目录
如果未做特别说明,文中的程序都是 Python3 代码。
QuantLib 金融计算——随机过程之 Heston 过程
载入模块
import QuantLib as ql
import pandas as pd
import numpy as np
import seaborn as sn
print(ql.__version__)
1.12
Heston 过程
著名的 Heston 模型描述了下列 SDE:
\[ \begin{aligned} d S_t & = \mu S_t d t + \sqrt { V_t } S_t d W_t^S \\ d V_t & = \kappa \left( \theta - V_t \right) d t + \sigma \sqrt { V_t } d W_t^V \\ d W_t^S d W_t^V & = \rho d t \end{aligned} \]
quantlib-python 中 Heston 过程的构造函数如下:
HestonProcess(riskFreeRate,
dividendYield,
s0,
v0,
kappa,
theta,
sigma,
rho)
其中,
-
riskFreeRate
:YieldTermStructureHandle
对象,描述无风险利率的期限结构; -
dividendYield
:YieldTermStructureHandle
对象,描述股息率的期限结构; -
s0
:QuoteHandle
对象,资产价格的起始值; -
v0
:浮点数,波动率的起始值; -
kappa
、theta
、sigma
:浮点数,描述波动率的 SDE 的参数; -
rho
:浮点数,模型中两个布朗运动之间的相关性
除了一些检查器之外,HestonProcess
没有提过其他特别的成员函数。
由于方程没有显式解,因此必须在 evolve
函数中使用算法进行离散化。quantlib-python 默认的离散化方法是 Quadratic Exponential Martingale 方法(或称 Quadratic Exponential 方法),具体的算法细节请参考(Andersen 和 Leif,2008)
由于 evolve
函数将离散化计算中对布朗运动的离散化以参数形式暴露了出来,使得用户可以容易地显现出随机波动率对资产价格序列的影响。下面的例子比较了一般 Black Scholes 过程和 Heston 过程,所模拟的资产价格除了波动率结构以外,都完全一致。
def testingStochasticProcesses2(seed):
refDate = ql.Date(27, ql.January, 2019)
riskFreeRate = 0.0321
dividendRate = 0.0128
spot = 52.0
cal = ql.China()
dc = ql.ActualActual()
rdHandle = ql.YieldTermStructureHandle(
ql.FlatForward(refDate, riskFreeRate, dc))
rqHandle = ql.YieldTermStructureHandle(
ql.FlatForward(refDate, dividendRate, dc))
spotHandle = ql.QuoteHandle(
ql.SimpleQuote(spot))
kappa = 1.2
theta = 0.08
sigma = 0.05
rho = -0.6
v0 = theta
hestonProcess = ql.HestonProcess(
rdHandle, rqHandle, spotHandle, v0,
kappa, theta, sigma, rho)
volHandle = ql.BlackVolTermStructureHandle(
ql.BlackConstantVol(refDate, cal, np.sqrt(v0), dc))
bsmProcess = ql.BlackScholesMertonProcess(
spotHandle, rqHandle, rdHandle, volHandle)
unifMt = ql.MersenneTwisterUniformRng(seed)
bmGauss = ql.BoxMullerMersenneTwisterGaussianRng(unifMt)
dt = 0.004
numVals = 250
dw = ql.Array(2)
x = ql.Array(2)
x[0] = spotHandle.value()
x[1] = v0
y = x[0]
htn = pd.DataFrame(
dict(
t=np.linspace(0, dt * numVals, numVals + 1),
price=np.nan,
vol=np.nan))
bsm = pd.DataFrame(
dict(
t=np.linspace(0, dt * numVals, numVals + 1),
price=np.nan,
vol=v0))
htn.loc[0, 'price'] = x[0]
htn.loc[0, 'vol'] = x[1]
bsm.loc[0, 'price'] = y
for j in range(1, numVals + 1):
dw[0] = bmGauss.next().value()
dw[1] = bmGauss.next().value()
x = hestonProcess.evolve(htn.loc[j, 't'], x, dt, dw)
y = bsmProcess.evolve(bsm.loc[j, 't'], y, dt, dw[0])
htn.loc[j, 'price'] = x[0]
htn.loc[j, 'vol'] = x[1]
bsm.loc[j, 'price'] = y
htn = htn.melt(
id_vars='t',
var_name='component',
value_name='path')
htn['type'] = 'stochastic vol'
bsm = bsm.melt(
id_vars='t',
var_name='component',
value_name='path')
bsm['type'] = 'constant vol'
htn_bsm = pd.concat([htn, bsm])
sn.relplot(
x='t',
y='path',
data=htn_bsm,
col='component',
hue='type',
kind="line",
height=8,
facet_kws=dict(sharey=False))
testingStochasticProcesses2(100)
参考文献
- Andersen, Leif. 2008. Simple and efficient simulation of the Heston stochastic volatility model. Journal of Computational Finance 11: 1–42.