hmmmm... wow, just wow

Poor Professor Solow... he wins a Nobel Prize and he's still stuck with the same size office as Professor Ergin. Who, in turn, has to come in and see two Nobel Prizes every morning. In conculsion: rocks to be Samuelson.

I forgot to blog about Solow and Samuelson's National Medals of Science... which are both up on the wall.

So, do you still have the "Master Model Builder" bio on Paul Samuelson (Larry Summers' uncle) in your binder? (And, I was wondering: did I show "Trillion Dollar Bet" with Samuelson in it -- it was all about the LTCM debacle -- the Scholes-Black formula)?

That was the video where Paul Samuelson expressed the hope that the mathematical theory of probability in statistics could be a skeleton key to help investers understand the nature of chance, perhaps to predict it better, or even perhaps to control it?

NARRATOR: Unknown to the economists of the 1930s, a French graduate student at the turn of the century, Louis Bachelier, had already exploited the structure of randomness in his doctoral thesis titled, "The Theory of Speculation." He compared the behavior of buyers and sellers to the random movements of particles suspended in fluid, anticipating key insights later developed by Einstein and the mathematics of probability. But Bachelier's accomplishments would go unnoticed for decades.

PAUL SAMUELSON: In the early 1950s I was able to locate by chance this unknown book, rotting in the library of the University of Paris, and when I opened it up it was as if a whole new world was laid out before me. In fact as I was reading it, I arranged to get a translation in English, because I really wanted every precious pearl to be understood.

NARRATOR: Using a series of equations, Bachelier created the first complete mathematical model of stock market fluctuations. He too believed stock prices moved at random and that it was impossible to make exact predictions about them. But then Bachelier said he had found a way to control risk, through an obscure financial contract called an option. He realized options could protect investors from market fluctuations and made the first attempt to figure out how to price them. But his superiors were unimpressed. His academic career faltered.

A stock option is a contract that gives you the right -- but not the obligation -- to buy or sell a stock at a pre-specified price (the exercise price) for a pre-specified time, that is, until the option "expires." If the option gives you the right to buy shares of a stock, it is a call option. If the option gives you the right to sell shares of a stock, it is a put option. Exactly how much you should pay for these contracts is determined using the Black-Scholes Formula.

Options are usually sold in sets of 100 (which would allow you to buy or sell 100 shares of the underlying stock at a certain price for the duration of the option).

There are two kinds of stock options: American-type and European-type. American-type stock options allow you to buy or sell the shares of the underlying stock at the exercise price ("exercise" the option) any time until the option expires. European-type stock options allow you to exercise the option only at its expiration date. The formula we provide here applies to a European-type call option. You can buy and sell options just like stocks; their value is determined by the likelihood that they will be "exercised" for a payoff ("in the money"). You can calculate the exact value of the call option using the Black-Scholes Formula (if you know what you're doing, of course).

The Black-Scholes equation:

C = SN(d1) - Le-rTN(d1 - ¦É sqrt(T))

C is the current call option value. This is the cost to purchase one European-type call option of a certain stock.

S is the current stock price. This is the price that the stock underlying the call option is currently trading at.

N(d1) is a fraction (whose value is between 0 and 1) determined by the price of the stock; the exercise price; the risk-free interest rate (the annualized continuously compounded rate on a safe asset with the same maturity as the option); the time to maturity of the call option, and the volatility of the underlying stock price.

d1 is derived from the following formula:

d1 = (ln(S/E) + (r+¦É2/2)T) / ¦É sqrt(T)

where:

ln = natural logarithm

S = price of the stock

E = exercise price

r = risk-free interest rate (the annualized, continuously compounded rate on a safe asset with the same maturity as the option)

¦É = the volatility of the stock - that is, the standard deviation of the annualized, continuously compounded rate of return on the stock

T = time to maturity of the option in years

L is the exercise price, the price at which you have the right to buy the stock when the call option expires.

e-rT is a term that adjusts the exercise price, L, by taking into account the time value of money. Here, "e" is 2.718, the base for the natural logarithm, used for continuous compounding; "r" is the risk-free interest rate; and "T" is the time until expiration of the call option.

N(d1 - ¦É sqrt(T)) is a fraction (whose value is between 0 and 1) determined by the price of the stock; the exercise price; the risk-free interest rate (the annualized continuously compounded rate on a safe asset with the same maturity as the call option), the time to maturity of the option, and the volatility of the underlying stock price. It differs from N(d1) in that one subtracts ¦É sqrt(T) (the volatility of the stock times the square root of the time till the call option expires) from d1 before the N function is performed.

The theory behind the formula

Derived by economists Myron Scholes, Robert Merton, and the late Fischer Black, the Black-Scholes Formula is a way to determine how much a call option is worth at any given time. The economist Zvi Bodie likens the impact of its discovery, which earned Scholes and Merton the 1997 Nobel Prize in Economics, to that of the discovery of the structure of DNA. Both gave birth to new fields of immense practical importance: genetic engineering on the one hand and, on the other, financial engineering. The latter relies on risk-management strategies, such as the use of the Black-Scholes formula, to reduce our vulnerability to the financial insecurity generated by a rapidly changing global economy.

Here's the theory behind the formula: When a call option on a stock expires, its value is either zero (if the stock price is less than the exercise price) or the difference between the stock price and the exercise price of the option. For example, say you buy a call option on XYZ stock with an exercise price of $100. If at the option's expiration date the price of XYZ stock is less than $100, the option is worthless. If, however, the stock price is greater than $100 -- say $120, then the call option is worth $20. The higher the stock price, the more the option is worth. The difference between the stock price and the exercise price is the "payoff" to the call option.

The Black-Scholes Formula was derived by observing that an investor can precisely replicate the payoff to a call option by buying the underlying stock and financing part of the stock purchase by borrowing. To understand this, consider our example of XYZ stock. Suppose that instead of owning the call option, you purchased a share of XYZ stock itself and borrowed the $100 exercise price. At the option's expiration date, you sell the stock for $120, you pay back the $100 loan, and you are left with the $20 difference less the interest on the loan. Note that at any price above the $100 exercise price, this equivalence exists between the payoff from the call option and the payoff from the so-called "replicating portfolio."

But what about before the call option expires? Believe it or not, you can still match its future payoff by creating a replicating portfolio. However, to do so you must buy a fraction of a share of the stock and borrow a fraction of the exercise price. How do you know what these fractions are? That is what the Black-Scholes Formula tells you.

It states that the price of the call option, C, is equal to a fraction -- N(d1) -- of the stock's current price, S, minus a fraction -- N(d1 - ¦É sqrt(T) -- of the exercise price. The fractions depend on five factors, four of which are directly observable. They are: the price of the stock; the exercise price of the option; the risk-free interest rate (the annualized, continuously compounded rate on a safe asset with the same maturity as the option); and the time to maturity of the option. The only unobservable is the volatility of the underlying stock price.

If the current stock price is way above the exercise price, these fractions are close to 1, and therefore the call option is approximately the difference between the stock's current price and the present discounted value of the exercise price. If, on the other hand, the current stock price is way below the exercise price, these fractions are close to zero, making the value of the call option very low.

After the discovery of Bachelier's work there suddenly came to the mind of all the eager workers the notion of what the Holy Grail was. There was the next step needed. It was to get the perfect formula to evaluate and to price options.

I showed that video, didn't I?

Anyway...

Cool sketch of the wing.

ROBERT SOLOW, Professor Emeritus, Massachusetts Institute of Technology: We thought it was a great day when Kennedy decided to give that speech at Yale and to talk about economic policy. That speech suggested that we had won over Kennedy. We had won the heart and mind of the president.

JFK: What is at stake in our economic decisions today is not some grand warfare of rival ideology which will sweep the country with passion, but the practical management of a modern economy. What we need is not labels and clichˆ©s, but more basic discussion of the sophisticated and technical questions involved in keeping a great economic machinery moving ahead.

NARRATOR: Kennedy's council of economic advisors had drafted his speech along Keynesian lines.

Samuelson and Solow -- wingmates....

Interesting.

Hope you're having fun back there!

TKD