The Wizards of Wall Street: Did Mathematics Change Finance? Paul Embrechts the Wizards of Wall Street: Did Mathematics Change Finance?

What field of science would one be talking about when hearing the following words: butterflies, eagles, rainbows, passports, Europeans, Americans, Asians, Parisians, Russians, knock-in, knock-out, barriers, swaps, swaptions, calls, puts, baskets, digitals, swings, down-and-under, . . .? This rather strange zoology of terms comes from the realm of modern finance. All the words above refer to so-called financial derivatives. Most of them are options. A derivative is a financial product written on (derived from) another financial product. The latter is typically referred to as the underlying. A typical example is a so-called European call option written on a stock. This product (derivative) gives the holder the right (not the obligation) to exchange for the underlying stock at a predetermined date and price. The date is referred to as maturity, the price as strike. The buyer pays the seller a premium for this right. For instance, suppose a bank on March 28, 2002 writes a European call with maturity 1 year and strike 160.– CHF on Swiss Re N, then it promises the buyer (holder) of the call to deliver him one year fromMarch 28, 2002 one stock Swiss Re N at the agreed price of 160.– CHF. A key question now concerns the calculation of a premium, fair for both buyer as well as seller. Swiss Re N closed on March 28 at 154.75 CHF. The Wizards of Wall Street mentioned in the title of this paper not only solved this problem, but also devised for the seller a perfect hedge; starting with the initial premium, they constructed a dynamic portfolio which allows the seller to exactly replicate the value of the call at maturity. In this paper, I will discuss some mathematical techniques used in solving the above problem. Special attention will be given to the conditions underlying the solution. The world of derivatives will be placed in its historic context. Besides a brief excursion into the realm of risk management, I will also make some comments on current and future research in the field.