HENRIËTTE PRAST traces                                                                                    Oct 2001
the history of gambling and
investigates the motivations of
gamblers past and present.

               he 17th-century mathematician and theolo-
               gian Blaise Pascal became fascinated by
               gambling as a phenomenon after being
               approached for help by a gambling addict
               who had got himself into financial difficulties.
Tradition has it that this meeting caused Mr. Pascal to
retreat to a monastery, where, ironically, he invented the                           IN THE LAP OF
roulette wheel. Pascal’s philosophising about gambling
also produced a rational explanation for the Christian
                                                                                         THE GODS
faith. If God exists, the religious have gambled well and         eventually elected to specialise in hazard, a game of
won themselves a place in heaven; if there is no God,             pure chance, which lost him 140,000 pounds within a
nothing is lost. Pagans, on the other hand, will end up in        year.
Hell, if God turns out to exist after all. According to              There are more indications suggesting that gambling
Pascal, this makes opting for the faith a rational bet.           addicts unconsciously go for losing. This very trait makes
    A recognised syndrome, gambling as an addiction has           them terrific opponents for those not burdened by an
inspired captivating stories, such as Arthur Schnitzler's         Oedipus complex. But if a trader senses an opportunity
Spiel im Morgengrauen and Dostoyewski’s The Gambler.              to rake in money, he had better undergo psychoanalysis
One important characteristic that distinguishes compul-           before tempting fate.
sive gambling from other addictions is that it does not
involve physical dependence. It is all in the head.               THE MARKET FOR MAGIC
    Psychoanalysis retraces the cause of a gambling
addiction to the victim’s childhood years, with an empha-         AND HOPE
sis on the Oedipus complex. The son’s wish to be the              Not all gamblers are addicted and suffering from an
mother’s object of love goes hand in hand with the desire         Oedipus complex. In fact, people have gambled since
to kill the father. Naturally, this makes for feelings of guilt   time immemorial, and even the venerable game of chess
and an ambivalent (love-hate) relationship with the               once began as a game of chance where dice determined
father. In psychoanalytical terms, fate is a surrogate for        the moves. The explanation psychologists adduce for
the father. The idea of fate or destination being mascu-          man’s natural propensity to gamble consists of two ele-
line and happiness feminine (Lady Luck), for that matter,         ments.
is at least as old as ancient mythology. Gambling is a               Firstly, man is always in need of stimuli. Hence, the
means to challenge fate and give vent to the said feel-           price of a lottery ticket should partially be regarded as
ings of ambivalence. No matter what, the outcome is               the amount someone is willing to fork out for the rush of
never satisfactory. If the gambler wins, it is as if he has       excitement he feels at the moment of the draw. That is
killed his father. He feels almighty, but despite this sensa -    why Adam Smith is mistaken by arguing that a lottery is
tion, or for this very reason, also quite guilty. If he loses,    nothing but a tax 'on all the fools in creation'. Smith posit-
he finds himself stuck with a financial debt.                     ed this theorem in response to the habit of the British
Psychoanalysts conclude from this that deep down gam-             Kings George I and George II to fill their coffers by set-
bling addicts prefer to lose. Indeed, it is less dramatic to      ting up lotteries.
incur large debts than it is to be rich but go through life          Secondly, man needs a magic world offering an
weighed down by guilt without ever having to do                   escape from the limitations of daily life whenever he feels
penance. Freud applies his views to a concrete case by            like it. Just as the adoration of a film star, pop idol or
finding a connection between the patricide in the                 fairytale-like princess, gambling presents a way out of
Brothers Karamazov and Dostoyevski’s own gambling                 reality to a dream world. Fantasising about what one
addiction as described in The Gambler.                            might do if one had ten million to spend affords much
    While indemonstrable, the Oedipus theory is supported         more satisfaction than calculating what is left for luxury
by the fact that the incidence of gambling addiction is           purchases after the daily shopping. The fact that the aris-
highest among men. Experimental research by behav-                tocracy and lowest social class alike have a more than
ioural psychologists has demonstrated that most casino            average proclivity to gamble fits this theory.1 The first cat-
frequenters refrain from following strategies that minimise       egory is bored and in dire need of a stimulus, while the
expected loss. The conduct of the 18th-century British            latter, having no prospect of moving up in society, is
aristocrat Charles Fox is a case in point. A wizard at            eager for a glimmer of hope.
whist, a game whose outcome can be influenced, Fox                   Playing cards for money, participating in a lottery and

                   www.w i l m o t t. com - Serving the Quant i t at i ve Fi n a n ce Co m m u n i t y
Henriëtte Prast In the Lap of the Gods                                                         Wilmott October 2001

the occasional visit to a casino are socially acceptable         could benefit from what had happened in the past.
these days. It is not all that long ago, though, that            Rubin only took risks once he was sure that, statistical-
these pastimes were viewed in quite a different light.           ly, the odds to win were sufficiently high.
Until well into the second half of the previous century,            Since, at two-hundred thousand dollars, Salomon
the church authorities used to caution believers about           underpaid him, Rubin took up a job with Merrill Lynch
the curse of gambling, and card playing was looked               in 1985. Despite the financial progress he was to make
upon as sinful in many a Protestant family. According            there – one million per year plus provision – Rubin
to psychologists specialising in the phenomenon,                 was in tears when he handed in his resignation. At
rather than from genuine concern about the welfare of            Salomon he had felt part of a family and a compara-
the common people, the church fathers’ stance in the             tively modest pay rise would have made him stay.
matter arose from their resolve to protect the church’s             On 29 April 1987, the Wall Street Journal reported
rationale. Gambling was a fearsome competitor of the             the gravest financial blow ever suffered on Wall Street
faith because it offered hope for a better life as well as       in a single transaction. Through a fault of Rubin’s,
creating the impression that a state of bliss could be           Merrill Lynch had lost 250 million dollars. Rubin was
attained with little effort. Indeed, the Protestant church       dismissed, but had no difficulty landing a job else-
teaches that the search for God is a laborious struggle.         where.
   The growing tolerance towards gambling seen in the               The big question, of course, is where things went
course of the 20th century is largely the result of the          wrong. Did Rubin take too many chances and, if so,
secularisation of society. The renewed interest in               why? Was it the change of ambience that made him
beliefs like Feng Shui underscores man’s need for                reckless? Apparently, Rubin was an emotional person.
magic and hope. Fortunately, those ashamed of being              Or should Rubin’s error be put down to hubris? After
susceptible to so much irrationality are always wel-             all, Rubin had been extremely successful. Had he
come to try their luck at the options exchange.                  become a full-blown gambling addict and had he want-
                                                                 ed to lose as a consequence? Did he unconsciously
ACED OUT                                                         begrudge Merrill Lynch a victory, because in his heart
                                                                 of hearts he wanted to be part of the Salomon clan?
Blackjack is the only gambling game that is not deter-
                                                                 Who is to say? Any hypothesis is more interesting than
mined by chance alone. The shrewd player can influ-
                                                                 the assumption that the market is unbeatable and that
ence the outcome by counting the cards.2 The fact that
                                                                 with hindsight Rubin might just as well have invested
most casino visitors unconsciously aim at losing (the
                                                                 in the index.
gambling addicts) or play for the thrill creates chances
for the rational gambler, the one who sits down at the
blackjack table well-prepared and poised to win. Howie
Rubin is a living example of this specimen. His track
record, described in Liar's Poker by Michael Lewis,
also shows the resemblance between the world of
gamblers and that of investors.
   Rubin was a chemist making less than twenty thou-
sand dollars a year. Bored with his daily routine, he
made up his mind after watching a television docu-
mentary on blackjack. He quit his job and left for Las
Vegas with three thousand dollars, which he augment-
ed to eighty thousand within two years’ time. With this
occupation, too, he grew bored, though, added to the
fact that it became increasingly hard for Rubin to gain
admittance to the casinos. One thing was certain in
any case: Rubin had found his vocation.
   After a two-year course at Harvard Business School,
Rubin joined Salomon Brothers in order to gamble on
Wall Street with other people’s money. He did fine. In
his first year he earned his employer $ 25 million, in
his second year $ 30 million. Rubin thought the atmos-
                                                                  1  The eagerness of the British Royal Family in horse
phere on the dealing floor much like that in Las Vegas.          betting is a case in point. In fact, the Duke of
In either case, a pokerface was indispensable, and               Edinburgh has been quoted as saying about his
where at the casino sipping at a gin-tonic was                   daughter ‘If it does not fart or eat hay, Anne is not
designed to suggest nonchalance, eating a cheese-                interested’. See Kitty Kelly (1997), The Royals, Warner
burger on the dealing floor created quite the same               Books.
                                                                   2 See Paul Wilmott on Quantitative Finance, John
   But the two occupations did not differ much in                Wiley and Sons (2000) for the winning strategy in this
nature either. Just as with blackjack, a clever dealer           game.

                  www.w i l m o t t. com - Serving the Quant i t at i ve Fi n a n ce Co m m u n i t y
                                                                              or indeed for convertible bond static data, have to be addressed in the
                                                                              general IT architecture surrounding FreeBound, and not at the level of
                                                                              the analytical engine itself.
                                                                              In order to turn FreeBound into a stand-alone application, ITO 33 has
                                                                              taken steps towards building this architecture. We have for instance

    TO 33 is an engineering company specialising in two, very often           developed a database tool that extracts CB static information from data
    inseparable, mathematical aspects of the derivatives pricing prob-        providers, stores it, and links it to the analytical engine. Also we have
    lem: the theoretical framework where the problem is formulated, and       developed a Front-Office Excel interface that allows the user to analyse
the algorithm to solve it.                                                    and monitor several Convertible Bonds (Theoretical value, implied
The two main products of the company, the convertible bond analyser           volatility, greeks, etc.) under live feeds.
FreeBound, and the incomplete market model for smiles in equity and
currency options, the I CARE project, perfectly illustrate this point.        II - SMILE PROBLEM : THE I CARE PROJECT
Although the theoretical model underlying convertible bond pricing can        The I CARE project is a major engineering program directed towards
be straightforward Black-Scholes, the complexity of the derivative            the smile problem. It combines financial theory and numerical imple-
instrument itself (combination of early conversion, early redemption          mentation at their highest level of sophistication. Its proximate aim is
and early put clauses, sometimes with variable triggers and strike lev-       the pricing of exotic structures (currency and equity barrier options,
els, discrete dividends, etc.) makes it so that accurate and efficient val-   convertible bonds, etc.) consistently with an implied volatility surface as
uation will require very sophisticated numerical schemes (finite differ-      may be inferred from vanilla option prices, through the Black-Scholes
ences or finite elements on adaptive grids). Not mentioning the credit        formula. Its deeper objective is a unified risk management framework
risk problem, a theoretical problem in its own right, which is probably       where:
the big problem of convertible bond pricing. As for the smiles, the theo-     a) the smile is explained through market incompleteness
retical models themselves carry complexity enough (jump diffusion, sto-       b) optimal dynamic hedge strategies are computed
chastic volatility) for the traditional ways of numerically solving the       c) the residual risk attaching to them (a dynamic Value at Risk, if you
Black-Scholes PDE to be no longer appropriate.                                will) is exhibited at last.
Instead of elaborating any further the theoretical and numerical compe-
tence of ITO 33, maybe the best thing to do in order to present the
                                                                              THE CARE-FREE APPROACH
company, is two produce two extracts of ITO 33’s own assessment of
                                                                              The only way to explain the smile in complete markets is the local
its methodology both in convertible bonds and smiles.
                                                                              deterministic volatility approach. Volatility is made a complete function
                                                                              of time and underlying in the Black-Scholes PDE, and the question is
I - CONVERTIBLE BONDS : FREEBOUND                                             asked what instance of such a function can make the PDE output
“AMERICA”                                                                     match the prices of the smiley options. A very interesting inverse prob-
The main strengths of our system is that we first make sure the con-          lem indeed, fraught with numerical difficulty, whose only motivation is
vertible bond problem is formulated in a sound theoretical framework,         “to keep up appearances”: to make it all look as if complete markets
and then take special care that the numerical solution very accurately        could cover the smile (or cover up for it).
reflects the theoretical solution.
On the theoretical side, this means we make it quite clear what default       THE CARE-FULL APPROACH
risk model has been superimposed on the traditional Black-Scholes             A more truthful approach is to look for alternatives to the Black-Scholes
model, and actually give the user the flexibility to himself choose           model, such as jump diffusion or stochastic volatility models (or a com-
among a variety of such models: Will the credit spread depend on              bination of both as the variety of smile shapes, both on longer term and
share value? Will the user wish to match a given market credit spread         shorter term option series, seems to indicate). Perfectly hedged
curve for a given spot value, then have this credit spread curve move         dynamic portfolios are no longer possible in this case, and the value of
with spot according to some predetermined schedule? Will the convert-         the option is no longer uniquely determined by the cost of the perfect
ible bond default entirely in case of a default event? Or will the holder     replicating strategy. Putting it differently, the market becomes incom-
recover a fraction of his investment? And will he be able to convert into     plete, and the risk neutral probability for expectation pricing is no
shares in case of default, etc. etc.?                                         longer unique.
Each one of this theoretical choices will lead to a different pricing equa-
tion which we will then have to solve numerically. On the numerical
side, our methodology is based on finite difference schemes for solving       INTRODUCING OUR HERO
of PDEs (Partial Differential Equations). The tree is the simplest            Truth about the option is no longer unique. Only in complete market
instance of such schemes, but is unfortunately unsuitable for instru-         models, which were blind to their own limitations, was the single output
ments as complex as the convertible bond: the theoretical value and           of “Theoretical Value” sufficient. As the story went, the option trader
greeks that it outputs are very poorly approximated. The scheme               would then question his choice of the volatility parameter, or, for that
underlying FreeBound makes use of an adaptive computational grid, so          matter, the validity of the whole Black-Scholes model, but this could
that events occurring during the lifetime of the instrument, such as          only take place outside the model.
coupons, dividends, early conversion, early redemption, or early put,         In incomplete markets, “truth about option value” is an undivided pack-
are always perfectly captured.                                                age. For you to believe it, not only does the pricing model have to pro-
Only if the theoretical pricing problem is properly framed, and the theo-     duce an output for option value, it has to produce the optimal replicat-
retical solution accurately estimated, will the convertible bond arbi-        ing strategy and the residual risk that attaches to the fact that this strat-
trageur make sure that his delta hedge is meaningful in terms of volatil-     egy is only optimal, that is to say, not perfect. Only then would you be
ity arbitrage. (The PDEs that FreeBound solves always derive from             able to appraise, as one package, the meaning of option value, the
theoretical arguments similar to the Black-Sholes argument of continu-        meaning of the hedge, and the meaning of the optimising criterion you
ous dynamic hedge and no arbitrage).                                          have chosen in the first place. To the extent that you care about the
How faithful to reality such a theoretical model will be is a different       option, the option gains value in your eyes.
question, and we certainly do not guarantee that our theoretical CB           This residual risk, we have christened H. E. R. O. or “Hedging Error at
value, or theoretical delta, will match the empirical delta or the empiri-    Replicating Optimum.” The optimal replicating strategy minimises the
cal price of the CB as both are observed in the market. We think it is        expected variance between the option payoff and the proceeds of the
up to the trader, or the CB quant, to fill in the gap between model and       self-financing hedge portfolio, and HERO is this minimum variance.
reality, and to be critical of the theoretical model he is using.             HERO can be computed for a single option trade, or indeed for a whole
FreeBound is just a tool after all, and the FreeBound user is at least        option book.
guaranteed that the theoretical benchmark is sound and quite reliable.
The delta (or indeed any greek) that is output by FreeBound will              The bigger the market incompleteness (bigger jumps in the underlying,
always be coherent with the scenario of CB theoretical values that is         more volatile volatility), or the harder a particular option to replicate
output by FreeBound.                                                          (very exotic structure), the more HEROic is the trade in itself. But
As such, FreeBound is just the analytical engine sitting at the heart of a    HERO is non linear. And you should expect the incremental HERO
general risk management system. We have taken care that FreeBound             experienced on some particular trade to differ from trader to trader,
is easily interfaceable with any such system. Consequently, general           depending on the option book each of them holds. Hence a trader who
questions that usually preoccupy the convertible bond analyst and sys-        outbids another on some particular option trade is not necessarily more
tem manager, such as the possibility of live feeds to generate inputs for     heroic
underlying share values, or implied volatility, or yield and spread curve,

To top