[Paper Review] Game pricing and double sequence of random variables

Yukio Hirashita|arXiv (Cornell University)|Mar 3, 2007

Stochastic processes and financial applications12 references3 citations

TL;DR

This paper proposes a novel game pricing model that determines the optimal investment proportion to maximize the long-term growth rate of capital in repeated investments, using a double sequence of random variables to model payoff expectations. The key contribution is a pricing formula that differs from the Black-Scholes model, showing that optimal growth is achieved when the expected growth rate equals the risk-free rate, not when the expected payoff equals the discounted strike price.

ABSTRACT

In this paper, we study a game with positive or plus infinite expectation and determine the optimal proportion of investment for maximizing the limit expectation of growth rate per attempt. With this objective, we introduce a new pricing method in which the price is different from that obtained by the Black-Scholes formula for a European option.

Motivation & Objective

• To determine the optimal investment proportion that maximizes the limit expectation of growth rate per attempt in a repeated game setting.
• To develop a new pricing mechanism for financial games or options that is independent of the Black-Scholes formula.
• To establish conditions under which the limit expectation of growth rate converges and is maximized.
• To demonstrate that the logarithmic utility function naturally emerges from repeated investment dynamics, even without assuming it a priori.
• To provide a framework for pricing the St. Petersburg game and European options using growth rate maximization instead of expected value.

Proposed method

• Introduces a double sequence of random variables {X_{N,n}} based on bounded step functions f_N(x) that converge to the payoff function a(x), enabling convergence analysis.
• Defines the growth rate per attempt as (M_n / M_0)^{1/n}, and uses the limit expectation E[X_{N,n}] as the key performance metric.
• Derives the limit expectation G_u(t) = exp(∫_I log(a(x)t/u - t + 1) dF(x)) as the long-term growth rate under fixed proportion investment t.
• Uses the condition G_u(t_u) = r + 1 (or e^r for continuous compounding) to determine the fair price u for a game with risk-free rate r.
• Applies Jensen’s inequality and integral inequalities to prove that G_u(t) is strictly decreasing in t and continuous in u, ensuring uniqueness of optimal t_u.
• Employs the essential infimum ξ and integrals H = ∫ dF(x)/a(x) and H_ξ = ∫ dF(x)/(a(x) - ξ) to characterize existence and bounds of optimal investment proportions.

Experimental results

Research questions

• RQ1 What is the optimal investment proportion t_u that maximizes the limit expectation of growth rate per attempt for a given game price u?
• RQ2 How does the proposed pricing method differ from the Black-Scholes formula in determining the fair price of a financial option?
• RQ3 Under what conditions does the limit expectation of growth rate exist and converge to a finite value?
• RQ4 Why does the logarithmic utility function emerge as optimal in repeated investment games, even without assuming it?
• RQ5 What is the fair price of the St. Petersburg game when the risk-free rate is 4%?

Key findings

• For the St. Petersburg game with 4% risk-free rate, the optimal price is u ≈ 5.1052, yielding a maximized growth rate of 1.04.
• When the risk-free rate is 4%, the optimal investment proportion for the St. Petersburg game is t_u ≈ 0.1658.
• For a lognormal-distributed game with S=100, σ=0.3, r=0.04, the optimal price is u ≈ 95.6132, achieving a growth rate of e^0.04 ≈ 1.0408.
• The Black-Scholes formula yields a higher price (u ≈ 21.2176) for a European put option, but results in a lower growth rate (1.0096) than the optimal price (u ≈ 17.8157) which achieves 1.0833.
• The optimal investment proportion for the European put option is t_u ≈ 0.5434 at the optimal price, compared to t_u ≈ 0.2278 at the Black-Scholes price.
• The paper proves that G_u(t) is continuous and strictly decreasing in u for u ∈ (0, E), ensuring existence and uniqueness of the optimal price when E = ∞ or E < ∞.