# 赫爾懷特模型

## 模型構成

$dr(t) = \left\{\theta(t) - \alpha(t) r(t)\right\}dt + \sigma(t) dW(t)$

• θ 是常數 ─ 瓦西塞克模型（Vasicek model
• θ 是跟時間相關的變數 ─ 即赫爾・懷特模型
• θ 還有 α 都是跟時間相關的變數 ─ 赫爾・懷特模型對瓦西塞克模型，又稱為擴展瓦西塞克模型（英：extended Vasicek model）。

$d\,f(r(t)) = \left [\theta(t) + u - \alpha(t)\,f(r(t))\right ]dt + \sigma_1(t)\, dW_1(t)\!$

$du = -bu\,dt + \sigma_2\,dW_2(t)$

θ 是由利率期間結構曲線計算而來，而α 代表的是利率的變動朝著θ 收斂回歸的速度，是由使用者自行設定的係數，一般由歷史資料推估而來。σ是由市場上所存在可以交易的利率交換選擇權（英：swaption）跟利率上限選擇權（英：caplet） 的波動校正項歷史資料所計算得知。

αθσ為常數，依伊藤引理可以證明以下方程成立。

$r(t) = e^{-\alpha t}r(0) + \frac{\theta}{\alpha} \left(1- e^{-\alpha t}\right) + \sigma e^{-\alpha t}\int_0^t e^{\alpha u}\,dW(u)$

$r(t) \sim N\left(e^{-\alpha t}r(0) + \frac{\theta}{\alpha} \left(1- e^{-\alpha t}\right), \frac{\sigma^2}{2\alpha} \left(1-e^{-2\alpha t}\right)\right)$

## 以此模型評價債券

$P(S,T) = A(S,T)e^{-B(S,T)r(S)}$

$B(S,T) = \frac{1-e^{-\alpha(T-S)}}{\alpha}$
$A(S,T) = \frac{P(0,T)}{P(0,S)}\exp($
$-B(S,T) \frac{\partial\log(P(0,t))}{dt} - \frac{\sigma^2(e^{-\alpha T}-e^{-aS})^2(e^{2aS}-1)}{4a^3})$

## 選擇權價格的評價

$V(t) = P(t,S)\mathbb{E}_S[V(S)| \mathcal{F}(t)]$

$F_V(t,T) = V(t)/P(t,S)$，故
$F_V(t,T) = \mathbb{E}_T[V(T)|\mathcal{F}(t)].\,$成立。

$V(S) = \left\{K - P(S,T)\right\}^+$

${E}_S[\left\{K-P(S,T)\right\}^{+}] = KN(-d_2) - F(t,S,T)N(d_1)$

$d_1 = \log{\frac{F}{K}} + \frac{\sigma_P^2S}{2}$
$d_2 = d_1 - \sigma_P \sqrt{S}$

$P(0,S)KN(-d_2) - P(0,T)N(-d_1)$

$\sqrt{S}\sigma_P =\frac{\sigma}{\alpha}\left\{1-e^{-\alpha(T-S)}\right\}\sqrt{\frac{1-e^{-2\alpha S}}{2a}}$

## 参考文獻

• John Hull and Alan White, "Using Hull-White interest rate trees," Journal of Derivatives, Vol. 3, No. 3 (Spring 1996), pp 26-36
• John Hull and Alan White, "Numerical procedures for implementing term structure models I," Journal of Derivatives, Fall 1994, pp 7-16
• John Hull and Alan White, "Numerical procedures for implementing term structure models II," Journal of Derivatives, Winter 1994, pp 37-48
• John Hull and Alan White, "The pricing of options on interest rate caps and floors using the Hull-White model" in Advanced Strategies in Financial Risk Management, Chapter 4, pp 59-67.
• John Hull and Alan White, "One factor interest rate models and the valuation of interest rate derivative securities," Journal of Financial and Quantitative Analysis, Vol 28, No 2, (June 1993) pp 235-254
• John Hull and Alan White, "Pricing interest-rate derivative securities", The Review of Financial Studies, Vol 3, No. 4 (1990) pp. 573-592