a single toss of a coin that the probability of getting a head is 50 percent. So if Mr. Oracle said the expected value was 10 or 1,000, we could neither prove nor disprove the accuracy of his statement.”

Well, what is the point of a making statements about expected values?” I asked, somewhat peevishly.

"The point is that with many observations, we can begin to test the accuracy of his predictions. For example, suppose he tells us that there is an equal likelihood of the value of Greater London stock being above 118 in one year or below 118. Then we ask him for the same information each day for, say, 30 days. On each of these days, the information is for one year from that day. If, after one year and 30 days, we discover that in approximately half the cases the stock price was above the corresponding number he gave us, then we have reason to believe Mr. Oracle. The closer the number is to 15, the more confident we can be in believing him.”

"I have just one more question.”

"Certainly,” Holmes replied equitably.

"These days there is a lot of talk about options. Can the same methods be used to give an arbitrage price for options?”

"Suppose that the admiral borrows 100 pounds at the treasury rate of 10 percent and uses the money to buy Gas Works stock. It is not reasonable, therefore, for him to agree to any forward price that is greater than 110 pounds.”

"Not under all circumstances, but general enough to get started. We will need to write down a few things,” Holmes said, putting more sheets of paper on his table. "Can you move your chair closer?”

After we were settled again, Holmes continued. "We are, of course, assuming that the price of Greater London Gas Works stock is given by a random walk. It will be easier if I explain it by wring down some of the details. Denote by S' the random variable describing the price of the stock in 12 months. Then, according to Mr. Oracle, the expected value of S' is 120.”

Holmes wrote this down as "E(S') – 120” explaining that the symbol "E” meant expected value. From what he had said before, I gathered that this was a type of mathematical abstraction of the concept of a long-term average. In some sense it reflected the actual or observable behavior of Greater London Gas Works stock on a year-to-year basis.

"Now suppose that the market is risk-neutral. This means that market participants do not expect any extra return for purchasing risky stock. In such a market the expected value of the stock in one year must be 110 pounds.”

"Why is that?” I asked, my head beginning to spin by this time.

"It's the efficient market hypothesis again. If the expected value was above 110, then everyone would borrow from the British Treasury and buy Gas Works stock. This would give them a positive expected return over the year, all that is asked for by risk-neutral investors. The effect would be to drive up the price of the stock until balance was restored. Similarly, if the expected value of Gas Works stock was below 110, the same people would take short positions in the stock for one year and invest the money in Treasury bonds. This time the effect would be to depress the price of the stock until equilibrium was restored once again.

This time Holmes wrote "E*(S') – 110,” explaining that the symbol "E*(S')” stood for the expected value in a risk-neutral market of the price S' of the stock in one year.”

"I see,” I said, putting as much confidence as possible into my voice. "We now have that E*(S'), the risk-neutral expected value of the stock price in one year, is equal to the arbitrage forward price we determined above. Is this just a fortunate coincidence for forward contracts?”

"Far from it. Suppose we wish to value a call option on Greater London Gas Works stock with a strike of 110 pounds and expiration in one year. Suppose also that the option is European-style, meaning that it cannot be exercised before that time. If the price of the stock is above 110 pounds in one year, the payoff from the option is the difference between the stock price and 110; otherwise there is no payoff.”

Continuing in his spidery handwriting, Holmes wrote this as "max(S' - 110, 0).” He then explained that, as for forward contracts, there are two natural contenders for the fair value of such an option—the expected value E(max(S' - 110, 0)) of the payoff under the observable expectation, and the expected value E*(max(S' - 110, 0)) of the payoff under the risk-neutral expectation.

"It is the latter expected value, discounted back to present time using the treasury rate, that is the fair value for the option,” Holmes said. "If it can be purchased for less than this, the purchaser can always make an arbitrage profit. Conversely, if it is sold for more than this amount, the vendor can always make such a profit. We call this value the arbitrage value of the option.”

"Can the same methods be used to give an arbitrage price for options?”    "We will need to write down a few things,” Holmes said, putting some more sheets of paper on his table. "Can you move your chair closer?”

"That is remarkable,” I exclaimed. "Do you mean to say, just as for the case of forward contracts, there is a value for such option contracts that does not depend upon anyone's opinion regarding the expected rate of return of the stock? And furthermore, in both cases the key to calculating this value is this strange concept of risk-neutral expectation?”

"That is precisely what I mean to say. However, this time we do need to impose further restrictions on the random walk describing the stock price. For if we do not do this, we cannot set up an arbitrage mechanism as we did for forward contacts.”

He then lay back in his armchair and closed his eyes. When he opened them again he had a faraway look.

"The best approach is to try to describe the returns of the prices as a Brownian motion, rather than the prices themselves. This simply means considering the ratios of prices over consecutive days instead of the prices themselves. I explained all this to Louis Bachelier. The result is called geometric Brownian motion. He is preparing his dissertation on this topic for the French Academy of Science. However, I think that he may decide to use what amounts to Brownian motion. But still, it will undoubtedly be a major contribution to the mathematics of option theory in both theory and practice. One of the problems with using Brownian motion is that you always get a positive probability of negative stock prices.”

"And this doesn't happen with geometric Brownian motion?” I asked tentatively.

"Correct. Whatever value you start with, you go up or down by certain positive ratios, and not by certain positive amounts.”

"How can this be applied to valuing options?”

"We need geometric Brownian motion to be able to bring about the arbitrage profit. Just as for the case of forwards discussed earlier, the principle is that if someone purchases a call option with a strike of 110 pounds for more than the current arbitrage value, the vendor can make an arbitrage profit. And if less, the purchaser can make such a profit. The only difference between this case and that of forwards is that it is more difficult to construct the arbitrage profit. For example, suppose that a call option is sold for more than its arbitrage value. It is, of course, possible to use part of the proceeds to form a portfolio consisting of Greater London Gas Works stock and treasury bonds in precise amounts, the value of which is equal to the arbitrage value of the option. Next, this portfolio must be adjusted at regular intervals according to precise mathematical rules that need not concern us here. It suffices to say that they depend on the stock price following a geometric Brownian motion. These rules are also needed to determine the exact proportions of the initial portfolio. Furthermore, the adjustments to the portfolio are to take place without contributing cash to or removing cash from the portfolio. The outcome will be a portfolio that has its value equal to the payoff of the option. This means that the portfolio can be used to pay any commitments due the purchaser of the option at its expiration. The excess of the proceeds from the initial sale of the option and the establishment of the portfolio is your arbitrage profit.”

"Do you mean to say there is a value for such option contracts that does not depend upon anyone's opinion regarding the expected rate of return of the stock?”

Holmes then sat back in his chair and neatly arranged the sheets of paper on his table. From these actions, I inferred that his masterly exposition of some of the mysteries of forwards and options was over. I thanked him warmly for taking the time to explain these things to me. Although he said nothing, I thought from his expression that he was pleased at my evident appreciation and admiration.

After bidding him farewell, my intention was to slip quietly from the room, for I was eager to get to the London Stock Exchange to learn the closing price for stock in Barings Bank. As I was opening the door, Holmes said quietly, "Now here is an interesting case. These documents that I have been examining from Charles Ponzi in America state that he promises to pay investors 50 percent interest every 45 days. What do you think, Watson?” The exchange could wait.

Author's note: I have searched the publications and laboratory notes of Robert Brown, but could find no mention of the name Sherlock Holmes. The same applies to the writings of Bachelier, who successfully presented his dissertation "Theory of Speculation” in 1900. In both cases, I assume that Holmes, with uncharacteristic modesty, asked these people to suppress any mention of his input. As to his reasons for this, I can only conjecture. There is no question that, rightly so, he regarded himself as the world's greatest consulting detective and that this is how he wanted to be remembered. Perhaps he thought that accolades in other areas might somehow cast a shadow on the brilliance of his achievements in criminal detection. These are matters on which I do not wish to dwell. This same desire applies to the fact that Brown had performed his revolutionary experiment in 1828, years before Holmes was born.

Regarding George Oracle, when I saw him a few days later I told him of my conversation with Sherlock Holmes. I placed particular emphasis on the fact that his statement regarding the expected value of Greater London Gas Works stock had caused me considerable frustration. It had raised my hopes of receiving 120 pounds for their sale to Admiral Smithies whereas, as Holmes explained to me, the fair price was just 110 pounds. He replied that all he could do was follow his family tradition and just give the information as it came to him. However, as an act of goodwill, he did tell me that Holmes' formula for pricing options would be forgotten for many years until it would be rediscovered in 1973 by two gentlemen named Fischer Black and Myron Scholes.

John Price is head of the mathematics department at Maharishi University of Management in Fairfield, Iowa, and is vice president of Integrated Energy Services.