Free lesson · Prediction strategies
Binary Kelly sizing and estimation error
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Start with the idea
A favorable bet can still be too large. Kelly sizing maximizes expected logarithmic wealth under a very specific known-probability model.
Symbols, units & horizon
- f*: unconstrained fraction of wealth spent on a long binary claim
- p: true probability assumed known
- a: purchase price in USD per $1 claim, 0
- costs and discounting omitted
- one terminal betting horizon
When and why to use this
Understand how price, forecast edge and wealth allocation connect, then stress probability error and event overlap.
A favorable bet can still be too large. Kelly sizing maximizes expected logarithmic wealth under a very specific known-probability model.
Let f be the fraction of wealth spent buying a binary claim. If it succeeds, each dollar spent earns net odds (1−a)/a; otherwise the spent fraction is lost.
The full formula assumes a known probability and a single isolated bet. Forecast uncertainty, correlated events, liquidity and a need to preserve cash justify explicit caps or smaller fractions. A negative computed fraction means skip a long-only purchase, not automatically short.
Binary Kelly sizing and estimation error
- Write expected log growth as p ln(1+f(1−a)/a)+(1−p)ln(1−f).
- Differentiate and set p(1−a)/(a+f(1−a))=(1−p)/(1−f).
- Cross-multiply and solve f(1−a)=p−a; cap to the allowed long-only range.
p=.60 and a=.50 imply f*=.10/.50=.20. A half-Kelly rule would spend .10 of wealth, before applying further portfolio limits.
Apply it in a strategy
- Understand how price, forecast edge and wealth allocation connect, then stress probability error and event overlap.
- Record the input timestamp, executable quantity, currency and horizon. Reconcile the result with a cash-flow or state table.
- Stress this failure condition: Overestimated probabilities make the log-optimal calculation overbet; correlated contracts invalidate independent sizing.
Research deliverable
Build and explain a binary kelly sizing and estimation error worksheet. Understand how price, forecast edge and wealth allocation connect, then stress probability error and event overlap.
Evidence boundary: Synthetic arithmetic and scenarios illustrate mechanics. They are not historical returns, a paper replication, or evidence of an executable edge. Research sources and their access limitations are recorded at the end of this module.
Python implementation
Self-contained teaching example. Python 3.10+; dependencies and input conventions are shown in the code and notation. Run in your own Python environment.
# Python 3.10+; standard library unless NumPy is imported below.
# Inputs and outputs use the units defined in this lesson. Synthetic teaching example.
def binary_kelly(probability,price,fraction=1):
if not 0<=probability<=1 or not 0<price<1 or not 0<=fraction<=1: raise ValueError("Invalid inputs")
return fraction*max(0,min(1,(probability-price)/(1-price)))
print(binary_kelly(.60,.50),binary_kelly(.60,.50,.5))Continue learning
Prediction Strategies: Logic, Sizing and Market Making — all lessons- Complete-set purchases and redemption
- Subset relations and executable bounds
- Bounds for joint and union events
- Binary Kelly sizing and estimation error
- Decision buffers for probability uncertainty
- Quoting revenue and adverse selection
- Event overlap and portfolio variance
- Evaluate the decision process, including failed fills
Quantitative finance and development glossary · Python resources and libraries · Research sources and limitations