凱利公式與其延伸之演繹

An Interpretation of the Kelly Criterion and Its Extensions

廖悠翔(Yu-Hsiang Liao)

指導教授 ： 呂育道

國立臺灣大學/管理學院/財務金融學研究所/碩士(2019年)

https://doi.org/10.6342/NTU201904255

摘要

本論文建立在過去動態投資組合理論的理論發展上，探討其實用價值。過去理論推導相關文獻幾乎沒有提到最佳風險資產權重公式的涵義及性質，對於股價隨機過程參數估計方法的選擇、實務上使用最適合的交易頻率等問題皆很少人討論。然而，資金控管無論對於短期交易乃至長期投資都有著重要的影響力，是金融機構非常重要的技術。本論文整理了凱利公式的相關文獻以及主流文獻中股票服從的隨機過程的動態投資組合理論的推導以及定理。其中，在過去文獻中涉及連續時間的理論都使用Hamilton–Jacobi–Bellman (HJB) equation 的技巧，將隨機控制問題轉化成解偏微分方程式。除此之外，本文也透過模擬的方法探討了對於參數的敏感度，包括幾何布朗運動與Heston 模型皆有的平均報酬(μ) 與波動度(σ) 對於高低估的不對稱敏感度以及只出現在Heston 模型中的參數(ρ, κ, σv) 對於最適風險資產投資比例的影響。最後，本文提到現實世界中無法存在的連續交易影響連續時間最佳解的最佳性的程度。進而推論在Heston 模型下，最佳交易頻率是由參數估計的準確度與最佳解中調整項對於平均效用的貢獻之間權衡得出的。

關鍵字

凱利公式 ； 動態投資組合問題 ； 資金控管 ； HJB 方程式 ； 敏感度分析 ； 交易頻率

並列摘要

This thesis uses the dynamic portfolio theories to explore its practical implications. We seldom find the meanings and properties behind formulas of optimal risky assets weight in the literature. How to choose the parameter estimators and the optimal trading frequency is rarely discussed. However, money management has a lot of influence on shortterm trading and longterm investing. Hence, it is a vital technique for financial institutions. This thesis compiles past works on the Kelly criterion and the mainstream stochastic processes of the stock price in dynamic portfolio problems. The technique used to develop the continuoustime theory in the literature is the Hamilton–Jacobi–Bellman equation. It converts the stochastic control problem into a deterministic PDE. This thesis also investigates the sensitivity of the expected return (μ) and the volatility (σ), which are in both the geometric brownian motion and Heston models and the additional parameter ρ, κ, σv present only in the Heston model. Lastly, how the impossibility of continuous trading in the real world affects the optimality of our solution is also presented. We then further infer that under the Heston model the decision of trading frequency should be a tradeoff between the accuracy of the estimated parameters and contribution of the adjusted terms in the optimal solution to the expected utility.

並列關鍵字

Kelly Criterion ； Dynamic Portfolio Problem ； Money Management ； HJB Equation ； Sensitivity Analysis ； Trading Frequency

參考文獻

[1] Bell, R., & Cover, T.M. (1988). Gametheoretic optimal portfolios. Management Science, 34(6), 724–733.

[2] Breiman, L. (1961). Optimal gambling systems for favorable games, In Proc. Fourth Berkeley Symp. on Math. Statist. and Prob., Vol. 1, Berkeley, CA (pp. 65–78). Berkeley, CA: University of California Press.

[3] Ethier, S.N. (2004). The Kelly system maximizes median fortune. J. Appl. Prob. 41, 1230–1236.

[4] Hakansson, N.H., & Miller, B.L. (1975). Compoundreturn meanvariance efficient portfolios never risk ruin. Management Science, 22(4), 391–400.

[5] Hurn, A.S., Lindsay, K.A., & McClelland, A.J. (2015). Estimating the parameters of stochastic volatility models using option price data. Journal of Business and Economic Statistics, 33(4), 579–594.