Sherlock
Holmes
and
The Case of the Missing Ten Pounds
In which
Sherlock Holmes explains forward pricing,
options theory and other financial arcana.
by John Price
With
apologies to Sir Arthur Conan Doyle.
Reprinted with permission from Derivatives
Strategy, October 1997 and
Derivatives and Financial Mathematics (Nova Science Publishers, 1997; ed. J. F. Price) .
"I’ve been
hoodwinked," I said, as soon as I entered the
sitting room. "And by a gentleman." Sherlock
Holmes glanced at me from above his magnifying glass
and then returned to 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 that
Holmes would seem 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. Secondly, 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 only moved it 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 so upset
you 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."
|
| "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 added quickly. "We decided that unless we
were both completely happy with the agreed price,
then 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."
"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 Westend, 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 twelve 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 twenty 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 that he 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 ten percent
and that the forward price for my stock should be
based on this. All he would agree to was that in
twelve 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
also as someone with remarkable deductive abilities.
What is the fair price, 110 pounds, 120 pounds,
or something else?"
Holmes stared vacantly
out of 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 upon 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 market
place.
 |
| "But
I put it to you Holmes, as a dear friend and
also as someone with remarkable deductive
abilities. What is the fair price, 110 pounds,
120 pounds, or something else?" |
"Assumptions? What
do you mean assumptions?" I said with some
surprise. "This is real money we are dealing
with here. Not some academic flights of fancy."
"But that’s
just it," Holmes replied. "As soon as
you started 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."
"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 now for the
current price of 100 pounds and invest the money
in treasury bonds. At the end of twelve 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 pound at the treasury
rate of 10 per cent 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. Thus, in
effect, he will have bought stock in one year with
a forward price of 110 pounds. Thus it is not reasonable
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
rather less enthusiasm.
 |
| "Suppose
that the Admiral borrows 100 pound at the
treasury rate of 10 per cent and uses the
money to buy Gas Works stock." |
"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 ten 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."
"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
said, 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 head left or right, or 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 to enough
to benefit. Hence, for all practical purposes, it
is a random walk. By letting the size of the time
intervals and the step sizes go to zero, 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 give 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 of 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 cycles 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 twelve 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 discomfit, 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 fifty percent. So even if Mr. Oracle said the
expected value was 10 or 1000, 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
that he tells us that there is an equal likelihood
of the value of Greater London stock being above
118 in one year, as it is to be below. Then we ask
him for the same information each day for, say,
thirty days. On each of these days, the information
is for one year from that day. If, after one year
and thirty 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 fifteen,
the more confident that 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?"
"Not under all circumstances,
but general enough to get started. 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?"
 |
"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?" |
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 twelve 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 which 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.
"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?"
 |
| "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?" |
"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
which 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
or removing any 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 to 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."
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 the 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 a very interesting case. These
documents that I have been examining from Charles
Ponzi in America state that he promises to pay investors
50 per cent interest every forty-five days. What
do you think, Watson?" The Exchange could wait.
_____________________
Author’s note. I
have searched the publications and laboratory notes
of Dr 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 that I do not wish
to dwell upon. 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 upon 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.
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