以衍生性商品避險進行統計套利

Statistical Arbitrage with Derivatives Hedging

楊子慧(Tzu-Hui Yang)

指導教授 ： 韓傳祥

國立清華大學/科技管理學院/計量財務金融系/碩士(2009年)

摘要

在完全市場的假設下，傳統的Black-Scholes-Merton 理論建構了標的資產與其衍生性商品價格間之無套利與風險中立的關係。透過動態避險投資組合的建立，可以完全規避衍生性商品的風險；但由於市場的不完備，風險無法完全規避，卻也因此特性帶來了統計套利的機會，意即透過適當的避險投資組合，雖無法保證每一次的交易都能獲利，但是只要重複執行這種交易就會有正的「平均」收益。
針對選擇權的風險，本研究提出一些「Model Free」的避險策略，包括Delta避險、停損避險與調整後停損避險策略，與隨機波動率Model Free（Stochastic Volatility Model Free）的避險策略，包括修正後的Delta避險與Delta-Gamma避險策略。利用美國與台灣指數選擇權的數據，希望找出在不完全的市場下，面對不同的情形（例如：不同到期日、價內程度、未來趨勢等），各種避險策略的表現，以及其是否具有統計套利的機會。實證結果發現，建立上述避險投資組合，在美國買權的實證研究中，大致具有統計套利的機會。使用Delta避險策略雖平均損益較低，但亦較為穩定；相反的，停損避險策略雖有較高的或收益，但其標準差亦較高；在漲勢下，使用調整後停損避險策略有較高的平均收益。

關鍵字

統計套利 ； Delta避險 ； Delta-Gamma避險 ； 停損避險 ； 調整後停損避險 ； 波動率指數

並列摘要

Under the assumption of market completeness, the traditional Black-Scholes-Merton theory establishes the relationship between non-arbitrage and risk neutrality in the underlying asset and its derivatives. By constructing the hedging portfolio, the risk of derivatives can be diminished. However, the real-world market is incomplete. It is impossible to hedge perfectly, but this also makes statistical arbitrage possible. Statistical arbitrage does not guarantee profit from each trade, but it makes possible profit “on the average” after repeating similar trade strategies several times.
In this research, we try to build several “model free” hedging strategies, such as Delta, stop-loss and adjusted stop-loss, and stochastic volatility “model free” hedging strategies, such as Delta-Gamma and corrected Delta, for index options. We test these strategies with data from the U.S. and Taiwan. Under several scenarios, e.g., different times to maturity, moneyness, future trends, we investigate (1) the performance of these hedging strategies and (2) the opportunity of statistical arbitrage. The empirical results in American call show that preceding strategies bring the opportunity of statistical arbitrage in substance. Delta hedging strategy is more stable but brings fewer profits on the average. On the other hand, stop-loss hedging strategy brings more profits on the average but is less stable. Adjusted stop-loss hedging strategy brings higher average return under the up trends.

參考文獻

Avellaneda, M. and Lee, J.-H. (2008) “Statistical Arbitrage in the U.S. Equities Market,”working paper, New York University.

Black, F. and Scholes, M. (1973) “The Pricing of Options and Corporate Liabilities,” Journal Political Economy, 81, 637-654.

Coleman, T.F., Li, Y. and Patron, M. (2003) “Discrete Hedging under Piecewise Linear Risk Minimization,”Journal of Risk , Vol. 5, pp. 39-65.

Frittelli, M. (2000) “The minimal entropy martingale measure and the valuation problem in incomplete markets,” Mathematical Finance, 10, 39-52.

Haynes H. M. Yung, and H. Zhang (2003). “An empirical investigation of the GARCH option pricing model: Hedging performance.” The Journal of Futures Markets, 23, 1191-1207.