Volatility Smiles and Stylised Facts in the Heston Model

Project on Volatility Smiles and Stylized Facts in the Heston Model
Project on
Volatility Smiles and Stylized Facts in the Heston
Model
Kamrul Hasan
Supervised by
Professor Dr. Jörn Saß
Department of Mathematics, TU Kaiserslautern
December 7, 2022

Introduction
Introduction
The Black-Scholes-Merton model
First analytic option pricing model
Often quite sucessful in explening the stock option prices
Known biases
Constant volatility

Introduction
Introduction
The Black-Scholes-Merton model
First analytic option pricing model
Often quite sucessful in explening the stock option prices
Known biases
Constant volatility
The Heston Model
Generalization of the Black-Scholes model
Randomness of the volatility is considered.
Closed-form solution
Stock price St and variance vt follow a Black-Scholes type
stochastic process and a CIR process respectively.

Introduction
Introduction
What we have observed in this project?
European call option prices
Simulation paths and the effect of changing the input
parameters
Volatility smile
Stylized facts

The Black Scholes Model and Formula

The Black-Scholes model
The stock price is defined by the following equation,
dSt = µStdt + σStdWt

The solution of the stock price,
St = S0e(µ−σ2
2
)t+σWt

St = S0e(µ−σ2
2
)t+σWt
Girsanov’s theorem is used to express the stock price St in the
following form,
dSt = rStdt + σStdW̃t

St = S0e(µ−σ2
2
)t+σWt
Girsanov’s theorem is used to express the stock price St in the
following form,
dSt = rStdt + σStdW̃t
log return follows normal distribution,
ln

St+∆t
St

∼ N

(r − σ2
2 )∆t, σ2∆t

The Black-Scholes Formula
The formula for the European call price,
Call Option = C(St, t)
= StN(d1) − N(d2)Ke−rτ

The Black-Scholes Formula
The formula for the European call price,
Call Option = C(St, t)
= StN(d1) − N(d2)Ke−rτ
d1 and d2 are defined by,
d1 =
ln( St
K
)+(r+σ2
2
τ)
σ
√
τ
and d2 = d1 − σ
√
τ

The Black-Scholes Europena call option Price
Given parameters
S r T σ
100 0.1 1 0.3

The Black-Scholes Europena call option Price
Given parameters
S r T σ
100 0.1 1 0.3
Strike price Call price
98 17.79
100 16.73

Effects of parameters and Volatility smile
Effects of changing the input parameters
(a) Changing the
call price with S
(b) Changing the
call price with T
(c) Changing the
call price with σ

Effects of changing the input parameters
(a) Changing the
call price with S
(b) Changing the
call price with T
(c) Changing the
call price with σ
Figure: Observation of the European call option price

Volatililty Smile in Option Pricing
(a) Implied volatility Vs
strike price of the
Black-Scholes model

Volatililty Smile in Option Pricing
(a) Implied volatility Vs
strike price of the
Black-Scholes model
(b) Implied volatility Vs
strike price of the Heston
model
Figure: Volatility smile of the Black-Scholes model and Heston model

Black-Scholes call price and simulated call price
Option pricing by Monte Carlo Simulation of the Black
Scholes Model
Simulation paths and call price of the Black Scholos Model
The stock price at T maturity,
ST = S0 exp[(r −
1
2
σ2
)t + σ
√
TN]

Scholes Model
ST = S0 exp[(r −
1
2
σ2
)t + σ
√
TN]
Generate a large number of stock price

Scholes Model
ST = S0 exp[(r −
1
2
σ2
)t + σ
√
TN]
Option price (expected value of payoff function)

Scholes Model
ST = S0 exp[(r −
1
2
σ2
)t + σ
√
TN]
Option price (expected value of payoff function)
Discount factor

Simulation paths of the Black Scholos Model
(a) Simulation paths for
N = 1000

Simulation paths of the Black Scholos Model
(a) Simulation paths for
N = 1000
(b) Simulation paths for
N = 10000
Figure: Simulation paths of the Black-Scholes Model

Compare Black-Scholes call price with simulation results
Call price for N = 1000 and σ = 0.3
Black Scholes
price
17.79
Simulated call
price
16.27

Compare Black-Scholes call price with simulation results
Black Scholes
price
17.79
Simulated call
price
16.27
Black Scholes
price
17.79
Simulated call
price
17.72

The Heston Model, Formula and Characteristics functions

The Heston model
dSt = µStdt +
√
vtStdz1(t)

The Heston model
dSt = µStdt +
√
vtStdz1(t)
The instantaneous variance vt follows a
Cox-Ingersoll-Ross(CIR) process and it is defined by,
dvt = κ(θ − vt)dt + σ
√
vtdz2(t)

The Heston model
dSt = µStdt +
√
vtStdz1(t)
√
vtdz2(t)
z1 and z2 are Wiener processes with correlation ρ which is
equivalently with covariance ρdt as follows,
dz1(t)dz2(t) = ρdt

The Heston model
dSt = µStdt +
√
vtStdz1(t)
√
vtdz2(t)
z1 and z2 are Wiener processes with correlation ρ which is
equivalently with covariance ρdt as follows,
dz1(t)dz2(t) = ρdt
The St under the risk neutral measure Q by using Girsanov’s
theorem by the following expression,
dSt = rStdt +
√
vtStd ˜
z1(t)

The Heston Model
The risk neutral log price process, ln St is defined by,
ln St = r − 1
2

dt +
√
vtdz1(t)

The Formula and Characteristics functions of Heston model
The Formula and Characteristics functions
European call option price is defined by,
C(S, v, t) = SP1 − KP(t, T)P2

The Formula and Characteristics functions of Heston model
The Formula and Characteristics functions
European call option price is defined by,
C(S, v, t) = SP1 − KP(t, T)P2
Where Pj , j = 1, 2 are risk neutral probabilities obtained by
inverting the characteristic function fj
Pj (x, v, T; ln(K)) = Pr(ln St ln K)
=
1
2
+
1
π
Z ∞
0
Re
he−iϕ ln(K)fj (x, v, T; ϕ)
iϕ
i
dϕ

Call price and simulated price under Heston model
European call option price under the Heston model
Correlation between
Brownian motions
European call option
price
0.1 14.32
-0.1 14.45

European call option price under the Heston model
Correlation between
Brownian motions
European call option
price
0.1 14.32
-0.1 14.45
S0 K r v0 κ θ λ σ T
100 98 0.1 0.04 2 0.04 0 0.3 1
Table: Parameters are used in the Heston Model

Price and variance paths of the Heston Model

(a) Variance simulation
paths for N = 1000

(a) Variance simulation
paths for N = 1000
(b) Price simulation paths
for N = 1000
Figure: Price and variance simulation paths of the Heston Model

Distribution of the Stock price and variance
(a) Stock price at maturity

Distribution of the Stock price and variance
(a) Stock price at maturity (b) Variance at maturity
Figure: Stock price and variance distribution of the Heston Model

Simulation price of the Heston Model
Number of paths (N) European call option price
1000 14.34
10000 14.33

Staylized facts
Stylized facts
Following stylized facts are often occur for financial time series:

Staylized facts
Stylized facts
Non-stationary

Staylized facts
Stylized facts
Non-stationary
Level of returns and volatility clustering.

Staylized facts
Stylized facts
Non-stationary
White noise

Staylized facts
Stylized facts
Non-stationary
White noise
Leptokurtic

Staylized facts
Stylized facts
Non-stationary
White noise
Leptokurtic
No arbitrage oppurtunity

Effects of σ and ρ
Effect of Volatility of Variance(σ)
Effect of σ:

Effect of σ:
σ effects the pick of the distribution of the log stock price.

Effect of σ:
σ effects the pick of the distribution of the log stock price.
For σ = 0, the variance process is dropped out.
Increasing σ rises the peak of the distribution.
Create heavy tails on both sides.

Figure: The effect of Volatility of Variance on Heston Density Function.

Effect of correlation(ρ)
Effect of ρ:

Effect of ρ:
ρ controls the heaviness of the tails.

Effect of ρ:
ρ controls the heaviness of the tails.
When ρ = 0, the skewness will be close to 0.
Forρ 0, the variance rises with the increase in stock price.
Forρ 0, the volatility rises if stock price decreases.

Figure: The effect of correlation on Heston Density Function.

Log-return distributions of both models
Log-return distribution of the Black-Scholes model

The log-return is defined as ln(St
S0
) and it can be expressed as,
ln

St
S0

=

r −
1
2
σ2

t + σ
√
TN

The log-return is defined as ln(St
S0
) and it can be expressed as,
ln

St
S0

=

r −
1
2
σ2

t + σ
√
TN
Figure: Log return distribution of the Black-Scholes model

Log-return distribution of the Heston model

The log return process is given by,
Return = log

St+dt
St

=

µt −
1
2
vt

dt +
√
vtWdt

The log return process is given by,
Return = log

St+dt
St

=

µt −
1
2
vt

dt +
√
vtWdt
Figure: Log return distribution of the Heston model

Squared return, absolute return and return of the Black-Scholes model
Squared return of the Black-Scholes model
Squared return:
Two lags (3 and 7) are significanly different from 0 in acf and
pacf.
R2
t is not an independent sequence.

Squared return of the Black-Scholes model
(a) Autocorrelation
function
(b) Partial autocorrelation
function
Figure: Autocorrelation and partial autocorrelation function of squared
return (R2
t )of the Black-Scholes model

Absolute return of the Black-Scholes model
Absolute return:
Lags 7, 12 and 14 in the pacf are more significantly different
from 0 than the acf of those lags.
Absolute returns,|Rt|, are independent sequence i.e. absolute
return,|Rt|, are uncorrelated.

Absolute return of the Black-Scholes model
(a) Autocorrelation
function
function
Figure: Autocorrelation and partial autocorrelation function of absolute
return of the Black-Scholes model

Return of the Black-Scholes model
Return:
Returns of the Black-Scholes model are uncorrelated.
In the pacf some lags are more significantly different from 0
than the acf of those lags.
Returns, Rt, are independent sequence of the Black-Scholes
model.

Return of the Black-Scholes model
(a) Autocorrelation
function
function
Figure: Autocorrelation and partial autocorrelation function of return of
the Black-Scholes model

Squared return, absolute return and return of the Heston model
Squared return of the Heston model
Squared return:
All lags except 5 and 6 of both functions are inside the
bounds.
Two lags (5 and 6) are significantly different from 0.
R2
t is not an independent sequence i.e. Squared retruns R2
t
are correlated.

Squared return of the Heston model
(a) Autocorrelation
function
function
Figure: Autocorrelation and partial autocorrelation function of squared
return (R2
t )of the Heston model

Absolute return of the Heston model
Absolute return:
All lags are inside the bounds.
Counter behavior is not observed in the both functions.
Absolute returns, |Rt|,are correlated.

Absolute return of the Heston model
(a) Autocorrelation
function
function
Figure: Autocorrelation and partial autocorrelation function of absolute
return of the Heston model

Return of the Heston model
Return:
Most of the lags of autocorrelation and partial autocorrelation
functions are inside the bounds.
Two lags (5 and 10) are significantly different from 0 in the
acf and pacf.
Rt is not an independent sequence i.e. returns (Rt)are
correlated .

Return of the Heston model
(a) Autocorrelation
function
function
Figure: Autocorrelation and partial autocorrelation function of return of
the Heston model

Conclusion
What we have learned today?

Conclusion
Heston model shows good smile effect.

Conclusion
Log return distributions of the Black-Scholes model and
Heston model represent counter behavior.

Conclusion
In the Heston model, σ effects the peak of the lnST and ρ
controls the heaviness of the tails of lnST .

Conclusion
Squared returns, absolute returns and returns are correlated in
the Heston model.

Conclusion
Squared returns, absolute returns and returns are correlated in
the Heston model.
In the Black-Scholes model, only squared retruns are
correlated. Absolute returns and returns are uncorrelated.

References
References
Stefano M. Iacus
Simulation and Inference for Stochastic Differential Equations
Springer-Verlag New York, 2008
Fisher Black, Myron Scholes
The Pricing of Options and Corporate Liabilities
Journal of Political Economy, Volume 3, 1973
Steven L. Heston
A Closed-Form Solution for Options with Stochastic Volatility
with Applications to Bond and Currency Options
The Review of Financial Studies, Volume 6, 1993
John Hull
Options, futures, and other derivatives
Pearson Education, New York, 2018

References
Rama Cont, Peter Tankov
Financial Modeling With Jump Processes
Champan Hall/CRC, 2004
Bernt Øksendal
Stochastic Differential Equations
Springer-Verlag Berlin Heidelberg,2003
Jörn Sass
Lecture notes in Financial Mathematics
University of Kaiserslautern, Summer (2022)
Ser-Huang Poon
A Practical Guide to Forecasting Financial Market Volatility
John Wiley and Sons limited (2005)

References
Tahmid Tamrin Suki, A B M Shahadat Hossain
A comparative analysis of the Black-Scholes-Merton model
and the Heston stochastic volatility model
J. Bangladesh Math. Soc., Vol. 39 (2019) 127-140
Fabrice Douglas Rouah
The Heston Model and Its Extensions in Matlab and C++
John Wiley and Sons, Inc., Hoboken, New Jersey (2013)
Hull J. and White A
The Pricing of Options on Assets with Stochastic Volatilities
Journal of Finance, 42 (1987), 281-300
Mitun Kumar Mondal, Md. Abdul Alim, Md. Faizur Rahman,
Md. Haider Ali Biswas
Mathematical Analysis of Financial Model on Market Price
with Stochastic Volatility
Journal of Mathematical Finance, 07(02):351-365

References
John C. Hull
Options, Futures and Other Derivatives
Global edition, Eighth edition, Pearson Education Limited
(2015-2016)
Rubinstein, M.
Non parametric Tests of Alternative Option Pricing Models
Using All Reported Trades and Quotes on the 30 Most Active
CBOE Option Classes from August 23, 1976 through August
31, 1978
Journal of Finance, 40, 455-480
Johnson H. and Shanno D.
Option Pricing when the Variance Is Changing. Journal of
Finance and Quantitative Analysis
22 (1987), 143-151
Wiggins J. B.
Option Values under Stochastic Volatility: Theory and

References
John, Options-trading-Black-Scholes-model
https://www.codearmo.com/python-tutorial/options-trading-
black-scholes-model
December 22, 2020
John, Calculating the Volatility Smile,
https://www.codearmo.com/python-tutorial/calculating-
volatility-smile
January 03, 2021
John, Implied Volatility for European Call with Python,
https://www.codearmo.com/blog/implied-volatility-european-
call-python
September 08, 2020
Implied Volatility,
https://py-vollib-
vectorized.readthedocs.io/en/latest/pkg ref/iv.html

Volatility Smiles and Stylised Facts in the Heston Model

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