Bankroll Management: Formalization of the Capital Buffer

Bankroll management is formalized as a task of optimal allocation of a finite resource (capital buffer) under conditions of stochastic uncertainty of results. Mathematically, the task reduces to maximizing the objective function U(B_T) under the constraint P(B_t ≤ 0, ∃t ≤ T) ≤ ε, where B_t is the value of the buffer at time t, T is the planning horizon, and ε is the acceptable probability of complete depletion of the resource. The solution to this problem depends on the choice of the utility function U: for logarithmic utility U(x) = ln(x), the fixed fraction strategy (Kelly criterion) is optimal. Formalization allows for converting intuitive ideas about resource management into a rigorous mathematical model.

The Kelly criterion determines the optimal capital fraction f* for each iteration that maximizes the expected logarithmic growth: f* = (p·b − q) / b, where p is the probability of a positive outcome, q = 1 − p is the probability of a negative one, and b is the payout multiplier. For systems with multiple outcomes, the formula is generalized: f* = argmax_f Σᵢ pᵢ·ln(1 + f·mᵢ), where mᵢ is the net multiplier of the i-th outcome. The use of the full Kelly criterion ensures maximum asymptotic growth, but is associated with high short-term volatility. In practice, a fractional Kelly (f = α·f*, α ∈ [0.25, 0.5]) is used, which reduces variance at the expense of sub-optimal but more stable growth.

Drawdown protection is implemented through a system of dynamic stop limits and adaptive position scaling. The maximum drawdown (MDD) is defined as MDD = max_{t∈[0,T]} (max_{s∈[0,t]} B_s − B_t) / max_{s∈[0,t]} B_s. The probability of reaching a given drawdown level d for a strategy with known parameters μ and σ is estimated via the reflection property of Brownian motion: P(MDD ≥ d) ≈ 2·Φ(−d·√n / σ), where Φ is the standard normal cumulative distribution function. A session "stop-loss" method is also applied: if B_t < (1 − L)·B₀, where L is the acceptable loss threshold (usually L = 0.2), operations are suspended until the next session.

Position sizing is an algorithm for dynamic adjustment of the input position size depending on the current state of the buffer and market conditions. Fixed percentage model: wₜ = f·Bₜ, where f is a fixed fraction and Bₜ is the current buffer. Volatility-adjusted model: wₜ = (f·Bₜ) / (σ_est / σ_target), where σ_est is the estimated current volatility and σ_target is the target volatility. Anti-martingale strategies (increasing position as the buffer grows) ensure geometric growth under positive mathematical expectation, whereas martingale strategies (increasing position upon losses) lead to guaranteed ruin over an infinite horizon.