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Behavioral Finance, Racetrack Betting and Options and Futures Trading. William T. Ziemba Alumni Professor of Financial Modeling and Stochastic Optimization Sauder School of Business University of British Columbia, Vancouver, Canada Mathematical Finance Seminar Stanford University
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Options and Futures Trading
William T. Ziemba
Alumni Professor of Financial Modeling and Stochastic Optimization
Sauder School of Business
University of British Columbia, Vancouver, Canada
Mathematical Finance Seminar
January 30, 2004
Investing in traditional financial markets has many parallels with racetrack & lottery betting
If you bet you can make three spades, that’s entertainment.
If you bet cotton will go up three points, that’s business.
See the difference?
Unbridled at Claiborne Farms, Paris, Kentucky
Conclusion: spend your money getting good mean estimates and use historical variances and covariances
Reference: Chopra and Ziemba (1993), Journal of Portfolio Management, reprinted in Ziemba-Mulvey (1998) Worldwide asset and liability management, Cambridge University Press
Results similar in multiple period models and the sensitivity is especially high in continuous time models. See examples in AIMR, 2003.
QW/Wi if horse i is 1st
Typical behavior of the betting fortune of the average bettor
Probability of being even or ahead for the typical win bettor in California
Griffith 1949 study - 1386 races in 1947, Churchill Downs, Belmont and Hialeah 1934 data similar
Number of entries, winners and winners times odds for every odds group
Odds (subjective) vs percent winners (objective)
Reprinted in HLZ (1994)
The effective track payback less breakage for various odds levels in California and New York, more than 300,000 races over various years and tracks, Ziemba and Hausch (1986) Betting at the Racetrack.
Expected return per dollar bet with and without the track take deducted for different odds levels in the Kentucky Derby 1903-1986 and in 35,285 races run during 1947-1975, from data in Snyder (1978)
The better races have flatter biases; see references in Tompkins, Ziemba and Hodges (2003) The favorite/longshot bias in S&P500 and FTSE100 index futures options: the return to bets and the cost of insurance.
Has the bias in the US changed with rebates and betting exchanges? Yes.
It seems to be more flat.
Data for essentially all North American races for the last 7+ years, about 2 million horses in about 300,000 races.
We have a good data set: essentially all of the S&P500 futures puts and calls, all years, all strikes.
Do the buyers of puts and calls on the Stock Index options behave like racetrack bettors?
Who buys and sells them
Average return per dollar bet vs. odds levels: 3-month stock index calls, 1985-2002
Average return per dollar bet vs. odds levels: 1-month stock index calls, 1985-2002
This is essentially all the data on the S&P500 futures options; the graph is smooth with a few kinks.
Possible peso effect: you are paying for an event that has not yet occurred but could occur.
But this data includes October 1987, October 1989 and 2000-2002 crashes, so has a lot in it already.
You sell the options with negative expected value and buy those with positive expected value and have various levels of hedging using S&P500 futures.
20 years of actual trading with real money and extensive simulation and other research suggests which ones and when to sell/buy/hedge in the various strategies to provide good, steady gains with minimal risk.
Jan 2002 to present Oct 2003 to present
Inefficiencies are possible since:
1) more complex wager
2) prob(horse places) > prob(horse wins) ==> favorites may be good bets
To investigate place bets we need:
1) determine place payoffs
2) their likelihood
3) expected place payoffs
4) betting strategy, if expected payoffs are positive
Bettors do not like place and show bets.
Use data in a simple market (win) to generate probabilities of outcomes
Then use those in a complex market (place and show) to find positive expectation bets
Then bet on them following the capital growth theory to maximize long run wealth
So in the long run, the log utility investor dominates.
Security rises if the bet sizes are decreased with fractional Kelly (negative power) but growth decreases as well. Half Kelly is -w-1.
Fractional Kelly f=1/1-a, a<0, in awa is exact for lognormal, approx otherwise. See MacLean-Ziemba, Time to Wealth (2002).
Half Kelly is half the Kelly wager plus cash.
Right graph has dosage filter; do not bet on horses that cannot run 1 1/4 miles on the first Saturday in May.
Horses with high dosage cannot really win the Kentucky Derby or Belmont Stakes
See Bain, Hausch and Ziemba (2004) An application of expert information to win betting on the Kentucky Derby, 1981-2001; merging of prices (odds) with expert opinion (breeding measured by dosage).
The first two are “stock market” people and are closer to full Kelly. See text in AIMR, 2003.
Keynes: Graph of the performance of the Chest Fund, 1927-1945 in AIMR, 2003
Growth of assets, various high performing funds in AIMR, 2003
See Thorp (1998) for analysis showing Buffett is close to full Kelly.
Buffett’s Sharpe is below Ford’s but if you delete upside, it’s better, see AIMR, 2003.
Thorp and Benter - the “gamblers” like smooth wealth paths using fractional Kelly.
Princeton Newport Partners, L.P., Cumulative Results, November 1968-December 1988 in AIMR, 2003
Hong Kong racing syndicate to 1994, see Benter’s paper in Hausch, Lo, Ziemba (1994)
Hong Kong racing syndicate to about 2001.
You can have 700 independent bets all with a 14% advantage and still lose 98% of your fortune
Half Kelly loses a lot of the growth much of the time in exchange for a smoother wealth path.
Probability of doubling and quadrupling before halving and relative growth rates versus fraction of wealth wagered for Blackjack (2% advantage, p=0.51 and q=0.49
Betting more than the Kelly bet is non-optimal as risk increases and growth decreases; betting double the Kelly leads to a growth rate of zero plus the riskfree asset.
LTCM was at this level or more, see AIMR, 2003.
Pi = public’s place bet on i=1,…n
P = Pi
If i and j are the first two horses, gross return/dollar bet on i to place is:
Si = public’s show bet on i=1,…n, S = Si
If i, j and k are the first 3 horses, gross return/dollar bet on i to show is:
Hausch, Ziemba, Rubinstein (1981) Management Science and Ziemba-Hausch books. Academic papers in 1994 Academic Press.
Toteboard and place payoffs
If horses 2 and 3 are the first two horses, the gross payoff/dollar bet to place on horse 2 is:
$4.20 (with breakage) returned for each $1 place bet on i.
Efficient win market gives qi = Wi/W.
place probabilities determined using just public’s win odds.
Let qij = probability i is first and j is second
Harville (1973) proposed:
qij = prob(i first)•prob(j first if i not entered) = qi •
There are biases in these formulas; see HLZ (1994) - they are ok for this system though as the biases tend to cancel.
Probability i and j are 1st & 2nd in either order
Place payoff on i if i & j are 1st & 2nd in either order
Expected place payoff
Expected gross return on a $1 bet to place on horse i =
r w.p. q
0 w.p. 1-q
Optimal capital growth model
If a wager has a positive expected return, how much should be wagered?
Assume a sequence of independent wagers with the following return per $1 wagered:
If rq>1, what is the optimal fraction of wealth to invest each period?
Optimal capital growth model (Kelly 1956, Breiman 1961, Algoet and Cover, 1988)
The expected rate of growth of wealth from wagering fraction f of wealth each period for n periods is:
Therefore, maximizing the asymptotic expected rate of growth is achieved by myopic use of a logarithmic utility function, i.e.,
With one wager, the optimal fraction to bet is f*=edge/odds=p-q.
f*=64% for place/show; suggests fractional Kelly.
Optimal capital growth model, assumes our bet influences the odds and we can bet on multiple horses.
Non concave program but it seems to converge.
Favorites are usually underbet to win; this is useful in situations like the place and show systems. These are high probability low payoff situations.
Top number is final speed number, other numbers are pace within the race.