*Nobel Prize for Economics*was awarded jointly to Myron Scholes (Fischer Black's co-author of the paper on option pricing) and to Robert C. Merton (another pioneer in the development of the valuation of stock-options) [104]. A Nobel Prize is not awarded posthumously but Fischer Black would undoubtedly have been a joint winner of the

*1997 Nobel Prize for Economics*had he lived. In their announcement of the 1997 Prize, the Nobel Committee paid tribute to Black's key role [104]. The Black-Scholes-Merton partial differential equation for the price of a financial asset was derived in their famous paper [15], using Itô's Lemma, with the economics coming in by way of the observation, due to Merton R. C., that, if all the randomness/riskiness of a portfolio had been eliminated, the portfolio must return the risk-free interest rate because there is no such thing as a

*free lunch*(i.e. an arbitrage opportunity) in financial markets.

In 1959, Fischer Black earned his bachelor's degree in physics. In 1964, he earned a PhD from Harvard University in applied mathematics. He started his career working for the management consultants Bolt, Beranek & Newman but he 'discovered' finance when he moved, in the late 1965, to the consulting firm of Arthur D. Little. A seminal influence in his transition was Jack Treynor, who was also at Arthur D. Little at the time and was at one time the editor of the *Financial Analysts Journal*. Treynor was one of the co-developers, along with W. F. Sharpe (Nobel Prize for Economics, 1990) [105], of the celebrated *Capital Asset Pricing Model* (CAPM) [107] and [108]. Black started working on the CAPM with respect to both empirical and theoretical work. One of the classic papers of Black, Jensen and Scholes on the testing of the CAPM is [12]. The key insight of the CAPM was that the excess return of an individual stock (over the risk-free rate) is proportional (the so-called *beta* of the stock) to the excess return of the stock-market. Black viewed the excess return on an individual stock as being linked to the riskiness of that stock, otherwise no-one would buy the stock. He extended this idea into pricing options.

In 1969, Black founded his own consulting firm, Associates in Finance.

The question of how call-options should be priced had been the subject of long intellectual debates, commencing from the early sixties and other economists (for example see the 1961 paper by Spenkle, [106]) had come close to the Black-Scholes Formula. Paul Samuelson of MIT (Nobel Prize for Economics, 1970) had attacked the problem, from both theoretical and empirical points of view. Black and Scholes worked intensely on this problem, from 1968 to 1971, in a friendly competition with a student of Samuelson's, Robert Merton.

From 1972-75, Black was at the University of Chicago, Graduate School of Business first as visiting professor and then as a full professor.

The greatest jewel in Black's crown is undoubtedly the celebrated Black-Scholes formula. In 1973, Black published with Myron Scholes their famous paper entitled *The Pricing of Options and Corporate Liabilities* [15] which derived and solved the Black-Scholes-Merton differential equation thereby solving the stock-option pricing problem [Note 1] . In their paper was a reference to the book by Cootner [88] containing the translation from French of Bachelier's 1900 doctoral thesis for the École Polytechnique.

The solution was to demand a new level of mathematical technique. The solution worked using a number of steps :-

- the model of the stock-price in continuous time was represented by a special type of differential equation (so called
*stochastic*differential equation) which allowed for randomness in stock-price. The differential equation for the stock-price,*S*(*t*), was:-

*dS*(*t*) =*μ**S*(*t*)*dt*+*σ**S*(*t*)*dW*(*t*)

*μ*is the rate of investment return on the stock-price,*σ*is the volatility of the investment return on the stock-price and*W*(*t*) is Brownian Motion.When the above differential equation was integrated, it gave a stock-price distribution that was lognormal (i.e. the log of

*S*(*t*) had a normal distribution),

*S*(*t*) =*S*(0) exp {(*μ*-^{1}/_{2}*σ*^{2})*t*+*σ**W*(*t*)}

- the use of Itô's Lemma [Note 2], [94] and [95], to derive how the price,
*f*(*S*,*t*), of the stock-option would change if the stock-price,*S*(*t*), changed (a kind of*beta*for the option-price versus the stock-price i.e. very similar CAPM), - using a portfolio consisting of the stock-option and a continuously adjusted amount of the stock, Black and Scholes determined the amount of stock (subsequently called the
*delta*of the option), such that a portfolio consisting of one stock-option, together with shorting an amount*delta*of stock, would eliminate the randomness/riskiness of the portfolio. This continuously adjusted amount of stock was ∂*f*/∂*S*, the*delta*of the option. It was realised that this scheme would only work if the amount of stock was continuously adjusted as*delta*varied i.e. the adjustment had to be*dynamic*. Merton pointed out to Black and Scholes that, if all randomness/riskiness was eliminated, the portfolio would have to return the risk-free interest rate, otherwise there would be an opportunity for a*free lunch*(an arbitrage opportunity). The fact that stock-option was shown to be a*redundant*security that could be*made*out of traded securities enabled the option to be priced (by adding up the prices of the traded securities) and proved to have profound consequences.

The famous paper

*The Pricing of Options and Corporate Liabilities*[15] has two ways of deriving the relevant partial differential equation. The more elegant way proceeds as above, namely, if all randomness/riskiness is eliminated from a portfolio, then, following Merton, the portfolio must return the risk-free rate. The other way, is the method originally used by Black and Scholes and is given in their paper as an

*"Alternative Method"*. It uses the key insight of the CAPM, namely that excess return of an individual stock is proportional to the excess return of the market and the same applies to a stock-option, although with a different coefficient of proportionality.

Putting

*a*=

*μ*

*S*(

*t*) and

*b*=

*σ*

*S*(

*t*) in Itô's Lemma and exploiting that there are no

*free lunches*in a randomless/riskless portfolio, gave rise to the famous partial differential equation:-

*f*(

*S*,

*t*) is the price of the stock-option,

*S*is the stock-price,

*σ*is the volatility of the return on the stock (the standard deviation of the stock's return) and

*r*is the risk-free interest rate.

This equation is satisfied by the traded assets themselves, for example, *f* (*S*, *t*) = *S*(*t*) and by *f* (*S*, *t*) = *A* *e*^{rt} but these do not have the European call-option boundary conditions (at time *T*) that Black and Scholes were interested in (i.e. *f* (*S*(*T*), *T*) = maximum(*S*(*T*) - *K*, 0)).

By 1969, Black and Scholes had the above differential equation. They tried to solve this partial differential equation with the European call-option boundary condition but could not solve it. But Black and Scholes had noticed the curious absence (in the differential equation) of the investment return, *μ*, of the stock-price or any parameter representing the degree of preference, as to risk, on the part of option purchasers. If there was no *μ*, they wondered what would happen if both the stock-price and the stock-option-price both returned the risk-free rate, *r*. Sprenkle [Note 3] [106] had published a formula for the price of a call-option but Sprenkle's formula contained *μ* and a parameter representing risk preferences on the part of purchasers. But when Black and Scholes set *μ* equal to the risk-free rate, *r*, and set purchasers to require a return from the stock-option also of the risk-free rate, *r*, i.e. the return was the risk-free rate, irrespective of the purchaser's risk preferences, they found that Sprenkle's formula, with these adjustments, satisfied the partial differential equation. As Black himself is quoted as saying *... they had their formula* .

The famous Black-Scholes formula for the solution is:-

where

*Journal of Political Economy*for publication, who responded by rejecting their paper. Convinced that their ideas had merit, they sent a copy to the

*Review of Economics and Statistics*, eliciting the same response. After making some revisions based on the comments of Merton H. Miller (Nobel Prize for Economics, 1990) and Eugene Fama they submitted their paper again to the

*Journal of Political Economy*, who finally accepted it.

It is clear that the unexpected aspect of the Black-Scholes-Merton differential equation was not at first accepted. The unexpected aspect was the complete absence of the rate of investment return, *μ*, of the stock-price or any parameter representing the degree of risk-preference on the part of the option purchaser. Other investigators, such as Sprenkle [106], had these, as unknowns, in their option-pricing formula which were very similar to the Black-Scholes Formula (B-S Formula). Although this at first seemed hard to understand (even to Black and Scholes) in due course, it was realised that they and Merton were right and the stock-option price was independent of the stock-price return or investor preferences, as they were all incorporated into *S*(*t*) i.e. they did not enter *f* (*S*, *t*) other than through *S*(*t*). The fact that an option could be *made* by continuously adjusting simpler ingredients (the stock and cash) had not been realised before and, in due course, this realisation had profound consequences. Black, Scholes and Merton could not have realised, at the time, the enormous repercussions that this realisation was to have.

The B-S Formula assumes that the stock is a non-dividend stock and that interest rates are constant but Merton showed, in 1973, how to extend the formula to embrace both a dividend which is a percentage of the stock-price [Note 4] (thus handing options on the market index) and interest rates that are stochastic [Note 5] [102].

In 1975, Black left the University of Chicago to teach at MIT Sloan School of Management.

Black 1976 Model [Note 6] (known as Black's 76-formula) was also able to price options-on-commodity-forwards/futures assuming interest rates were non-stochastic [Note 7].

Indeed Black's 76-formula if slightly modified, by replacing *e*^{-rT} by *P*_{T}(*t*) (the price of a T-bond), enabled the pricing of European options on anything that had a log-normal distribution for the forward price under the appropriate measure, even if interest rates were stochastic.

It was not unreasonable to assume the forward T-bond-price [Note 8] was lognormal, so options on T-bonds could be priced. The formula [Note 9] for was initially thought to be something of an approximation when interest rates were stochastic. However advances in financial mathematics explored the effect of taking expectations with respect to special probabilities - the so-called *forward-measure* probabilities - under which the forward-price of assets (the ratio of the price of the asset to the price of the T-bond) were martingales. The expected value of the forward-price at time *T*, was shown to be equal to the current forward-price and this enabled Black's formula to be shown to be exact under log-normal assumptions for the forward-price (of the underlying asset) under the *forward-measure*. Thus, under appropriate assumptions, Black's extended 76-formula was shown to be exact, even allowing for interest rates to be stochastic.

Black's extended 76-formula enabled the valuation of *caps* (i.e. an upper bound for the applicable forward interest rates) on interest rates (assumed to be log-normal under the *'forward measure')* and swaptions (option on the swap rate, being the coupon rate on a par bond commencing at the exercise date of the option) assuming the forward swap rate is log-normal under the so-called *forward annuity measure*.

In 1984, Black who always had a practical streak, was invited to join Goldman Sachs on Wall Street. He accepted and left MIT. In 1986, he became a partner at Goldman Sachs and, in due course, became the Director of the Quantitative Strategies Group. He remained at Goldman Sachs until his untimely death in 1995.

Black was also active in the field in interest rates research where one of the problems was how to model the yield curve. As the redemption yields, on different duration bonds, vary with duration to redemption, the yield curve was, in mathematical terms, an infinite dimensional stochastic process (albeit with many constraints) and not the one-dimensional stochastic process formed by the stock-price.

Initial modelling of the yield curve was simplified by assuming that only one Brownian Motion drove the whole of the yield curve, although this assumption was known to lack realism and was in due extended to two or more Brownian Motion's driving the yield curve.

The Black-Derman-Toy model [Note 10], of 1990, [68] for interest rates gave a good fit to both the T-bond prices and their redemption yield volatilities. The Black-Karasinki model [Note 11] of 1991, [69] for interest rates assumed log-normality of the short interest rate and had one more parameter than the Black-Derman-Toy model.

Black was unusual on Wall Street being a soft-spoken thinker among fast talking merchant bankers and traders [Note 12]. His work encompassed investigations not just into option pricing but also into asset pricing (CAPM), design of financial markets, organisation of financial institutions and services, portfolio management (particularly of pension funds), portfolio insurance, taxation, economic behaviour, business cycles, monetary and banking theory.

Only someone with a subtle appreciation of history could have written a paper entitled, *'How we came up with the Option Formula* [56]'. Only someone with a well-developed sense of humour could have written a paper entitled *The Holes in Black-Scholes* [51] and followed it up with a paper entitled *How to Use the Holes in Black-Scholes* [57]. Black knew better than anyone that the assumptions made in deriving the B-S Formula were only borne out approximately in the market-place [51], for example, the stock-price could jump, and the distribution of stock-returns was fatter tailed than log-normal, the volatility of the stock-price return was not constant and the so-called *smile* (seen in the stock-option-prices quoted by the market) could not be explained by the *vanilla* B-S formula.

Black embodied the ideas of a true scholar and was quite ready to abandon a position, and admit being in error, when this was demonstrated convincingly to him. He was in the forefront of recognising the importance of computer technology and good trading/engineering systems and thought that trying to model reality was more important than closed form analytical solutions. He was widely consulted because he gave his objective, even if forthright, assessment of the work of others but, even when this was criticism, he did not make enemies as it was recognised that he had no 'side' to him.

In 1994, Black was diagnosed with throat cancer. Surgery at first appeared successful and Black was well enough to attend, in October 1994, the annual meeting of the *International Association of Financial Engineers* where he received their reward as *Financial Engineer of the Year*. But the cancer returned and Black died on 30 August 1995.

The Chicago Board of Trade had started trading options in 1973. The Black-Scholes formula, of the same date, was a memorable landmark in the history of finance. At a deep level, it created an understanding of options that paved the way for the large scale expansion of options markets. Without the Black-Scholes formula, only a small set of brave people would have been comfortable with using options in their day-to-day life. After the work of Black, Scholes and Merton, options, and their pricing, were reduced to simplicity and clarity.

This entire effort marked the coming of age of the field of *financial mathematics* which had been pioneered some seventy years before, by the Frenchman Louis Bachelier, the father of financial mathematics. Inter alia, Bachelier, had shown in his thesis [88] the close connection between random walks and the Fourier heat equation, something that was expanded on by Kac, in 1951, [98] and by Feynman [89], where it was shown that the solution of Fourier's equation could be expressed as the distribution function of a random variable arising of a large number of random walks each with *n* steps (and with each step size proportional to √(*t*/*n*)) and by letting *n* become very large (i.e. *n* tends to infinity in the limit).

Harrison, Krebs, Pliska [90], [91] and [92] showed that the Black-Scholes formula could be derived using an approach derived from probability theory and the theory of martingales. It was recognised that, in a *complete* market with no *free lunch*, there had to be a unique martingale measure, corresponding to the so-called traded numeraire portfolio, under which the process formed by the ratio of all traded portfolios to the numeraire portfolio, was a martingale process.

For example, if the traded portfolio was taken as the T-bond, price *P*_{T}(*t*), then, if a traded portfolio could replicate the option, then the ratio of the option price *O*(*t*) to *P*_{T}(*t*) had to be a martingale process under the probability measure induced by *P*_{T}(*t*) [Note 13], if there was to be no arbitrage in financial markets. Thus Harrison, Krebs and Pliska advanced financial mathematics by introducing the martingale approach to option pricing.

Today, these extended methods are used to price *exotic* derivative instruments as well as the *vanilla* options of the 1970s and 1980s. The financial derivatives industry today, which trades trillions of pounds a year, is built on the mathematical methods which were pioneered by Black, Scholes, Merton and Bachelier.

**Article by:** David O Forfar Heriot-Watt University.

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