雙變數樹狀模型之建造與複雜度

On the Construction and Complexity of Bivariate Lattices

王釧茹(Chuan-Ju Wang)

指導教授 ： 呂育道

國立臺灣大學/電機資訊學院/資訊工程學研究所/博士(2011年)

https://doi.org/10.6342/NTU.2011.02720

摘要

衍生性金融商品是一種財務工具或契約,其價值是根據標的物之資產價值來決 定。由於此種金融商品在金融市場上扮演了重要的角色,能夠有效率且精準地 評價此類商品即成為十分重要的研究議題。1973年,Black and Scholes 在資產 服從幾何布朗運動及無風險利率和資產波動度皆為常數之假設下,提出了一個 開創性的定價公式。然而許多的實證研究結果卻顯示,被廣泛應用在模擬資產 價格的幾何布朗運動無法完整捕捉資產價格的動態。因此,許多替代的隨機 過程一一被提出,例如著名的跳躍擴散過程(jump-diffusion process);另一方 面,許多隨機利率模型以及隨機波動度模型亦被提出討論。
多數的衍生性金融商品沒有解析解評價公式,因此它們必須使用數值方法 (如樹狀模型)來評價。樹狀模型的評價結果會隨著期數的增加而逼近到正確 的價格。然而,衍生性金融商品之價值函數之非線性誤差可能導致價格收斂緩 慢,甚至大幅振盪。
本論文首先針對跳躍擴散過程提出一個準確且有效率的樹狀評價模型。此 模型不但能夠使評價結果收斂平緩,解決價格大幅震盪之問題,其時間複雜度 亦低於目前其他現有方法。本文提供之數值結果證實了此樹狀模型之優越的準 確度及效率。
另一方面,針對隨機利率模型,本論文的第二部份指出當利率模型允許利 率以無上限的方式增長時,先前文獻提出之樹狀模型皆有可能出現非法機率。由於絕大多數的隨機利率模型皆允許利率以無上限的的方式增長,如何解決此 問題就變得十分重要。本論文首先提出一個雙變數樹狀模型,此模型能夠保證 不論利率以何種方式增長,所有機率皆在有效範圍之內。接著本論文證明在 利率以(超)多項式成長之情況下,所提出的雙變數樹狀模型之大小亦會以 (超)多項式之方式成長。最後,本論文將證明我們所提出的雙變數樹狀模型 之最佳性。

關鍵字

樹狀模型 ； 複雜度 ； 衍生性金融商品

並列摘要

Derivatives are popular financial instruments whose values depend on
other more fundamental financial assets (called the underlying
assets). As they play essential roles in financial markets, evaluating
them efficiently and accurately is critical. In 1973, Black and Scholes arrived at their ground-breaking analytical pricing formula, which assumes that the underlying asset follows a lognormal diffusion process with a constant risk-free interest rate and a constant volatility of the underlying asset.
However, the lognormal diffusion process, which has been widely used
to model the underlying asset's price dynamics, does not capture the
empirical findings satisfactorily. Therefore, many alternative processes have been proposed, and a very popular one is the jump-diffusion process.
Additionally, since interest rates do not stay constant in the
real world, many stochastic interest rate models are put forward.

Most derivatives have no analytical formulas once one goes beyond the
most basic setup; therefore, they must be priced by numerical methods
like lattices.
A lattice divides the time interval between the derivative's initial
date and the maturity date into $n$ equal time steps.
The pricing results converge to the theoretical values when the number
of time steps increases.
Unfortunately, the nonlinearity error introduced by the nonlinearity
of the value function may cause the pricing results to converge slowly
or even oscillate significantly.

This dissertation first proposes an accurate and efficient lattice for
the jump-diffusion process.
The proposed lattice is accurate because its structure can suit the
derivatives’ specifications so that the pricing results converge
smoothly.
To our knowledge, no other lattices for the jump-diffusion process
have successfully solved the oscillation problem.
In addition, the time complexity of our lattice is lower than those of
existing lattice methods by at least half an order.
Numerous numerical calculations confirm the superior performance of
our lattice to existing methods in terms of accuracy, speed, and
generality.

As for the stochastic interest models, the second part of this
dissertation shows that, when the interest rate models allow rates to
grow without bounds in magnitude, previous work on the lattice methods
shares a fundamental flaw: invalid transition probabilities.
As the overwhelming majority of stochastic interest rate models share
this property, a solution to the problem becomes important.
This thesis presents the first bivariate lattice that guarantees valid
probabilities even when interest rates can go without bounds.
Also, we prove that the proposed bivariate lattice grows
(super)polynomially in size if the interest rate model allows rates to
grow (super)polynomially.
Finally, we show the optimality of our bivariate lattice.

並列關鍵字

lattice ； complexity ； derivative

參考文獻

[2] K. Amin. Jump Diffusion Option Valuation in Discrete Time. Journal of Finance, 48(5):1833–1863, 1993.

[5] G. Bierwag. The Ho-Lee Binomial Stochastic Process and Duration. Journal of Fixed Income, 6(2):76–87, 1996.

[6] F. Black, E. Derman, and W. Toy. A One-Factor Model of Interest Rates and Its Application to Treasury Bond Options. Financial Analysts Journal, 46(1):33–39, 1990.

[7] F. Black and P. Karasinski. Bond and Option Pricing When Short Rates Are Lognormal. Financial Analysts Journal, 47(4):52–59, 1991.

[8] F. Black and M. Scholes. The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81(3):637–654, 1973.