資產評價隨機波動模型研究

Essays in Asset Pricing under Stochastic Volatility

郭家豪(Guo Jia-Hau)

指導教授 ： 洪茂蔚

國立臺灣大學/管理學院/國際企業學研究所/博士(2007年)

https://doi.org/10.6342/NTU.2007.10144

摘要

隨機波動模型自九零年代初期提出以來，深受學術界及實務界的重視，價格的隨機波動性證據在許多實證研究中廣泛出現，也得到一致性的確認其在資產評價上的顯著貢獻，然而，隨機波動模型在資產評價上的應用仍然有許多瓶頸，有待學術研究之突破發展。本論文的兩個部分，乃分別就隨機波動模型，在不同狀況條件及標的下，如何進行資產評價，進一步分析研究隨機波動性會對資產價值產生哪些影響。
論文的第一個部分是研究隨機波動模型在股酬交換評價上的應用，主要是提出了一個考慮隨機波動性及隨機利率的股酬交換評價模式之封閉解並藉此了解隨機過程之間相關係數對股酬交換(equity swap)之交換率(swap rate)的影響。探討有關之相關係數除了前文獻論述到的利率與標的物報酬率之間的相關係數外，我們更進一步檢視標的物報酬率與其波動性之間的相關性係數。我們的遠期中立評價模式除了應用在固定名目本金(constant notional principal)或可變名目本金(variable notional principal)之股酬交換外，也使用在隱含有遠期生效選擇權(forward start option)的上限型股酬交換(capped equity swap)。本研究證實雖然波動性是否放寬為隨機並不影響固定名目本金或可變名目本金股酬交換之評價，然而，對上限型股酬交換卻具有重要性。
而論文的第二個部分是研究隨機波動模型如何擴展到美式選擇權上的應用，考慮在可能提早履約的情境下，標的物之隨機波動性與代表突發衝擊事件之跳躍模式，如何來影響提早履約溢酬的問題。主要是將Bates在1991年和1996年使用到美式選擇權二次近似解方法(此法由Barone-Adesi和Whaley於1987首度提出)，對固定波動性的假設作進一步放寬至隨機波動性，並多考慮了波動性出現跳躍模型的可能性。本文研究顯示提早履約溢酬對短期且價外之選擇權價值可能非常顯著，因此一般實證研究上將短期價外之美式選擇權當作不具提早履約溢酬之歐式選擇權的做法，可能有必要重新檢視其適當性。

關鍵字

隨機波動 ； 權益交換 ； 美式選擇權 ； 二次近似解 ； 遠期選擇權

並列摘要

THESIS ABSTRACT
Stochastic volatility models have enjoyed an excellent reputation both theoretically and practically since introduced in the early 1990s. Lots of empirical studies provide evidence that the volatility of the price return is stochastic. The significant contribution of stochastic volatility models in asset pricing is consistently confirmed. However, there are still a few bottlenecks in asset pricing for the application of stochastic volatility models. Lots of problems remain unsolved. We consider different asset pricing problems in the two parts of the thesis, and provide the analytic solutions under stochastic volatility. We further analyze the impact of stochastic volatility on asset pricing.
The purpose of the first part is to consider the problem of pricing equity swaps in a stochastic volatility and stochastic interest rates economy. This article adds to the literature on equity swaps by presenting an equity swap pricing model that allows for non-deterministic volatility and by exploiting the relation between the swap rate and the volatility variation of underlying equity returns. The pricing formulae consider not only the correlation between interest rates and underlying equity returns but also the correlation between volatility shocks and underlying equity returns. Closed form solutions for a variety of equity swaps with constant or variable notional principal in the stochastic volatility and stochastic interest rate model are derived from the forward-neutral pricing model. No matter whether the notional principal of the equity swap is constant or variable, its swap rate in the stochastic volatility case is shown to be the same as that in the deterministic volatility case. Nevertheless, it is not the case for capped equity swaps. A capped equity swap is composed of a normal equity swap and a series of forward-start European call options. Stochastic volatility plays an important role on the valuation of capped equity swaps.
The problem of pricing American options using the quadratic approximation method with stochastic volatility and jumps is considered in the second part of the thesis. Compared to Monte Carlo simulations or other time-consumption numerical techniques, it is particularly valuable to extend the existing efficient solutions for American options from constant to stochastic volatility. Our results show that deep out of money American options with short-maturities should not be over-simplified to be treated as the European ones. Early exercise premiums are also found to be very sensitive to the changes in interest rates and dividend rates.

並列關鍵字

Stochastic Volatility ； Equity Swap ； American Option ； Quadratic Approximation ； Forward-Start Option

參考文獻

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Andersen, L. and R. Brotherton-Ratcliffe, 1998, The equity option volatility smile-An implicit finite-difference approach, The Journal of Computational Finance 1 (2), 5-38.

Bakshi, G., C. Cao and Z. Chen, 1997, Empirical performance of alternative option pricing models, Journal of Finance 53, 499-547.