## Volatility arbitrage

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Thist story was originally published in the May edition of Scientific American. We are posting it in light of recent news involving Lehman Brothers and Risk free option trading using arbitrage trailer Lynch. Beginning last year, an investor could buy or sell a contract whose value depended entirely on fluctuations in temperature or accumulations of rain, hail or snow. These weather derivatives might pay out, for example, if the amount of rainfall at the Los Angeles airport ranged between 17 and 27 inches from October through April.

They are a means for an insurer to help provide for future claims by policyholders or a farmer to protect against crop losses. Or the contracts might allow a heating oil supplier to cope with a cash shortfall from a warmer than risk free option trading using arbitrage trailer winter by purchasing a heating degree-day floor—a contract that would compensate the company if the temperature failed to fall risk free option trading using arbitrage trailer 65 degrees as often as expected.

Weather derivatives mark an example of the growing reach of a discipline called financial engineering. This bailiwick of high-speed computing and the intricate mathematical modeling of mathematicians, physicists and economists can help mitigate the vagaries of running a global business.

It entails the custom packaging of securities to provide price insurance against a drop in either the yen or the thermometer. The uncertainties of a market crash or the next monsoon can be priced, divided into marketable chunks and sold to someone who is willing to bear that risk—in exchange for a fee or a future stream of payments.

Jarrow, a professor of finance at Cornell University. The engineering of financial instruments has emerged in response to turbulence during recent decades in ever more interconnected world markets: The creative unleashing of new products continues with increasingly sophisticated forms of securities and derivatives—options, futures and other contracts derived from an underlying asset, financial index, interest or currency exchange rate.

New derivatives will help electric utilities protect against price and capacity **risk free option trading using arbitrage trailer** in newly deregulated markets. Credit derivatives let banks pass off to other parties the risk of default on a loan.

Securities that would help a business cope with the year bug have even been contemplated. This ferment of activity takes place against a tainted background. Concerns have also focused on the integrity of the mathematical modeling techniques that make derivatives trading possible. Despite the tarnish, financial engineering received a valentine of sorts in October.

Scholes and Robert C. Merton, two of the creators of the options-pricing model that has helped fuel the explosion of activity in the derivatives markets. Options represent the right but not the obligation to buy or sell stock or some other asset at a given price on or before a certain date. Another major class of derivatives, called forwards and futures, obligates the buyer to purchase an asset at a set price and time. Swaps, yet another type of derivative, allow companies to exchange cash flows—floating-interest- rate for fixed-rate payments, for instance.

Financial engineering uses these building blocks to create custom instruments that might provide a retiree with a guaranteed minimum return on an investment or allow a utility to fill its future power demands through contractual arrangements instead of constructing a new plant. Creating complicated financial instruments requires accurate pricing methods for the derivatives that make up their constituent parts. It is relatively easy to establish the price of a futures contract.

When the cost of wheat rises, the price risk free option trading using arbitrage trailer the futures contract on the commodity increases by the same relative amount. Thus, the relationship is linear. For options, there is no such simple link between the derivative and the underlying asset.

Einstein and Options The proper valuation of options had perplexed economists for most of this century. Remarkably, one component of the formula that he conceived for this purpose anticipated a model that Albert Einstein later used in his theory of Brownian motion, the random movement of particles through fluids. Other academic thinkers, including Nobelist Paul Samuelson, tried to attack the risk free option trading using arbitrage trailer. They foundered in the difficult endeavor of calculating a risk premium: The insight shared by Black, Scholes and Merton was that an estimate of a risk premium was not needed, because it is contained in the quoted stock price, a critical input in risk free option trading using arbitrage trailer option formula.

The market causes the price of a riskier risk free option trading using arbitrage trailer to trade further below its expected future value than a more risk free option trading using arbitrage trailer equity, and that difference serves as a discount for inherent riskiness. Here is a simplified example: To hedge against risks in changes in share price, the investor can buy two options for every share he or she owns; the profit then will counter the loss.

Hedging creates a risk-free portfolio, one whose return is the same as that of a treasury bill. As the share price changes over time, the investor must alter the composition of the portfolio—the ratio of the number of shares of stocks to the number of options—to ensure that the holdings remain without risk.

The Black-Scholes formula, in fact, is elicited from a partial differential equation demonstrating that the fair price for an option is the one that would bring a risk-free return within such a hedging portfolio. Variations on the hedging strategy outlined by Black, Scholes and Merton have proved invaluable to financial- center banks and a range of other institutions that can use them to protect portfolios against market vagaries—ensuring against a steep decline in stocks, for instance.

With the right option, investors can bet or hedge on any kind of uncertainty, from the volatility up-and-down movement of the market to the odds of catastrophic weather. When it did reach the Journal of Political Economy inits impact on the financial markets was immediate. Within months, their formula was being programmed into calculators. This number, however, could be estimated from the ups and downs of past prices.

Investors who buy options are basically purchasing volatility—either to speculate on or to protect against market turbulence. The more ups and downs in the market, the more the option is worth. An investor who speculates with a call—an option to buy a stock—can lose only the cost of purchase, called a premium, if the stock risk free option trading using arbitrage trailer to reach the price at which the buyer can exercise the right to purchase it. In contrast, if the stock shoots above the exercise price, the potential for profit is unlimited.

Similarly, the investor who hedges with options also anticipates rough times ahead and so may buy protection against a drop in the market. Physicists on Wall Street Although it can be reduced to operations on a pocket calculator, the mathematics behind the Black-Scholes equation is stochastic calculus, a descendant from the work of Bachelier and Einstein.

These equations were by no means the standard fare in most business administration programs. Enter the Wall Street rocket scientists: Moving from synchrotrons to trading rooms does not always result in such a seamless transition. But a physicist must remember that **risk free option trading using arbitrage trailer** can go up as well as go down.

Or else they may train physicists, engineers and mathematicians before moving on to Wall Street. As part of their studies, financial engineers in training learn about the progression of mathematical modeling beyond the original work of Black, Scholes and Merton. The basic Black-Scholes formula made unrealistic assumptions about how the market operates.

It takes a fixed interest rate as an input, but of course interest rates change, and that influences the value of an option—particularly an option on a bond. The formula also assumes that changes in the growth rate of stock prices fall into a normal statistical distribution, a bell curve in which events cluster around the mean.

Thus, it fails to take into account extraordinary events such as the or stock market crashes. Black, Scholes and Merton—and legions of quants—have spent the ensuing years refining many of the original ideas. Emanuel Derman, head of the quantitative strategies group at Goldman Sachs, is a physicist-turned-quant whose job over the past 13 years has been to tackle the imperfections of the Black- Scholes equation.

Derman, a native of Cape Town, South Africa, received his doctorate from Columbia University in for a thesis on the weak interaction among subatomic particles. In the late s Derman decided to leave academia: So in he went to Bell Laboratories in New Jersey, where he worked on a computer language tailored for finance. In Goldman Sachs hired him to develop methods of modeling interest rates.

He has worked there since, except for a year spent at Salomon Brothers. At Goldman, he met the recently **risk free option trading using arbitrage trailer** Fischer Black, and the two began working with another colleague, William W.

Toy, on a method of valuing bond options. Risk free option trading using arbitrage trailer remembers Black as a bluntly truthful man with punctilious writing habits who wore a Casio Data Bank watch. The Black-Scholes equation was to finance what Newtonian mechanics was to physics, Derman asserts. Nobody knows what to do next except extend it. Finance differs from physics in that no mathematical model can capture the multitude of ever mutating economic factors that cause major market perturbations— the recent Asian collapse, for instance.

Outside the domain of Wall Street, the parallels between physical concepts and finance are sometimes taken more literally by academics. Ilinski replaces an electromagnetic field, which controls the interaction of charged particles, with a so-called arbitrage field that can describe changes in option and stock prices.

Trading that brings the value of the stock and the option portfolio into line is called arbitrage. He goes on to show how his arbitrage field model elucidates opportunities for profit that were not envisaged by the original Black-Scholes equation. Ilinski is a member of the nascent field of econophysics, which held its first conference last July in Budapest. Nevertheless, literal parallelism between physics and finance has gained few adherents. Ilinski recognizes the controversial nature of his labors.

Whether risk free option trading using arbitrage trailer Richard Feynman or Fischer Black, the use of mathematical models to value and hedge securities is an exercise in estimation. Model risk comes in many forms. So can inaccurate assumptions underlying the model—failing to take into account the volatility of interest rates during an exchange- rate crisis, for instance. Many models do not cope well with sudden alterations in the relation among market variables, such as a change in the normal trading range between the U.

Langsam, a former mathematician who develops and tests models for fixed-income securities at Morgan Stanley. Morgan Stanley and other firms adopt various means of testing, such as determining how well their models value derivatives for which there is a known price. One modeler at the session—Derman of Goldman Sachs—defended his craft. He compared models to gedanken thought experiments, risk free option trading using arbitrage trailer are unempirical but which help physicists contemplate the world more clearly: We are sort of investigating imaginary worlds and trying to get some value out of them and see which one best approximates our own.

The creators of these newfangled instruments place the losses in broader perspective. The market value of the outstanding derivatives contracts themselves represents only a few percentage points of the overall figure but an amount that may still total a few trillion dollars.

More mundane investments can also hurt investors. Derivatives make the news because, like an airplane crash, their losses can prove sudden and dramatic. The contracts can involve enormous leverage. A derivatives investor may put up only a fraction of the value of an underlying asset, such as a stock or a bond.