00_cqf_lecture.md

Market-Based Valuation of Equity Options

CQF Lecture, 09. April 2018, London

Dr. Yves J. Hilpisch, The Python Quants GmbH

Resources

Short link to this Gist:

• http://bit.ly/cqf_april_2018

General resources:

• http://tpq.io
• http://hilpisch.com
• http://certificate.tpq.io
• http://twitter.com/dyjh

Abstract

This lecture covers numerical methods for the market-based valuation of equity options. The lecture is mainly based on the book Derivatives Analytics with Python (http://dawp.tpq.io).

Slides

You find the slides under http://tpq.io/p/cqf_lecture_april_2018.html

Python

This Gist contains the files needed to replicate all the results shown during the lecture. The code base has been updated to Python 3.6.

01_cqf_lecture.ipynb

02_cqf_demo.ipynb

03_cqf_demo.ipynb

BSM_imp_vol.py

#
# Valuation of European Call Options in BSM Model
# and Numerical Derivation of Implied Volatility
# 03_stf/BSM_imp_vol.py
#
# (c) Dr. Yves J. Hilpisch
# from Hilpisch, Yves (2014): Python for Finance, O'Reilly.
#
from math import log, sqrt, exp
from scipy import stats
from scipy.optimize import fsolve

class call_option(object):
    ''' Class for European call options in BSM Model.
    
    Attributes
    ==========
    S0 : float
        initial stock/index level
    K : float
        strike price
    t : datetime/Timestamp object
        pricing date
    M : datetime/Timestamp object
        maturity date
    r : float
        constant risk-free short rate
    sigma : float
        volatility factor in diffusion term
        
    Methods
    =======
    value : float
        return present value of call option
    vega : float
        return vega of call option
    imp_vol : float
        return implied volatility given option quote
    '''
    
    def __init__(self, S0, K, t, M, r, sigma):
        self.S0 = float(S0)
        self.K = K
        self.t = t
        self.M = M
        self.r = r
        self.sigma = sigma

    def update_ttm(self):
        ''' Updates time-to-maturity self.T. '''
        if self.t > self.M:
            raise ValueError("Pricing date later than maturity.")
        self.T = (self.M - self.t).days / 365.

    def d1(self):
        ''' Helper function. '''
        d1 = ((log(self.S0 / self.K)
            + (self.r + 0.5 * self.sigma ** 2) * self.T)
            / (self.sigma * sqrt(self.T)))
        return d1
        
    def value(self):
        ''' Return option value. '''
        self.update_ttm()
        d1 = self.d1()
        d2 = ((log(self.S0 / self.K)
            + (self.r - 0.5 * self.sigma ** 2) * self.T)
            / (self.sigma * sqrt(self.T)))
        value = (self.S0 * stats.norm.cdf(d1, 0.0, 1.0)
            - self.K * exp(-self.r * self.T) * stats.norm.cdf(d2, 0.0, 1.0))
        return value
        
    def vega(self):
        ''' Return Vega of option. '''
        self.update_ttm()
        d1 = self.d1()
        vega = self.S0 * stats.norm.pdf(d1, 0.0, 1.0) * sqrt(self.T)
        return vega

    def imp_vol(self, C0, sigma_est=0.2):
        ''' Return implied volatility given option price. '''
        option = call_option(self.S0, self.K, self.t, self.M,
                             self.r, sigma_est)
        option.update_ttm()
        def difference(sigma):
            option.sigma = sigma
            return option.value() - C0
        iv = fsolve(difference, sigma_est)[0]
        return iv

es50_imp_vol.py

#
# Black-Scholes-Merton Implied Volatilities of
# Call Options on the EURO STOXX 50
# Option Quotes from 30. September 2014
# Source: www.eurexchange.com, www.stoxx.com
# 03_stf/ES50_imp_vol.py
#
# (c) Dr. Yves J. Hilpisch
# Derivatives Analytics with Python
#
import numpy as np
import pandas as pd
from BSM_imp_vol import call_option
import matplotlib as mpl
import matplotlib.pyplot as plt
mpl.rcParams['font.family'] = 'serif'

# Pricing Data
pdate = pd.Timestamp('30-09-2014')

#
# EURO STOXX 50 index data
#

# URL of data file
es_url = 'http://www.stoxx.com/download/historical_values/hbrbcpe.txt'
# column names to be used
cols = ['Date', 'SX5P', 'SX5E', 'SXXP', 'SXXE',
        'SXXF', 'SXXA', 'DK5F', 'DKXF', 'DEL']
# reading the data with pandas
es = pd.read_csv(es_url,  # filename
                 header=None,  # ignore column names
                 index_col=0,  # index column (dates)
                 parse_dates=True,  # parse these dates
                 dayfirst=True,  # format of dates
                 skiprows=4,  # ignore these rows
                 sep=';',  # data separator
                 names=cols)  # use these column names
# deleting the helper column
del es['DEL']
S0 = es['SX5E']['30-09-2014']
r = -0.05

#
# Option Data
#
data = pd.read_csv('http://hilpisch.com/es50_option_data.csv', index_col=0)
data['Date'] = data['Date'].apply(lambda x: pd.Timestamp(x))
data['Maturity'] = data['Maturity'].apply(lambda x: pd.Timestamp(x))

#
# BSM Implied Volatilities
#


def calculate_imp_vols(data):
    ''' Calculate all implied volatilities for the European call options
    given the tolerance level for moneyness of the option.'''
    data['Imp_Vol'] = 0.0
    tol = 0.30  # tolerance for moneyness
    for row in data.index:
        t = data['Date'][row]
        T = data['Maturity'][row]
        ttm = (T - t).days / 365.
        forward = np.exp(r * ttm) * S0
        if (abs(data['Strike'][row] - forward) / forward) < tol:
            call = call_option(S0, data['Strike'][row], t, T, r, 0.2)
            data['Imp_Vol'][row] = call.imp_vol(data['Call'][row])
    return data

#
# Graphical Output
#
markers = ['.', 'o', '^', 'v', 'x', 'D', 'd', '>', '<']


def plot_imp_vols(data):
    ''' Plot the implied volatilites. '''
    maturities = sorted(set(data['Maturity']))
    plt.figure(figsize=(10, 6))
    for i, mat in enumerate(maturities):
        dat = data[(data['Maturity'] == mat) & (data['Imp_Vol'] > 0)]
        plt.plot(dat['Strike'].values, dat['Imp_Vol'].values,
                 'b%s' % markers[i], label=str(mat)[:10])
    plt.grid(True)
    plt.legend()
    plt.xlabel('strike')
    plt.ylabel('implied volatility')
    plt.show()

gbm_helper.py

#
# Analyzing Returns from Geometric Brownian Motion
# 03_stf/GBM_returns.py
#
# (c) Dr. Yves J. Hilpisch
# Derivatives Analytics with Python
#
import math
import numpy as np
import pandas as pd
import scipy.stats as scs
import statsmodels.api as sm
import matplotlib as mpl
import matplotlib.pyplot as plt
plt.style.use('seaborn')
mpl.rcParams['font.family'] = 'serif'

#
# Helper Function
#


def dN(x, mu, sigma):
    ''' Probability density function of a normal random variable x.

    Parameters
    ==========
    mu : float
        expected value
    sigma : float
        standard deviation

    Returns
    =======
    pdf : float
        value of probability density function
    '''
    z = (x - mu) / sigma
    pdf = np.exp(-0.5 * z ** 2) / math.sqrt(2 * math.pi * sigma ** 2)
    return pdf

# Return Sample Statistics and Normality Tests

def print_statistics(data):
    print("RETURN SAMPLE STATISTICS")
    print("---------------------------------------------")
    print("Mean of Daily  Log Returns %9.6f" % np.mean(data['returns']))
    print("Std  of Daily  Log Returns %9.6f" % np.std(data['returns']))
    print("Mean of Annua. Log Returns %9.6f" % (np.mean(data['returns']) * 252))
    print("Std  of Annua. Log Returns %9.6f" % \
                (np.std(data['returns']) * math.sqrt(252)))
    print("---------------------------------------------")
    print("Skew of Sample Log Returns %9.6f" % scs.skew(data['returns']))
    print("Skew Normal Test p-value   %9.6f" % scs.skewtest(data['returns'])[1])
    print("---------------------------------------------")
    print("Kurt of Sample Log Returns %9.6f" % scs.kurtosis(data['returns']))
    print("Kurt Normal Test p-value   %9.6f" % \
                scs.kurtosistest(data['returns'])[1])
    print("---------------------------------------------")
    print("Normal Test p-value        %9.6f" % \
                scs.normaltest(data['returns'])[1])
    print("---------------------------------------------")
    print("Realized Volatility        %9.6f" % data['rea_vol'].iloc[-1])
    print("Realized Variance          %9.6f" % data['rea_var'].iloc[-1])

#
# Graphical Output
#

# daily quotes and log returns
def quotes_returns(data):
    ''' Plots quotes and returns. '''
    plt.figure(figsize=(10, 6))
    plt.subplot(211)
    data['index'].plot()
    plt.ylabel('daily quotes')
    plt.grid(True)
    plt.axis('tight')

    plt.subplot(212)
    data['returns'].plot()
    plt.ylabel('daily log returns')
    plt.grid(True)
    plt.axis('tight')

# histogram of annualized daily log returns
def return_histogram(data):
    ''' Plots a histogram of the returns. '''
    plt.figure(figsize=(10, 6))
    x = np.linspace(min(data['returns']), max(data['returns']), 100)
    plt.hist(np.array(data['returns']), bins=50, normed=True)
    y = dN(x, np.mean(data['returns']), np.std(data['returns']))
    plt.plot(x, y, linewidth=2)
    plt.xlabel('log returns')
    plt.ylabel('frequency/probability')
    plt.grid(True)

# Q-Q plot of annualized daily log returns
def return_qqplot(data):
    ''' Generates a Q-Q plot of the returns.'''
    plt.figure(figsize=(10, 6))
    sm.qqplot(data['returns'], line='s')
    plt.grid(True)
    plt.xlabel('theoretical quantiles')
    plt.ylabel('sample quantiles')

# realized volatility
def realized_volatility(data):
    ''' Plots the realized volatility. '''
    plt.figure(figsize=(10, 6))
    data['rea_vol'].plot()
    plt.ylabel('realized volatility')
    plt.grid(True)

# mean return, volatility and correlation (252 days moving = 1 year)
def rolling_statistics(data):
    ''' Calculates and plots rolling statistics (mean, std, correlation). '''
    plt.figure(figsize=(11, 8))

    plt.subplot(311)
    mr = pd.rolling_mean(data['returns'], 252) * 252
    mr.plot()
    plt.grid(True)
    plt.ylabel('returns (252d)')
    plt.axhline(mr.mean(), color='r', ls='dashed', lw=1.5)

    plt.subplot(312)
    vo = pd.rolling_std(data['returns'], 252) * math.sqrt(252)
    vo.plot()
    plt.grid(True)
    plt.ylabel('volatility (252d)')
    plt.axhline(vo.mean(), color='r', ls='dashed', lw=1.5)
    vx = plt.axis()

    plt.subplot(313)
    co = pd.rolling_corr(mr, vo, 252)
    co.plot()
    plt.grid(True)
    plt.ylabel('correlation (252d)')
    cx = plt.axis()
    plt.axis([vx[0], vx[1], cx[2], cx[3]])
    plt.axhline(co.mean(), color='r', ls='dashed', lw=1.5)