Tl;dr: The problem is that we have no way to bet on joint outcomes. If we add bets on joint outcomes, then the market is complete, we can combine the two outcomes into a single joint outcome, and Kelly criteria should work. To properly break Kelly, we need bets which resolve at different times.
This hits on a critical point which is fundamental to mathematical finance, but virtually unknown outside of it: complete markets. A “complete market” is one in which we can place any possible bet on whatever random variables are involved.
For instance, if we have a stock market with nothing but a single stock, and we’re betting on the stock’s price in the next time-step, then that’s an incomplete market: we have no way to place a bet which pays $1 if the price ends up within some window, and $0 otherwise. On the other hand, if we add in the full option chain (call options at every possible price), then the market is complete. We can pick a portfolio of options to make any possible bet on the stock’s price next timestep.
Mathematically, incomplete markets are a mess. You can’t get the bet you actually want, so you’re stuck trying to approximate it with the available bets, and that approximation gets messy.
On the other hand, if you do have complete markets, then you can combine everything into a single random variable and just use the Kelly criterion.
Chapter 6 of Cover & Thomas’ “Elements of Information Theory” gives good info on the Kelly criterion, how to derive it, and the relations between prices/probabilities and entropy/rate of return.
For math finance, the class I took back in college used Shreve’s “Stochastic Calculus for Finance II”. I wouldn’t necessarily recommend that just to learn about this, but it’s a good source for brownian motion, some basic measure theory, and the core theory of asset pricing.
Typically complete markets come up in discussing the fundamental theorem of asset pricing. The first part of the theorem says that any arbitrage-free set of asset prices has a “risk-neutral measure”, i.e. a market-implied set of probabilities. The second part says those probabilities are unique iff the market is complete—if some bets can’t be placed, then there are multiple possible market-implied probabilities. Any book which covers the fundamental theorem should have at least some coverage of complete markets.
Finally, if you’re looking for something more applied, Hull’s “Options, Futures and Other Derivatives” is the usual starting point.
Tl;dr: The problem is that we have no way to bet on joint outcomes. If we add bets on joint outcomes, then the market is complete, we can combine the two outcomes into a single joint outcome, and Kelly criteria should work. To properly break Kelly, we need bets which resolve at different times.
This hits on a critical point which is fundamental to mathematical finance, but virtually unknown outside of it: complete markets. A “complete market” is one in which we can place any possible bet on whatever random variables are involved.
For instance, if we have a stock market with nothing but a single stock, and we’re betting on the stock’s price in the next time-step, then that’s an incomplete market: we have no way to place a bet which pays $1 if the price ends up within some window, and $0 otherwise. On the other hand, if we add in the full option chain (call options at every possible price), then the market is complete. We can pick a portfolio of options to make any possible bet on the stock’s price next timestep.
Mathematically, incomplete markets are a mess. You can’t get the bet you actually want, so you’re stuck trying to approximate it with the available bets, and that approximation gets messy.
On the other hand, if you do have complete markets, then you can combine everything into a single random variable and just use the Kelly criterion.
Can you recommend me a good textbook that covers these things? I know basic econ (Krugman and Wells) and a bunch of probability and game theory.
Chapter 6 of Cover & Thomas’ “Elements of Information Theory” gives good info on the Kelly criterion, how to derive it, and the relations between prices/probabilities and entropy/rate of return.
For math finance, the class I took back in college used Shreve’s “Stochastic Calculus for Finance II”. I wouldn’t necessarily recommend that just to learn about this, but it’s a good source for brownian motion, some basic measure theory, and the core theory of asset pricing.
Typically complete markets come up in discussing the fundamental theorem of asset pricing. The first part of the theorem says that any arbitrage-free set of asset prices has a “risk-neutral measure”, i.e. a market-implied set of probabilities. The second part says those probabilities are unique iff the market is complete—if some bets can’t be placed, then there are multiple possible market-implied probabilities. Any book which covers the fundamental theorem should have at least some coverage of complete markets.
Finally, if you’re looking for something more applied, Hull’s “Options, Futures and Other Derivatives” is the usual starting point.
Thank you!