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A Mathematician on Wall Street

Column 1 Option Theory: What I Knew and When I Knew It – Part I by Edward O. Thorp Copyright 2001 One of the themes of this column will be how and to what extent markets are inefficient, and how you might profit from this. Let’s begin by going back in time to the early days of quantitative finance. Paul Cootner’s book, The Random Character of Stock Market Prices, M.I.T. Press, 1964, presented much of the work that had been done on the random walk theory of stock prices and on the problem of warrant pricing. The warrant valuation problem was essentially the problem of valuing options and, more generally, derivatives. Progress was substantial but the Black-Scholes breakthrough would not appear until 1973. Meanwhile, in 1965 Eugene Fama proposed that markets were well described as “efficient,” with all-knowing rational participants who acted quickly on their information, causing securities prices to properly and rapidly adjust to correctly reflect current knowledge. I arrived on this scene with a unique perspective. In 1959-60, I had discovered that the casino game of blackjack could be beaten, and I devised and demonstrated a mathematical system to do so, based on keeping track of which cards had been played. Announced in December of 1960 and in January of 1961 (Proc. N.A.S.) and published in detail in my Beat The Dealer (1962; revised 1966), the system showed that the blackjack “market” was “inefficient.” In a similar investigation of other gambling games, I discovered how to beat roulette by physical prediction (1955-61) and, with Claude Shannon (of Information Theory fame) built the first wearable computer (1961), whose function was to successfully predict roulette outcomes. The predictions of the computer gave us the huge positive expectation of 44%. Shannon and I then used the computer successfully in Las Vegas to win small sums. The casino gambling “market” had yet another “inefficiency.” For more, see Thorp (1969, 1998) and http://wearables.www.media.mit.edu/wearables/. Click “history,” then click “a brief history of wearable computing timeline.” I investigated several other gambling games with some additional successes, and by 1964 I began to consider the greatest gambling game of all time, the stock market. Whereas I thought of card counting in blackjack as a million dollar idea, my stock market explorations would lead to a hundred million dollar idea. Here’s a brief history of what happened. 1964 I spent an intensive summer introducing myself to the markets, investments and finance. 1964-1965 I discovered, and in some cases rediscovered, curious minor technical patterns in stock prices. 1965

At the start of the summer I resumed intensive study of the market and discovered warrants. I understood at once that there was a qualitative link between the price behavior of a warrant and its underlying stock and that it could be more or less quantified and that it could likely be an exploitable market inefficiency. (See Chapter 2, Beat the Market). 1965-1966 I met economist Sheen Kassouf in the fall of 1965 when we both arrived as founding faculty members at the new Irvine campus of the University of California. Kassouf was already hedging warrants and we collaborated, publishing much of our work in Beat the Market. We understood static hedging, dynamic hedging, and delta hedging, in particular market neutral delta hedging. We documented a historical market inefficiency: warrants with two years or less to expiration tended to be very overpriced. We used a static hedge to simplify historical simulations for the reader. We used dynamic hedging in our own portfolios, making adjustments in moderate sized steps both to limit risk and to limit the cost of doing so. In practice my hedge ratio, or “delta,” was determined both by the macrostructure of the payoff (wide range of protection against large price changes in either direction) and the microstructure (market neutrality against small changes). It became increasingly clear to me as I thought and invested during 1966-1968 that the clearest proof of an exploitable market inefficiency is to construct a low (ideally zero) risk (hedged) package that has little covariance with the market, yet produces a substantial “risk adjusted excess return.” 1967 Beat the Market appeared. Kassouf and I ended our collaboration with the completion of Beat the Market in late 1966, and went our separate business ways. I had become familiar with Cootner’s 1964 book and various warrant valuation models based on integration. I had earlier concluded that using the log normal distribution for stock price changes and computing the expected terminal value of a warrant, led to a reasonable candidate for a warrant formula. I learned from reading Cootner that several people had already attempted this, in various ways. The version I liked, which I give in Thorp (1969), was, as I learned only in 1975 from Larry Fisher, first derived by Boness. However I do not see this in the Boness paper that is in Cootner, nor elsewhere in my copy of Cootner (the original 1964 hardcover), but did note Sprenkle’s related attempt (Cootner page 466). “My” formula had a growth rate M for the stock and, in the case where the warrant or option short sale proceeds are credited to the account at once, a discount rate d for the present value. Note: M = m + v 2 2 where m is the log normal drift parameter and v is the volatility. E ( S (t ) ) = S (0) exp( Mt ) is the expected value of the stock at time t if S (0) is the initial price. 1967-1968 I puzzled over the two parameters M and d , and speculate that in a risk neutral world I can set them both equal to r , the riskless rate corresponding to the time until expiration (which, as it happens, gives the future Black-Scholes formula). I also note that a continually adjusted “delta neutral” hedged portfolio is riskless and so should have its value discounted at the rate r. This also suggests the simple idea of just setting M

and d both equal to r. I mention a couple of these 1967 arguments long after the fact in Thorp (1975). Occam’s razor and these plausibility arguments suggested to me that if there was a simple formula, this (the future Black-Scholes formula) is it. I didn’t see how to prove the formula but I decided to go ahead and use it to invest, because there was in 1967-68 an abundance of vastly overpriced (in the sense of Beat the Market) OTC options. I used the formula to sell short the most extremely overpriced. I had limited capital and margin requirements were unfavorable so I shorted the options (typically at two to three times fair value) “naked,” i.e. without hedging with the underlying stock. As it happened, small company stocks were up 84% in 1967 and 36% in 1968 (Ibbotson), so naked shorts of options were a disaster. Amazingly, I ended up breaking even overall, on about $100,000 worth of about 20 different options sold short at various times from late ’67 through ’68. The formula has proven itself in action. I estimated volatility from the values of log( H / L), where H and L are the monthly highs and lows, using the last twelve months which were easily and quickly obtained from my library of S&P monthly stock guides. A previously published math paper (Anderson, 1951?) gives properties of this volatility estimator, including expected value and standard deviation. My recollection as I write now is that E log( H / L), the expected value of H / L , was SQRT (8 / PI ) ∗ SIGMA where SIGMA is the volatility parameter in the assumed underlying lognormal distribution. The use of the volatility estimator log( H / L) prefigures a 1977? paper on this by Parkinson and a 1980 paper by Garman. Garman combines this and other volatility estimators into a grand overall estimator. 1967-68 Comment: My naked calls sold short were probably the world’s first actual investment use of the future BS formula. END – Column 1.

References [1] [2] [3] [4] [5] [6] [7] [8] [9]

Black, Fisher and Myron Scholes. “The Pricing of Options and Corporate Liabilities.” J. Political Economy 81, 1973, 637-54 Cootner, Paul (editor), The Random Character of Stock Market Prices, M.I.T. Press, 1964. Thorp, Edward O. “Fortune’s Formula: A Winning Strategy for Blackjack.” Notices of the American Mathematical Society, Dec. 1960: 935-936. _____. “A Favorable Strategy for Twenty-One.” Proceedings of the National Academy of Sciences, 47, No. 1, (1961) :110-112. _____. Beat The Dealer. New York: Random House, 1962. _____. Beat The Dealer, 2nd Ed., New York: Vintage, 1966. _____. “Optimal Gambling Systems for Favorable Games.” Review of the International Statistical Institute, 37, 1969, 273-293. _____. “The Invention of the First Wearable Computer.” Proceedings of the Second International Symposium on Wearable Computers, Pittsburgh, PA, October 19-20, 1998. Thorp, Edward O. and S.T. Kassouf. Beat the Market. New York: Random House, 1967

Option Theory: What I Knew and When I Knew It – Part 2 by Edward O. Thorp Copyright 2001 In November 1969 l and a partner, Jay Regan, launched what I believe was the world’s first market neutral hedge fund. We called it Convertible Hedge Associates (CHA), and later changed its name to Princeton-Newport Partners (PNP). It used warrants, OTC options and convertible bonds and preferreds, along with the underlying common stock, to construct delta neutral dynamically adjusted hedges. (Listed options and publication of the Black-Scholes formula were still almost four years in the future). Since “the formula” was, to me, highly plausible but not proven, we used in addition a variety of techniques and screens, all of which the proposed mispriced security had to pass: [1] the formula (available for options and warrants, suitably modified for when and to what extent the economic value of short sale proceeds are actually available; generally not until expiration; in those days the brokers pocketed it.) [2] scatter diagrams of prices: derivative versus stock or derivative versus derivative, over time (e.g. Figure 2.2 of Beat the Market). [3] cross-sectional scatter plots on standardized coordinate diagrams (at a fixed time, such as that day’s closing prices) to compare derivatives within a class (e.g. Figure 10.2 of Beat the Market). 1969-1972 The first market neutral hedge fund, which consisted of a collection of derivatives hedges, each of which was (dynamically) approximately delta neutral, prospered. See track records in (Thorp, 71, 75, 00). Early 1973 The CBOE announced it would soon begin trading exchange-listed options. We at Convertible Hedge Associates were electrified (figuratively and literally) by the news. This could facilitate a major expansion of our business. I had an HP 9830A desktop computer which was easy to program in BASIC, was math user friendly, and drove a pen plotter with which we drew magnificent color coded graphs. I had the “integral formula” programmed and drawing option and warrant curves when, out of the blue I got a letter and an article from someone called Fisher Black. He said “I am an admirer of your work” and explained that his approach was like Beat the Market but he and Scholes took another step: they explored the (analytic) consequences of the no arbitrage principle as applied to our (dynamically adjusted) delta neutral hedge, noting that such a hedge should then return the (appropriate time period) riskless rate on net equity invested. I sat down, programmed in his formula and drew option curves. Shock! The graphs disagreed with my graphs. It couldn’t be. But then I realized I was graphing the “short warrants or options, long stock” version of my formula. This version assumed that the interest from the proceeds from the short sale of the warrant or option was captured by the broker, not the investor, as was the practice then for the warrants and over the counter options which I had been trading. But the proceeds from shorting listed options (only calls on a limited number of large companies were initially available when the C.B.O.E. opened in 1973) would be credited to the investor on settlement date for the trade. Thus one needed to pre-multiply by exp( − r (t ∗ − t ) ) to discount the expected terminal value of the warrant to expected present value. Now the graphs were identical!

1973 What I already had, in fact, was not just the Black-Scholes formula but a more general pair of formulas, with the Black-Scholes formula as the limiting case. One of them incorporated a parameter to account for the loss to the broker of some or all, as the case might be, of the interest earned by the short sale proceeds (SSP interest) on the warrant (or option) short versus stock long hedge. This family of curves started with the Black-Scholes curve (all SSP interest available) and moved continuously higher as the fraction of available SSP interest dropped, with the highest curve being my old warrant curve, corresponding to no SSP interest available. The other formula covered the warrant (or option) long versus stock short hedge. This one parameter family started with the Black-Scholes curve and had successively lower curves as the economic value of the stock SSP interest available to the investor was reduced. The equations for highest and lowest curves are presented in Thorp (1973). That was written a few weeks after I got the Black-Scholes paper and was immediate because I already knew these formulas. In the original version of Thorp (1973) I had a section showing how I had found the Black-Scholes formula by setting M and d equal to r in the formula for the expected value of the warrant or option (as discussed in the previous column). But I had to delete this to fit my abstract into the spaced allowed. The two formulae create a “band” around the Black-Scholes value, within which the delta neutral hedger cannot expect to achieve the riskless rate. This band widens further when one adjusts the required pair of stock and option (or warrant) prices to cover (expected) transactions costs, present and future. As years passed, industry practice changed with competitive pressures and investors tended to gain some of the interest from their short sale proceeds, splitting this economic benefit with their broker-dealer. Currently, in the U.S. some hedge funds and other institutional investors get an interest credit equal to Fed Funds (a proxy for the “riskless rate” r ) minus seventy five basis points (0.75% annualized) or better. So the pair of one parameter families has remained relevant. Yet, even today they have not, as far as I know, been discussed in the literature. This is curious, given their practical value for so many users of the Black-Scholes formula. 1973 Planning ahead for the opening of the CBOE, I had prepared a catalog of standardized call option diagrams (see Beat the Market, chapter 6 for standardized variables), of (option price)/(exercise price) versus (stock price)/(exercise price). For stocks which paid no dividends during the life of the call option, for each of a range of r and v (volatility) pairs there was one set of curves for various times until expiration. These “universal” Black-Scholes curves covered all cases where our hedge was short CBOE listed calls (full cash credit at once for SSP) versus the underlying common stock long. We knew how to use numerical methods to calculate correct values for the option price in cases where the stock paid dividends during the life of the option, but is was usually sufficient to use easy approximations which covered most cases and could be incorporated as a quick correction directly on the graphical plot. Remember, this was 1973 when computing power was comparatively limited, scarce and expensive. With market prices continually changing and the number of options expanding rapidly, plus the need to monitor a substantial list of warrants and convertibles, graphical short cut methods were valuable in this era. We simply plotted the latest recent (stock, option) price pairs on the appropriate r, v diagram and looked to see whether it was far enough above or below the appropriate curve to offer a profitable

hedge. Delta, the hedge ratio, corresponded to the slope of the tangent and could be immediately read off the picture. We expanded the r, v catalog of diagrams as needed.

Option Theory: What I Knew and When I Knew It – Part 3 by Edward O. Thorp Copyright 2001 For the reverse case of stock short and options (or warrants) long, we used one of my formulas extending the Black-Scholes model to draw another catalog of curves, also correcting for dividends as needed. As computing time allowed, we drew custom graphs for the more important individual situations. For instance, I have from the fall of 1973 three separate pages of custom curves for hedging Chrysler warrants, accurately corrected for dividends. One set covered stock long versus warrants short, the second dealt with stock long versus listed call options short, and the third showed stock short versus options or warrants long. These latter did not allow for early exercise, so they were lower bounds only. We knew how to numerically solve the problem for early exercise but didn’t because our limited computer power was better used elsewhere. 1973 After seeing the Black-Scholes derivation, I explored the power series approach in 1973-74. After seeing my write-up in Thorp (76), Black told me that they were already aware of some of the power series ideas in 1969-70. 1974 Using power series, I developed what we dubbed internally as DOP, the “diversified option portfolio.” This extended and generalized the idea of delta hedging and measured the risk of arbitrarily complex hedges that were constructed from any or all of the available derivative securities (e.g. options, warrants, convertibles) on a single underlying common. We tried to achieve excess expectation while minimizing the exposure to the various power series terms. We considered not only delta neutrality but we also reduced our exposure to what people later called “the Greeks,” and for which they later used names like gamma, vega, theta, etc. Again, this idea was well ahead of the literature and, the important point for us, gave us an edge over the practitioners with whom we competed. 1973 fall Using the integral method, we easily solved and programmed the numerical calculation of warrant values for dividend-paying stocks. We had been generating custom graphs for each listed warrant and approximating the dividend correction. Now we had it exactly. 1974 We discovered the analytic solution (in terms of multivariate normal distributions) to the problem of call options on dividend-paying stocks, assuming that early exercise is not optimal (which is true unless the dividends are “large” or one of them is “close” to the expiration date). However we preferred our numerical solution because it covered all cases and was computationally efficient. Robert Geske discovered and published (1979, 1981) the same analytic solution in what, if I recall correctly, he called his “compound option model.” 1974 fall We were told that the CBOE would start trading American puts “soon.” After the Black-Scholes formula, this was the central unsolved problem in option theory. Because

of the ease and power of the integral model, I was able in an hour to conceive and outline the solution to the problem for my programmer. It used recursive numerical integration of the log normal probability density function for the stock, using the appropriate riskless rate for the expected growth rate and for the discount to present value, as described earlier. All the boundary conditions were incorporated and the time and space steps used for the backward integration were chosen fine enough to give the desired accuracy. This solution was complete: it incorporated dividends and determined the early exercise region. We drew graphs and printed tables. At a one-on-one dinner meeting to which Fisher Black invited me on May 14th or 15th, 1975, just prior to the University of Chicago Center for Research in Securities Prices (CRSP) meeting in Chicago, I brought along my solution to the American put problem and had placed a folder of graphs on the table to show him. Then he said no one had solved the problem, and asked what I thought about the “numerical solution to p.d.e.” approach. While I was giving my view that the approach worked but one had to be careful in choosing the sizes and relative sizes of the time and space steps (I had already looked at it and seen how to do it, but chose instead to use the integral method as “easier.”) I realized I had a fiduciary duty to my investors to keep our secrets, and quietly put my folder with the world’s first American put curves back in my briefcase. Schwartz and Parkinson each published solutions in 1977 that were “cousins” to our integral method version. 1974 In my classes at U.C. Irvine, I taught that there were three roads to option theory: [1] [2]

[3]

The integral model, “all powerful” for producing numerical (and some analytical) solutions. Stochastic differential equations (Black-Scholes 1973, Merton 1992), the most elegant and technically demanding approach. Useful for producing analytic formulas when they existed but they did not always lend themselves as easily to solving problems numerically. Power series: very useful for special problems.

I didn’t “teach” a fourth method, the Monte Carlo approach, since it seemed obvious and, though appropriate for solving various options problems, had much wider applicability. Later, Bill Sharpe’s suggestion led the finance world to the development of a fifth method, the binary model. This is closely related to the integral model, just as discrete binary random walks in the limit tend to Brownian motion. 1975-1985 We were able to stay ahead using the integral model, then later converting to the binary model for greater computational speed, but as the financial literature advanced much of our theoretical edge in option theory slowly vanished. The Black-Scholes methodology revolutionized finance, “everyone” adopted it, and listed option spreads narrowed until only the competitors with the lowest costs could still extract excess risk adjusted returns. However in practice we were able to stay ahead in derivatives trading through our computer programs and applications, especially in convertible bonds and in the analysis of an expanding crop of new derivative products. There is a lot more to this account. I have cartons of rough notes, and research ideas which I explored, both in derivative theory and practice and in other areas of

finance. Keeping what we discovered secret, while benefiting from ongoing published academic work, was a major factor in producing some $250 million in profits for CHA/PNP. 2000 Convertibles and other derivative hedging is still profitable. The derivatives based market inefficiencies exploited in Beat the Market have expanded vastly in number and size and account for a significant part of today’s hedge fund industry. A salient example is the $8 billion Chicago based Citadel Investments with its domestic and offshore partnerships. Under its principal general partner, Kenneth Griffin, now in his early thirties, it recently celebrated its tenth anniversary with a 30% annual compound rate of return for the decade. It’s widely considered to be the most valuable hedge fund business in the world. Another concept, first discovered at Princeton-Newport in December 1979 or January 1980, is the core idea of what is now called statistical arbitrage. The more primitive “pairs trading” had already been discovered at PNP and used in minor ways in the late 70s. Based on the approximately Brownian motion structure of stock prices, the idea has led to a set of techniques for “draining energy” (i.e. money) from the ceaselessly excessive (see Schiller) fluctuations in stock prices. Note on the integral model: The key is the observation that growth and discount rates can all be set to r , the riskless rate. Cox-Ross later proved in 1976 that this is correct (see Merton, 1992, pp 334ff). As soon as I saw the Black-Scholes proof in 1973, I felt certain that this consequence of their result applied not only to call options but generally to all derivative problems using log normal diffusion for the underlying security. I then immediately implemented this in numerically solving by iterated numerical integration, backward from the terminal value, in “small” time steps. Fortunately for me, this method was ours alone to practice from 1973 to 1976, and even after the Cox-Ross proof, we didn’t know of other practitioners who adopted it.

References [10]

[11] [12]

[13]

[14]

[15]

Thorp, Edward O. “Portfolio Choice and the Kelly Criterion.” Proceedings of the 1971 Business and Economics Section of the American Statistical Association, 1971, 215-224. Reprinted in Stochastic Optimization Models in Finance, edited by W.T. Ziemba, S.L. Brumelle, and R.G. Vickson, Academic Press, 1975, 599620. _____. “Extensions of the Black-Scholes Option Model.” Contributed Papers 39th Session of the International Statistical Institute, Vienna, Austria, August 1973, 1029-1036. _____. “Options in Institutional Portfolios, Theory and Practice.” Proceedings, Seminar on the Analysis of Security Prices, Volume 20, No. 1, May 15-16, 1975, 229-252. Center for Research in Security Prices, Graduate School of Business, University of Chicago. _____. “Common Stock Volatilities in Option Formulas.” Proceedings, Seminar on the Analysis of Security Prices, Vol. 21, No. 1, May 13-14, 1976, 235-276. Center for Research in Security Prices, Graduate School of Business, University of Chicago. _____. “The Kelly Criterion in Blackjack Sports Betting and the Stock Market.” Finding the Edge: Mathematical Analysis of Casino Games, eds. Olaf Vancura, Judy A. Cornelius and William R. Eadington, University of Nevada, Reno Bureau of Business & Economic Res., 2000. Thorp, Edward O. and S.T. Kassouf. Beat the Market. New York: Random House, 1967.

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A Mathematician on Wall Street

Column 1 Option Theory: What I Knew and When I Knew It – Part I by Edward O. Thorp Copyright 2001 One of the themes of this column will be how and to what extent markets are inefficient, and how you might profit from this. Let’s begin by going back in time to the early days of quantitative finance. Paul Cootner’s book, The Random Character of Stock Market Prices, M.I.T. Press, 1964, presented much of the work that had been done on the random walk theory of stock prices and on the problem of warrant pricing. The warrant valuation problem was essentially the problem of valuing options and, more generally, derivatives. Progress was substantial but the Black-Scholes breakthrough would not appear until 1973. Meanwhile, in 1965 Eugene Fama proposed that markets were well described as “efficient,” with all-knowing rational participants who acted quickly on their information, causing securities prices to properly and rapidly adjust to correctly reflect current knowledge. I arrived on this scene with a unique perspective. In 1959-60, I had discovered that the casino game of blackjack could be beaten, and I devised and demonstrated a mathematical system to do so, based on keeping track of which cards had been played. Announced in December of 1960 and in January of 1961 (Proc. N.A.S.) and published in detail in my Beat The Dealer (1962; revised 1966), the system showed that the blackjack “market” was “inefficient.” In a similar investigation of other gambling games, I discovered how to beat roulette by physical prediction (1955-61) and, with Claude Shannon (of Information Theory fame) built the first wearable computer (1961), whose function was to successfully predict roulette outcomes. The predictions of the computer gave us the huge positive expectation of 44%. Shannon and I then used the computer successfully in Las Vegas to win small sums. The casino gambling “market” had yet another “inefficiency.” For more, see Thorp (1969, 1998) and http://wearables.www.media.mit.edu/wearables/. Click “history,” then click “a brief history of wearable computing timeline.” I investigated several other gambling games with some additional successes, and by 1964 I began to consider the greatest gambling game of all time, the stock market. Whereas I thought of card counting in blackjack as a million dollar idea, my stock market explorations would lead to a hundred million dollar idea. Here’s a brief history of what happened. 1964 I spent an intensive summer introducing myself to the markets, investments and finance. 1964-1965 I discovered, and in some cases rediscovered, curious minor technical patterns in stock prices. 1965

At the start of the summer I resumed intensive study of the market and discovered warrants. I understood at once that there was a qualitative link between the price behavior of a warrant and its underlying stock and that it could be more or less quantified and that it could likely be an exploitable market inefficiency. (See Chapter 2, Beat the Market). 1965-1966 I met economist Sheen Kassouf in the fall of 1965 when we both arrived as founding faculty members at the new Irvine campus of the University of California. Kassouf was already hedging warrants and we collaborated, publishing much of our work in Beat the Market. We understood static hedging, dynamic hedging, and delta hedging, in particular market neutral delta hedging. We documented a historical market inefficiency: warrants with two years or less to expiration tended to be very overpriced. We used a static hedge to simplify historical simulations for the reader. We used dynamic hedging in our own portfolios, making adjustments in moderate sized steps both to limit risk and to limit the cost of doing so. In practice my hedge ratio, or “delta,” was determined both by the macrostructure of the payoff (wide range of protection against large price changes in either direction) and the microstructure (market neutrality against small changes). It became increasingly clear to me as I thought and invested during 1966-1968 that the clearest proof of an exploitable market inefficiency is to construct a low (ideally zero) risk (hedged) package that has little covariance with the market, yet produces a substantial “risk adjusted excess return.” 1967 Beat the Market appeared. Kassouf and I ended our collaboration with the completion of Beat the Market in late 1966, and went our separate business ways. I had become familiar with Cootner’s 1964 book and various warrant valuation models based on integration. I had earlier concluded that using the log normal distribution for stock price changes and computing the expected terminal value of a warrant, led to a reasonable candidate for a warrant formula. I learned from reading Cootner that several people had already attempted this, in various ways. The version I liked, which I give in Thorp (1969), was, as I learned only in 1975 from Larry Fisher, first derived by Boness. However I do not see this in the Boness paper that is in Cootner, nor elsewhere in my copy of Cootner (the original 1964 hardcover), but did note Sprenkle’s related attempt (Cootner page 466). “My” formula had a growth rate M for the stock and, in the case where the warrant or option short sale proceeds are credited to the account at once, a discount rate d for the present value. Note: M = m + v 2 2 where m is the log normal drift parameter and v is the volatility. E ( S (t ) ) = S (0) exp( Mt ) is the expected value of the stock at time t if S (0) is the initial price. 1967-1968 I puzzled over the two parameters M and d , and speculate that in a risk neutral world I can set them both equal to r , the riskless rate corresponding to the time until expiration (which, as it happens, gives the future Black-Scholes formula). I also note that a continually adjusted “delta neutral” hedged portfolio is riskless and so should have its value discounted at the rate r. This also suggests the simple idea of just setting M

and d both equal to r. I mention a couple of these 1967 arguments long after the fact in Thorp (1975). Occam’s razor and these plausibility arguments suggested to me that if there was a simple formula, this (the future Black-Scholes formula) is it. I didn’t see how to prove the formula but I decided to go ahead and use it to invest, because there was in 1967-68 an abundance of vastly overpriced (in the sense of Beat the Market) OTC options. I used the formula to sell short the most extremely overpriced. I had limited capital and margin requirements were unfavorable so I shorted the options (typically at two to three times fair value) “naked,” i.e. without hedging with the underlying stock. As it happened, small company stocks were up 84% in 1967 and 36% in 1968 (Ibbotson), so naked shorts of options were a disaster. Amazingly, I ended up breaking even overall, on about $100,000 worth of about 20 different options sold short at various times from late ’67 through ’68. The formula has proven itself in action. I estimated volatility from the values of log( H / L), where H and L are the monthly highs and lows, using the last twelve months which were easily and quickly obtained from my library of S&P monthly stock guides. A previously published math paper (Anderson, 1951?) gives properties of this volatility estimator, including expected value and standard deviation. My recollection as I write now is that E log( H / L), the expected value of H / L , was SQRT (8 / PI ) ∗ SIGMA where SIGMA is the volatility parameter in the assumed underlying lognormal distribution. The use of the volatility estimator log( H / L) prefigures a 1977? paper on this by Parkinson and a 1980 paper by Garman. Garman combines this and other volatility estimators into a grand overall estimator. 1967-68 Comment: My naked calls sold short were probably the world’s first actual investment use of the future BS formula. END – Column 1.

References [1] [2] [3] [4] [5] [6] [7] [8] [9]

Black, Fisher and Myron Scholes. “The Pricing of Options and Corporate Liabilities.” J. Political Economy 81, 1973, 637-54 Cootner, Paul (editor), The Random Character of Stock Market Prices, M.I.T. Press, 1964. Thorp, Edward O. “Fortune’s Formula: A Winning Strategy for Blackjack.” Notices of the American Mathematical Society, Dec. 1960: 935-936. _____. “A Favorable Strategy for Twenty-One.” Proceedings of the National Academy of Sciences, 47, No. 1, (1961) :110-112. _____. Beat The Dealer. New York: Random House, 1962. _____. Beat The Dealer, 2nd Ed., New York: Vintage, 1966. _____. “Optimal Gambling Systems for Favorable Games.” Review of the International Statistical Institute, 37, 1969, 273-293. _____. “The Invention of the First Wearable Computer.” Proceedings of the Second International Symposium on Wearable Computers, Pittsburgh, PA, October 19-20, 1998. Thorp, Edward O. and S.T. Kassouf. Beat the Market. New York: Random House, 1967

Option Theory: What I Knew and When I Knew It – Part 2 by Edward O. Thorp Copyright 2001 In November 1969 l and a partner, Jay Regan, launched what I believe was the world’s first market neutral hedge fund. We called it Convertible Hedge Associates (CHA), and later changed its name to Princeton-Newport Partners (PNP). It used warrants, OTC options and convertible bonds and preferreds, along with the underlying common stock, to construct delta neutral dynamically adjusted hedges. (Listed options and publication of the Black-Scholes formula were still almost four years in the future). Since “the formula” was, to me, highly plausible but not proven, we used in addition a variety of techniques and screens, all of which the proposed mispriced security had to pass: [1] the formula (available for options and warrants, suitably modified for when and to what extent the economic value of short sale proceeds are actually available; generally not until expiration; in those days the brokers pocketed it.) [2] scatter diagrams of prices: derivative versus stock or derivative versus derivative, over time (e.g. Figure 2.2 of Beat the Market). [3] cross-sectional scatter plots on standardized coordinate diagrams (at a fixed time, such as that day’s closing prices) to compare derivatives within a class (e.g. Figure 10.2 of Beat the Market). 1969-1972 The first market neutral hedge fund, which consisted of a collection of derivatives hedges, each of which was (dynamically) approximately delta neutral, prospered. See track records in (Thorp, 71, 75, 00). Early 1973 The CBOE announced it would soon begin trading exchange-listed options. We at Convertible Hedge Associates were electrified (figuratively and literally) by the news. This could facilitate a major expansion of our business. I had an HP 9830A desktop computer which was easy to program in BASIC, was math user friendly, and drove a pen plotter with which we drew magnificent color coded graphs. I had the “integral formula” programmed and drawing option and warrant curves when, out of the blue I got a letter and an article from someone called Fisher Black. He said “I am an admirer of your work” and explained that his approach was like Beat the Market but he and Scholes took another step: they explored the (analytic) consequences of the no arbitrage principle as applied to our (dynamically adjusted) delta neutral hedge, noting that such a hedge should then return the (appropriate time period) riskless rate on net equity invested. I sat down, programmed in his formula and drew option curves. Shock! The graphs disagreed with my graphs. It couldn’t be. But then I realized I was graphing the “short warrants or options, long stock” version of my formula. This version assumed that the interest from the proceeds from the short sale of the warrant or option was captured by the broker, not the investor, as was the practice then for the warrants and over the counter options which I had been trading. But the proceeds from shorting listed options (only calls on a limited number of large companies were initially available when the C.B.O.E. opened in 1973) would be credited to the investor on settlement date for the trade. Thus one needed to pre-multiply by exp( − r (t ∗ − t ) ) to discount the expected terminal value of the warrant to expected present value. Now the graphs were identical!

1973 What I already had, in fact, was not just the Black-Scholes formula but a more general pair of formulas, with the Black-Scholes formula as the limiting case. One of them incorporated a parameter to account for the loss to the broker of some or all, as the case might be, of the interest earned by the short sale proceeds (SSP interest) on the warrant (or option) short versus stock long hedge. This family of curves started with the Black-Scholes curve (all SSP interest available) and moved continuously higher as the fraction of available SSP interest dropped, with the highest curve being my old warrant curve, corresponding to no SSP interest available. The other formula covered the warrant (or option) long versus stock short hedge. This one parameter family started with the Black-Scholes curve and had successively lower curves as the economic value of the stock SSP interest available to the investor was reduced. The equations for highest and lowest curves are presented in Thorp (1973). That was written a few weeks after I got the Black-Scholes paper and was immediate because I already knew these formulas. In the original version of Thorp (1973) I had a section showing how I had found the Black-Scholes formula by setting M and d equal to r in the formula for the expected value of the warrant or option (as discussed in the previous column). But I had to delete this to fit my abstract into the spaced allowed. The two formulae create a “band” around the Black-Scholes value, within which the delta neutral hedger cannot expect to achieve the riskless rate. This band widens further when one adjusts the required pair of stock and option (or warrant) prices to cover (expected) transactions costs, present and future. As years passed, industry practice changed with competitive pressures and investors tended to gain some of the interest from their short sale proceeds, splitting this economic benefit with their broker-dealer. Currently, in the U.S. some hedge funds and other institutional investors get an interest credit equal to Fed Funds (a proxy for the “riskless rate” r ) minus seventy five basis points (0.75% annualized) or better. So the pair of one parameter families has remained relevant. Yet, even today they have not, as far as I know, been discussed in the literature. This is curious, given their practical value for so many users of the Black-Scholes formula. 1973 Planning ahead for the opening of the CBOE, I had prepared a catalog of standardized call option diagrams (see Beat the Market, chapter 6 for standardized variables), of (option price)/(exercise price) versus (stock price)/(exercise price). For stocks which paid no dividends during the life of the call option, for each of a range of r and v (volatility) pairs there was one set of curves for various times until expiration. These “universal” Black-Scholes curves covered all cases where our hedge was short CBOE listed calls (full cash credit at once for SSP) versus the underlying common stock long. We knew how to use numerical methods to calculate correct values for the option price in cases where the stock paid dividends during the life of the option, but is was usually sufficient to use easy approximations which covered most cases and could be incorporated as a quick correction directly on the graphical plot. Remember, this was 1973 when computing power was comparatively limited, scarce and expensive. With market prices continually changing and the number of options expanding rapidly, plus the need to monitor a substantial list of warrants and convertibles, graphical short cut methods were valuable in this era. We simply plotted the latest recent (stock, option) price pairs on the appropriate r, v diagram and looked to see whether it was far enough above or below the appropriate curve to offer a profitable

hedge. Delta, the hedge ratio, corresponded to the slope of the tangent and could be immediately read off the picture. We expanded the r, v catalog of diagrams as needed.

Option Theory: What I Knew and When I Knew It – Part 3 by Edward O. Thorp Copyright 2001 For the reverse case of stock short and options (or warrants) long, we used one of my formulas extending the Black-Scholes model to draw another catalog of curves, also correcting for dividends as needed. As computing time allowed, we drew custom graphs for the more important individual situations. For instance, I have from the fall of 1973 three separate pages of custom curves for hedging Chrysler warrants, accurately corrected for dividends. One set covered stock long versus warrants short, the second dealt with stock long versus listed call options short, and the third showed stock short versus options or warrants long. These latter did not allow for early exercise, so they were lower bounds only. We knew how to numerically solve the problem for early exercise but didn’t because our limited computer power was better used elsewhere. 1973 After seeing the Black-Scholes derivation, I explored the power series approach in 1973-74. After seeing my write-up in Thorp (76), Black told me that they were already aware of some of the power series ideas in 1969-70. 1974 Using power series, I developed what we dubbed internally as DOP, the “diversified option portfolio.” This extended and generalized the idea of delta hedging and measured the risk of arbitrarily complex hedges that were constructed from any or all of the available derivative securities (e.g. options, warrants, convertibles) on a single underlying common. We tried to achieve excess expectation while minimizing the exposure to the various power series terms. We considered not only delta neutrality but we also reduced our exposure to what people later called “the Greeks,” and for which they later used names like gamma, vega, theta, etc. Again, this idea was well ahead of the literature and, the important point for us, gave us an edge over the practitioners with whom we competed. 1973 fall Using the integral method, we easily solved and programmed the numerical calculation of warrant values for dividend-paying stocks. We had been generating custom graphs for each listed warrant and approximating the dividend correction. Now we had it exactly. 1974 We discovered the analytic solution (in terms of multivariate normal distributions) to the problem of call options on dividend-paying stocks, assuming that early exercise is not optimal (which is true unless the dividends are “large” or one of them is “close” to the expiration date). However we preferred our numerical solution because it covered all cases and was computationally efficient. Robert Geske discovered and published (1979, 1981) the same analytic solution in what, if I recall correctly, he called his “compound option model.” 1974 fall We were told that the CBOE would start trading American puts “soon.” After the Black-Scholes formula, this was the central unsolved problem in option theory. Because

of the ease and power of the integral model, I was able in an hour to conceive and outline the solution to the problem for my programmer. It used recursive numerical integration of the log normal probability density function for the stock, using the appropriate riskless rate for the expected growth rate and for the discount to present value, as described earlier. All the boundary conditions were incorporated and the time and space steps used for the backward integration were chosen fine enough to give the desired accuracy. This solution was complete: it incorporated dividends and determined the early exercise region. We drew graphs and printed tables. At a one-on-one dinner meeting to which Fisher Black invited me on May 14th or 15th, 1975, just prior to the University of Chicago Center for Research in Securities Prices (CRSP) meeting in Chicago, I brought along my solution to the American put problem and had placed a folder of graphs on the table to show him. Then he said no one had solved the problem, and asked what I thought about the “numerical solution to p.d.e.” approach. While I was giving my view that the approach worked but one had to be careful in choosing the sizes and relative sizes of the time and space steps (I had already looked at it and seen how to do it, but chose instead to use the integral method as “easier.”) I realized I had a fiduciary duty to my investors to keep our secrets, and quietly put my folder with the world’s first American put curves back in my briefcase. Schwartz and Parkinson each published solutions in 1977 that were “cousins” to our integral method version. 1974 In my classes at U.C. Irvine, I taught that there were three roads to option theory: [1] [2]

[3]

The integral model, “all powerful” for producing numerical (and some analytical) solutions. Stochastic differential equations (Black-Scholes 1973, Merton 1992), the most elegant and technically demanding approach. Useful for producing analytic formulas when they existed but they did not always lend themselves as easily to solving problems numerically. Power series: very useful for special problems.

I didn’t “teach” a fourth method, the Monte Carlo approach, since it seemed obvious and, though appropriate for solving various options problems, had much wider applicability. Later, Bill Sharpe’s suggestion led the finance world to the development of a fifth method, the binary model. This is closely related to the integral model, just as discrete binary random walks in the limit tend to Brownian motion. 1975-1985 We were able to stay ahead using the integral model, then later converting to the binary model for greater computational speed, but as the financial literature advanced much of our theoretical edge in option theory slowly vanished. The Black-Scholes methodology revolutionized finance, “everyone” adopted it, and listed option spreads narrowed until only the competitors with the lowest costs could still extract excess risk adjusted returns. However in practice we were able to stay ahead in derivatives trading through our computer programs and applications, especially in convertible bonds and in the analysis of an expanding crop of new derivative products. There is a lot more to this account. I have cartons of rough notes, and research ideas which I explored, both in derivative theory and practice and in other areas of

finance. Keeping what we discovered secret, while benefiting from ongoing published academic work, was a major factor in producing some $250 million in profits for CHA/PNP. 2000 Convertibles and other derivative hedging is still profitable. The derivatives based market inefficiencies exploited in Beat the Market have expanded vastly in number and size and account for a significant part of today’s hedge fund industry. A salient example is the $8 billion Chicago based Citadel Investments with its domestic and offshore partnerships. Under its principal general partner, Kenneth Griffin, now in his early thirties, it recently celebrated its tenth anniversary with a 30% annual compound rate of return for the decade. It’s widely considered to be the most valuable hedge fund business in the world. Another concept, first discovered at Princeton-Newport in December 1979 or January 1980, is the core idea of what is now called statistical arbitrage. The more primitive “pairs trading” had already been discovered at PNP and used in minor ways in the late 70s. Based on the approximately Brownian motion structure of stock prices, the idea has led to a set of techniques for “draining energy” (i.e. money) from the ceaselessly excessive (see Schiller) fluctuations in stock prices. Note on the integral model: The key is the observation that growth and discount rates can all be set to r , the riskless rate. Cox-Ross later proved in 1976 that this is correct (see Merton, 1992, pp 334ff). As soon as I saw the Black-Scholes proof in 1973, I felt certain that this consequence of their result applied not only to call options but generally to all derivative problems using log normal diffusion for the underlying security. I then immediately implemented this in numerically solving by iterated numerical integration, backward from the terminal value, in “small” time steps. Fortunately for me, this method was ours alone to practice from 1973 to 1976, and even after the Cox-Ross proof, we didn’t know of other practitioners who adopted it.

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[13]

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Thorp, Edward O. “Portfolio Choice and the Kelly Criterion.” Proceedings of the 1971 Business and Economics Section of the American Statistical Association, 1971, 215-224. Reprinted in Stochastic Optimization Models in Finance, edited by W.T. Ziemba, S.L. Brumelle, and R.G. Vickson, Academic Press, 1975, 599620. _____. “Extensions of the Black-Scholes Option Model.” Contributed Papers 39th Session of the International Statistical Institute, Vienna, Austria, August 1973, 1029-1036. _____. “Options in Institutional Portfolios, Theory and Practice.” Proceedings, Seminar on the Analysis of Security Prices, Volume 20, No. 1, May 15-16, 1975, 229-252. Center for Research in Security Prices, Graduate School of Business, University of Chicago. _____. “Common Stock Volatilities in Option Formulas.” Proceedings, Seminar on the Analysis of Security Prices, Vol. 21, No. 1, May 13-14, 1976, 235-276. Center for Research in Security Prices, Graduate School of Business, University of Chicago. _____. “The Kelly Criterion in Blackjack Sports Betting and the Stock Market.” Finding the Edge: Mathematical Analysis of Casino Games, eds. Olaf Vancura, Judy A. Cornelius and William R. Eadington, University of Nevada, Reno Bureau of Business & Economic Res., 2000. Thorp, Edward O. and S.T. Kassouf. Beat the Market. New York: Random House, 1967.

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