Options and Volatility Smile

An equity option represents the right to buy (“call” option) or sell (“put” option) a unit of underlying stock at a pre-specified price (strike) at a predetermined maturity date (European option) or at any time up to the predetermined date (American option).

Option writer sells an option and option holder buys an option.

For a European call option on an index with strike 8,000 and index level of 8200 at maturity, the option holder receives the difference of $200 from option writer. This is called the instrinsic value or payoff of the option from the holder’s point of view.

The payoff function for a call option is $$ h_{T}(S,K) = max[S_{T}-K, 0] \tag{Eq. 1}$$

where T = maturity date, $\ S_T $ is the index level at maturity and K is the strike price.

In-the-money: a call (put) is in-the-money when S > K (S < K)
At-the-money: call or put is at-the-money when $\ S \approx K $
Out-of-the-money: a call is out-of-the-money when S < K (S > K)

A fair present value (is different than payoff) of a European call option is given by Black-Scholes formula:

$\ C_{0}^{*} = C^{BSM}(S_{0},K,T,r,\sigma) \tag{Eq. 2}$

$\S_{0} $ current index level (spot)
K strike price of the option
T time-to-maturity of the option
r risk-free short rate
$\ \sigma $ volatility or the std dev of the index returns

$\ C^{BSM} = S_{t} . N(d_{1}) – e^{r(T-t)} . K. N(d_{2})\tag{Eq. 3} $


$\displaystyle N(d) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{d} e^{-\frac{1}{2}x^{2}} dx$

$\displaystyle d1 = \frac{\log\frac{S_{t}}{K} + (r + \frac{\sigma^2}{2})(T-t)}{\sigma\sqrt{T-t}}$

$\displaystyle d2 = \frac{\log\frac{S_{t}}{K} + (r – \frac{\sigma^2}{2})(T-t)}{\sigma\sqrt{T-t}}$


# BSM option valuation
import math
import numpy as np
import matplotlib as mpl
import matplotlib.pyplot as plt
from scipy.integrate import quad
from scipy.optimize import brentq
from scipy.interpolate import interp2d
from mpl_toolkits.mplot3d import Axes3D
from matplotlib import cm
## Helper functions ##
def dN(x):
    ''' Probability density function of standard normal random variable x.'''
    return math.exp(-0.5 * x ** 2) / math.sqrt(2 * math.pi)

def N(d):
    ''' Cumulative density function of standard normal random variable x. '''
    return quad(lambda x: dN(x), -20, d, limit=50)[0]

def d1f(St, K, t, T, r, sigma):
    ''' Black-Scholes-Merton d1 function.
        Parameters see e.g. BSM_call_value function. '''
    d1 = (math.log(St / K) + (r + 0.5 * sigma ** 2)
        * (T - t)) / (sigma * math.sqrt(T - t))
    return d1

def BSM_call_value(St, K, t, T, r, sigma):
    ''' Calculates Black-Scholes-Merton European call option value
    St: float
    stock/index level at time t
    K: float
    strike price
    t: float
    valuation date
    T: float
    date of maturity/time-to-maturity if t = 0; T > t
    r: float
    constant, risk-less short rate
    sigma: float
    call_value: float
    European call present value at t
    d1 = d1f(St, K, t, T, r, sigma)
    d2 = d1 - sigma * math.sqrt(T - t)
    call_value = St * N(d1) - math.exp(-r * (T - t)) * K * N(d2)
    return call_value

def BSM_put_value(St, K, t, T, r, sigma):
    ''' Calculates Black-Scholes-Merton European put option value.
    St: float
    stock/index level at time t
    K: float
    strike price
    t: float
    valuation date
    T: float
    date of maturity/time-to-maturity if t = 0; T > t
    r: float
    constant, risk-less short rate
    sigma: float
    put_value: float
    European put present value at t
    put_value = BSM_call_value(St, K, t, T, r, sigma) - St + math.exp(-r * (T - t)) * K
    return put_value
# Test Option Payoff and Time value
K = 8000 # Strike price
T = 1.0 # time-to-maturity
r = 0.025 # constant, risk-free rate
vol = 0.2 # constant volatility

# Generate spot prices 
S = np.linspace(4000, 12000, 150)
h = np.maximum(S - K, 0) # payoff of the option
C = [BSM_call_value(Szero, K, 0, T, r, vol) for Szero in S] #BS call option values

plt.plot(S, h, 'b-.', lw=2.5, label='payoff') # plot inner value at maturity
plt.plot(S, C, 'r', lw=2.5, label='present value') # plot option present value
plt.xlabel('index level $S_0$')
plt.ylabel('present value $C(t=0)$')

The present value of the option is always higher than the undiscounted payoff, the difference being the time value. In other words, the option’s present value is composed of payoff plus the time value. Time value indicates that there is always a chance of option going in-the-money or more in-the-money during that time.

Simulating Returns

The Geometric Brownian motion model of the BS equation is given by

$$\displaystyle dS_{t} = rS_{t}dt + \sigma S_{t} dt dZ_{t}\tag{Eq.4}$$

The discretized version is

$$\displaystyle S_{t} = S_{t – \Delta t} e^{(r – \frac{1}{2}\sigma^2) \Delta t + \sigma \sqrt{\Delta t} z_{t}}\tag{Eq.5}$$

where t $\in {(\Delta t, 2\Delta t,…..,T)}$

Using the above discretized version, we will simulate the spot prices with $S_{0}$=100, T=10, r = 0.05 and $\sigma$=0.2

import pandas as pd

# model parameters
S0 = 100.0 # initial index level
T = 10.0 # time horizon
r = 0.05 # risk-less short rate
vol = 0.2 # instantaneous volatility
# simulation parameters
#generate a pd array with business dates, ignores holidays
gbm_dates = pd.DatetimeIndex(start='10-05-2007',end='10-05-2017',freq='B')
M = len(gbm_dates) # time steps
I = 1 # index level paths
dt = 1 / 252. # 252 business days a year
df = math.exp(-r * dt) # discount factor

# stock price paths
rand = np.random.standard_normal((M, I)) # random numbers
S = np.zeros_like(rand) # stock matrix
S[0] = S0 # initial values
for t in range(1, M): # stock price paths using Eq.5
    S[t] = S[t - 1] * np.exp((r - vol ** 2 / 2) * dt + vol * rand[t] * math.sqrt(dt))
#create a pd dataframe with date as index and a column named "spot"
gbm = pd.DataFrame(S[:, 0], index=gbm_dates, columns=['spot']) 
gbm['returns'] = np.log(gbm['spot'] / gbm['spot'].shift(1)) #log returns
# Realized Volatility 
gbm['realized_var'] = 252 * np.cumsum(gbm['returns'] ** 2) / np.arange(len(gbm))
gbm['realized_vol'] = np.sqrt(gbm['realized_var'])
print gbm.head()
gbm = gbm.dropna()
               spot   returns realized_var realized_vol 
2007-10-08 98.904295 -0.011018 0.030589    0.174898
2007-10-09 98.444867 -0.004656 0.018026    0.134261
2007-10-10 97.696364 -0.007632 0.016911    0.130041
2007-10-11 98.280594 0.005962  0.014922    0.122158
plt.figure(figsize=(9, 6))
plt.ylabel('daily quotes')
plt.ylabel('daily log returns')

Implied Volatility is the value of $\sigma$ that solves Eq. 2 given the option market quote $C_{0}^{*}$

Volatility surface is the plot of the implied volatilities for different option strikes and different option maturities on the same underlying (an option chain).

Vol Surfaces exhibit :
Smiles: option implied volatilities exhibit a smile form, i.e. for calls the OTM implied volatilities are higher than the ATM ones; sometimes they rise again for ITM options
term structure:smiles are more pronounced for short-term options than for longer-term options; a phenomenon sometimes called volatility term structure

To demonstrate Vol Surface, I will use an option chain on AAPL stock as of 5/11/2017. I have downloaded this data from a reputable vendor, you can find this file here AAPL_BBG_vols

headers = ['Date','Strike','Call_Bid','Call_Ask','Call','Maturity','Put_Bid','Put_Ask','Put']
dtypes = {'Date': 'str', 'Strike': 'float', 'Call_Bid': 'float', 'Call_Ask': 'float',
parse_dates = ['Date', 'Maturity']

data = pd.read_csv('AAPL_BBG_vols.csv',skiprows=1,header=None, names=headers, dtype=dtypes, parse_dates=parse_dates)
data['BS_Imp_Vol'] = 0.0
data['TTM'] = 0.0
r = 0.0248 #risk-free rate
S0 = 153.97 # spot price as of 5/11/2017
div = 0.0182
for i in data.index:
    t = data['Date'][i]
    T = data['Maturity'][i]
    K = data['Strike'][i]
    Put = data['Put'][i]
    time_to_maturity = (T - t).days/365.0
    data.loc[i, 'TTM'] = time_to_maturity
    def func_BS(sigma):
        return BSM_put_value(S0, K, 0, time_to_maturity, r, sigma) - Put

    bs_imp_vol = brentq(func_BS, 0.03, 1.0)
    data.loc[i,'BS_Imp_Vol'] = bs_imp_vol

Once we have the implied volatilities, we will generate a grid of strikes and maturities and use Cubic interpolation to derive the missing implied volatilities needed for a smooth surface.

ttm = data['TTM'].tolist()
strikes = data['Strike'].tolist()
imp_vol = data['BS_Imp_Vol'].tolist()
f = interp2d(strikes,ttm,imp_vol, kind='cubic')

plot_strikes = np.linspace(data['Strike'].min(), data['Strike'].max(),25)
plot_ttm = np.linspace(0, data['TTM'].max(), 25)
fig = plt.figure()
ax = fig.gca(projection='3d')
X, Y = np.meshgrid(plot_strikes, plot_ttm)
Z = np.array([f(x,y) for xr, yr in zip(X, Y) for x, y in zip(xr,yr) ]).reshape(len(X), len(X[0]))
surf = ax.plot_surface(X, Y, Z, rstride=1, cstride=1, cmap=cm.coolwarm,linewidth=0.1)
ax.set_zlabel('Implied Volatility')
fig.colorbar(surf, shrink=0.5, aspect=5)

Valar Dohaeris!

Here is my first post, just a quick one really. I had these ideas and code I have been playing around with for a while and wanted to put that into something concrete. I plan on posting a few things I have learned on the job or self-learned, things that I find interesting or otherwise. Stay tuned.