bond_simulation_tutorial.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 債券利率蒙地卡羅模擬教學\n",
    "\n",
    "本 Jupyter Notebook 旨在教學演示如何使用 Python 進行債券利率的蒙地卡羅模擬。\n",
    "程式碼中的變數名稱、公式與註解都對標 'equation.md' 文件，以利於學習與對照。\n",
    "\n",
    "## 內容包含：\n",
    "\n",
    "1.  **四種利率模型**：\n",
    "    *   **Vasicek 模型**：用於模擬無風險利率 (drf)，具有均值回歸特性。\n",
    "    *   **Cox-Ingersoll-Ross (CIR) 模型**：Vasicek 的變體，確保利率為正。\n",
    "    *   **幾何布朗運動 (GBM) 模型**：一個常用於股價的簡化利率模型。\n",
    "    *   **有風險利率模型**：在無風險利率上疊加一個隨機的信用利差。\n",
    "2.  **債券價格近似計算**：\n",
    "    *   根據模擬出的利率路徑，使用「修正存續期間」與「價格曲度」來近似計算債券價格的變化。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1. 載入所需函式庫\n",
    "\n",
    "首先，我們載入 `numpy` 用於數值計算，`matplotlib.pyplot` 用於繪圖，以及 `datetime` 用於處理日期。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from datetime import datetime"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2. 利率模擬函數\n",
    "\n",
    "此區塊定義了四種不同的利率模型函數，用於生成未來的利率路徑。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2.1 Vasicek 模型\n",
    "\n",
    "**隨機微分方程 (SDE):** `dr_t = k(θ - r_t)dt + σdW_t`\n",
    "\n",
    "此模型描述了一個具有「均值回歸」特性的利率。利率 `r_t` 會傾向於回歸到其長期平均水平 `θ`，回歸的速度由 `k` 控制，而 `σ` 則代表其隨機波動的幅度。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def simulate_vasicek(r_0, k, theta, sigma, T, dt, n_simulations, dW_t):\n",
    "    \"\"\"\n",
    "    使用 Vasicek 模型模擬利率路徑。\n",
    "    \n",
    "    :param r_0: float, 初始利率 (r_t 在 t=0 的值)。\n",
    "    :param k: float, 均值回歸速度 (kappa)，代表利率回復到長期均值的速度。\n",
    "    :param theta: float, 長期平均利率或均衡水平 (θ)。\n",
    "    :param sigma: float, 波動率 (σ)，代表利率隨機波動的幅度。\n",
    "    :param T: float, 總模擬時長(年)。\n",
    "    :param dt: float, 每個時間步長(年)。\n",
    "    :param n_simulations: int, 模擬路徑的數量。\n",
    "    :param dW_t: np.ndarray, 預先生成的維納過程增量 (dW_t)，代表隨機衝擊。\n",
    "    :return: np.ndarray, 模擬的利率路徑。\n",
    "    \"\"\"\n",
    "    num_steps = int(T / dt)\n",
    "    r_paths = np.zeros((n_simulations, num_steps + 1))\n",
    "    r_paths[:, 0] = r_0\n",
    "\n",
    "    for t in range(1, num_steps + 1):\n",
    "        r_paths[:, t] = r_paths[:, t-1] + k * (theta - r_paths[:, t-1]) * dt + sigma * dW_t[:, t-1]\n",
    "        r_paths[:, t] = np.maximum(0.0001, r_paths[:, t]) # 確保利率為正\n",
    "        \n",
    "    return r_paths"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2.2 Cox-Ingersoll-Ross (CIR) 模型\n",
    "\n",
    "**隨機微分方程 (SDE):** `dr_t = k(θ - r_t)dt + σ√r_t dW_t`\n",
    "\n",
    "CIR 模型是 Vasicek 的變體，其主要優點是能確保利率恆為正值（在 `2kθ > σ²` 的條件下）。波動項 `σ√r_t` 表示利率越高，其波動也越大。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def simulate_cir(r_0, k, theta, sigma, T, dt, n_simulations, dW_t):\n",
    "    \"\"\"\n",
    "    使用 Cox-Ingersoll-Ross (CIR) 模型模擬利率路徑。\n",
    "    \n",
    "    :param r_0: float, 初始利率。\n",
    "    :param k: float, 均值回歸速度。\n",
    "    :param theta: float, 長期平均利率。\n",
    "    :param sigma: float, 波動率。\n",
    "    :param T: float, 總模擬時長(年)。\n",
    "    :param dt: float, 每個時間步長(年)。\n",
    "    :param n_simulations: int, 模擬路徑的數量。\n",
    "    :param dW_t: np.ndarray, 預先生成的維納過程增量。\n",
    "    :return: np.ndarray, 模擬的利率路徑。\n",
    "    \"\"\"\n",
    "    num_steps = int(T / dt)\n",
    "    r_paths = np.zeros((n_simulations, num_steps + 1))\n",
    "    r_paths[:, 0] = r_0\n",
    "\n",
    "    for t in range(1, num_steps + 1):\n",
    "        sqrt_r = np.sqrt(np.maximum(0, r_paths[:, t-1]))\n",
    "        r_paths[:, t] = r_paths[:, t-1] + k * (theta - r_paths[:, t-1]) * dt + sigma * sqrt_r * dW_t[:, t-1]\n",
    "        r_paths[:, t] = np.maximum(0.0001, r_paths[:, t]) # 再次確保利率為正\n",
    "        \n",
    "    return r_paths"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2.3 幾何布朗運動 (GBM) 模型\n",
    "\n",
    "**隨機微分方程 (SDE):** `dr_t = μr_t dt + σr_t dW_t`\n",
    "\n",
    "GBM 是金融領域中一個非常經典的模型，常用於模擬股價。它假設資產的報酬率服從常態分佈。雖然在利率模擬中不如均值回歸模型常用，但作為一個基礎模型仍有其教學價值。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def simulate_gbm(r_0, mu, sigma, T, dt, n_simulations, dW_t):\n",
    "    \"\"\"\n",
    "    使用幾何布朗運動 (GBM) 模型模擬利率路徑。\n",
    "    \n",
    "    :param r_0: float, 初始利率。\n",
    "    :param mu: float, 長期漂移趨勢。\n",
    "    :param sigma: float, 年化波動率。\n",
    "    :param T: float, 總模擬時長(年)。\n",
    "    :param dt: float, 每個時間步長(年)。\n",
    "    :param n_simulations: int, 模擬路徑的數量。\n",
    "    :param dW_t: np.ndarray, 預先生成的維納過程增量。\n",
    "    :return: np.ndarray, 模擬的利率路徑。\n",
    "    \"\"\"\n",
    "    num_steps = int(T / dt)\n",
    "    r_paths = np.zeros((n_simulations, num_steps + 1))\n",
    "    r_paths[:, 0] = r_0\n",
    "\n",
    "    for t in range(1, num_steps + 1):\n",
    "        drift = (mu - 0.5 * sigma**2) * dt\n",
    "        diffusion = sigma * dW_t[:, t-1]\n",
    "        r_paths[:, t] = r_paths[:, t-1] * np.exp(drift + diffusion)\n",
    "        r_paths[:, t] = np.maximum(0.0001, r_paths[:, t]) # 確保利率為正\n",
    "        \n",
    "    return r_paths"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2.4 有風險利率模型\n",
    "\n",
    "**公式:**\n",
    "1.  `drs_t = drf_t + spread_t`\n",
    "2.  `d(spread_t) = σ_s dW_s`\n",
    "3.  `dW_s = ρ * dW_f + √(1 - ρ²) * dZ_t`\n",
    "\n",
    "此模型在無風險利率 `drf_t` (例如由 Vasicek 模型生成) 的基礎上，疊加一個隨機變動的信用利差 `spread_t`，從而得到有風險利率 `drs_t`。\n",
    "\n",
    "信用利差的變動 `dW_s` 與無風險利率的變動 `dW_f` 之間存在相關性 `ρ`，這使得模型能更真實地反映市場情況（例如，經濟惡化時，無風險利率可能下降，而信用利差卻會擴大）。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def simulate_risky_rate(drf_paths, dW_f, initial_spread, sigma_s, rho, T, dt, n_simulations):\n",
    "    \"\"\"\n",
    "    模擬有風險利率 (drs)，即在無風險利率(drf)上疊加信用利差(spread)。\n",
    "    \n",
    "    :param drf_paths: np.ndarray, 已模擬好的無風險利率路徑。\n",
    "    :param dW_f: np.ndarray, 生成 drf_paths 所使用的隨機衝擊。\n",
    "    :param initial_spread: float, 初始信用利差。\n",
    "    :param sigma_s: float, 信用利差的波動率。\n",
    "    :param rho: float, drf 與 spread 變動之間的相關係數。\n",
    "    :param T: float, 總模擬時長(年)。\n",
    "    :param dt: float, 每個時間步長(年)。\n",
    "    :param n_simulations: int, 模擬路徑的數量。\n",
    "    :return: tuple (np.ndarray, np.ndarray), 分別為模擬的有風險利率路徑和信用利差路徑。\n",
    "    \"\"\"\n",
    "    num_steps = int(T / dt)\n",
    "    \n",
    "    # 1. 生成一個獨立的隨機衝擊 dZ_t\n",
    "    dZ_t = np.random.normal(0, 1, (n_simulations, num_steps)) * np.sqrt(dt)\n",
    "    \n",
    "    # 2. 根據相關係數 rho，合成信用利差的隨機衝擊 dW_s\n",
    "    dW_s = rho * dW_f + np.sqrt(1 - rho**2) * dZ_t\n",
    "    \n",
    "    # 3. 模擬信用利差的路徑\n",
    "    spread_paths = np.zeros((n_simulations, num_steps + 1))\n",
    "    spread_paths[:, 0] = initial_spread\n",
    "    for t in range(1, num_steps + 1):\n",
    "        spread_paths[:, t] = spread_paths[:, t-1] + sigma_s * dW_s[:, t-1]\n",
    "        spread_paths[:, t] = np.maximum(0.0001, spread_paths[:, t]) # 確保利差為正\n",
    "        \n",
    "    # 4. 有風險利率 = 無風險利率 + 信用利差\n",
    "    drs_paths = drf_paths + spread_paths\n",
    "    \n",
    "    return drs_paths, spread_paths"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3. 債券價格計算函數\n",
    "\n",
    "在得到利率路徑後，我們需要一個方法來估算債券價格的變化。這裡我們使用基於「修正存續期間 (Modified Duration)」和「曲度 (Convexity)」的二階泰勒展開式來近似計算。\n",
    "\n",
    "**近似公式:** `ΔP/P ≈ -D_mod * Δy + (C_mod / 2) * (Δy)²`\n",
    "\n",
    "其中：\n",
    "-   `ΔP/P` 是價格的變動百分比。\n",
    "-   `D_mod` 是修正存續期間，衡量價格對利率變動的敏感度（一階）。\n",
    "-   `C_mod` 是曲度，用於修正存續期間的線性估計誤差（二階）。\n",
    "-   `Δy` 是殖利率的變化量，我們在此用模擬的利率 `r` 來替代。\n",
    "\n",
    "**注意：** 在我們的系統中，資料庫提供的 `convT` 值是 `P * C_mod / 2`，因此我們需要先將其轉換回公式中所需的 `C_mod / 2` 項。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def calculate_price_paths(r_paths, D_mod, convT, initial_price=100.0):\n",
    "    \"\"\"\n",
    "    使用修正存續期間(D_mod)和價格曲度(convT)來近似計算債券價格路徑。\n",
    "    \n",
    "    :param r_paths: np.ndarray, 模擬的利率路徑 (y)。\n",
    "    :param D_mod: float, 債券的修正存續期間 (Modified Duration)。\n",
    "    :param convT: float, 來自資料庫的價格曲度值 (P * C_mod / 2)。\n",
    "    :param initial_price: float, 債券的初始價格。\n",
    "    :return: np.ndarray, 模擬的債券價格路徑。\n",
    "    \"\"\"\n",
    "    num_steps = r_paths.shape[1] - 1\n",
    "    price_paths = np.zeros_like(r_paths)\n",
    "    price_paths[:, 0] = initial_price\n",
    "    \n",
    "    # 從 convT 反解出公式中需要的 (C_mod / 2) 項\n",
    "    convexity_factor = convT / initial_price\n",
    "    \n",
    "    for t in range(1, num_steps + 1):\n",
    "        delta_y = r_paths[:, t] - r_paths[:, t-1]\n",
    "        prev_price = price_paths[:, t-1]\n",
    "        \n",
    "        # 計算價格變動百分比 (ΔP/P)\n",
    "        price_change_percentage = -D_mod * delta_y + convexity_factor * (delta_y ** 2)\n",
    "        \n",
    "        # 計算新價格 P(t) = P(t-1) * (1 + ΔP/P)\n",
    "        price_paths[:, t] = prev_price * (1 + price_change_percentage)\n",
    "        \n",
    "    return price_paths"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 4. 主執行區塊\n",
    "\n",
    "現在我們將前面定義的函數組合起來，進行完整的模擬流程。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 4.1 設定模擬參數\n",
    "\n",
    "首先，我們定義債券的基本資料和模擬的通用參數。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 債券基本資料 (此處為範例，實際應用中從資料庫讀取)\n",
    "bond_data = {\n",
    "    'name': 'CGB10Y',\n",
    "    'avgYld': 2.2,      # 平均殖利率 (%)\n",
    "    'mdurT': 8.9,       # 修正存續期間 (D_mod)\n",
    "    'convT': 45.5,      # 價格曲度 (P * C_mod / 2)\n",
    "    'maturity': datetime(2034, 5, 15)\n",
    "}\n",
    "\n",
    "# 模擬通用參數\n",
    "n_simulations = 100      # 模擬次數\n",
    "dt = 1/252               # 時間步長 (年)，假設一年252個交易日\n",
    "\n",
    "# 計算剩餘到期年限，作為模擬總時長 T\n",
    "time_to_maturity = (bond_data['maturity'] - datetime.now()).days / 365.25\n",
    "T = max(time_to_maturity, 0.1) # 確保至少模擬一小段時間\n",
    "\n",
    "num_steps = int(T / dt)\n",
    "time_points = np.linspace(0, T, num_steps + 1) # 模擬的時間點數列\n",
    "\n",
    "# 初始利率 (將百分比轉為小數)\n",
    "r_0 = bond_data['avgYld'] / 100.0\n",
    "\n",
    "print(f\"債券: {bond_data['name']}\")\n",
    "print(f\"剩餘到期年限 (T): {T:.2f} 年\")\n",
    "print(f\"模擬次數: {n_simulations}\")\n",
    "print(f\"時間步長 (dt): {dt:.4f} 年\")\n",
    "print(f\"總步數: {num_steps}\")\n",
    "print(f\"初始利率 (r_0): {r_0*100:.2f}%\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 4.2 設定各模型參數\n",
    "\n",
    "為每個利率模型設定具體的參數值。這些值通常需要透過歷史數據進行校準。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 1. Vasicek & CIR 模型參數 (將百分比轉為小數)\n",
    "k_vasicek = 0.3\n",
    "theta_vasicek = 2.2 / 100.0\n",
    "sigma_vasicek = 0.5 / 100.0\n",
    "\n",
    "k_cir = 0.2\n",
    "theta_cir = 2.5 / 100.0\n",
    "sigma_cir = 0.4 / 100.0\n",
    "\n",
    "# 2. GBM 模型參數\n",
    "mu_r = 0.05 / 100.0\n",
    "sigma_r = 0.6 / 100.0\n",
    "\n",
    "# 3. 有風險利率模型參數\n",
    "initial_spread = 0.8 / 100.0\n",
    "sigma_s = 0.3 / 100.0\n",
    "rho = 0.6"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 4.3 生成隨機衝擊\n",
    "\n",
    "為了讓模擬結果可以重現，我們使用 `np.random.seed` 設定隨機種子。然後，為每個需要獨立隨機性的模型預先生成維納過程的增量 `dW_t`。\n",
    "\n",
    "`dW_t` 是一個服從 `N(0, dt)` 的隨機變數，等價於 `sqrt(dt) * N(0, 1)`。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.random.seed(42) # 設定隨機種子以確保結果可重現\n",
    "dW_vasicek = np.random.normal(0, 1, (n_simulations, num_steps)) * np.sqrt(dt)\n",
    "dW_cir = np.random.normal(0, 1, (n_simulations, num_steps)) * np.sqrt(dt)\n",
    "dW_gbm = np.random.normal(0, 1, (n_simulations, num_steps)) * np.sqrt(dt)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 4.4 執行模擬\n",
    "\n",
    "呼叫前面定義的函數，生成利率路徑和對應的債券價格路徑。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "print(\"開始執行利率模擬...\")\n",
    "# 1. 模擬無風險利率 drf (Vasicek)\n",
    "drf_paths = simulate_vasicek(r_0, k_vasicek, theta_vasicek, sigma_vasicek, T, dt, n_simulations, dW_vasicek)\n",
    "\n",
    "# 2. 模擬有風險利率 drs (在 drf 基礎上增加信用利差)\n",
    "drs_paths, spread_paths = simulate_risky_rate(drf_paths, dW_vasicek, initial_spread, sigma_s, rho, T, dt, n_simulations)\n",
    "\n",
    "# 3. 模擬綜合利率 dr (GBM)\n",
    "dr_paths = simulate_gbm(r_0, mu_r, sigma_r, T, dt, n_simulations, dW_gbm)\n",
    "\n",
    "# 4. 模擬 CIR 利率 dr_cir\n",
    "dr_cir_paths = simulate_cir(r_0, k_cir, theta_cir, sigma_cir, T, dt, n_simulations, dW_cir)\n",
    "print(\"利率模擬完成。\")\n",
    "\n",
    "print(\"開始計算債券價格路徑...\")\n",
    "# 5. 計算對應的債券價格路徑\n",
    "price_drf_paths = calculate_price_paths(drf_paths, bond_data['mdurT'], bond_data['convT'])\n",
    "price_drs_paths = calculate_price_paths(drs_paths, bond_data['mdurT'], bond_data['convT'])\n",
    "price_dr_paths = calculate_price_paths(dr_paths, bond_data['mdurT'], bond_data['convT'])\n",
    "price_dr_cir_paths = calculate_price_paths(dr_cir_paths, bond_data['mdurT'], bond_data['convT'])\n",
    "print(\"價格計算完成。\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 5. 結果可視化\n",
    "\n",
    "將模擬結果繪製成圖表，以便直觀地理解利率和價格的動態變化。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 5.1 設定 Matplotlib 中文顯示\n",
    "\n",
    "為了讓圖表能正確顯示中文標題和標籤，我們需要設定 Matplotlib 的字體。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "try:\n",
    "    plt.rcParams['font.sans-serif'] = ['Microsoft JhengHei'] # For Windows\n",
    "    plt.rcParams['axes.unicode_minus'] = False\n",
    "except:\n",
    "    print(\"未找到 'Microsoft JhengHei' 字體，圖表中的中文可能無法正常顯示。\")\n",
    "    print(\"您可以嘗試安裝 'Microsoft JhengHei' 或替換為系統中已有的中文字體，例如 'SimHei' (黑體) 或 'KaiTi' (楷體)。\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 5.2 利率模擬路徑圖\n",
    "\n",
    "下圖展示了四種模型生成的前 50 條利率路徑。可以觀察到不同模型的特徵：\n",
    "- **Vasicek** 和 **CIR** 顯示出向長期均值回歸的趨勢。\n",
    "- **GBM** 則沒有明顯的均值回歸特性，路徑發散較大。\n",
    "- **有風險利率** 的路徑整體高於無風險利率，反映了信用利差的存在。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig_rates, axes_rates = plt.subplots(2, 2, figsize=(14, 10), sharex=True)\n",
    "fig_rates.suptitle(f'四種利率模型的模擬路徑 (前 {min(50, n_simulations)} 條)', fontsize=16)\n",
    "\n",
    "axes_rates[0, 0].plot(time_points, drf_paths[:50, :].T * 100, lw=0.5)\n",
    "axes_rates[0, 0].set_title('1. 無風險利率 (drf) - Vasicek')\n",
    "axes_rates[0, 0].set_ylabel('利率 (%)')\n",
    "axes_rates[0, 0].grid(True, linestyle='--', alpha=0.6)\n",
    "\n",
    "axes_rates[0, 1].plot(time_points, drs_paths[:50, :].T * 100, lw=0.5)\n",
    "axes_rates[0, 1].set_title('2. 有風險利率 (drs) - Vasicek + Spread')\n",
    "axes_rates[0, 1].grid(True, linestyle='--', alpha=0.6)\n",
    "\n",
    "axes_rates[1, 0].plot(time_points, dr_paths[:50, :].T * 100, lw=0.5)\n",
    "axes_rates[1, 0].set_title('3. 綜合利率 (dr) - GBM')\n",
    "axes_rates[1, 0].set_xlabel('時間 (年)')\n",
    "axes_rates[1, 0].set_ylabel('利率 (%)')\n",
    "axes_rates[1, 0].grid(True, linestyle='--', alpha=0.6)\n",
    "\n",
    "axes_rates[1, 1].plot(time_points, dr_cir_paths[:50, :].T * 100, lw=0.5)\n",
    "axes_rates[1, 1].set_title('4. CIR 利率 (dr_cir)')\n",
    "axes_rates[1, 1].set_xlabel('時間 (年)')\n",
    "axes_rates[1, 1].grid(True, linestyle='--', alpha=0.6)\n",
    "\n",
    "fig_rates.tight_layout(rect=[0, 0, 1, 0.96])\n",
    "plt.savefig(\"tutorial_rate_paths.png\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 5.3 債券價格模擬路徑圖\n",
    "\n",
    "下圖展示了與上述利率路徑相對應的債券價格變化。我們可以看到，當利率上升時，債券價格下降，反之亦然，這符合債券的基本特性。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig_prices, axes_prices = plt.subplots(2, 2, figsize=(14, 10), sharex=True, sharey=True)\n",
    "fig_prices.suptitle(f'對應的債券價格模擬路徑 (前 {min(50, n_simulations)} 條)', fontsize=16)\n",
    "\n",
    "axes_prices[0, 0].plot(time_points, price_drf_paths[:50, :].T, lw=0.5)\n",
    "axes_prices[0, 0].set_title('基於 drf (Vasicek) 的價格')\n",
    "axes_prices[0, 0].set_ylabel('債券價格')\n",
    "axes_prices[0, 0].grid(True, linestyle='--', alpha=0.6)\n",
    "\n",
    "axes_prices[0, 1].plot(time_points, price_drs_paths[:50, :].T, lw=0.5)\n",
    "axes_prices[0, 1].set_title('基於 drs (Risky) 的價格')\n",
    "axes_prices[0, 1].grid(True, linestyle='--', alpha=0.6)\n",
    "\n",
    "axes_prices[1, 0].plot(time_points, price_dr_paths[:50, :].T, lw=0.5)\n",
    "axes_prices[1, 0].set_title('基於 dr (GBM) 的價格')\n",
    "axes_prices[1, 0].set_xlabel('時間 (年)')\n",
    "axes_prices[1, 0].set_ylabel('債券價格')\n",
    "axes_prices[1, 0].grid(True, linestyle='--', alpha=0.6)\n",
    "\n",
    "axes_prices[1, 1].plot(time_points, price_dr_cir_paths[:50, :].T, lw=0.5)\n",
    "axes_prices[1, 1].set_title('基於 dr_cir (CIR) 的價格')\n",
    "axes_prices[1, 1].set_xlabel('時間 (年)')\n",
    "axes_prices[1, 1].grid(True, linestyle='--', alpha=0.6)\n",
    "\n",
    "fig_prices.tight_layout(rect=[0, 0, 1, 0.96])\n",
    "plt.savefig(\"tutorial_price_paths.png\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 5.4 最終價格分佈圖\n",
    "\n",
    "此直方圖顯示了在模擬期結束時，所有模擬路徑的最終債券價格分佈。這對於評估風險至關重要，例如計算 VaR (Value at Risk)。\n",
    "\n",
    "從圖中可以看出，不同利率模型導致的最終價格分佈形狀和範圍有顯著差異。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.figure(figsize=(10, 6))\n",
    "plt.hist(price_drf_paths[:, -1], bins=50, alpha=0.7, label='基於 drf (Vasicek)', density=True)\n",
    "plt.hist(price_drs_paths[:, -1], bins=50, alpha=0.7, label='基於 drs (Risky)', density=True)\n",
    "plt.hist(price_dr_paths[:, -1], bins=50, alpha=0.7, label='基於 dr (GBM)', density=True)\n",
    "plt.hist(price_dr_cir_paths[:, -1], bins=50, alpha=0.7, label='基於 dr_cir (CIR)', density=True)\n",
    "plt.title('模擬結束時的債券價格分佈')\n",
    "plt.xlabel('最終價格')\n",
    "plt.ylabel('機率密度')\n",
    "plt.legend()\n",
    "plt.grid(True, linestyle='--', alpha=0.6)\n",
    "plt.savefig(\"tutorial_final_price_dist.png\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 6. 簡單統計分析\n",
    "\n",
    "最後，我們對兩種較為重要的模型（無風險和有風險）的最終價格進行統計分析，計算其平均值、標準差和 5% 分位數，並估算 95% VaR。\n",
    "\n",
    "**VaR (Value at Risk)**：在險價值，衡量在給定的信賴水準（此處為 95%）和持有期間內，預期的最大潛在損失。此處的 VaR 是從初始價格 100 計算的損失。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "final_prices_drf = price_drf_paths[:, -1]\n",
    "final_prices_drs = price_drs_paths[:, -1]\n",
    "\n",
    "print(\"--- 最終價格統計分析 ---\")\n",
    "print(f\"模型: 基於 drf (Vasicek)\")\n",
    "print(f\"  平均最終價格: {np.mean(final_prices_drf):.4f}\")\n",
    "print(f\"  價格標準差: {np.std(final_prices_drf):.4f}\")\n",
    "print(f\"  5% 分位數價格: {np.percentile(final_prices_drf, 5):.4f}\")\n",
    "print(f\"  95% VaR (從初始價100計算的潛在最大損失): {100 - np.percentile(final_prices_drf, 5):.4f}\")\n",
    "\n",
    "print(f\"模型: 基於 drs (Risky)\")\n",
    "print(f\"  平均最終價格: {np.mean(final_prices_drs):.4f}\")\n",
    "print(f\"  價格標準差: {np.std(final_prices_drs):.4f}\")\n",
    "print(f\"  5% 分位數價格: {np.percentile(final_prices_drs, 5):.4f}\")\n",
    "print(f\"  95% VaR (從初始價100計算的潛在最大損失): {100 - np.percentile(final_prices_drs, 5):.4f}\")\n",
    "print(\"------------------------\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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