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\"\"<\/figure>\n\n\n\n

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\"\"<\/figure>\n\n\n\n

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\"\"<\/figure>\n\n\n\n

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\u2463 Python \u4ee3\u7801\uff0c\u5b8c\u6574\u53ef\u8dd1<\/strong><\/h2>\n\n\n\n

\u4e0b\u9762\u8fd9\u6bb5\u4ee3\u7801\u662f\u5b8c\u6574\u7684\uff0c\u4e00\u884c\u6ca1\u5220\uff0c\u4ece\u5b9a\u4e49\u516c\u5f0f\u5230\u753b\u51fa\u62df\u5408\u66f2\u7ebf\uff0c\u4f60\u590d\u5236\u8fc7\u53bb\u5c31\u80fd\u8dd1\u3002\u6838\u5fc3\u5c31\u4e09\u6b65\uff0c\u5b9a\u4e49 SABR \u6ce2\u52a8\u7387\u516c\u5f0f\uff0c\u628a\u53c2\u6570\u6821\u51c6\u5230\u6837\u672c\u5e02\u573a\u4ef7\uff0c\u753b\u51fa\u62df\u5408\u597d\u7684\u5fae\u7b11\u66f2\u7ebf\u3002<\/p>\n\n\n\n

import numpy as np\nfrom scipy.optimize import least_squares\nimport matplotlib.pyplot as plt\n\ndef sabr_vol(f, k, t, alpha, beta, rho, nu):\n\u00a0 \u00a0 \"\"\"Hagan's approximate SABR implied Black volatility.\"\"\"\n\u00a0 \u00a0 if abs(f - k) < 1e-12: \u00a0# ATM case\n\u00a0 \u00a0 \u00a0 \u00a0 return alpha * (f ** (beta - 1)) * (\n\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 1 + (((1 - beta)**2 * alpha**2) \/ (24 * f**(2 - 2*beta)) +\n\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0(rho * beta * nu * alpha) \/ (4 * f**(1 - beta)) +\n\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0((2 - 3 * rho**2) * nu**2) \/ 24) * t)\n\u00a0 \u00a0 z = (nu \/ alpha) * ((f * k) ** ((1 - beta) \/ 2)) * np.log(f \/ k)\n\u00a0 \u00a0 x = np.log((np.sqrt(1 - 2 * rho * z + z**2) + z - rho) \/ (1 - rho))\n\u00a0 \u00a0 A = ((1 - beta)**2 * alpha**2) \/ (24 * (f * k)**(1 - beta))\n\u00a0 \u00a0 B = (rho * beta * nu * alpha) \/ (4 * (f * k)**((1 - beta)\/2))\n\u00a0 \u00a0 C = (2 - 3 * rho**2) * nu**2 \/ 24\n\u00a0 \u00a0 factor = 1 + (A + B + C) * t\n\u00a0 \u00a0 if abs(z) < 1e-8:\n\u00a0 \u00a0 \u00a0 \u00a0 vol = alpha * (f ** (beta - 1)) * factor\n\u00a0 \u00a0 else:\n\u00a0 \u00a0 \u00a0 \u00a0 vol = (alpha * z \/ x) * (1 \/ (f * k)**((1 - beta)\/2)) * factor\n\u00a0 \u00a0 return vol\n\ndef objective(params, f, t, strikes, market_vols, beta):\n\u00a0 \u00a0 alpha, rho, nu = params\n\u00a0 \u00a0 model_vols = np.array([sabr_vol(f, k, t, alpha, beta, rho, nu) for k in strikes])\n\u00a0 \u00a0 return model_vols - market_vols\n\n# Example data (SPY-like equity options, 3-month expiration)\nf = 400.0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0# Forward price\nt = 0.25 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 # Time to expiration in years\nbeta = 0.5 \u00a0 \u00a0 \u00a0 \u00a0 # Common choice for equities\nstrikes = np.array([350, 375, 400, 425, 450])\nmarket_vols = np.array([0.28, 0.24, 0.20, 0.22, 0.26]) \u00a0# Observed implied vols\n\ninitial_guess = [0.25, -0.3, 0.6]\nresult = least_squares(objective, initial_guess,\n\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0args=(f, t, strikes, market_vols, beta),\n\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0bounds=([0.01, -0.99, 0.1], [1.0, 0.99, 2.0]))\nalpha, rho, nu = result.x\nprint(\"Calibrated - Alpha:\", round(alpha, 2), \"Rho:\", round(rho, 2), \"Nu:\", round(nu, 2))\n\n# Generate fitted vols\nfitted_vols = np.array([sabr_vol(f, k, t, alpha, beta, rho, nu) for k in strikes])\n\n# Plot\nplt.figure(figsize=(10, 6))\nplt.plot(strikes, market_vols * 100, 'o-', label='Market Implied Vol', linewidth=2)\nplt.plot(strikes, fitted_vols * 100, 's--', label='SABR Fitted Smile', linewidth=2)\nplt.xlabel('Strike Price')\nplt.ylabel('Implied Volatility (%)')\nplt.title('SABR Calibration: Market vs Fitted Volatility Smile')\nplt.legend()\nplt.grid(True)\nplt.show()<\/code><\/pre>\n\n\n\n
\u8dd1\u5b8c\u7684\u8f93\u51fa\u662f\u8fd9\u6837\u7684\u3002<\/p>\n\n\n\n
Calibrated - Alpha: 1.0 Rho: -0.04 Nu: 2.0<\/code><\/pre>\n\n\n\n
\u8fd9\u6bb5\u793a\u4f8b\u7528\u7684\u662f\u7c7b SPY \u7684\u80a1\u7968\u671f\u6743\uff0c\u4e09\u4e2a\u6708\u5230\u671f\uff0c\u8fdc\u671f\u4ef7 400\uff0c\u6267\u884c\u4ef7\u4ece 350 \u5230 450 \u4e94\u4e2a\u6863\uff0c\u5bf9\u5e94\u7684\u5e02\u573a\u9690\u542b\u6ce2\u52a8\u7387\u662f 0.28\u30010.24\u30010.20\u30010.22\u30010.26\u3002\u4e00\u773c\u80fd\u770b\u51fa\uff0c\u4e2d\u95f4\u4f4e\u3001\u4e24\u8fb9\u9ad8\uff0c\u8fd9\u5c31\u662f\u5fae\u7b11\u7684\u5f62\u72b6\u3002<\/p>\n\n\n\n
\u2464 \u4e09\u4e2a\u8bfb\u6570\uff0c\u8fd8\u539f\u5e02\u573a\u5728\u60f3\u4ec0\u4e48<\/strong><\/h2>\n\n\n\n
\u6821\u51c6\u51fa\u6765\u7684\u4e09\u4e2a\u6570\uff0c\u5404\u8bb2\u4e86\u4e00\u4ef6\u4e8b\u3002<\/p>\n\n\n\n
Alpha \u662f 0.99\uff0c\u63a5\u8fd1\u4e0a\u9650\u3002\u8fd9\u8bf4\u660e\u57fa\u7840\u6ce2\u52a8\u7387\u504f\u9ad8\uff0c\u6a21\u578b\u5f97\u628a\u8fd9\u4e2a\u8d77\u70b9\u5b9a\u9ad8\uff0c\u624d\u591f\u5f97\u7740\u6574\u6761\u5fae\u7b11\u7684\u9ad8\u5ea6\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u5e02\u573a\u8ba4\u4e3a\u5f53\u524d\u6574\u4f53\u6ce2\u52a8\u4e0d\u4f4e\u3002<\/p>\n\n\n\n
Rho \u662f -0.04\uff0c\u4e00\u4e2a\u5f88\u8f7b\u7684\u8d1f\u6570\u3002\u8bf4\u660e\u504f\u659c\u5f88\u5e73\u8861\uff0c\u5e02\u573a\u6ca1\u6709\u5728\u8fd9\u4e2a\u5230\u671f\u65e5\u4e0a\uff0c\u628a\u4e0b\u884c\u6050\u60e7\u5b9a\u5f97\u6bd4\u4e0a\u884c\u98ce\u9669\u91cd\u591a\u5c11\u3002<\/p>\n\n\n\n
Nu \u662f 1.99\uff0c\u540c\u6837\u63a5\u8fd1\u4e0a\u9650\u3002\u8fd9\u662f”\u6ce2\u52a8\u7387\u7684\u6ce2\u52a8\u7387”\uff0c\u6570\u503c\u9ad8\u610f\u5473\u7740\u5fae\u7b11\u66f2\u7ebf\u7684\u66f2\u7387\u5f88\u5f3a\uff0c\u5e02\u573a\u7ed9\u5927\u6da8\u548c\u5927\u8dcc\u90fd\u8d4b\u4e86\u4e0d\u4f4e\u7684\u6982\u7387\uff0c\u4e5f\u5c31\u662f\u80a5\u5c3e\u3002<\/p>\n\n\n\n
\"\"<\/figure>\n\n\n\n
\u628a\u4e09\u4e2a\u6570\u5408\u8d77\u6765\u8bfb\uff0c\u662f\u8fd9\u4e48\u4e00\u5e45\u56fe\u666f\uff0c\u5e02\u573a\u9884\u671f\u6ce2\u52a8\u4f1a\u5f88\u5927\uff0c\u5c3e\u90e8\u98ce\u9669\u663e\u8457\uff0c\u4f46\u5bf9\u65b9\u5411\u6ca1\u6709\u660e\u663e\u504f\u5411\u3002\u6709\u4e86\u8fd9\u5957\u53c2\u6570\uff0c\u4f60\u5c31\u80fd\u7ed9\u4efb\u610f\u6267\u884c\u4ef7\u63d2\u503c\u51fa\u9690\u542b\u6ce2\u52a8\u7387\uff0c\u6216\u8005\u7ed9\u81ea\u5b9a\u4e49\u671f\u6743\u505a\u8ddf\u5e02\u573a\u4e00\u81f4\u7684\u5b9a\u4ef7\u3002<\/p>\n\n\n\n
\u2465 \u5fae\u7b11\u548c\u504f\u659c\uff0c\u5c31\u662f\u5e02\u573a\u7684\u6050\u60e7\u66f2\u7ebf<\/strong><\/h2>\n\n\n\n
\u4e00\u6761\u6821\u51c6\u597d\u7684\u5fae\u7b11\u66f2\u7ebf\uff0c\u8bb2\u7684\u5176\u5b9e\u662f\u5f53\u4e0b\u7684\u60c5\u7eea\u3002<\/p>\n\n\n\n
\u504f\u659c\u9661\u5ced\uff0c\u8bf4\u660e\u62a2\u7740\u4e70\u4e0b\u884c\u4fdd\u62a4\u7684\u4eba\u591a\uff0c\u8fd9\u5728\u7ecf\u6d4e\u62c5\u5fe7\u671f\u5f88\u5e38\u89c1\u3002\u66f2\u7387\u660e\u663e\uff0c\u8bf4\u660e\u5e02\u573a\u8ba4\u4e3a\u5927\u6da8\u5927\u8dcc\u7684\u6982\u7387\u90fd\u4e0d\u4f4e\u3002\u800c\u4e0d\u540c\u5230\u671f\u65e5\u7684\u5fae\u7b11\u5bbd\u5ea6\u8fd8\u4e0d\u4e00\u6837\uff0c\u4e34\u8fd1\u91cd\u5927\u4e8b\u4ef6\u65f6\uff0c\u77ed\u671f\u7684\u5fae\u7b11\u4f1a\u7a81\u7136\u53d8\u5bbd\uff0c\u957f\u671f\u7684\u5219\u66f4\u53cd\u6620\u7ed3\u6784\u6027\u5224\u65ad\u3002\u89c2\u5bdf SABR \u53c2\u6570\u600e\u4e48\u968f\u65f6\u95f4\u6f14\u53d8\uff0c\u80fd\u5e2e\u4f60\u5206\u6e05\uff0c\u8fd9\u662f\u4e00\u6b21\u6027\u51b2\u51fb\uff0c\u8fd8\u662f\u8d8b\u52bf\u6027\u7684\u53d8\u5316\u3002<\/p>\n\n\n\n
\u5bf9\u80a1\u7968\u6295\u8d44\u8005\u6765\u8bf4\uff0c\u8fd9\u4e2a\u4fe1\u53f7\u80fd\u8865\u4e0a\u57fa\u672c\u9762\u5206\u6790\u7684\u76f2\u533a\u3002\u5982\u679c\u4e0b\u884c\u504f\u659c\u5728\u6ca1\u6709\u516c\u53f8\u7ea7\u5229\u7a7a\u7684\u60c5\u51b5\u4e0b\u7a81\u7136\u53d8\u9661\uff0c\u80cc\u540e\u53ef\u80fd\u662f\u66f4\u5927\u7684\u7ec4\u5408\u5bf9\u51b2\u76d8\uff0c\u6216\u8005\u5b8f\u89c2\u5c42\u9762\u7684\u9ebb\u70e6\u3002\u4f01\u4e1a\u8d22\u52a1\u90e8\u95e8\u4e5f\u80fd\u7528\u5b83\u6765\u6311\u5bf9\u51b2\u65f6\u673a\uff0c\u538b\u4f4e\u6210\u672c\u3002<\/p>\n\n\n\n
\u2466 \u5982\u4f55\u7528\u8fd9\u4e2aSABR<\/strong><\/h2>\n\n\n\n
\u7ec4\u5408\u7ecf\u7406\u62ff\u6821\u51c6\u597d\u7684\u8868\u9762\uff0c\u7ed9\u590d\u6742\u671f\u6743\u5934\u5bf8\u5b9a\u51c6\u786e\u7684\u4ef7\uff0c\u7b97\u51fa\u66f4\u53ef\u9760\u7684\u5e0c\u814a\u5b57\u6bcd\u53bb\u5bf9\u51b2\u3002\u98ce\u63a7\u90e8\u95e8\u9760\u5b83\u505a\u538b\u529b\u6d4b\u8bd5\u548c\u98ce\u9669\u4ef7\u503c\u8ba1\u7b97\uff0c\u4e0d\u518d\u5047\u8bbe\u6ce2\u52a8\u7387\u670d\u4ece\u6b63\u6001\uff0c\u5c3e\u90e8\u4f30\u8ba1\u66f4\u51c6\uff0c\u8d44\u672c\u7f13\u51b2\u548c\u5e94\u6025\u9884\u6848\u4e5f\u5c31\u66f4\u5408\u7406\u3002<\/p>\n\n\n\n
\u4f01\u4e1a\u8fd9\u4e00\u4fa7\uff0c\u53d1\u53ef\u8f6c\u503a\u3001\u7ba1\u517b\u8001\u91d1\u3001\u505a\u80a1\u4efd\u56de\u8d2d\u7684\uff0c\u90fd\u80fd\u4ece\u66f4\u51c6\u7684\u4f30\u503c\u8f93\u5165\u91cc\u53d7\u76ca\u3002\u7ba1\u5916\u6c47\u548c\u5229\u7387\u655e\u53e3\u7684\u8d22\u52a1\u56e2\u961f\uff0c\u53ef\u4ee5\u6311\u90a3\u4e9b\u5e02\u573a\u5b9a\u4ef7\u66f4\u5212\u7b97\u7684\u6267\u884c\u4ef7\u548c\u5230\u671f\u65e5\u3002\u79c1\u52df\u548c\u98ce\u6295\u5728\u4ea4\u6613\u7ed3\u6784\u91cc\u8d8a\u6765\u8d8a\u591a\u78b0\u5230\u671f\u6743\u6761\u6b3e\uff0c\u61c2\u6ce2\u52a8\u7387\u5efa\u6a21\uff0c\u8c08\u5224\u548c\u6295\u540e\u90fd\u66f4\u4e3b\u52a8\u3002<\/p>\n\n\n\n
\u54ea\u6015\u4f60\u957f\u671f\u6295\u8d44\u3001\u624b\u4e0a\u6ca1\u6709\u4efb\u4f55\u671f\u6743\uff0c\u8fd9\u5957\u601d\u8def\u4e5f\u6709\u7528\u3002\u5f53 SABR \u8868\u9762\u5b9a\u4ef7\u7684\u6781\u7aef\u884c\u60c5\uff0c\u8ddf\u7ecf\u6d4e\u57fa\u672c\u9762\u660e\u663e\u5bf9\u4e0d\u4e0a\uff0c\u53ef\u80fd\u5c31\u662f\u4e2a\u6709\u8010\u5fc3\u7684\u8d44\u91d1\u8fdb\u573a\u7684\u597d\u65f6\u70b9\u3002\u53cd\u8fc7\u6765\uff0c\u5fae\u7b11\u5f02\u5e38\u5e73\u5766\uff0c\u53cd\u800c\u662f\u8be5\u8b66\u60d5\u5e02\u573a\u8fc7\u4e8e\u677e\u61c8\u7684\u4fe1\u53f7\u3002<\/p>\n\n\n\n
\u2467 \u4e94\u6761\u503c\u5f97\u8bb0\u4f4f\u7684\u6559\u8bad<\/strong><\/h2>\n\n\n\n
\u7b2c\u4e00\uff0c\u5e02\u573a\u5bf9\u98ce\u9669\u5b9a\u4ef7\u662f\u4e0d\u5bf9\u79f0\u3001\u52a8\u6001\u7684\u3002<\/strong>\u5f88\u591a\u8d44\u4ea7\u7c7b\u522b\u91cc\u6301\u7eed\u7684\u8d1f\u504f\u659c\u63d0\u9192\u4f60\uff0c\u9632\u8dcc\u662f\u6709\u6ea2\u4ef7\u7684\u3002\u505a\u7ec4\u5408\u65f6\u628a\u8fd9\u4e00\u70b9\u653e\u8fdb\u53bb\uff0c\u7528\u5206\u6563\u3001\u9009\u62e9\u6027\u5bf9\u51b2\u6216\u4e0d\u5bf9\u79f0\u6536\u76ca\u7ed3\u6784\uff0c\u97e7\u6027\u4f1a\u66f4\u597d\u3002<\/p>\n\n\n\n
\u7b2c\u4e8c\uff0c\u6ce2\u52a8\u7387\u662f\u5747\u503c\u56de\u5f52\u7684\uff0c<\/strong>\u4e2d\u95f4\u5939\u6742\u5c16\u5cf0\u8109\u51b2\u3002\u80fd\u6293\u5230 nu \u7684\u6a21\u578b\uff0c\u5e2e\u4f60\u63d0\u524d\u51c6\u5907\u8fd9\u4e9b\u8109\u51b2\uff0c\u800c\u4e0d\u662f\u5f53\u5b83\u4eec\u662f\u5b8c\u5168\u7684\u610f\u5916\u3002<\/p>\n\n\n\n
\u7b2c\u4e09\uff0c\u6821\u51c6\u597d\u7684\u6a21\u578b\u7ed9\u7684\u662f\u7ed3\u6784\uff0c<\/strong>\u4f46\u53c2\u6570\u5fc5\u987b\u653e\u5230\u5b8f\u89c2\u80cc\u666f\u3001\u884c\u4e1a\u8fdb\u5c55\u3001\u516c\u53f8\u50ac\u5316\u5242\u91cc\u53bb\u89e3\u8bfb\u3002\u91cf\u5316\u7684\u8868\u9762\u914d\u4e0a\u5b9a\u6027\u7684\u5224\u65ad\uff0c\u51b3\u7b56\u624d\u7a33\u3002<\/p>\n\n\n\n
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