QuantLib 金融计算——数学工具之求解器

如果未做特别说明,文中的程序都是 Python3 代码。

QuantLib 金融计算——数学工具之求解器

载入模块

import QuantLib as ql
import scipy
from scipy.stats import norm

print(ql.__version__)
1.12

概述

QuantLib 提供了多种类型的一维求解器,用以求解单参数函数的根,

\[f(x)=0 \]

其中 \(f : R \to R\) 是实数域上的函数。

QuantLib 提供的求解器类型有:

  • Brent
  • Bisection
  • Secant
  • Ridder
  • Newton(要求提供成员函数 derivative,计算导数)
  • FalsePosition

这些求解器的构造函数均为默认构造函数,不接受参数。例如,Brent 求解器实例的构造语句为 mySolv = Brent()

调用方式

求解器的成员函数 solve 有两种调用方式:

solve(f,
      accuracy,
      guess,
      step)

solve(f,
      accuracy,
      guess,
      xMin,
      xMax)
  • f:单参数函数或函数对象,返回值为一个浮点数。
  • accuracy:浮点数,表示求解精度 \(\epsilon\),用于停止计算。假设 \(x_i\) 是根的准确解,
    • \(|f(x)| < \epsilon\)
    • \(|x - x_i| < \epsilon\) 时停止计算。
  • guess:浮点数,对根的初始猜测值。
  • step:浮点数,在第一种调用方式中,没有限定根的区间范围,算法需要自己搜索,确定一个范围。step 规定了搜索算法的步长。
  • xMinxMax:浮点数,左右区间范围

根求解器在量化金融中最经典的应用是求解隐含波动率。给定期权价格 \(p\) 以及其他参数 \(S_0\)\(K\)\(r_d\)\(r_f\)\(\tau\),我们要计算波动率 \(\sigma\),满足

\[f(\sigma) = \mathrm{blackScholesPrice}(S_0 , K, r_d , r_f , \sigma , \tau, \phi) - p = 0 \]

其中 Black-Scholes 函数中 \(\phi = 1\) 代表看涨期权;\(\phi = −1\) 代表看跌期权。

非 Newton 算法(不需要导数)

下面的例子显示了如何加一个多参数函数包装为一个单参数函数,并使用 QuantLib 求解器计算隐含波动率。

例子 1

# Black-Scholes 函数
def blackScholesPrice(spot,
                      strike,
                      rd,
                      rf,
                      vol,
                      tau,
                      phi):
    domDf = scipy.exp(-rd * tau)
    forDf = scipy.exp(-rf * tau)
    fwd = spot * forDf / domDf
    stdDev = vol * scipy.sqrt(tau)

    dp = (scipy.log(fwd / strike) + 0.5 * stdDev * stdDev) / stdDev
    dm = (scipy.log(fwd / strike) - 0.5 * stdDev * stdDev) / stdDev

    res = phi * domDf * (fwd * norm.cdf(phi * dp) - strike * norm.cdf(phi * dm))

    return res


# 包装函数
def impliedVolProblem(spot,
                      strike,
                      rd,
                      rf,
                      tau,
                      phi,
                      price):
    def inner_func(v):
        return blackScholesPrice(spot, strike, rd, rf, v, tau, phi) - price

    return inner_func


def testSolver1():
    # setup of market parameters
    spot = 100.0
    strike = 110.0
    rd = 0.002
    rf = 0.01
    tau = 0.5
    phi = 1
    vol = 0.1423

    # calculate corresponding Black Scholes price

    price = blackScholesPrice(spot, strike, rd, rf, vol, tau, phi)
    # setup a solver
    mySolv1 = ql.Bisection()
    mySolv2 = ql.Brent()
    mySolv3 = ql.Ridder()

    accuracy = 0.00001
    guess = 0.25

    min = 0.0
    max = 1.0

    myVolFunc = impliedVolProblem(spot, strike, rd, rf, tau, phi, price)

    res1 = mySolv1.solve(myVolFunc, accuracy, guess, min, max)
    res2 = mySolv2.solve(myVolFunc, accuracy, guess, min, max)
    res3 = mySolv3.solve(myVolFunc, accuracy, guess, min, max)

    print('{0:<35}{1}'.format('Input Volatility:', vol))
    print('{0:<35}{1}'.format('Implied Volatility Bisection:', res1))
    print('{0:<35}{1}'.format('Implied Volatility Brent:', res2))
    print('{0:<35}{1}'.format('Implied Volatility Ridder:', res3))


testSolver1()
# Input Volatility:                  0.1423
# Implied Volatility Bisection:      0.14229583740234375
# Implied Volatility Brent:          0.14230199334812577
# Implied Volatility Ridder:         0.1422999996313447

Newton 算法(需要导数)

Newton 算法要求为根求解器提供 \(f(\sigma)\) 的导数 \(\frac{\partial f}{\partial \sigma}\)(即 vega)。下面的例子显示了如何将导数添加进求解隐含波动率的过程。为此我们需要一个类,一方面提供作为一个函数对象,另一方面要提供成员函数 derivative

例子 2

class BlackScholesClass:
    def __init__(self,
                 spot,
                 strike,
                 rd,
                 rf,
                 tau,
                 phi,
                 price):
        self.spot_ = spot
        self.strike_ = strike
        self.rd_ = rd
        self.rf_ = rf
        self.phi_ = phi
        self.tau_ = tau
        self.price_ = price
        self.sqrtTau_ = scipy.sqrt(tau)
        self.d_ = norm

        self.domDf_ = scipy.exp(-self.rd_ * self.tau_)
        self.forDf_ = scipy.exp(-self.rf_ * self.tau_)
        self.fwd_ = self.spot_ * self.forDf_ / self.domDf_
        self.logFwd_ = scipy.log(self.fwd_ / self.strike_)

    def blackScholesPrice(self,
                          spot,
                          strike,
                          rd,
                          rf,
                          vol,
                          tau,
                          phi):
        domDf = scipy.exp(-rd * tau)
        forDf = scipy.exp(-rf * tau)
        fwd = spot * forDf / domDf
        stdDev = vol * scipy.sqrt(tau)

        dp = (scipy.log(fwd / strike) + 0.5 * stdDev * stdDev) / stdDev
        dm = (scipy.log(fwd / strike) - 0.5 * stdDev * stdDev) / stdDev

        res = phi * domDf * (fwd * norm.cdf(phi * dp) - strike * norm.cdf(phi * dm))

        return res

    def impliedVolProblem(self,
                          spot,
                          strike,
                          rd,
                          rf,
                          vol,
                          tau,
                          phi,
                          price):
        return self.blackScholesPrice(
            spot, strike, rd, rf, vol, tau, phi) - price

    def __call__(self,
                 x):
        return self.impliedVolProblem(
            self.spot_, self.strike_, self.rd_, self.rf_,
            x,
            self.tau_, self.phi_, self.price_)

    def derivative(self,
                   x):
        # vega
        stdDev = x * self.sqrtTau_
        dp = (self.logFwd_ + 0.5 * stdDev * stdDev) / stdDev
        return self.spot_ * self.forDf_ * self.d_.pdf(dp) * self.sqrtTau_


def testSolver2():

    # setup of market parameters
    spot = 100.0
    strike = 110.0
    rd = 0.002
    rf = 0.01
    tau = 0.5
    phi = 1
    vol = 0.1423

    # calculate corresponding Black Scholes price
    price = blackScholesPrice(
        spot, strike, rd, rf, vol, tau, phi)
    solvProblem = BlackScholesClass(
        spot, strike, rd, rf, tau, phi, price)

    mySolv = ql.Newton()

    accuracy = 0.00001
    guess = 0.10
    step = 0.001

    res = mySolv.solve(
        solvProblem, accuracy, guess, step)

    print('{0:<20}{1}'.format('Input Volatility:', vol))
    print('{0:<20}{1}'.format('Implied Volatility:', res))


testSolver2()
# Input Volatility:   0.1423
# Implied Volatility: 0.14230000000000048

导数的使用明显提高了精度。

扩展阅读

《QuantLib 金融计算》系列合集

posted @ 2018-10-23 21:16  xuruilong100  阅读(1395)  评论(0编辑  收藏  举报