Do Stochastic Volatility and/or Jumps Improve the Consistency of Risk-Neutral Distributions Implicit in the S&P 500 Index Option Market and the Cash Index Market?

隨機波動度及價格跳躍過程是否可增進史丹普500指數選擇權市場與其現貨市場間風險中立機率值之一致性？

林月能(Yueh-Neng Lin)

《財務金融學刊》 14卷1期 (2006/04) Pp. 35-75

https://doi.org/10.6545/JFS.2006.14(1).2

摘要

本文比較隱含在選擇權與現貨價格中的風險中立機率分配是否因加入隨機波動度或價格挑躍而更為接近，來探討對史丹普指數選擇權定償的相對重要性。選擇權價格之風險中立分配係使用核迴歸從隱含波幅曲面估得，而現貨價格之風險中立分配為使用消距離測度方法獲得。相關統計檢定顯示隨機波動度對30天期選擇權的定價最為有利，而事賣外加入價格跳躍則可增進對300天期選擇權價格的說明。此外，除30天期隨機波動度加上價格跳躍的模型外，本文發現選擇權投資者較現貨交易者更驅避風險，說明了文獻上隱含波動度經常大於實現波動度的現象。

關鍵字

隨機波動度 ； 價格跳躍 ； 風險中立機率分配 ； 熵距離測度 ； 風險驅避

並列摘要

This paper examines the relative importance of stochastic volatility and price jumps in option pricing by exploiting the differences in option-based and index return-based risk-neutral densities. As evinced by the persistent smirked volatility smile, this study extracts the risk-neutral densities by exploiting the information embedded in the implied volatilities that are smoothed by a kernel regression. On the other hand, a canonical valuation approach is adopted to identify the risk-neutral density from the observed index returns. Statistical tests and implied risk aversion are constructed to investigate the differences between two risk-neutral densities. The 30-day S&P 500 options under stochastic volatility are found efficiently priced. By using a longer horizon of underlying returns, however, we are able to partly reconcile the differences between the index and option-implied risk-neutral densities after adding a jump component to the index dynamics. Option investors are found more risk averse than stock traders except for the 30-day jump-diffusion with stochastic volatility. The finding may help illustrate the puzzle that implicit volatilities are greater than subsequent realized volatilities.

並列關鍵字

stochastic volatility ； price jumps ； risk-neutral distributions ； canonical valuation ； risk aversion

參考文獻

Aït-Sahalia, Yacine,Andrew W. Lo(1998).Nonparametric estimation of state-price densities implicit in financial asset prices.Journal of Finance.53,499-547.

Aït-Sahalia, Yacine,Andrew W. Lo(2000).Nonparametric risk management and implied risk aversion.Journal of Econometrics.94,9-51.

Aït-Sahalia, Yacine,Yubo Wang,Francis Yared(2001).Do option markets correctly price the probabilities of movement of the underlying asset?.Journal of Financial Econometrics.102,67-110.

Andersen, Torben G.,Luca Benzoni,Jesper Lund(2002).An empirical investigation of continuous-time models for equity returns.Journal of Finance.57,1239-1284.

Bahra, Bhupinder(1997).Implied risk-neutral probability density functions from option prices: Theory and application.Working paper, Bank of England, ISSN 1368-5562.