Graph showing growth rate G on Y axis with increasing bet size f on X axis. A red dot represents the vertex of this curve and optimal bet size f.

Kelly Position Sizing for Options Using Geometric Brownian Motion and Monte Carlo Simulation

Dec 3, 2022

Summary
This project implements the Kelly Criterion for options positions, aiming to maximize the expected value of log wealth. It involves generating geometric Brownian motion paths, calculating terminal spot prices, and determining the return on premium at expiration for each path. By assessing various bet sizes, it identifies the optimal bet size (f*) that maximizes the growth rate. Results include growth rate curves, sampling distributions of optimal bet sizes, and mean returns at different forecasted volatilities, providing insights into position sizing strategies for options trading. References include Edward O. Thorp's work on the Kelly Criterion.
Github Repo.

This project focuses on the implementation of the Kelly Criterion, aimed at maximizing the expected value of log wealth for options positions. It leverages models from an existing repository that includes a Black-Scholes-Merton (BSM) model and an Option position model.

Requirements

The implementation requires the BSM model and Option position model from the specified repository to calculate optimal bet sizes and project returns on options positions.

Approach

The method involves simulating N geometric Brownian motion paths using inputs like annualized volatility, current spot price, risk-free rate, days to expiration, and N itself. Terminal spot prices are calculated from these paths to derive the return on premium at expiration for different option position scenarios, ultimately determining the optimal bet size (f*) that maximizes the expected growth rate (G).

Results

Analysis shows a growth rate curve G(f) peaking at approximately 14.5% bet size, based on a Monte Carlo simulation of 100,000 GBM paths, under specific market conditions.

Sampling distribution of optimal bet size (f*) based on Monte Carlo simulations.

More Results

Extended analysis with 100,000 paths showed varying mean returns and optimal bet sizes across different volatility forecasts.

Mean Return at Various GBM Volatility Forecasts

Implied Vol 	Forecast Vol 	Mean Return
59% 		200% 		-235.17%
59% 		70% 		-22.80%
59% 		59% 		-4.03%
59% 		50% 		10.95%
59% 		20% 		59.43%

Optimal Bet Size f* at Various GBM Volatility Forecasts

Implied Vol 	Forecast Vol 	f*
59% 		200% 		0.0%
59% 		70% 		0.0%
59% 		59% 		0.0%
59% 		50% 		15.20%
59% 		20% 		71.10%