Generalized CRR 樹---一個容易建構、評價效率高且精確的樹狀結構

Abstract

大多數的衍生性金融商品都沒有簡易地評價公式，而必須使用數值方法評價。然而 採用數值方法評價時，會引進distribution error 和nonlinearity error，這兩種 誤差會讓數值評價法收斂速度緩慢，甚至造成巨幅震盪。為了解決這個問題，Figlewski 和Gao(1999)提出一個新的樹狀結構，adaptive mesh model (以下簡稱AMM)。但是AMM 的結構十分複雜，所以很難根據不同的衍生性商品量身打造，建構出適合的AMM。 我的研究計畫要開發一個新的樹狀結構：generalized CRR tree model。該模型可 快速並精確地評價各式不同衍生性金融商品。本樹狀結構可針對不同的衍生性金融商 品的規格量身打造，故可以大幅地減少 nonlinearity error。以標準選擇權為例，該 選擇權的報酬函數在標的物價格接近履約價且時間接近到期日時，會出現巨幅非線性 的變化，為了減少評價誤差，可調整generalized CRR tree，讓它有一個節點能在到 期日時觸及履約價格。本計劃打算發展一套建構generalized CRR tree 的方法，讓它 有足夠的自由度能夠觸及重要的節點或是價格水平(price level)，進而可讓評價結果 迅速收斂，而不會出現巨幅震盪。此外，Frishling (2002) 指出評價支付固定股息的 股票選擇權時，因股票價格隨機過程會因支付股息而不連續，所以沒有可精確評價的 封閉公式解，也不易使用數值方法求解，而generalized CRR tree model 則可輕易地 解決此問題。此外，generalized CRR tree model 的主體是由CRR tree 構成，而Dai and Lyuu (2006)指出有許多用CRR tree 評價選擇權的問題，無論期數設得多高都可 使用組合數學方法快速解決，而這些組合數學方法都可應用到generalized CRR tree model，所以使用generalized CRR tree model 評價時，可藉由將期數調高，來減少 distribution error，又可使用組合數學快速評價。 Generalized CRR tree model 可在structural form 的假定下，處理信用風險的問 題。Structural form 假定公司的資產價值服從對數常態隨機過程，當公司資產的價值 在到期日時無法償還債務(見Merton (1974)模型)，或是當公司的資產在到期日之前滑 落到某個門檻以下(見Black and Cox (1976)討論的 first passage model)，則認定 該公司發生違約事件。由於在structural form 中，通常假定公司的資產服從對數常 態分配，所以可用generalized CRR tree model 模擬公司資產價值的變化，其中公司 資產因債務償還或分配現金股利而滑落的現象，可用generalized CRR tree 模擬股票 配發股利而滑落的技術來處理，而造成違約事件的門檻則可類比為障礙選擇權中的障 礙價格，故可用評價障礙選擇權的技術來處理。近來探討structural form 的學術文 獻，多假定利率服從特定的利率期限結構，例如Kim, Ramaswamy and Sundaresan (1993) 採用CIR 模型，Longstaff and Schwartz(1995)採用Vasieck 模型，我也計畫將 Hull-White 三元樹利率模型(見Hull and White (1996))整合到 generalized CRR tree model 中，以增加 generalized CRR tree model 處理信用風險評價的精確度。 參考文獻: 1. Black, F., and J. Cox. 「Valuing Corporate Securities: Liabilities: Some Effects of Bond Indenture Provisions. Journal of Finance, 31 (1976) pp. 351-367. 2. Dai, T.-S., Y. D. Lyuu, and L. M. Liu. 「Developing Efficient Option Pricing Algorithms by Combinatorial Techniques.」 The 2006 International Conference on Scientific Computing, Las Vegas, USA, Jun. 2006. 3. Figlewski, S., and Gao, B. 「The Adaptive Mesh Model: A New Approach to Efficient Option Pricing.」 Journal of Financial Economics, 53 (1999), pp. 313--351. 4. Frishling, V. 「A Discrete Question,」 Risk, 15 (2002), pp.115--6. 5. Hull, J., and A. White, 「Using Hull-White Interest Rate Trees,」 Journal of Derivatives, (1996) pp 26-36. 6. Merton, R. 「On the Pricing of Corporate Debt: The Risk Structure of Interest Rate.」 Journal of Finance, 29 (1974) pp. 449-470. 7. Kim, J., K. Ramaswamy, and S. Sundaresan. 「Does Default Risk in Coupons Affect tge Valuation of Corporate Bonds? A Contingent Claims Model.」 Financial Management, 22 (1993) pp. 117-131. 8. Longstaff, F., and E. Schwartz. 「Valuing Risky Debt: A New Approach.」 Journal of Finance, 50 (1995), 151-178.

Most derivatives do not have simple valuation formulas and must be priced by numerical methods. However, the distribution error and the nonlinearity error introduced by many numerical methods make the pricing results converge slowly or even oscillate significantly. To solve this problem, Figlewski and Gao (1999) propose a novel tree model, the adaptive mesh model (AMM), However, it has a complicated structure, which makes it hard to implement and still harder to tailor to different derivatives. My project plans to construct a flexible tree model, the generalize CRR tree model, for pricing a wide range of derivatives efficiently and accurately. This model reduces the nonlinearity error sharply by adjusting its structure to suit the derivative's specification. For example, the payoff function of a vanilla option is highly nonlinear near the exercise price at maturity. Then the generalized CRR tree can be adjusted to hit the exercise price at maturity date for pricing vanilla options. In this research project, I plan to develop a systematical approach to construct a generalized CRR tree that has enough degree of freedoms to match all the critical points (or price levels) for pricing the option; consequently, the pricing results converge smoothly and quickly. It can also precisely price the stock options with dividend payout, which is hard to be priced accurately and efficiently by analytical forms or by numerical methods (see Frishling (2002)). As an added benefit, pricing of some European-style options on the generalized CRR tree with a large number of time steps can be made extremely efficient by combinatorial tools (see Dai and Lyuu (2006)), which is not possible with most traditional tree models. Therefore, the generalized CRR tree can efficiently reduce the distribution error by picking a large number of time steps. I also plan to extend the generalized CRR tree model to implement the structural credit model. A structural credit model models the process of the firm value. The firm defaults when its value can not meet loan repayment at maturity date (see Merton (1974)) or its value is below a default boundary (i.e., the first passage model discussed in Black and Cox (1976)). Since the firm value is usually assumed to follow the lognormal diffusion process, it can be simulated by the generalized CRR tree model, where the decrease of the firm value due to the loan repayment can be modeled as the decrease of the stock price due to the dividend payout, and the default boundary can be modeled as the barrier (of the knock-out barrier options). Recently, some structural credit models assume that the interest rate follows certain stochastic process like CIR model (See Kim, Ramaswamy and Sundaresan (1993)), Vasicek model (see Longstaff and Schwartz (1995)). My tree model will try to adopt the Hull-White term structure model by incorporating the Hull and White trinomial interest tree (see Hull and White (1996)) into the generalized CRR tree model. Reference: 1. Black, F., and J. Cox. 「Valuing Corporate Securities: Liabilities: Some Effects of Bond Indenture Provisions. Journal of Finance, 31 (1976) pp. 351-367. 2. Dai, T.-S., Y. D. Lyuu, and L. M. Liu. 「Developing Efficient Option Pricing Algorithms by Combinatorial Techniques.」 The 2006 International Conference on Scientific Computing, Las Vegas, USA, Jun. 2006. 3. Figlewski, S., and Gao, B. 「The Adaptive Mesh Model: A New Approach to Efficient Option Pricing.」 Journal of Financial Economics, 53 (1999), pp. 313--351. 4. Frishling, V. 「A Discrete Question,」 Risk, 15 (2002), pp.115--6. 5. Hull, J., and A. White, 「Using Hull-White Interest Rate Trees,」 Journal of Derivatives, (1996) pp 26-36. 6. Merton, R. 「On the Pricing of Corporate Debt: The Risk Structure of Interest Rate.」 Journal of Finance, 29 (1974) pp. 449-470. 7. Kim, J., K. Ramaswamy, and S. Sundaresan. 「Does Default Risk in Coupons Affect tge Valuation of Corporate Bonds? A Contingent Claims Model.」 Financial Management, 22 (1993) pp. 117-131. 8. Longstaff, F., and E. Schwartz. 「Valuing Risky Debt: A New Approach.」 Journal of Finance, 50 (1995), 151-178.