關於選擇權定價的三則短論

Abstract

選擇權定價是當代財務科學最重要也最引人入勝的的課題之一，各種複雜的價格模型與演算方法均可能派上用場，讓不同背景但同樣精於計量方法的學者在此一展身手。 在實務上，定價也是多種套利交易策略的核心技術。為求精確的定價，通常需要使用較複雜且具有結構的模型，其中之一就是連續時間隨機波動率模型。然而，這類模型中重要的狀態變數-波動率實際上是不可觀察的，同時整個系統的概率函數也無法以解析形式表達。 這一系列論文探討利用隨機波動率模型進行選擇權定價相關的統計問題，特別將波動率不可觀察這一問題納入考量。基於 GARCH模型會收斂到隨機波動率模型這一性質，只要建構一個僅可部分觀察的GARCH模型，其概率函數可運用蒙地卡羅馬可夫鍊方法計算，且為原有模型之良好近似，因此可用於相關的統計推論，包括估計與定價。在此脈絡下，既有做法的一些不足與缺失均可獲得改善。 最後針對選擇權定價中的損失函數也作一討論。簡言之，雖然定價理論並未對選擇損失函數設下絕對標準，但損失函數必須基於足以表現選擇權價格資訊內涵的統計量(如隱含波動率)，而由於定價均基於均衡的論述，由模型獲得的該統計量僅需作為實繼價格獲得的統計量進行均數回歸的目標，而非價格的絕對參照。

Option pricing may be one of the most important and fascinating topics in modern finance. Complex models and algorithms can be applied here so that researchers and practitioners may bring their quantitative skills into full play. In practice, pricing is also the core of different types of arbitrage strategies. For more precise pricing, structural models are generally necessary where continuous time stochastic volatility model can be one of the candidates. However, volatility, the most important state variable, is in fact unobservable and the likelihood cannot be available in close form for the stochastic volatility models. The sets of articles explore related statistical issues about option pricing with stochastic volatility models. Especially, the unobservability of volatilities is taken into considerations. By the fact that a GARCH model would converge weakly to the corresponding stochastic volatility model, statistical inference including estimation and pricing can be made based on a specially designed partially observed GARCH model whose likelihood will be obtained through MCMC methods. In this context, some drawbacks from the current practices can be improved. Finally, an investigation on the loss functions for option pricing is also made. Although the pricing theory does not restrict to any specific loss functions, the statistics that correspond to the information contents, for example implied volatilities, should be used as a basis for the construction of loss functions. Furthermore, due to the fact that pricing is generally based on some equilibrium conditions, the model implied statistics would just play as the target of mean reversion of the real price implied process, instead of an absolute reference of prices.