局部波動度模型之樹狀結構計算

Abstract

局部波動度模型為一種選擇權定價模型，該模型假設瞬間波動度為股價與時間的函數，其優點在於它既能捕捉到真實市場所觀察到的波動度微笑現象，同時亦能保留了在布萊克－休斯模型中無偏好的特性，因此，局部波動度模型是一種常用的選擇權評價模型。樹狀結構這種數值方法常被應用於局部波動度模型的計算上，特別是在選擇權定價和定價模型的參數校正上。大部份局部波動度模型之樹狀結構都是可接合樹，其形式為二項樹或是三項樹。以往文獻提出的兼容波動度微笑現象的局部波動度模型之樹狀結構，都無法保證無效的節點股價和轉移機率不會出現。為處理該問題，過去的樹狀結構都會直接變更節點股價和轉移機率，但這麼一來，樹狀結構計算出來的隱含波動度曲面與真實的隱含波動度曲面就會產生不一致，因而造成選擇權價格計計算上的偏差。

本論文的貢獻在於成功建構兩種高效並保證合法並的局部波動度模型樹狀結構，換句話說，本論文中的兩種新提出的樹狀結構皆為可接合樹，並且，它們節點股價和轉移機率皆為合法。第一種樹狀結構為針對可分離局部波動度模型之二項樹，第二種則為針對局部波動度模型之三項樹。以往的針對局部波動度模型之二項樹都會有不合法節點股價與轉移機率的傾向，就算是波動度曲面為一常數(如同在布萊克－休斯模型中的假設)，該問題依然存在。本論文發現這問題的一個可能根本原因：二項樹含有發散不動點。雖然經過數十年的研究，針對局部波動度模型，建構合法之二項樹依然困難重重，因此，我們把焦點放在可分離局部波動度模型，為之建構保證合法之二項樹，這種二項樹我們稱它為水線樹。水線樹的特點在於，樹的上半部份會與股價作動差匹配，而樹的下半部份與股價的自然對數作動差匹配，這種創新的結構確保了水線樹只含有收斂不動點。 本論文提出的第二種樹狀結構，為針對局部波動度模型而建構的三項樹，只要波動度曲面為一個有上限的函數，該三項樹上的所有節點股價與轉移機率皆為合法。作為實務上一個很重要的應用，我們將展示本論文所提出的三項樹，能對含有波動度微笑現象之雙界限選擇權作定價，且其收歛速度非常快。本論文也以實驗方法，檢驗了該兩種樹狀結構有很好的數值結果。

The local-volatility (LV) model for option pricing assumes the instantaneous volatility is a function of the stock price and time. This model is popular because it captures the volatility smile observed in practice as well as retains the preference freedom of the Black-Scholes model. A tree for the LV model is called an LV tree. It is the basis for option pricing and model calibration under the LV model. Most LV trees are recombining, i.e., they are binomial or trinomial trees. Past attempts to construct a smile-consistent LV tree all resort heuristics to deal with invalid asset prices and transition probabilities. These trees may not match the implied volatility surface. As a result, the options cannot be priced accurately.

This dissertation aims at constructing two efficient and valid LV trees, i.e., both trees are recombining and have positive stock prices and valid transition probabilities. The first tree is a binomial tree for separable LV model and the second one is a trinomial tree for general LV model. Past attempts to construct a binomial LV tree are prone to having invalid stock prices and transition probabilities. In fact, this problem occurs even when the volatility surface is flat as in the Black-Scholes model. This dissertation unearths a potentially fundamental reason for that failure: the binomial trees contain repelling fixed points. As efficient and valid binomial trees for general LV models remain elusive despite decades of research, we turn to separable LV models. An efficient and valid binomial tree is then built for such models. Our novel tree is named the waterline LV tree because its upper part (the part that is above the water, so to speak) matches the moments of the price, whereas the lower part matches the moments of its logarithmic price. This break from traditional trees ensures that only attracting fixed points remain. The second efficient and valid trees in this dissertation is the trinomial LV tree. It is a heuristics-free tree for general LV model as long as the LV surface has an upper limit. As an important application, our trinomial LV tree can price double-barrier options with fast convergence even in the presence of volatility smile. Numerical results of both waterline and trinomial LV trees confirm their excellent performances.