{"id":4419,"date":"2026-07-07T07:44:00","date_gmt":"2026-07-06T23:44:00","guid":{"rendered":"https:\/\/laoyulaoyu.com\/?p=4419"},"modified":"2026-06-28T16:49:17","modified_gmt":"2026-06-28T08:49:17","slug":"90%e7%9a%84%e9%87%8f%e5%8c%96%e4%ba%ba%e6%b2%a1%e7%94%a8%e6%98%8e%e7%99%bdheston%ef%bc%9a%e7%89%b9%e5%be%81%e5%87%bd%e6%95%b0%e6%89%8d%e6%98%afalpha%e7%9a%84%e7%9c%9f%e6%ad%a3%e6%9d%a5%e6%ba%90","status":"publish","type":"post","link":"https:\/\/laoyulaoyu.com\/index.php\/2026\/07\/07\/90%e7%9a%84%e9%87%8f%e5%8c%96%e4%ba%ba%e6%b2%a1%e7%94%a8%e6%98%8e%e7%99%bdheston%ef%bc%9a%e7%89%b9%e5%be%81%e5%87%bd%e6%95%b0%e6%89%8d%e6%98%afalpha%e7%9a%84%e7%9c%9f%e6%ad%a3%e6%9d%a5%e6%ba%90\/","title":{"rendered":"90%\u7684\u91cf\u5316\u4eba\u6ca1\u7528\u660e\u767dHeston\uff1a\u7279\u5f81\u51fd\u6570\u624d\u662fAlpha\u7684\u771f\u6b63\u6765\u6e90"},"content":{"rendered":"\n

\u4f5c\u8005\uff1a\u8001\u4f59\u635e\u9c7c<\/a><\/p>\n\n\n\n

\u539f\u521b\u4e0d\u6613\uff0c\u8f6c\u8f7d\u8bf7\u6807\u660e\u51fa\u5904\u53ca\u539f\u4f5c\u8005\u3002<\/mark><\/strong><\/p>\n\n\n\n

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

\u672c\u6587\u7ea6 6200 \u5b57 | \u9605\u8bfb\u9884\u8ba1\u9700 14 \u5206\u949f\uff0c\u5efa\u8bae\u5148\u6536\u85cf<\/p>\n\n\n\n

\n

\u5199\u5728\u524d\u9762\u7684\u8bdd\uff1a<\/strong>\u8fd9\u7bc7\u6587\u7ae0\u7cfb\u7edf\u68b3\u7406\u4e86\u7279\u5f81\u51fd\u6570 \u03d5(t) \u5728\u91cf\u5316\u671f\u6743\u5b9a\u4ef7\u4e2d\u7684\u6838\u5fc3\u5730\u4f4d<\/mark>\uff0c\u4ee5\u53ca\u5b83\u5982\u4f55\u652f\u6491\u4e09\u4e2a\u6301\u7eed\u6027Alpha\u6765\u6e90\uff1a\u65b9\u5dee\u98ce\u9669\u6ea2\u4ef7\u3001\u6ce2\u52a8\u7387\u9762\u9519\u4f4d\u5957\u5229\u3001\u7c97\u7cd9\u5ea6\u76f8\u5bf9\u4ef7\u503c\uff0c\u8fd9\u6761\u6280\u672f\u94fe\u4eca\u5929\u5c31\u7ed9\u5927\u5bb6\u8bb2\u6e05\u695a\u3002<\/p>\n<\/blockquote>\n\n\n\n

\u91cf\u5316\u57fa\u91d1\u7684\u6570\u5b66Alpha\u4ece\u54ea\u6765<\/strong><\/h2>\n\n\n\n

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\u671f\u6743\u5b9a\u4ef7\u7684\u6838\u5fc3\u516c\u5f0f\u5927\u5bb6\u90fd\u719f\u6089\uff1a\u4f60\u9700\u8981\u628a\u6536\u76ca\u51fd\u6570\u548c\u98ce\u9669\u4e2d\u6027\u5bc6\u5ea6 q(s) \u4e58\u8d77\u6765\u505a\u79ef\u5206\u3002\u95ee\u9898\u5728\u4e8e\uff0c\u4e00\u65e6\u4f60\u7684\u6a21\u578b\u9700\u8981\u523b\u753b\u6ce2\u52a8\u7387\u805a\u96c6\u3001\u8df3\u8dc3\u3001\u539a\u5c3e\u8fd9\u4e9b\u771f\u5b9e\u5e02\u573a\u7684\u7279\u5f81\uff0cq(s) \u5c31\u6ca1\u6709\u89e3\u6790\u89e3\u4e86\u3002<\/p>\n\n\n\n

\u4f46\u7279\u5f81\u51fd\u6570 \u03d5(t) = E[e^{itX}] \u6709\u3002\u5b83\u662f\u5bc6\u5ea6\u51fd\u6570\u7684\u5085\u91cc\u53f6\u53d8\u6362\uff0c\u80fd\u5b8c\u6574\u63cf\u8ff0\u4e00\u4e2a\u968f\u673a\u8fc7\u7a0b\u7684\u5168\u90e8\u6982\u7387\u4fe1\u606f\uff1a\u5404\u9636\u77e9\u3001\u7d2f\u79ef\u91cf\u3001\u5c3e\u90e8\u884c\u4e3a\uff0c\u5168\u538b\u7f29\u5728\u4e00\u4e2a\u590d\u503c\u51fd\u6570\u91cc\u3002<\/p>\n\n\n\n

\u8fd9\u91cc\u6709\u4e2a\u5173\u952e\u7684\u4e0d\u5bf9\u79f0\u6027\uff1a\u03d5(t) \u6709\u89e3\u6790\u89e3\u7684\u6a21\u578b\u5bb6\u65cf\uff0c\u5bf9\u5e94\u7684\u5bc6\u5ea6\u51fd\u6570 f(x) \u901a\u5e38\u6ca1\u6709<\/strong>\u3002\u6b63\u662f\u8fd9\u4e2a\u4e0d\u5bf9\u79f0\uff0c\u6210\u4e86\u540e\u9762\u6240\u6709Alpha\u6765\u6e90\u7684\u6570\u5b66\u57fa\u7840\u3002<\/p>\n\n\n\n

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

\u56fe1\uff5c\u5de6\uff1a\u5bc6\u5ea6\u51fd\u6570 f(x) \u5728\u590d\u6742\u6a21\u578b\u4e0b\u65e0\u89e3\u6790\u89e3\uff1b\u53f3\uff1a\u7279\u5f81\u51fd\u6570 \u03d5(t) \u5728\u9891\u57df\u6709\u5c01\u95ed\u5f62\u5f0f\u3002\u5085\u91cc\u53f6\u53d8\u6362\u8fde\u63a5\u4e24\u7aef\uff0c\u4f46\u53ea\u6709 \u03d5(t) \u7aef\u53ef\u89e3\u6790\u8ba1\u7b97\u3002<\/p>\n\n\n\n

Kwok\u3001Leung\u548cTang\u90fd\u5728\u4ed6\u4eec\u7684\u8bba\u6587\u91cc\u8bf4\u5f97\u5f88\u76f4\u63a5\uff1a”L\u00e9vy\u8fc7\u7a0b\u53ef\u4ee5\u5b8c\u5168\u7531\u5176\u7279\u5f81\u51fd\u6570\u63cf\u8ff0\u3002\u8ba1\u7b97\u5085\u91cc\u53f6\u53d8\u6362\u7684\u79ef\u5206\u53ef\u80fd\u66f4\u5bb9\u6613\uff0c\u56e0\u4e3a\u7279\u5f81\u51fd\u6570\u6bd4\u5bc6\u5ea6\u51fd\u6570\u672c\u8eab\u66f4\u5bb9\u6613\u5904\u7406\u3002”<\/p>\n\n\n\n

\u4e09\u4e2a\u6027\u8d28\u8ba9\u8fd9\u4e2a\u66ff\u6362\u51e0\u4e4e\u9002\u7528\u4e8e\u6240\u6709\u573a\u666f\uff1a<\/p>\n\n\n\n

    \n
  • \u6536\u655b\u6027\u4fdd\u8bc1<\/strong>\uff1a|\u03d5(t)| \u2264 1 \u5bf9\u6240\u6709\u5b9e\u6570 t \u6210\u7acb\uff0c\u79ef\u5206\u603b\u662f\u6536\u655b\u7684\u3002<\/li>\n\n\n\n
  • \u5206\u5e03\u4fe1\u606f\u5b8c\u6574<\/strong>\uff1a\u6bcf\u4e2a\u7d2f\u79ef\u91cf \u03ba\u2099 \u662f \u03d5(t) \u5728 t=0 \u5904\u7684\u7b2c n \u9636\u5bf9\u6570\u5bfc\u6570\uff0c\u4e0d\u7528\u5bc6\u5ea6\u51fd\u6570\u5c31\u80fd\u6062\u590d\u5206\u5e03\u5f62\u72b6\u3002<\/li>\n\n\n\n
  • \u7ed3\u6784\u53ef\u7ee7\u627f<\/strong>\uff1a\u03d5(t) \u5728\u6574\u4e2a\u4eff\u5c04\u8df3\u6269\u6563\u6a21\u578b\u65cf\u91cc\u90fd\u4fdd\u6301\u6307\u6570\u4eff\u5c04\u7ed3\u6784\u3002<\/li>\n<\/ul>\n\n\n\n

    \u4e8c\u3001\u4eff\u5c04\u8df3\u6269\u6563\uff1a\u4e00\u4e2a\u6846\u67b6\u7edf\u4e00\u6240\u6709\u4e3b\u6d41\u6a21\u578b<\/strong><\/h2>\n\n\n\n

    Duffie\u3001Pan\u548cSingleton\u57282000\u5e74\u53d1\u5728 Econometrica<\/em> \u7684\u8bba\u6587\u505a\u4e86\u4e00\u4ef6\u7edf\u4e00\u7684\u4e8b\u3002\u5bf9\u4e8e\u4efb\u4f55”\u4eff\u5c04\u8df3\u6269\u6563”\uff08AJD\uff09\u6a21\u578b\uff0c\u53ea\u8981\u6f02\u79fb\u9879\u3001\u6269\u6563\u7cfb\u6570\u548c\u8df3\u8dc3\u5f3a\u5ea6\u90fd\u662f\u72b6\u6001\u5411\u91cf X \u7684\u4eff\u5c04\u51fd\u6570\uff0c\u6761\u4ef6\u671f\u671b\u5c31\u6ee1\u8db3\uff1a<\/p>\n\n\n\n

    E[e^{u\u00b7X_T} | X_t] = exp(\u03b1(u, \u03c4) + \u03b2(u, \u03c4)\u00b7X_t)<\/code><\/pre>\n\n\n\n
    \u5176\u4e2d \u03b1 \u548c \u03b2 \u89e3\u4e00\u7ec4\u590d\u503cRiccati\u5e38\u5fae\u5206\u65b9\u7a0b\u3002\u4ee4 u = it \u5c31\u80fd\u6062\u590d\u4efb\u610fAJD\u6a21\u578b\u7684\u7279\u5f81\u51fd\u6570\u3002<\/p>\n\n\n\n
    \u8fd9\u662f\u4ec0\u4e48\u610f\u601d\uff1fHeston\u968f\u673a\u6ce2\u52a8\u7387\u6a21\u578b\u3001Bates\u5e26\u8df3\u8dc3\u7684\u968f\u673a\u6ce2\u52a8\u7387\u6a21\u578b\u3001\u591a\u56e0\u5b50\u80a1\u503a\u6a21\u578b\u3001\u5229\u7387\u4eff\u5c04\u671f\u9650\u7ed3\u6784\u6a21\u578b\uff0c\u5168\u5c5e\u4e8e\u8fd9\u4e00\u7c7b<\/strong>\u3002\u5bf9\u6bcf\u4e2a\u6a21\u578b\uff0c\u03d5(t) \u53ea\u9700\u7b97\u4e00\u6b21\uff0c\u5c31\u80fd\u7528\u4e8e\u5168\u884c\u6743\u4ef7\u3001\u5168\u5230\u671f\u65e5\u7684\u5b9a\u4ef7\u3002<\/p>\n\n\n\n
    Duffie\u7b49\u4eba\u5728\u8bba\u6587\u91cc\u8bf4\u5f97\u5f88\u6e05\u695a\uff1a”\u5e94\u7528\u793a\u4f8b\u5305\u62ec\u56fa\u5b9a\u6536\u76ca\u5b9a\u4ef7\u6a21\u578b\uff08\u542b\u8fdd\u7ea6\u7684\u5f3a\u5ea6\u6a21\u578b\uff09\u4ee5\u53ca\u5e7f\u6cdb\u7684\u671f\u6743\u5b9a\u4ef7\u5e94\u7528\u3002”<\/p>\n\n\n\n
    \u4e09\u3001Carr-Madan 1999\uff1a\u8ba9\u5b9e\u65f6\u5b9a\u4ef7\u6210\u4e3a\u53ef\u80fd\u7684\u7b97\u6cd5<\/strong><\/h2>\n\n\n\n
    \u77e5\u9053 \u03d5(t) \u6709\u89e3\u6790\u89e3\u53ea\u662f\u7b2c\u4e00\u6b65\u3002\u8981\u505a\u5b9e\u76d8\u4ea4\u6613\uff0c\u4f60\u8fd8\u9700\u8981\u4e00\u4e2a\u7b97\u6cd5\uff0c\u5feb\u5230\u80fd\u5bf9\u6574\u6761\u671f\u6743\u94fe\u5b9a\u4ef7\u3002<\/p>\n\n\n\n
    Carr\u548cMadan\u57281999\u5e74\u7684\u8bba\u6587\u89e3\u51b3\u4e86\u4e00\u4e2a\u5177\u4f53\u7684\u79ef\u5206\u969c\u788d\uff1aCall\u671f\u6743\u4ef7\u683c\u4f5c\u4e3a\u5bf9\u6570\u884c\u6743\u4ef7 k = log(K) \u7684\u51fd\u6570\u4e0d\u662f\u5e73\u65b9\u53ef\u79ef\u7684\uff0c\u5085\u91cc\u53f6\u53d8\u6362\u4e0d\u6536\u655b\u3002\u4ed6\u4eec\u7684\u89e3\u6cd5\u662f\u52a0\u4e00\u4e2a\u963b\u5c3c\u56e0\u5b50\uff1a<\/p>\n\n\n\n
    c_T(k) = e^{\u03b1k} \u00b7 C_T(k), \u00a0\u03b1 > 0\n\u03c8_T(v) = e^{\u2212rT} \u03d5_T(v \u2212 (\u03b1+1)i) \/ (\u03b1\u00b2 + \u03b1 \u2212 v\u00b2 + i(2\u03b1+1)v)<\/code><\/pre>\n\n\n\n
    \u7136\u540e\u7528FFT\uff08\u5feb\u901f\u5085\u91cc\u53f6\u53d8\u6362\uff09\u5bf9\u6574\u4e2a\u884c\u6743\u4ef7\u7f51\u683c\u505a\u53cd\u6f14\uff0cN\u4e2a\u884c\u6743\u4ef7\u7684\u5b9a\u4ef7\u53ea\u9700 O(N log N) \u6b21\u8fd0\u7b97\uff0c\u800c\u4f20\u7edf\u6c42\u79ef\u6cd5\u662f O(N\u00b2)\u3002<\/p>\n\n\n\n
    Carr\u548cMadan\u539f\u6587\u8bf4\uff1a”\u6211\u4eec\u5728\u53cd\u6f14\u9636\u6bb5\u4f7f\u7528FFT\uff0c\u5141\u8bb8\u4f7f\u7528\u771f\u5b9e\u6a21\u578b\u8fdb\u884c\u5b9e\u65f6\u5b9a\u4ef7\u3001\u6807\u8bb0\u548c\u5bf9\u51b2\uff0c\u5373\u4f7f\u5bf9\u4e8e\u5305\u542b\u6570\u5343\u4e2a\u671f\u6743\u7684\u7ec4\u5408\u4e5f\u662f\u5982\u6b64\u3002”<\/p>\n\n\n\n
    \u901f\u5ea6\u5dee\u8ddd\u6709\u591a\u5927\uff1fCarr-Madan\u8bba\u6587\u62a5\u544a\u975eFFT\u89e3\u6790\u65b9\u6cd5\u5bf9160\u4e2a\u671f\u6743\u8981”\u7ea66.5\u79d2”\u3002COS\u65b9\u6cd5\u8bba\u6587\u62a5\u544a N<150 \u65f6”\u4e0d\u52300.5\u6beb\u79d2”\u5c31\u8fbe\u52308\u4f4d\u7cbe\u5ea6\u3002\u6574\u4e2a\u7ec4\u5408\u7684\u6821\u51c6\u5468\u671f\u4ece\u5206\u949f\u7ea7\u964d\u5230\u79d2\u7ea7\u3002<\/p>\n\n\n\n
    \u901f\u5ea6\u5373\u4f18\u52bf<\/strong>\uff1a\u6821\u51c6\u9891\u7387\u7684\u5dee\u8ddd\u662f\u5b9e\u76d8\u6267\u884c\u4f18\u52bf\u7684\u6765\u6e90\u3002\u8dd1\u5f97\u8d77\u8fde\u7eed\u6821\u51c6\u7684\u4ea4\u6613\u53f0\uff0c\u548c\u53ea\u80fd\u6bcf\u5c0f\u65f6\u8dd1\u4e00\u6b21\u7684\uff0c\u5728\u5feb\u884c\u60c5\u91cc\u5b8c\u5168\u662f\u4e24\u4e2a\u6e38\u620f\u3002<\/p>\n\n\n\n
    \"\"<\/figure>\n\n\n\n
    \u56fe2\uff5cCarr-Madan FFT \u5b9a\u4ef7\u7ba1\u7ebf\uff1a\u6a21\u578b\u53c2\u6570 \u2192 \u7279\u5f81\u51fd\u6570 \u03d5(t) \u2192 \u963b\u5c3c\u4fee\u6b63 \u2192 FFT \u53cd\u6f14 \u2192 \u5168\u884c\u6743\u4ef7\u671f\u6743\u4ef7\u683c\u3002\u4e00\u6b21FFT\u8c03\u7528\u540c\u65f6\u8f93\u51faN\u4e2a\u884c\u6743\u4ef7\u7684\u5b9a\u4ef7\u3002<\/p>\n\n\n\n
    \u56db\u3001Heston 1993\uff1a\u884c\u4e1a\u6807\u51c6\u7684\u7279\u5f81\u51fd\u6570<\/strong><\/h2>\n\n\n\n
    Heston\u57281993\u5e74 Review of Financial Studies<\/em> \u7684\u8bba\u6587\uff0c\u7ed9\u51fa\u4e86\u7b2c\u4e00\u4e2a\u968f\u673a\u6ce2\u52a8\u7387\u6a21\u578b\u7684\u5c01\u95ed\u5f62\u5f0f\u7279\u5f81\u51fd\u6570\u3002\u6a21\u578b\u8bbe\u5b9a\uff1a<\/p>\n\n\n\n
    dS\/S = \u03bc dt + \u221av dW\u2081\ndv = \u03ba(\u03b8 \u2212 v) dt + \u03c3_v \u221av dW\u2082\nd\u27e8W\u2081, W\u2082\u27e9 = \u03c1 dt<\/code><\/pre>\n\n\n\n
    \u901a\u8fc7Feynman-Ka\u010d\u5b9a\u7406\u63a8\u5bfc\uff0c\u7279\u5f81\u51fd\u6570\u5448\u6307\u6570\u4eff\u5c04\u5f62\u5f0f\uff0c\u89e3\u4e00\u4e2a\u5173\u4e8e {\u03ba, \u03b8, \u03c3_v, \u03c1, v\u2080} \u4e94\u4e2a\u53c2\u6570\u7684ODE\u5c31\u6709\u89e3\u6790\u89e3\u3002\u8fd9\u5c31\u662f\u63d2\u8fdbCarr-Madan FFT\u7ba1\u7ebf\u91cc\u7684\u90a3\u4e2a\u51fd\u6570\u3002<\/p>\n\n\n\n
    \u6821\u51c6\u8fc7\u7a0b\u5c31\u662f\u6700\u5c0f\u5316FFT\u8ba1\u7b97\u7684\u6a21\u578b\u4ef7\u683c\u4e0e\u5e02\u573a\u4e2d\u4ef7\u4e4b\u95f4\u7684\u52a0\u6743\u5e73\u65b9\u5dee\u3002Cui\u7b49\u4eba2017\u5e74\u7684\u5de5\u4f5c\u7ed9\u51fa\u4e86Heston\u671f\u6743\u4ef7\u683c\u5bf9\u6240\u6709\u4e94\u4e2a\u53c2\u6570\u7684\u89e3\u6790\u68af\u5ea6\uff0c\u66ff\u4ee3\u6570\u503c\u6709\u9650\u5dee\u5206\uff0c\u68af\u5ea6\u8ba1\u7b97\u901f\u5ea6\u63d0\u5347\u7ea610\u500d\u3002\u4ed6\u4eec\u8fd8\u4fee\u4e86\u4e00\u4e2a\u751f\u4ea7\u73af\u5883\u91cc\u7684\u5751\uff1a\u539f\u59cbHeston \u03d5(t) \u91cc\u7684\u590d\u5e73\u65b9\u6839\u6709\u652f\u5272\u4e0d\u8fde\u7eed\u6027\uff0c\u4f1a\u5bfc\u81f4\u67d0\u4e9b\u53c2\u6570\u7ec4\u5408\u4e0b\u6570\u503c\u4e0d\u7a33\u5b9a\u3001\u5e0c\u814a\u5b57\u6bcd\u7b97\u9519\u3002\u4ed6\u4eec\u7684\u91cd\u65b0\u8868\u8ff0\u6d88\u9664\u4e86\u8fd9\u4e2a\u6545\u969c\u6a21\u5f0f\u3002<\/p>\n\n\n\n
    \u751f\u4ea7\u73af\u5883\u63d0\u9192<\/strong>\uff1aCui\u7b49\u4eba\u6307\u51fa\uff0c\u5f15\u7528Guillaume\u548cSchoutens\u7684\u8bdd\uff0c”\u4e0d\u540c\u7684\u6821\u51c6\u7a0b\u5e8f\u53ef\u80fd\u5bfc\u81f4\u5947\u5f02\u671f\u6743\u4ef7\u683c\u548c\u5bf9\u51b2\u6bd4\u7387\u51fa\u73b0\u5b9e\u8d28\u6027\u5dee\u5f02”\u3002\u4e24\u4efd\u6821\u51c6\u53ef\u80fd\u62df\u5408\u51fa\u5b8c\u5168\u76f8\u540c\u7684\u6ce2\u52a8\u7387\u9762\uff0c\u4f46\u5728\u969c\u788d\u671f\u6743\u4ef7\u683c\u4e0a\u5206\u6b67\u5f88\u5927\u3002\u8fd9\u662f\u5b9a\u4ef7\u5947\u5f02\u671f\u6743\u65f6\u7684\u7ed3\u6784\u6027\u6a21\u578b\u98ce\u9669\u3002<\/p>\n\n\n\n
    \u4e94\u3001\u65b9\u5dee\u98ce\u9669\u6ea2\u4ef7\uff1a\u6700\u624e\u5b9e\u7684Alpha\u6765\u6e90<\/strong><\/h2>\n\n\n\n
    \u65b9\u5dee\u4e92\u6362\uff08Variance Swap\uff09\u662f\u4e00\u4e2aOTC\u5408\u7ea6\uff0c\u5230\u671f\u65f6\u652f\u4ed8\u5df2\u5b9e\u73b0\u65b9\u5dee\u4e0e\u56fa\u5b9a\u4e92\u6362\u5229\u7387\u7684\u5dee\u989d\uff0c\u5165\u573a\u96f6\u6210\u672c\u3002\u4e92\u6362\u5229\u7387\u7b49\u4e8e\u98ce\u9669\u4e2d\u6027\u65b9\u5dee\u671f\u671b E_Q[\u222b\u2080^T \u03c3\u00b2_t dt]\uff0c\u53ef\u4ee5\u4ece\u671f\u6743\u4ef7\u683c\u5408\u6210\u3002\u65b9\u5dee\u98ce\u9669\u6ea2\u4ef7\uff08VRP\uff09\u5c31\u662f\u8fd9\u4e2a\u5408\u6210\u5229\u7387\u4e0e\u540e\u7eed\u5b9e\u9645\u65b9\u5dee\u7684\u5dee\u3002<\/p>\n\n\n\n
    5.1 \u5b9e\u8bc1\u8bc1\u636e\u94fe<\/strong><\/h3>\n\n\n\n
    Bakshi\u548cKapadia 2003\u5e74\u7528\u6807\u666e500\u6307\u6570\u671f\u6743\u6784\u5efa\u4e86Delta\u5bf9\u51b2\u7684Call\u671f\u6743\u7ec4\u5408\uff0c\u9996\u6b21\u7ed9\u51fa\u76f4\u63a5\u5b9e\u8bc1\u8bc1\u636e\uff1a”Delta\u5bf9\u51b2\u7b56\u7565\u7684\u6536\u76ca\u4f4e\u4e8e\u96f6”\u4e14”\u6ce2\u52a8\u7387\u8d8a\u9ad8\u65f6\u4f4e\u5f97\u8d8a\u591a”\u3002\u8fd9\u4e2a\u8d1f\u6ce2\u52a8\u7387\u98ce\u9669\u6ea2\u4ef7\u7684\u53d1\u73b0\uff0c\u6210\u4e86\u505a\u7a7a\u65b9\u5dee\u4ea4\u6613\u7684\u5b9e\u8bc1\u57fa\u7840\u3002<\/p>\n\n\n\n
    Carr\u548cWu 2009\u5e74\u5728 Review of Financial Studies<\/em> \u628a\u89c4\u6a21\u653e\u5927\u4e86\u3002\u4ed6\u4eec\u75285\u4e2a\u4e3b\u8981\u80a1\u6307\u548c35\u53ea\u4e2a\u80a17\u5e74\u7684\u671f\u6743\u6570\u636e\u5408\u6210\u65b9\u5dee\u4e92\u6362\u5229\u7387\uff0c\u4e0d\u5047\u8bbe\u4efb\u4f55\u53c2\u6570\u5316\u52a8\u6001\u3002\u6838\u5fc3\u53d1\u73b0\uff1a”\u6536\u76ca\u65b9\u5dee\u7684\u5e73\u5747\u98ce\u9669\u6ea2\u4ef7\u5bf9\u6807\u666e500\u3001\u6807\u666e100\u548c\u9053\u743c\u65af\u663e\u8457\u4e3a\u8d1f\u3002”\u8d1f\u53f7\u610f\u5473\u7740\u65b9\u5dee\u5356\u65b9\uff0c\u4e5f\u5c31\u662f\u4e92\u6362\u7684\u7a7a\u5934\uff0c\u7cfb\u7edf\u6027\u6536\u5230\u6ea2\u4ef7\u3002<\/p>\n\n\n\n
    5.2 \u5bf9\u51b2\u57fa\u91d1\u5728\u505a\u4ec0\u4e48<\/strong><\/h3>\n\n\n\n
    Bondarenko 2004\u5e74\u7684\u8bba\u6587\uff082006 AFA\u6ce2\u58eb\u987f\u4f1a\u8bae\u5ba3\u8bfb\uff09\u5206\u6790\u4e86\u5bf9\u51b2\u57fa\u91d1\u622a\u9762\u6570\u636e\uff0c\u53d1\u73b0\uff1a”\u65b9\u5dee\u98ce\u9669\u88ab\u5b9a\u4ef7\uff0c\u5176\u98ce\u9669\u6ea2\u4ef7\u4e3a\u8d1f\u4e14\u7ecf\u6d4e\u4e0a\u975e\u5e38\u5927\u3002\u5927\u90e8\u5206\u5bf9\u51b2\u57fa\u91d1\u5bf9\u65b9\u5dee\u6536\u76ca\u5448\u8d1f\u655e\u53e3\uff0c\u6697\u793a\u5b83\u4eec\u5e38\u89c4\u6027\u5730’\u505a\u7a7a’\u65b9\u5dee\u98ce\u9669\u3002\u65b9\u5dee\u98ce\u9669\u56e0\u5b50\u89e3\u91ca\u4e86\u5bf9\u51b2\u57fa\u91d1\u5386\u53f2\u6536\u76ca\u7684\u76f8\u5f53\u4e00\u90e8\u5206\u3002”<\/p>\n\n\n\n
    \u7ef4\u57fa\u767e\u79d1”\u65b9\u5dee\u98ce\u9669\u6ea2\u4ef7”\u8bcd\u6761\u5f15\u7528Carr-Wu (2009)\u8bf4\uff1a”\u57282008\u5e74\u91d1\u878d\u5371\u673a\u524d\u7684\u51e0\u5e74\u91cc\uff0c\u6eda\u52a8\u505a\u7a7a\u65b9\u5dee\u662f\u5bf9\u51b2\u57fa\u91d1\u548c\u5176\u4ed6\u673a\u6784\u6295\u8d44\u8005\u7684\u70ed\u95e8\u4ea4\u6613\u3002”<\/p>\n\n\n\n
    \u6267\u884c\u65b9\u5f0f\uff1a\u4ee5FFT\u7b97\u51fa\u7684\u4e92\u6362\u5229\u7387\u505a\u7a7a1\u4e2a\u6708\u671f\u65b9\u5dee\u4e92\u6362\uff0c\u6bcf\u65e5Delta\u5bf9\u51b2\uff0c\u6708\u5ea6\u5c55\u671f\u3002\u6301\u7eed\u6027\u4f18\u52bf\u6765\u81eaVRP\u3002\u5c3e\u90e8\u98ce\u9669\u662f\u6ce2\u52a8\u7387\u8df3\u5347\u5bfc\u81f4\u5927\u7684\u76ef\u5e02\u635f\u5931\u3002\u4ed3\u4f4d\u7ba1\u7406\u6b63\u786e\u7684\u57fa\u91d1\uff0c\u7528\u540c\u4e00\u5957 \u03d5(t) \u57fa\u7840\u8bbe\u65bd\u7ed9\u5c3e\u90e8\u5bf9\u51b2\u671f\u6743\u5b9a\u4ef7\uff0c\u5728\u6267\u884c\u524d\u5bf9\u7167\u81ea\u5df1\u7684\u6a21\u578b\u4ef7\u9a8c\u8bc1\u5bf9\u51b2\u6210\u672c\u3002<\/p>\n\n\n\n
    5.3 VRP\u8fd8\u80fd\u9884\u793a\u80a1\u7968\u6536\u76ca<\/strong><\/h3>\n\n\n\n
    Bollerslev\u3001Tauchen\u548cZhou 2009\u5e74\u5728 Review of Financial Studies<\/em> \u53d1\u73b0\uff0c\u98ce\u9669\u4e2d\u6027\u65b9\u5dee\u4e0e\u5df2\u5b9e\u73b0\u65b9\u5dee\u7684\u5dee”\u80fd\u89e3\u91ca1990\u5e74\u540e\u80a1\u5e02\u6536\u76ca\u65f6\u95f4\u5e8f\u5217\u53d8\u5316\u4e2d\u4e0d\u53ef\u5ffd\u7565\u7684\u4e00\u90e8\u5206\uff0c\u9ad8\uff08\u4f4e\uff09\u6ea2\u4ef7\u9884\u793a\u9ad8\uff08\u4f4e\uff09\u672a\u6765\u6536\u76ca\u3002”<\/p>\n\n\n\n
    \u5173\u952e\u65b9\u6cd5\u8bba\u70b9\uff1a”\u6211\u4eec\u7684\u5b9e\u8bc1\u7ed3\u679c\u5173\u952e\u4f9d\u8d56\u4e8e\u4f7f\u7528’\u6a21\u578b\u65e0\u5173’\u800c\u975eBlack-Scholes\u7684\u9690\u542b\u6ce2\u52a8\u7387\u3002”\u6a21\u578b\u65e0\u5173\u9690\u542b\u65b9\u5dee\u901a\u8fc7Carr-Madan-Wu\u5bf9\u6570\u5408\u7ea6\u8ba1\u7b97\uff0c\u8fd9\u672c\u8eab\u5c31\u662f \u03d5(t) \u5bfc\u51fa\u671f\u6743\u4ef7\u683c\u7684\u79ef\u5206\u3002\u7528Black-Scholes ATM\u9690\u542b\u6ce2\u52a8\u7387\u66ff\u6362\u540e\uff0c\u6536\u76ca\u9884\u793a\u80fd\u529b\u6d88\u5931\u3002\u4fe1\u606f\u5728 \u03d5(t) \u7684\u5f62\u72b6\u91cc\uff0c\u4e0d\u53ea\u662f\u67d0\u4e2a\u5355\u70b9\u7684\u503c\u3002<\/p>\n\n\n\n
    \u5b9e\u64cd\u542b\u4e49<\/strong>\uff1a\u8dd1\u5b9e\u65f6 \u03d5(t) \u6821\u51c6\u7684\u4ea4\u6613\u53f0\u53ef\u4ee5\u6bcf\u65e5\u8ba1\u7b97\u6a21\u578b\u65e0\u5173VRP\uff0c\u4f5c\u4e3a\u4ed3\u4f4d\u7ba1\u7406\u7684\u989d\u5916\u4fe1\u53f7\u3002Bollerslev\u7b49\u4eba\u53d1\u73b0\u9884\u793a\u80fd\u529b”\u5728\u4e2d\u95f4\u7684\u5b63\u5ea6\u6536\u76ca\u671f\u9650\u4e0a\u7279\u522b\u5f3a\uff0c\u8d85\u8fc7\u4e86P\/E\u6bd4\u7387\u3001\u8fdd\u7ea6\u5229\u5dee\u548c\u6d88\u8d39\u8d22\u5bcc\u6bd4\u7387\u7b49\u5176\u4ed6\u6d41\u884c\u5148\u884c\u6307\u6807”\u3002<\/p>\n\n\n\n
    \u516d\u3001\u7c97\u7cd9\u6ce2\u52a8\u7387\uff1a\u77ed\u671f\u671f\u6743\u4e0a\u7684\u6a21\u578b\u4f18\u52bf<\/strong><\/h2>\n\n\n\n
    \u7ecf\u5178Heston\u6a21\u578b\u5728\u77ed\u671f\u9650\u4e0a\u6709\u4e00\u4e2a\u5df2\u77e5\u7684\u5931\u6548\u70b9\u3002Gatheral\u3001Jaisson\u548cRosenbaum 2018\u5e74\u5728 Quantitative Finance<\/em>\uff082014\u5e74\u9996\u6b21\u6302\u5728SSRN\uff09\u5b9e\u8bc1\u4e86\u6ce2\u52a8\u7387\u5bf9\u6570”\u57fa\u672c\u4e0a\u8868\u73b0\u4e3aHurst\u6307\u6570 H \u7ea6\u4e3a0.1\u7684\u5206\u6570\u5e03\u6717\u8fd0\u52a8”\u3002H=0.1 \u610f\u5473\u7740\u6ce2\u52a8\u7387\u589e\u91cf\u5f3a\u53cd\u76f8\u5173\uff0c\u8def\u5f84\u6bd4\u6807\u51c6\u5e03\u6717\u8fd0\u52a8\uff08H=0.5\uff09\u7c97\u7cd9\u5f97\u591a\u3002\u65e5\u672c\u94f6\u884c\u7684\u7c97\u7cd9\u6ce2\u52a8\u7387\u7efc\u8ff0\u786e\u8ba4\uff1a”Hurst\u53c2\u6570\u4f30\u8ba1\u503c\u7a33\u5b9a\u57280.1\u5de6\u53f3\uff080.082-0.142\uff09\uff0c\u4e0d\u8bba\u8d44\u4ea7\u7c7b\u578b\u3002”<\/p>\n\n\n\n
    \u6a21\u578b\u7c7b\u578b<\/th>Hurst\u6307\u6570H<\/th>ATM\u504f\u5ea6\u884c\u4e3a (T\u21920)<\/th>\u80fd\u5426\u62df\u5408\u77ed\u671f\u9650<\/th><\/tr>
    \u6807\u51c6\u5e03\u6717\u8fd0\u52a8<\/td>0.5<\/td>\u8d8b\u5e73<\/td>\u4e0d\u80fd<\/td><\/tr>
    \u7c97\u7cd9\u6ce2\u52a8\u7387<\/td>~0.1<\/td>T^{H-1\/2} \u7206\u70b8<\/td>\u80fd<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n
    \"\"<\/figure>\n\n\n\n
    \u56fe3\uff5c\u4e0a\uff1aH=0.5 \u6807\u51c6\u5e03\u6717\u8fd0\u52a8\u751f\u6210\u7684\u6ce2\u52a8\u7387\u8def\u5f84\uff0c\u8868\u9762\u5e73\u6ed1\uff1b\u4e0b\uff1aH=0.1 \u7c97\u7cd9\u6ce2\u52a8\u7387\u8def\u5f84\uff0c\u5448\u952f\u9f7f\u72b6\u5267\u70c8\u8df3\u53d8\u3002\u771f\u5b9e\u5e02\u573a\u6ce2\u52a8\u7387\u7684Hurst\u6307\u6570\u7a33\u5b9a\u57280.1\u9644\u8fd1\u3002<\/p>\n\n\n\n
    \u6807\u51c6\u6a21\u578b\u7684ATM\u504f\u5ea6\u5728\u671f\u9650\u8d8b\u4e8e\u96f6\u65f6\u53d8\u5e73\u3002\u7c97\u7cd9\u6ce2\u52a8\u7387\u6a21\u578b\u7684\u504f\u5ea6\u4ee5 T^{H-1\/2} \u7206\u70b8\uff0c\u6b63\u597d\u5339\u914d\u77ed\u671f\u9650SPX\u671f\u6743\u4e2d\u89c2\u5bdf\u5230\u7684\u5e42\u5f8b\u3002\u8fd9\u610f\u5473\u7740\u7ecf\u5178Heston\u5728\u7ed3\u6784\u4e0a\u65e0\u6cd5\u6b63\u786e\u62df\u5408\u8fd1\u6708\u6ce2\u52a8\u7387\u9762\u3002<\/p>\n\n\n\n
    El Euch\u548cRosenbaum 2019\u5e74\u5728 Mathematical Finance<\/em> \u63a8\u5bfc\u51fa\uff0c\u7c97\u7cd9Heston\u7684log-price\u7279\u5f81\u51fd\u6570\u4ecd\u7136\u5b58\u5728\u534a\u5c01\u95ed\u5f62\u5f0f\uff0c\u4f46\u5b83\u6ee1\u8db3\u7684\u4e0d\u662f\u6807\u51c6Riccati ODE\uff0c\u800c\u662f\u4e00\u4e2a\u7531 \u03b1 = H + 1\/2 \u53c2\u6570\u5316\u7684\u5206\u6570\u9636Riccati\u65b9\u7a0b\u3002FFT\u7ba1\u7ebf\u7167\u7528\uff0c\u989d\u5916\u5f00\u9500\u5728\u89e3\u5206\u6570\u9636ODE\u3002<\/p>\n\n\n\n
    \u8fd9\u4e2a\u6a21\u578b\u4f18\u52bf\u5e26\u6765\u7684Alpha\u5f88\u76f4\u63a5\uff1a\u7528\u7c97\u7cd9Heston\u6821\u51c6\u7684\u4ea4\u6613\u53f0\uff0c\u9762\u5bf9\u8fd8\u5728\u7528\u7ecf\u5178Heston\u7684\u4ea4\u6613\u5bf9\u624b\u65b9\uff0c\u5728\u8fd1\u6708\u671f\u6743\u4e0a\u6709\u66f4\u51c6\u786e\u7684 \u03d5(t)\u3002\u8003\u8651\u52300DTE\uff08\u96f6\u65e5\u5230\u671f\uff09\u671f\u6743\u5728SPX\u6210\u4ea4\u91cf\u4e2d\u7684\u5360\u6bd4\uff0cCboe\u786e\u8ba42023\u5e74\u5e95\u7ea645%\u30012024\u5e74\u7ea647%\uff0c\u8fd9\u4e2a\u6a21\u578b\u5dee\u8ddd\u5728\u6700\u6d41\u52a8\u7684\u5e02\u573a\u90e8\u5206\u4ea7\u751f\u5b9a\u4ef7\u504f\u5dee\u4fe1\u53f7\u3002<\/p>\n\n\n\n
    \u4e03\u3001”\u4e70\u7c97\u7cd9\u3001\u5356\u5e73\u6ed1”\uff1a\u5e74\u53166%\u7684\u7c97\u7cd9\u5ea6\u7b56\u7565<\/strong><\/h2>\n\n\n\n
    Glasserman\u548cHe 2020\u5e74\u5728 Quantitative Finance<\/em> \u53d1\u8868\u4e86\u7b2c\u4e00\u4e2a\u76f4\u63a5\u57fa\u4e8e\u9690\u542b\u6ce2\u52a8\u7387Hurst\u6307\u6570\u6784\u5efa\u7684\u4ea4\u6613\u7b56\u7565\u7684\u7cfb\u7edf\u56de\u6d4b\u3002\u7b56\u7565\u5f88\u7b80\u5355\uff1a\u505a\u591a\u9690\u542b\u6ce2\u52a8\u7387\u6700”\u7c97\u7cd9”\u7684\u80a1\u7968\uff0c\u505a\u7a7a\u6700”\u5e73\u6ed1”\u7684\u80a1\u7968\u3002<\/p>\n\n\n\n
    \u7ed3\u679c\uff1a”\u505a\u591a\u6700\u7c97\u7cd9\u6ce2\u52a8\u7387\u80a1\u7968\u3001\u505a\u7a7a\u6700\u5e73\u6ed1\u6ce2\u52a8\u7387\u80a1\u7968\u7684\u7b56\u7565\u53d6\u5f97\u7edf\u8ba1\u663e\u8457\u7684\u5e74\u5316\u8d85\u989d\u6536\u76ca6%\u6216\u66f4\u591a\uff0c\u5177\u4f53\u53d6\u51b3\u4e8e\u65f6\u95f4\u6bb5\u548c\u7b56\u7565\u7ec6\u8282\u3002\u7b56\u7565\u7684\u76c8\u5229\u80fd\u529b\u65e0\u6cd5\u88ab\u6807\u51c6\u56e0\u5b50\u89e3\u91ca\u3002”<\/p>\n\n\n\n
    \u4ed6\u4eec\u8fd8\u53d1\u73b0\u4e86\u4e00\u4e2a\u91cd\u8981\u7684\u6539\u8fdb\u70b9\uff1a”\u57fa\u4e8e\u9690\u542b\u800c\u975e\u5df2\u5b9e\u73b0\u7c97\u7cd9\u5ea6\u6392\u5e8f\u65f6\uff0c\u7b56\u7565\u76c8\u5229\u80fd\u529b\u66f4\u5f3a\u3002”\u9690\u542b\u7c97\u7cd9\u5ea6\u901a\u8fc7ATM\u504f\u5ea6\u5728\u4e0d\u540c\u884c\u6743\u4ef7\u548c\u5230\u671f\u65e5\u4e0a\u7684\u8870\u51cf\u7387\u6765\u8861\u91cf\uff0c\u8fd9\u662f\u7c97\u7cd9\u6ce2\u52a8\u7387\u6a21\u578b\u4e0b \u03d5(t) \u77ed\u671f\u6e10\u8fd1\u6027\u8d28\u7684\u76f4\u63a5\u4f53\u73b0\u3002\u5b83\u6bd4\u4ece\u5386\u53f2\u9ad8\u9891\u6570\u636e\u4f30\u8ba1\u7684\u7c97\u7cd9\u5ea6\u63d0\u4f9b\u4e86\u66f4\u5e72\u51c0\u7684\u4fe1\u53f7\u3002<\/p>\n\n\n\n
    \u5b9e\u76d8\u5b9e\u73b0\u8bb0\u5f55<\/strong><\/h3>\n\n\n\n
    Harel Jacobson\uff0c\u73b0Capstone Investment Advisors\u7684FX\u6ce2\u52a8\u7387\u4ea4\u6613\u5458\uff0c2021\u5e74\u516c\u5f00\u53d1\u8868\u4e86\u5c06Glasserman-He\u6846\u67b6\u5e94\u7528\u4e8eFX\u6ce2\u52a8\u7387\u7684\u8be6\u7ec6\u63cf\u8ff0\u3002\u4ed6\u5199\u9053\uff1a”\u53d72018\u5e74\u90a3\u7bc7\u53eb’Buy Rough, Sell Smooth’\u7684\u8bba\u6587\u542f\u53d1\uff0c\u6211\u5f00\u59cb\u7422\u78e8\u6709\u6ca1\u6709\u66f4\u597d\u7684\u65b9\u6cd5\u66ff\u4ee3\u4f20\u7edf\u7684’\u5df2\u5b9e\u73b0-\u9690\u542b’\u6ce2\u52a8\u7387\u4ea4\u6613\u3002”<\/p>\n\n\n\n
    \u4ed6\u7684\u6838\u5fc3\u53d1\u73b0\uff1a\u6d41\u52a8\u6027\u8f83\u4f4e\u3001\u8def\u5f84\u66f4\u7c97\u7cd9\u7684\u8d27\u5e01\u5bf9\u786e\u5b9e\u8dd1\u8d62\u6d41\u52a8\u6027\u5f3a\u3001\u8def\u5f84\u66f4\u5e73\u6ed1\u7684\u8d27\u5e01\u5bf9\uff0c\u4e3b\u8981\u96c6\u4e2d\u5728\u66f2\u7ebf\u524d\u7aef\u3002<\/p>\n\n\n\n
    Harbourfront Quant Substack\u603b\u7ed3\u540c\u4e00\u7b56\u7565\u65f6\u786e\u8ba4\uff1a”\u4e24\u79cd\u5ea6\u91cf\u90fd\u652f\u6301\u505a\u591a\u7c97\u7cd9\u6ce2\u52a8\u7387\u80a1\u7968\u3001\u505a\u7a7a\u5e73\u6ed1\u6ce2\u52a8\u7387\u80a1\u7968\u7684\u7b56\u7565\uff1b\u4f46\u57fa\u4e8e\u9690\u542b\u7c97\u7cd9\u5ea6\u6392\u5e8f\u6536\u76ca\u66f4\u9ad8\uff0c\u5bf9\u63a7\u5236\u5176\u4ed6\u56e0\u5b50\u66f4\u7a33\u5065\u3002\u7279\u522b\u662f\u5bf9\u63a7\u5236\u6d41\u52a8\u6027\u548cATM\u504f\u5ea6\u6c34\u5e73\u540e\u4ecd\u7136\u6210\u7acb\u3002”<\/p>\n\n\n\n
    \u516b\u3001SVI\uff1a\u7f8e\u6797\u7528\u4e86\u4e94\u5e74\u7684\u5185\u90e8\u5de5\u5177<\/strong><\/h2>\n\n\n\n
    \u7c97\u7cd9\u5ea6\u7684\u4f18\u52bf\u5728\u6a21\u578b\u5c42\u3002\u8fd8\u6709\u4e00\u4e2a\u540c\u6837\u6709\u8bb0\u5f55\u7684\u4fe1\u606f\u4f18\u52bf\u5728\u6ce2\u52a8\u7387\u9762\u63d2\u503c\u5c42\u3002<\/p>\n\n\n\n
    SVI\uff08Stochastic Volatility Inspired\uff09\u662f\u9690\u542b\u6ce2\u52a8\u7387\u5fae\u7b11\u7684\u4ece\u4e1a\u8005\u6807\u51c6\u53c2\u6570\u5316\u65b9\u6cd5\u3002Gatheral\u548cJacquier 2014\u5e74\u5728 Quantitative Finance<\/em> \u660e\u786e\u8bf4\uff1a”SVI\u53c2\u6570\u5316\u6700\u521d\u7531\u7f8e\u6797\u57281999\u5e74\u8bbe\u8ba1\uff0c\u968f\u540e\u5728Gatheral (2004)\u4e2d\u516c\u5f00\u53d1\u8868\u3002”<\/p>\n\n\n\n
    1999\u52302004\u8fd9\u4e94\u5e74\uff0c\u7f8e\u6797\u884d\u751f\u54c1\u4ea4\u6613\u53f0\u62e5\u6709\u65e0\u5957\u5229\u7684\u6ce2\u52a8\u7387\u9762\u53c2\u6570\u5316\u5de5\u5177\uff0c\u7ade\u4e89\u5bf9\u624b\u8fd8\u5728\u7528\u66f4\u7c97\u7cd9\u7684\u63d2\u503c\u65b9\u6cd5\u3002SVI\u51fd\u6570\u5f62\u5f0f\uff1a<\/p>\n\n\n\n
    w^SVI(k) = a + b[\u03c1(k \u2212 m) + \u221a((k \u2212 m)\u00b2 + \u03c3\u00b2)]<\/code><\/pre>\n\n\n\n
    \u4e0d\u662f\u968f\u4fbf\u5199\u7684\u3002Gatheral\u63a8\u5bfc\u51faHeston\u9690\u542b\u6ce2\u52a8\u7387\u5fae\u7b11\u7684\u5927\u671f\u9650\u6781\u9650\u6070\u597d\u5c31\u662fSVI\u3002\u7279\u5f81\u51fd\u6570\u7684\u5927\u53c2\u6570\u6e10\u8fd1\u884c\u4e3a\u51b3\u5b9a\u4e86\u5fae\u7b11\u7684\u7ffc\u90e8\u884c\u4e3a\uff0cSVI\u76f4\u63a5\u6355\u6349\u4e86\u8fd9\u4e9b\u6e10\u8fd1\u3002<\/p>\n\n\n\n
    Gatheral-Jacquier (2014)\u63a8\u5bfc\u4e86\u89e3\u6790\u7684\u65e0\u5957\u5229\u6761\u4ef6\uff1a\u65e0\u8776\u5f0f\u5957\u5229\u8981\u6c42\u9690\u542b\u5bc6\u5ea6\u5904\u5904\u975e\u8d1f\uff08Call\u671f\u6743\u4ef7\u683c\u5bf9\u884c\u6743\u4ef7\u51f8\uff09\uff1b\u65e0\u65e5\u5386\u4ef7\u5dee\u5957\u5229\u8981\u6c42\u603b\u65b9\u5dee\u968f\u671f\u9650\u9012\u589e\u3002\u4e24\u8005\u90fd\u8f6c\u5316\u4e3aSVI\u53c2\u6570\u7684\u663e\u5f0f\u7ea6\u675f\u3002\u8fdd\u53cd\u7ea6\u675f\u7684\u6821\u51c6\u9762\u4f1a\u4ea7\u751f\u865a\u5047\u7684\u5b9a\u4ef7\u504f\u5dee\u4fe1\u53f7\uff0c\u8ba9\u4f60\u5bf9\u7740\u53ea\u5728\u6a21\u578b\u91cc\u5b58\u5728\u3001\u5e02\u573a\u4e2d\u4e0d\u5b58\u5728\u7684\u9519\u4f4d\u4ea4\u6613\u3002<\/p>\n\n\n\n
    \u4e5d\u3001COS\u65b9\u6cd5\uff1a\u7279\u5f81\u51fd\u6570\u5b9a\u4ef7\u5947\u5f02\u671f\u6743<\/strong><\/h2>\n\n\n\n
    \u6b27\u5f0f\u671f\u6743\u6709\u5e72\u51c0\u7684\u5085\u91cc\u53f6\u7ed3\u6784\u3002\u969c\u788d\u671f\u6743\u3001\u4e9a\u5f0f\u671f\u6743\u3001\u767e\u6155\u5927\u671f\u6743\u9700\u8981\u5b8c\u6574\u8def\u5f84\u5206\u5e03\u3002Fang\u548cOosterlee 2009\u5e74\u5728 SIAM Journal<\/em> \u5f00\u53d1\u4e86COS\u65b9\u6cd5\uff0c\u5229\u7528\u7279\u5f81\u51fd\u6570\u4e0e\u8f6c\u79fb\u5bc6\u5ea6\u5085\u91cc\u53f6\u4f59\u5f26\u7ea7\u6570\u7cfb\u6570\u4e4b\u95f4\u7684\u5173\u7cfb\uff1a<\/p>\n\n\n\n
    f(y|x) \u2248 (2\/(b\u2212a)) \u03a3_{k=0}^{N} Re[\u03d5(k\u03c0\/(b\u2212a)) \u00b7 e^{\u2212ik\u03c0a\/(b\u2212a)}] \u00b7 cos(k\u03c0(y\u2212a)\/(b\u2212a))<\/code><\/pre>\n\n\n\n
    \u5176\u4e2d [a, b] \u662f\u622a\u65ad\u533a\u95f4\uff0c\u4ece\u5206\u5e03\u7684\u7d2f\u79ef\u91cf\u8bbe\u5b9a\uff0c\u800c\u7d2f\u79ef\u91cf\u672c\u8eab\u5728 t=0 \u5904\u4ece \u03d5(t) \u89e3\u6790\u8ba1\u7b97\u3002Fang\u548cOosterlee\u8bf4\uff1a”\u5927\u591a\u6570\u60c5\u51b5\u4e0bCOS\u65b9\u6cd5\u7684\u6536\u655b\u901f\u5ea6\u662f\u6307\u6570\u7ea7\u7684\uff0c\u8ba1\u7b97\u5f00\u9500\u662f\u7ebf\u6027\u7684\u3002”O(N)\uff0c\u6bd4Carr-Madan FFT\u7684 O(N log N) \u8fd8\u5feb\u3002<\/p>\n\n\n\n
    \u5341\u3001Alpha\u4e3a\u4ec0\u4e48\u80fd\u6301\u7eed\u5b58\u5728<\/strong><\/h2>\n\n\n\n
    Carr-Madan\u8bba\u65871999\u5e74\u516c\u5f00\u3002Duffie-Pan-Singleton 2000\u5e74\u3002Heston\u7279\u5f81\u51fd\u65701993\u5e74\u3002SVI 2004\u5e74\u3002\u7c97\u7cd9\u6ce2\u52a8\u73872018\u5e74\u3002\u4e3a\u4ec0\u4e48Alpha\u8fd8\u5728\uff1f<\/p>\n\n\n\n
    Bondarenko 2004\u5e74\u7684\u8bba\u6587\u7ed9\u51fa\u4e86\u76f4\u63a5\u7b54\u6848\uff1a\u5bf9\u51b2\u57fa\u91d1\u662f\u65b9\u5dee\u5356\u65b9\uff0c\u4e0d\u662f\u4e70\u65b9<\/strong>\u3002\u6ea2\u4ef7\u6301\u7eed\u5b58\u5728\uff0c\u56e0\u4e3a\u5b83\u4ee3\u8868\u4e86\u5bf9\u65e0\u6cd5\u5bf9\u51b2\u7684\u5de6\u5c3e\u98ce\u9669\u7684\u8865\u507f\u3002\u671f\u6743\u4ea4\u6613\u5546\u901a\u5e38\u79ef\u7d2f\u51c0\u7a7a\u5934\u6ce2\u52a8\u7387\u655e\u53e3\uff0c\u56e0\u4e3a\u7ec8\u7aef\u7528\u6237\u5bfb\u6c42\u4e0b\u884c\u4fdd\u62a4\uff1b\u4ed6\u4eec\u628a\u8fd9\u4e2a\u7ed3\u6784\u6027\u5931\u8861\u4ee5\u9ad8\u4e8e\u516c\u5e73\u4ef7\u503c\u7684\u6ea2\u4ef7\u5b9a\u4ef7\uff0c\u8865\u507f\u4e0d\u53ef\u5bf9\u51b2\u7684\u6b8b\u4f59\u90e8\u5206\u3002Carr\u548cWu (2009)\u786e\u8ba4\uff1a”\u80a1\u7968\u6536\u76ca\u4e0e\u6536\u76ca\u65b9\u5dee\u4e4b\u95f4\u6709\u8bb0\u5f55\u7684\u8d1f\u76f8\u5173\u5728\u4ea7\u751f\u6ea2\u4ef7\u4e2d\u8d77\u4e86\u91cd\u8981\u4f5c\u7528\u3002”<\/p>\n\n\n\n
    \u6a21\u578b\u524d\u6cbf\u4f18\u52bf\u662f\u7b2c\u4e8c\u5c42\u3002\u6bcf\u7bc7\u8bba\u6587\u90fd\u6253\u5f00\u4e86\u4e00\u4e2a\u591a\u5e74\u7a97\u53e3\u671f\uff0c\u6709\u6570\u5b66\u57fa\u7840\u8bbe\u65bd\u5b9e\u73b0\u8fd9\u4e9b\u7ed3\u679c\u7684\u5b9e\u8df5\u8005\u76f8\u5bf9\u8fd8\u5728\u8dd1\u7ecf\u5178\u6a21\u578b\u7684\u4eba\u6709\u4fe1\u606f\u4f18\u52bf\u3002<\/p>\n\n\n\n
    \u57fa\u7840\u8bbe\u65bd\u62a4\u57ce\u6cb3\u662f\u7b2c\u4e09\u5c42\u3002Cui\u7b49\u4eba\u7684\u89e3\u6790\u68af\u5ea6\u65b9\u6cd5\u3001\u7a33\u5b9a\u7684\u652f\u5272\u5904\u7406\u3001Fang-Oosterlee\u7684COS\u65b9\u6cd5\u3001Gao-Hyndman 2025\u5e74\u7684\u8bef\u5dee\u754c\uff0c\u5408\u5728\u4e00\u8d77\u4ee3\u8868\u4e86\u4e00\u4e2a\u591a\u5e74\u5de5\u7a0b\u9879\u76ee\u624d\u80fd\u6b63\u786e\u5b9e\u73b0\u7684\u5de5\u4f5c\u3002\u6570\u5b66\u7ed3\u679c\u662f\u516c\u5f00\u7684\u3002\u5b9e\u73b0\uff0c\u7a33\u5b9a\u3001\u5feb\u901f\u3001\u65e0\u5957\u5229\u3001\u6709\u9a8c\u8bc1\u8bef\u5dee\u754c\u548c\u4ea7\u54c1\u7ea7\u6570\u503c\u5904\u7406\u7684\u5b9e\u73b0\uff0c\u4e0d\u662f\u3002<\/p>\n\n\n\n
    \n
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