MSc in Finance
Author: Rikke Knold Christensen, 280702
Department of Business Studies
Academic advisor: Elisa Nicolato
Stochastic Volatility Models and the use of the
Fourier Transformation method as an option pricing tool
Aarhus School of Business
01-09-2011
Content
CHAPTER 1: INTRODUCTION .................................................................................................................................... 2
1.1
1.2
1.3
1.4
PREFACE ............................................................................................................................................................... 2
PROBLEM STATEMENT .......................................................................................................................................... 3
METHODOLOGY .................................................................................................................................................... 4
DELIMITATION ...................................................................................................................................................... 6
CHAPTER 2: THE BASIC THEORY OF OPTION PRICING .................................................................................. 7
2.1 STOCHASTIC PROCESSES ....................................................................................................................................... 7
2.2 THE BLACK-SCHOLES MODEL ............................................................................................................................ 10
2.3 EXTENSIONS TO THE BLACK-SCHOLES MODEL ................................................................................................... 16
2.3.1
EMPIRICAL EVIDENCE OF A STOCHASTIC VOLATILITY..................................................................................... 16
CHAPTER 3: STOCHASTIC VOLATILITY MODELS ........................................................................................... 21
3.1 GENERAL STOCHASTIC VOLATILITY MODELS..................................................................................................... 21
3.1.1
DERIVING A GENERAL PDE FOR STOCHASTIC VOLATILITY MODELS ............................................................... 24
3.2 THE HESTON STOCHASTIC VOLATILITY MODEL ................................................................................................. 27
3.2.1
THE PROCESS OF THE HESTON MODEL............................................................................................................ 27
3.2.2
PDE FOR THE HESTON MODEL ....................................................................................................................... 30
3.2.3
CHARACTERISTIC FUNCTIONS AND THE FOURIER TRANSFORMATION ............................................................ 31
3.2.4
IMPLEMENTATION OF THE FOURIER TRANSFORM ON THE HESTON MODEL ..................................................... 33
CHAPTER 4: DATA ...................................................................................................................................................... 36
CHAPTER 5: FITTING THE IMPLIED VOLATILITY SURFACE ....................................................................... 38
5.1
5.2
5.3
5.3.1
5.3.2
5.3.3
5.4
5.5
5.6
MODEL CALIBRATION ......................................................................................................................................... 38
SETTING THE INITIAL PARAMETERS ..................................................................................................................... 40
SIMULATION OF THE HESTON PROCESS ............................................................................................................... 43
MONTE CARLO SIMULATION .......................................................................................................................... 44
MILSTEIN SCHEME ......................................................................................................................................... 45
MIXING SOLUTION .......................................................................................................................................... 46
COMPARISON OF SIMULATED AND FOURIER PRICES ............................................................................................ 48
THE IMPLIED VOLATILITY SURFACE OF THE S&P500 INDEX OPTIONS.................................................................. 50
THE HESTON MODEL’ SENSITIVITY TO ITS PARAMETERS ..................................................................................... 53
CHAPTER 6: CONCLUSION ...................................................................................................................................... 59
LIST OF LITERATURE ............................................................................................................................................... 62
1
Chapter 1: Introduction
1.1
Preface
In the year of 1973 one of the most recognized models for stock options prices were introduced by
Fischer Black, Myron Scholes and Robert Merton (Black and Scholes, 1973). The model, which
today is referred to as the Black-Scholes Model, was the first model to present a closed form
solution to option pricing. Because of its simplicity and easy implementation the model gained an
increasing influence on the way that traders priced and hedged the options traded on the market
throughout the years. However, after the stock market crash in 1987, the characteristics of the stock
market changed. As a result of this, one of the main implications of the Black-Scholes Model,
namely that log-returns are normally distributed, was no longer visible in the market. Empirical
studies have later shown that the stock returns from 1987 and forward display a skewed leptokurtic
distribution with heavy tails. Furthermore, the stock returns also show clustering in periods, which
is evidence of a stochastic volatility. This is a clear violation of the constant volatility assumption
underlying the Black-Scholes Model, and stock returns therefore cannot be characterized as
constant over time.
In order to better understand the way that options are priced, it can be useful to examine the socalled implied volatility surface. The volatility surface is a collection of implied volatilities, which
is essentially the volatility that generates market prices when inserted into the Black-Scholes
formula. Among practitioners it is known as “…the wrong number in the wrong formula to get the
right price…”(Rebonato, 1999) As empirical evidence has shown, as mentioned, the market depicts
a stochastic volatility, which means that the implied volatility depends on both the strike price and
expiration of the option – and is not constant. The volatility surface and the examination of stock
returns can therefore be used to analyse 1) why options are priced the way that they are and 2) how
they should be priced if they were priced fairly (Gatheral, 2006, page xxiii).
One of the most recognized stochastic volatility models was introduced by Heston (Heston, 1993)
in 1993, which much like the Black-Scholes Model provides a (semi)-closed form solution for
pricing options. The model is one of the most widely used models that exist on the market, and the
model particularly introduces the mentioned (semi)-closed form solution for pricing European call
options based on a volatility measure related to a mean reverting square root process 1. The volatility
is thereby assumed to be non-constant and to follow a random process. There exist several methods
for estimating the stock price of an option through the Heston model, and these methods include
1
The mean square root process was first introduced by Cox, Ingersoll and Ross in 1985 (Gatheral, 2006, page 15)
2
simulation and Fourier Transformation. The different approaches have each their advantages and
disadvantages. The simulation approach has become very popular for option pricing as empirical
studies have shown that the simulation accuracy increases by the number of simulations performed;
however, this also means that the implementation time for the process increases. As a result of this,
there is an increasing interest for the use of Fourier Transformation for pricing options. This method
provides an estimate of the stock option price by the solution of an integral – which thereby makes
it computationally easy compared to the simulation approach.
1.2
Problem Statement
The main purpose of this paper is to analyse a chosen model to fit the implied volatility surface of
the market and to test the valuation method of Fourier Transformation. In order to analyse the
model, the basic theory behind option pricing is first presented and the shortcomings of the BlackScholes Model is analysed. As some of the real world implications of this model are quite severe,
the Black-Scholes Model can be considered insufficient for describing the nature of the financial
markets (to a certain extent). This raises the need for a model that incorporates stochastic volatility
in order to better fit the data observed in the market. The chosen model to be fitted on the implied
volatility surface is the Heston stochastic volatility model. Only one stochastic volatility model has
been chosen, in order to avoid the paper to become too comprehensive.
There will be two valuation methods implemented to the Heston Model: 1) a simulation and 2)
Fourier Transformation. This will enable an evaluation and discussion of the use of the Fourier
Transformation as an option pricing tool, as it can be directly compared to the simulation approach.
This lead to the overall problem statement:
How can a model with stochastic volatility be derived to fit the implied volatility surface of the market?
To answer the problem statement, the following research questions have been defined:
1) What is the basic theory behind option pricing?
2) What is stochastic volatility and why is it needed?
3) What are the characteristics of the chosen model to be fitted (the Heston Model)?
4) How can the Heston Model be fitted to the implied volatility surface?
3
a. How well is the implied volatility estimated from the Simulation vs. Fourier
Transformation
b. How well does the Fourier Transformation approach perform as an option pricing
tool?
5) How well does the Heston Model capture the behaviour of the market?
On the basis of the Heston Model the thesis will be build on the historic data from the S&P500
Index, which consists of 500 large-cap company stock options. The index is one of the most
followed indexes and it is widely said to be the best single gauge of the American equities market
(http://www.standardandpoors.com/indices/sp-500/en/us/?indexId=spusa-500-usduf--p-us-l--). This
makes the index very liquid, which is essential for the analysis and the required data.
In the end, the thesis will discuss the differences between the empirical volatility surface and the
model fitted volatility surface. The thesis will thereby have analysed the Heston model’s capability
to capture the behaviour of the S&P 500 options and discuss the use of Fourier Transformation as
an option pricing tool.
1.3
Methodology
The main structure of the thesis is divided into two parts – a theoretical part and a part where the
theory needed for the analysis of the Heston Model and Fourier Transformation is applied, as shown
in figure 1.3.1.
Part 1 consists of Chapter 2 and 3 where the theoretical framework of the thesis is introduced. The
basic theory behind pricing option is introduced, and the ideas behind stochastic processes,
including the Brownian and Geometric Brownian motions is shortly presented. Afterwards, the
famous Black & Scholes Model will be outlined and the rationale behind the risk neutral world will
be introduced. However, empirical studies have shown that the Black-Scholes Model has several
shortcomings, and is therefore insufficient in pricing options to a certain degree. Because of this, the
theoretical extensions to the Black-Scholes Model and the existence of stochastic volatility will be
discussed, introducing the implied volatility and the existence of the volatility smile.
In Chapter 3 the basics for a better option pricing model will be established by the introduction of
stochastic volatility models. This alternative to the Black-Scholes Model will be discussed and the
4
chosen stochastic volatility model, the Heston Model, will be presented. After this, the theory
behind using Characteristic Functions and Fourier Transformation will be presented in order to
solve the valuation problem and fit the Heston Model to the implied volatility surface.
Figure 1.3.1 – Methodology of the thesis
Theoretical aspect of the thesis
Chapter 2 - Basic Option Pricing
Chapter 3 - Stochastic Volatility Models
Implementation
Chapter 4 - Data basis
Chapter 5 - The implied volatility surface
Conclusion
Remarks on the performance of the model and the Fourier Transformation
Source: Own contribution
The second part of the thesis implements the theory discussed in the previous paragraphs and
applies a stochastic volatility model to the market data. First, the data used in the thesis will be
presented in Chapter 4, and the chosen underlying index, the S&P500, will be shortly introduced.
Thereafter, the data basis for the thesis will be established and the use of a proxy for the risk free
interest rate will be discussed. In Chapter 5 the actual fitting of the Heston Model to the implied
volatility surface will be conducted. The calibration of the Heston Model parameters will be
performed by the use of Fourier Transformation, where the parameters will be fitted to the market
data. The evaluation of the Fourier Transformation will then be performed by simulating the Heston
process by using a Monte Carlo simulation, and comparing the two methods. In the Monte Carlo
simulation the Milstein Scheme will be introduced and applied to the variance process in order to
discretize the process. However, the Heston Model is very sensitive with respect to its parameters,
and therefore a discussion of this will be performed. As an extension of this the sensitiveness of the
parameters will then be tested in order to illustrate the effect of changing the parameters on the
5
implied volatilities. In the end a comparison of the empirical and the Heston fitted volatility surface
will serve as an estimate the performance of the Heston model. Finally Chapter 6 will give a
conclusion on the thesis and the findings that has been made.
1.4
Delimitation
It is assumed that the reader is familiar with some of the basic theory underlying option pricing, and
this will therefore only be presented shortly as a recap and a basic for the development of the
stochastic volatility models. Furthermore, the analysis will only be conducted for European call
options, which means that American options and put options are delimited as they are beyond the
scope of the thesis.
In the introduction of stochastic volatility models, some alternative models other than the Heston
Model will shortly be presented. These models will be presented in different notations compared to
the original sources, in order to make the comparison of the models easier. Further analysis and
derivation will not be conducted on these models. Since the purpose of this thesis is to fit the
Heston Model to the implied volatility surface of the S&P500 index, the primary focus will be on
the Heston Model and Fourier Transformation. As the thesis thereby have a practical aspect, the
Heston Model and the theory underlying this will be presented in a way that a practical approach
and use of the model is possible for the reader of the thesis.
6
Chapter 2: The Basic Theory of Option Pricing
The purpose of the following section is to introduce the basic theory that is necessary for
developing a model to fit the volatility surface of the S&P 500 index. The section will more
specifically introduce the basic theory behind option pricing, and discuss how it applies to the real
world.
The section will begin with explaining the basic theory underlying the pricing of options, which
will be followed by the introduction of one of the most used option pricing models throughout the
years, the Black-Scholes model. In spite of this, the Black-Scholes model suffers from severe real
world implications, which makes the model inadequate for option pricing to a certain extent. The
discussion of the real world implications will lead to the introduction of the implied volatility and
volatility smile, which are empirical proofs that some of the basic assumptions underlying the
Black-Scholes model are violated in the real world. This establishes the need for an alternative
option pricing model that better describes the evolvement of the stock options on the market.
2.1
Stochastic processes
The first step in determining the price of an option is to understand how the underlying of a stock
option evolves over time. Any variable that changes in an uncertain way over time is said to follow
a stochastic process. There exist two types of stochastic processes, one in discrete time and one in
continuous time. The difference between the two is that a discrete time stochastic process can only
be measured on specific times while a continuous time process can be measured continuously.
One of the most common processes that are used to describe the evolvement of a stock price over
time is the so-called Geometric Brownian Motion. The Geometric Brownian Motion is a special
case of a Wiener Process or a Brownian Motion2, which has a mean of zero and a variance of 1 each
year. In differential form the Geometric Brownian Motion describes the changes in a variable Z as
shown below (Hull, 2008, page 266):
𝑑𝑆 = 𝜇𝑆𝑑𝑡 + 𝜎𝑆𝑑𝑍
Equation 2.1.1
where S is characterized as the stock price at time t, μ is the drift term or the expected return of the
stock option and σ is the volatility of the stock price. The term d corresponds to an infinitesimal
2
The Geometric Brownian Motion is also called a Generalized Wiener Process
7
small change of the variable, which means that the process is approximately continuous when dt
approaches 0.
The Geometric Brownian Motion has the advantage that the stock price will always remain positive
as long as the initial value of the stock price is positive. This is an advantage over the standard
Brownian Motion where the change in the stock price process, dS will move towards zero as the
stock price approaches zero. Another advantage for the Geometric Brownian Motion is that the
expected percentage return of a given stock option is independent of the stock price itself. This is
due to the fact that the process itself depends on the level of the stock price.
It should be noticed that the Geometric Brownian Motion is defined as a continuous-variable,
continuous time stochastic process, while stock prices can only be measured in discrete time. The
measurement of stock prices and the corresponding changes of these are restricted, because stock
options are only tradable when the exchange is open. Therefore, the prices and price changes can
only be observed at these points in time (Hull, 2008, page 259). In discrete time the return of the
stock price (or in other words the relative change of the stock price) can be defined as the relative
change in the stock price over a short period of time, ∆t, like shown below:
∆𝑆
𝑆
= 𝜇𝑑𝑡 + 𝜎𝑑𝑍
Equation 2.1.2
Where the process like before is determined by a drift3, μdt, and a stochastic rate4, σdZ.
The term Z is a Generalized Wiener Process, where μ and σ are constants, if the following
properties are fulfilled (Hull, 2008, page 261 and 263):
PROPERTY 1: The change of ∆Z during a small period of time, ∆t, is:
∆𝑍 = 𝜖√∆𝑡, 𝑤ℎ𝑒𝑟𝑒 𝜀 ~ 𝜙(0,1)
From the first property it follows that the expected value or mean of the variable Z should be equal
to zero as derived in the following:
𝐸(𝑍𝑡 ) = 𝐸(𝜀√𝑡) ⇔ 𝐸(𝑍𝑡 ) = 0, because ε ~ N(0,1)
3
4
Mean change per unit time
Variance per unit time
8
The variance is also easily derived as it can be shown that the variance is equal to t:
2
𝑉𝑎𝑟(𝑍𝑡 ) = 𝑉𝑎𝑟(𝜀 √𝑡) ⇔ 𝐸 [(𝜀√𝑡) ]
𝑉𝑎𝑟(𝑍𝑡 ) = 𝑡
→
𝜎 = √𝑡
As a consequence hereof, the standard deviation of the variable Z is equal to the square root of time.
PROPERTY 2: The values of ∆Z1 and ∆Z2 for any two different short intervals of time, ∆t1 and ∆t2,
are independent of each other.
It follows from this property, that ∆Z must follow a so-called Markov process. The Markov
property is a special case of a stochastic process, where the only relevant variable for predicting of
the future is the present variable (Hull, 2008, page 259). The process therefore implies that all
relevant information from the past prices is already incorporated into the present value, and
therefore the Markov process is also known as a process without memory. This property is also
consistent with empirical evidence from the real world as the competition in a market contributes to
ensuring the weak form of the market efficiency5. In addition the process is also assumed to be a
martingale process, which means that conditional expectation of the variable Z at any time t, is the
value of the stock today.
PROPERTY 3: The initial value of Z is equal to zero.
𝑍(0) = 0
PROPERTY 4: (Quote, Hull, 2006, page 263):
𝑇ℎ𝑒 𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑎𝑡ℎ 𝑓𝑜𝑙𝑙𝑜𝑤𝑒𝑑 𝑏𝑦 𝑍 𝑖𝑛 𝑎𝑛𝑦 𝑡𝑖𝑚𝑒 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙 𝑖𝑠 𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑒
PROPERTY 5: (Quote, Hull, 2006, page 263):
𝑇ℎ𝑒 𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑍 𝑒𝑞𝑢𝑎𝑙𝑠 𝑎𝑛𝑦 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑢𝑎𝑟
𝑣𝑎𝑙𝑢𝑒 𝑖𝑛 𝑎𝑛𝑦 𝑡𝑖𝑚𝑒 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙 𝑖𝑠 𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑒
5
Weak form for market efficiency is where the information set includes the history of prices and returns, and only this
(Campbell et al, 1997, page 22)
9
2.2
The Black-Scholes Model
The Black-Scholes model was first introduced by Fischer Black and Myron Scholes in the famous
article “The Pricing of Options and Corporate Liabilities” in 1973. This, together with Robert
Mertons article “Theory of Rational Option Pricing” (1973), began the biggest breakthrough in
option pricing that the world has ever seen. The Black-Scholes Model (also known as the BlackScholes-Merton Model) is considered to be one of the most influential models on the market for
option pricing (Hull, 2008, page 277). The model was the first of its kind to suggest a closed-form
solution to pricing stock options. Because of the ease of implementation and its close fit to the
observed market prices (compared to other models that were on the market at that time), the model
gained recognition and quickly became very popular among traders and researchers. As a
consequence of this, the model had a huge influence on the way that traders price their options, and
still does this day.
The partial differential equation of the Black-Scholes Model must be satisfied by the price of any
derivative dependent on a stock option (not paying out dividends) (Hull, 2008, page 285).
The basic assumptions underlying the Black-Scholes Model are shortly listed below (Hull, 2008,
page 286-287):

The stock prices follows a Geometric Brownian Motion with a constant mean and variance

Short selling of securities is permitted – with the full use of proceeds, and all securities are
perfect divisible

No transaction costs or taxes

No dividends

No riskless arbitrage opportunities in the market

Continuous security trading

Constant risk free rate, r, which is the same for all maturities
Many of the abovementioned assumptions underlying the Black-Scholes Model are in fact violated
to some degree in the real world. However, the consequences and severity of the violations vary
across the assumptions. One of the obvious violations is the one concerning no transaction costs. In
the real world there are often taxes on the derivative (this depends on the country in question) as
well as transaction costs. As a result of this it must be assumed that this will already be incorporated
10
into the final price of the stock option. The assumption concerning no dividends is without any
doubt also often violated in the real world. However, this can easily be relaxed, as the BlackScholes equation can be modified to take dividends into account, see equation 2.2.6. The
assumption concerning no arbitrage opportunities is also questionable, as some people earn their
living on finding such opportunities in the market. However, whenever an arbitrage opportunity
exists, the market will quickly rebalance itself again and therefore the arbitrage opportunity will
disappear relatively quickly. It is also assumed that the securities are continuously traded, which
means that the delta hedging6 of the portfolio is done continuously. However, this is not exactly true
in the real world, where the transaction cost often have an influence on the hedging frequency – the
lower the transaction cost, the higher the frequency of hedging, which contributes to making the
security trading discrete (Wilmott, 2005, page 146). Lastly the risk free interest rate, r, is assumed
to be constant, but in reality the interest rate is stochastic and cannot be known in advance.
The basic idea underlying the Black-Scholes Model is simply to perfectly hedge a portfolio, so that
risk is eliminated. This is known as a so-called delta-hedge. The final return of the portfolio will
then equal the return of the derivative at expiration, no matter how the price of the underlying
asset/stock evolves. This is done by constructing a riskless portfolio from a position in the
derivative and a position in the stock. The return from the portfolio must then equal the risk free
rate in order to fulfil the assumption of no arbitrage opportunities in the market. This is made
possible because stock price movements have a similar affect on stock price and the price of the
derivative. Thereby the price of the underlying stock will be perfectly correlated with the price of
the derivative in any short period of time. This makes it possible to construct a portfolio where a
possible gain or loss in the stock position is offset by a gain or loss from the derivative position –
thereby eliminating the risk and hedging the portfolio. However, it is important to emphasize that
the constructed portfolio is only riskless in a very short period of time, and for the portfolio to
remain riskless, it is necessary to continually rebalance the portfolio. This will be explained in more
detail in the following, where the construction of the riskless portfolio that enables the BlackScholes differential equation will be derived.
The value of a stock option is determined by the stock price (S), the time (t), the standard deviation
(σ), the drift (µ), the strike/exercise price (K), the time to maturity (T) and the risk free interest rate
6
Delta-hedging will be explained later
11
(r). In the following, the value of the stock option will be referred to as f(S,t) instead of f(S, t, σ, µ,
K, T, r) for simplicity.
As mentioned earlier, the value of the portfolio must consist of a long position in the stock option
and a short position (∆-quantity) in the underlying derivative. The value of the portfolio is then
defined as:
Π(𝑡) = 𝑓(𝑆, 𝑡) − Δ𝑆
In the next time interval, t + dt, the value of the portfolio depends on the change in the value of the
stock option as well as the change in the underlying derivative. Before rebalancing the portfolio,
this can be written as the following:
𝑑Π(𝑡) = 𝑑𝑓(𝑆, 𝑡) − Δ𝑑𝑆
When determining the price of an option, it is useful to consider the Itô process. The general Itô
process is a Generalized Wiener Process and can be defined as (Hull, 2008, page 265):
𝑑𝑆 = 𝑎(𝑆, 𝑡) + 𝑏(𝑆, 𝑡)𝑑𝑍
Equation 2.2.1
which states that the change in the underlying asset is determined by a drift, a, and a variance, b. As
a result of this, any derivative is a function of the time and the stochastic variable underlying the
derivative. In relation to option pricing an important result was found in 1951, where it was shown
that a function of a variable and the time follows the following process:
𝑑𝑓(𝑆, 𝑡) =
𝜕𝑓
𝜕𝑓
𝜕2 𝑓
1
𝑑𝑡 + 𝜕𝑆 𝑑𝑆 + 2 𝑏 2 (𝑆, 𝑡) 𝜕𝑆2 𝑑𝑡
𝜕𝑡
Equation 2.2.2
where 𝑏 2 = 𝜎 , since the diffusion term is not depending on S. This above equation is known as
Itô’s Lemma. When inserted into the portfolio equation, the value of the portfolio changes then
equals:
𝜕𝑓
𝜕𝑓
1 2
𝜕 2𝑓
𝑑Π(𝑡) =
𝑑𝑡 +
𝑑𝑆 + 𝜎 (𝑆, 𝑡) 2 𝑑𝑡 − Δ𝑑𝑆
𝜕𝑡
𝜕𝑆
2
𝜕𝑆
𝜕𝑓
1
𝜕2 𝑓
𝜕𝑓
⇕ 𝑑Π(𝑡) = ( 𝜕𝑡 + 2 𝜎 2 (𝑆, 𝑡) 𝜕𝑆2 ) 𝑑𝑡 + (𝜕𝑆 − Δ) 𝑑𝑆
12
When rearranged, the first part of the term can be characterized as the deterministic part or the drift
(the term in front of dt) whereas the latter part is a stochastic part, that generates the risk of the
portfolio. It is this part that must be eliminated in order to make the portfolio riskless. The risk
generating part of the previous equation is therefore defined as:
𝜕𝑓
( − Δ) 𝑑𝑆
𝜕𝑆
According to the Black-Scholes the uncertainty (the risk) can be eliminated if the delta quantity in
the small period of time from t to t+dt is set to the following (otherwise known as a delta-hedging
of the portfolio):
𝜕𝑓
( − ∆) 𝑑𝑆 = 0
𝜕𝑆
⇕ ∆=
𝜕𝑓
𝜕𝑆
which delta-hedges the portfolio. However, as the delta-hedge is not constant (depends on both S
and t) it is, as mentioned, necessary to rebalance the portfolio continuously in order to keep the
portfolio riskless. However, if continuous trading is allowed (like assumed in the Black-Scholes
Model) this is made possible, and the dynamics of the constructed riskless portfolio can be defined
as:
𝜕𝑓 1 2
𝜕 2𝑓
𝑑Π(𝑡) = ( + 𝜎 (𝑆, 𝑡) 2 ) 𝑑𝑡
𝜕𝑡 2
𝜕𝑆
If there exist no arbitrage opportunities in the market, the riskless portfolio must yield the risk free
interest rate, r. This essentially means that the investor doesn’t require any excess return to take on
the risk of the portfolio. If this is not the case, the return from holding the portfolio is higher than
the return of a risk free asset, and thereby an arbitrage opportunity will exist and it will be possible
to make money by buying the portfolio for lend money. Therefore it must hold that:
𝑑Π = 𝑟Π𝑑𝑡
Inserting the equations into the above formula, dividing by the time step, dt, and rearranging will
then equate the Black-Scholes differential equation as written below:
13
𝜕𝑓 1 2
𝜕 2𝑓
𝜕𝑓
+ 𝜎 (𝑆, 𝑡) 2 + 𝑟 𝑆 − 𝑟𝑓 = 0
𝜕𝑡 2
𝜕𝑆
𝜕𝑆
Therefore it can be concluded that for the market to have no arbitrage opportunities, the BlackScholes equation must hold for any derivative depending on a non-dividend paying stock option.
There are several possible ways to solve the Black-Scholes partial differential equation. It can either
be solved directly with the closed form solution, or through numeric methods or approximations.
One of the possible methods is the risk neutral valuation, which is also known as the martingale
approach. In terms of a risk neutral valuation it is important to emphasize that the Black-Scholes
equation does not depend on the value of the expected return, µ. The underlying reason for this is
that the value of the expected return depends on the risk preferences of the investor. This means that
the more risk averse the investor is, the higher the return, µ, that the investor demands also is.
However, as the Black-Scholes equation was eliminated for risks by choosing a delta quantity to be
hedged, the investor is assumed to be risk neutral. Risk neutral investors do not require a premium
for the added risk of an investment, and therefore the market price of risk is equal to zero (Hull,
2008, page 626). The expected return of all investments under a risk neutral assumption is therefore
the risk free rate of interest, r. The only risk related to the option pricing will therefore already be
incorporated into the volatility.
The solution to the Black-Scholes equation is the same, whether or not a risk neutral world is
assumed. This means that the solution to the equation will yield the same price regardless if it is
done under risk neutral assumptions or under the assumptions of the Black-Scholes Model. This is
validated by the Feynman-Kac theorem, which when applied to the Black-Scholes equation gives:
𝑓(𝑆, 𝑡) = 𝐸𝑡 [𝑒 −𝑟(𝑇−𝑡) 𝑓(𝑆 ∗ , 𝑇)] = 𝑒 −𝑟(𝑇−𝑡) 𝐸𝑡 [𝑓(𝑆 ∗ , 𝑇)]
Equation 2.2.3
where the process S* is computed in relation to the expected value of the option as:
𝑑𝑆 ∗ = 𝑟𝑆 ∗ 𝑑𝑡 + 𝜎𝑆 ∗ 𝑑𝑍, 𝑤ℎ𝑒𝑟𝑒 𝑆 ∗ (𝑡) = 𝑆
Equation 2.2.4
Here the term Z refers to a Geometric Brownian Motion under risk neutrality. The only difference
from this process and the process under real world assumptions is that the drift term is given by the
14
risk free rate, r, instead of µ, as discussed earlier. To account for dividends, µ should be replaced by
r-q.
Under risk neutrality the explicit formula for the price of a European call option is defined as the
following:
𝑐(𝑆, 𝑇) = 𝑆0 ∙ 𝑁(𝑑1 ) − 𝐾𝑒 (−𝑟∙𝑇) ∙ 𝑁(𝑑2 )
Equation 2.2.5
𝑐(𝑆, 𝑇) = 𝑒 (−𝑟∙𝑇) (𝑆0 𝑒 (𝑟∙𝑇) ∙ 𝑁(𝑑1 ) − 𝐾 ∙ 𝑁(𝑑2 ))
𝑤ℎ𝑒𝑟𝑒,
𝑑1 =
𝑆 𝑒 (𝑟∙𝑇)
𝜎2
ln ( 0 𝐾 ) + ( 2 ) ∙ 𝑇
𝜎√𝑇
𝑑2 = 𝑑1 − 𝜎√𝑇
In the above formulas for the Black-Scholes Model the term S corresponds to the spot price of the
underlying asset at time t (in the above formula this corresponds to time t = 0), K is the
strike/exercise price, r is the risk free interest rate and σ is the volatility of the underlying asset. The
term N(x) is the cumulative probability distribution function for a standardized normal distribution
(Hull, 2008, page 291). If the first equation in equation 2.2.5 is rewritten to the second equation, the
term N(x) can more easily be explained, as N(d2) then becomes is the probability that the option in
question will be exercised in the risk neutral world (Hull, 2008, page 292).
The original model as presented by Black and Scholes in 1973 did not take into account the
dividends that many options pay in the real world. However, with only a couple of slight
adjustments, the model can be modified to do so as well as being able to price American options
(this latter will not be further discussed here, as it is not relevant for this thesis). Taking dividends
into account the explicit formulas for the Black-Scholes Model are as follows:
𝑐(𝑆, 𝑇) = 𝑒 (−𝑟∙𝑇) (𝑆0 ∙ 𝑒 (𝑟−𝑞)𝑇 ∙ 𝑁(𝑑1 ) − 𝐾 ∙ 𝑁(𝑑2 ))
𝑤ℎ𝑒𝑟𝑒,
𝑑1 =
Equation 2.2.6
𝑆 𝑒 (𝑟−𝑞)𝑇
𝜎2
ln ( 0 𝐾
)+(2)∙𝑇
𝜎√𝑇
𝑑2 = 𝑑1 − 𝜎√𝑇
15
Later in the thesis the above found equations for European call options will be used to value call
options on the S&P500 in the mixing solution.
2.3
Extensions to the Black-Scholes Model
Empirical studies have in many years underlined the real world implications for the Black-Scholes
Model as implied in the preface.
In the following paragraph stochastic volatility will be presented. Empirical studies have shown that
the volatility of stock option returns are not constant over time, which was one of the main
implications of the Black-Scholes Model. Instead empirical data shows that the volatility depicts a
curve, a so-called volatility smile, which insinuates the presence of a stochastic volatility in the
market, which will be illustrated in the following. This will among others be done by introducing
the term implied volatility and implied volatility surface.
2.3.1 Empirical evidence of a stochastic volatility
In practice the term implied volatility is defined as the volatility that yields the market price of a
given option when inserted into the Black-Scholes pricing equation (Wilmott, 2005, page 183-184).
However, as the volatility is the only unknown parameter in the Black-Scholes formula the
volatility also becomes a measurement of the market’s expectance of the future volatility. This
means that the market price is set on the basis of the volatility that the market expects of the
underlying asset/stock in the future.
Traders often quote the implied volatility of an option instead of its price (Hull, 2008, page 297), as
the implied volatility is usually less volatile than the actual option price itself. This enables the
traders to compare the options across strike prices, the underlying, observation times and maturities,
in order to estimate an appropriate implied volatility for another option.
One of the most discussed assumptions underlying the Black-Scholes Model is the one concerning
the constant volatility – this means that in the Black-Scholes model the implied volatility is actually
assumed to be constant. In the Black-Scholes framework, the volatility is an increasing function of
the stock option price – this implies that a higher volatility, ceteris paribus, will result in a higher
value of the option, at least in theory. However, when determining whether a constant volatility fits
the observed data on the market, one may look at the empirical evidence from the market data.
16
First, it is natural to explore whether the log returns of a stock options follows a Geometric
Brownian Motion and thereby are normally distributed, as was one of the main assumption of the
Black-Scholes Model.
The normal distribution (also known as the Gaussian distribution) is one of the most widely used
distributions in the literature, and its probability density function is given by:
𝑓(𝑥) =
1
√2𝜋𝜎
𝑒
−
(𝑥−𝜇)2
2𝜎2
Equation 2.3.1.1
The normal probability density distribution is characterized as being symmetric around its mean, µ,
and having a skewness of zero and a kurtosis of three (where the kurtosis is a measurement of the
peakness of the distribution).
Throughout the years there has been conducted several empirical studies in order to test whether or
not the market data can be explained by the normal probability density function, and thereby
follows a normal distribution. However, empirical studies of financial time-series data have shown
that the market data doesn’t exhibit log-normally distributed returns. As an extension of this the
implications of the Black-Scholes Model will in the following be illustrated and tested on the
observed data for the S&P500 index. From Figure 2.3.1.1 it is clear to see that the daily returns of
the S&P 500 index returns doesn’t follow a normal distribution as the observed distribution display
fat tails and a high central peak when compared to the normal distribution.
Figure 2.3.1.1
Source: Own contribution
17
These are characteristics of mixtures of distributions with different variances, which is in conflict
with the Black-Scholes assumption.
Furthermore when looking at the log returns of the S&P 500 index options on a QQ-plot of the
S&P500 daily log returns compared to the normal distribution, it is clear to see that the distributions
differ (figure 2.3.1.2). If
the S&P500 log returns Figure 2.3.1.2
did actually depict a lognormal distribution, the
blue line in the figure
would
be
straight.
However, this is not the
case, and it can be seen
that the tails of the
observed
distribution
differs from the normal
distribution.
This
suggests that a stochastic
variance
should
be
considered in order to
explain the development
of a stock option price
over a period of time.
Source: Own contribution
Before the so-called Black Monday in 1987, the options market did actually depict a somewhat
constant volatility, which is also the assumption underlying the Black-Scholes Model. However, on
the day of Black Monday the market experienced an enormous volatility and extreme losses, and
therefore the market began to recognize that the existing models were not able to forecast and take
into account extreme events and a true stochastic volatility7.
7
This was also known as Crashophia (Hull, 2008, page 395)
18
Figure 2.3.1.3
Figure
2.3.1.3
S&P500
daily
shows
log
the
returns
plotted over a period of almost
30 years (from January 2nd 1980
to 31st December 2010), and
here the extreme rise in the
volatility in 1987 is clear. From
the graph it is also evident that
small movements are followed
by small movements and large
movements by large. This is
evidence of volatility clustering,
which implies that the volatility
is
also
auto-correlated
(Gatheral, 2006, page 1-3). The
autocorrelation
Source: Own contribution
occurs
as
a
result of the mean-reversion of
volatility.
In the Black-Scholes framework the implied volatility is assumed constant, and if this were to hold
in the real world, the implied volatility of options on a given spot price would be constant across
different maturities and strike prices. Empirically this has been shown to be violated in the real
world (Lewis, 2005, page 1-2). When plotting the implied volatilities from the S&P500 index
options across moneyness (the relationship between the strike price and the spot price) this is
clearly evident. If the implied volatility were in fact constant, the figures would depict straight lines.
Instead the figures show a smile or a smirk – this is the so-called volatility smile. From the
empirical studies of the volatility smile, it has been found that the implied volatility is often higher
for an ITM (in-the-money) stock option and OTM (out-of-the-money) option than for an ATM (atthe-money) stock option (http://www.ivolatility.com/help/14.html, visited on 14-08-2011). This is
partly due to the non-normality of the returns which contributes in creating the volatility smiles. The
fat tails in the true distribution of the returns increase the probability of extreme events. This could
for instance be a drop in the returns, and therefore the distributor of an ITM or OTM option must be
19
compensated for the increased potential risk for movements in the underlying stock. This
compensation is visible, as the higher implied volatility is, ceteris paribus, the higher the price of
the stock option. The most typical volatility smile is depicted where the implied volatility is a
decreasing function of the strike price – meaning that a higher strike price should yield a lower
volatility. In the figures plotted below, the implied volatility is a function of the moneyness, as
mentioned, and the implied volatility is in all cases a decreasing function of the moneyness and
thereby of the strike price.
Figure 2.3.1.4 – Plot of implied volatilities over log moneyness
Source: Own contribution
The volatility smile or smirk in equity options can also be explained by the leverage of the
underlying company (Hull, 2008, page 395). If the equity in the company declines, the leverage will
20
increase. This makes the equity more risky, as the debt part of the company increases relative to the
equity part, which increases the risk of the company defaulting (the risk taking by the stock holders
increase). This rationale can be transferred to the implied volatility, as the expected risk in the
market will decrease as the price of the stock option increases. This calls for a negatively
correlation between the return of the stock option and the volatility of the underlying stock – which
is also called the leverage effect.
It has now been proven that constant volatility models cannot be used to describe the actual market
for stock options, and therefore actors in the market have a demand for new models that incorporate
a stochastic volatility.
Chapter 3: Stochastic Volatility Models
In the following paragraph the theory of stochastic volatility models will be addressed. The section
will begin with a short presentation of the various volatility models that exist on the market today.
Here the different models’ characteristics will be shortly presented and their real world applications
will be discussed. Furthermore the general theory regarding stochastic differential equations will
shortly be presented, as this theory will then directly be transferred to the chosen model. After this,
the focus of the remaining section will be on the selected stochastic volatility model developed by
Heston (1993). The theory behind the model will first be introduced and the general theory behind
characteristic functions will be explained. This will lead to the presentation of the Heston model’s
characteristic functions and the Fourier transformation that can be used to price options.
3.1
General Stochastic Volatility Models
As the insufficiency of the Black-Scholes Model became clear in the market a development to
address the need for a better model began which lead to the introduction of stochastic volatility
models. In the process of trying to find a model that more closely adjusted itself to the complexity
of the real world, several models have been developed throughout the years.
Option pricing models in general have some features in common, as they all have to make
assumptions concerning the underlying process, the interest rate process and the market price of
21
factor risk (Bakshi et al 1997, page 2003-2004). The choices of assumptions are endless, and vary
across the different models that exist on the market, where some are more realistic than others.
However, when determining what model is most suitable for the task at hand, it is important to
consider the model’s complexity and implemental costs in respect to its accuracy – and whether a
more complex model will add significant information to the analysis. The more complex the model
is, the more time and resources are needed to make the model “run”. Simply changing a model from
constant volatility to a model that incorporates stochastic volatility can approximately double the
time needed to run a simulation of the option price. It is therefore an important trade-off that must
be considered when the choice of option pricing model must be made.
Option pricing models range from constant volatility models, stochastic volatility models and
interest rate models etc. where jumps can also be incorporated. As a direct comparison to the BlackScholes Model there exist two alternative models, the jump diffusion option pricing models and the
stochastic volatility models8. To provide an overview of the most influential models on the market,
the following list below has been composed9. The list is organised after “year of publication” and
the models can then be directly compared across the asset process and the volatility process.
Table 1: Option pricing models
Author(s)
Black & Scholes (1973)
Hull & White (1987)
Stein & Stein (1991)
Heston (1993)
Merton (1976)
Description
The Black-Scholes Model
Stochastic volatility
model
Stochastic volatility
model
Stochastic volatility
model
Jump diffusion model
Process (dS)
dS = µSdt + σSdZ
dS = µSdt + σSdZ
dS = µSdt + σSdZ
Process (dσ) + correlation (ρ)
None – constant variance
dσ = σ(αdt +γdZ)
dσ2 = aσ2dt + bσ2dZ,
ρ=0
dσ = β(α-σ)dt + γdZ
dσ = -κ(σ-θ)dt + bσdZ ρ = 0
dS = µSdt + σSdZ
dσ2 = κ(θ-σ)dt + bσdZ,
ρ≠0
dS = (α-λκ)dt +
σSdZ + Sdp
None – constant variance, ρ =
0
Source: Own contribution
Remark: The model parameters have been changed so that the models are more easily compared to each other and to
the Black-Scholes Model. For further information see the original sources.
After the market crash in 1987 the market realized that there was a need for models that took into
account the stochastic volatility that were present in the market – as a result of this Hull & White
(1987) introduced their model for stochastic volatility. As one of the earliest option pricing models
8
Interest rate models will not be further discussed, as they are not relevant for the thesis
It should be noted that the list is far from exhaustive, as interest rate models as well as stochastic volatility models
with jump diffusions among others are not included.
9
22
that incorporated a stochastic volatility, this model is also one of the simplest. The model is based
on similar assumptions as the Black-Scholes Model, and is also based on a Geometric Brownian
Motion with a correlation of zero between the variance and asset processes. However, the Hull &
White model has the disadvantage that the assumed volatility is not mean-reverting, which
empirical evidence has shown to be the case in the market. Stein & Steins stochastic volatility
model from 1991 incorporated exactly this, by assuming a so-called Ornstein-Uhlenbeck process
(Schoebel et al, 1998, page 1). Even though this model is more precise in the some way than the
earlier models, the main disadvantage of the model is that the variance can be negative (negative
volatility). This is rejected as true by empirical studies and pure logic. Another model, known as the
Heston model was introduced in 1993. This model assumed a correlation between the underlying
asset process and the variance process, and the volatility was furthermore assumed to follow a
square root process, similar to a CIR-process, see more in section 3.2. Generally it can be said that
one of the key disadvantages of all the stochastic volatility models is that they are unable to predict
extreme events in the evolvement of the stock option price (Gatheral, 2006, page 50-52 and Bakshi
et al, 1997). To correct for this, one can incorporate a jump in the model, which was first introduced
by Merton in 1976 (with his jump diffusion model).
Throughout the years there have been discussions on the performance of the different models
relative to each other. Bakshi et al performed in 1997 an empirical study of some of the most widely
used option pricing models in order to test the models ability to price options. The chosen models
included a stochastic volatility model (SV), a jump diffusion model and a stochastic volatility
random-jump model (SVJ). The models were then tested under three yardsticks10, in order to judge
the accuracy and the empirical performance of the given model in comparison to the Black-Scholes
Model. The study concluded that taking stochastic volatility into account was the “first-order”
importance when improving the Black-Scholes Model (Bakshi et al, 1997, page 2042). Additional,
it was shown that adding a random jump to the stochastic volatility model did actually improve the
performance of the model, and its ability to price short-term options. However, for hedging
purposes it showed that the SV model performed better.
For the purpose of this thesis the Heston model has been chosen to fit the volatility surface of the
S&P 500 index options. It has been concluded that the accuracy of the Heston model compared to
10
1) Internal consistency of implied parameters/volatility with relevant time-series data, 2) out-of-sample pricing and 3)
hedging
23
the Black-Scholes model is significant and as the purpose of the thesis is also to illustrate the
Fourier transformation method for pricing options, the choice of including a jump in the Heston
model has been considered to be irrelevant.
3.1.1 Deriving a general PDE for stochastic volatility models
In the following a pricing equation will be constructed under the assumption of stochastic volatility.
The theory behind this will later be transferred to the Heston stochastic volatility model. The
approach used in the construction will be based on Gatheral (2006) who uses a methodology closely
related to the one used in the derivation of the Black-Scholes Model.
The general stochastic differential equation of a stock option is defined as (Gatheral, 2006, page 4):
𝑑𝑆𝑡 = 𝜇𝑡 𝑆𝑡 𝑑𝑡 + √𝑣𝑡 𝑆𝑡 𝑑𝑍1
Equation 3.1.1.1
where the variance, vt, must satisfy the following:
𝑑𝑣𝑡 = 𝑎(𝑆𝑡 , 𝑣𝑡 , 𝑡)𝑑𝑡 + 𝑏(𝑆𝑡 , 𝑣𝑡 , 𝑡)𝑑𝑍2 ,
Equation 3.1.1.2
where 𝑏(𝑆𝑡 , 𝑣𝑡 , 𝑡) = 𝜂𝛽 √𝑣𝑡 𝑆𝑡 (𝑆𝑡 , 𝑣𝑡 , 𝑡) for simplicity. The term μ in equation 3.1.1.1 is the drift of
the stock price return and the term η is the volatility of the volatility (or the volatility of the
variance). Many stochastic volatility models also share another common feature, and that is the
mean reversion of the volatility. Mean reversion more specifically refers to the effect that the
variable has to revert back to a long-run average – the mean reversion of the volatility therefore
concerns the drift of the volatility itself or the drift of the underlying stock to which the volatility is
related. The speed of this process is referred to as the mean reversion speed. Empirical studies show
that mean reversion is present in real world prices, as shown in section 2.3.1, and therefore this
feature is vital for the stochastic volatility models to display, in order to capture the true evolvement
of the stock prices over time. In order to incorporate a mean reverting volatility into the above
general stochastic volatility formula, the drift must be defined more specifically as: 𝑎(𝑆𝑡 , 𝑣𝑡 , 𝑡) =
𝜅(𝜃 − 𝜎 2 ). This means that the drift must be depending on the so-called mean reversion speed, κ,
which is determined by the average level of the stock variable here called θ. This makes the drift, α,
pull the level of the stock variable towards the average, θ, on the long run. Conversely, this also
24
means that the volatility of the underlying stock will be pulled towards the average of the function
over time. In equation 3.1.1.1 and 3.1.1.2 there are two Wiener processes, Z1 and Z2, that must be
correlated with each other as in the real world – in other words the stock price return and the
changes in the variance must be correlated which each other, as written in the following:
(𝑑𝑍1 𝑑𝑍2 ) = 𝜌𝑑𝑡
Equation 3.1.1.3
It is here important to emphasize that equation 3.1.1.2 is a general differential equation for pricing
stock options. The equation itself is very similar to the one assumed in the derivation of the BlackScholes Model, and in the limit where η  0 equation 3.1.1.2 actually transforms into the BlackScholes formula. This is a clear advantage to stochastic volatility models, as practitioners are
usually familiar with the theory and intuition underlying the Black-Scholes Model. Hence, the
intuition of the stochastic volatility model is closely related to the one of the Black-Scholes formula.
It should also be noticed, that there is no assumption made concerning the terms a and b in the
general stochastic volatility model – this means that the functional form of the variance in the model
is unspecified, and therefore the stochastic volatility models can differ from each other here
(Gatheral, 2006, page 4).
The only randomness that is present in the Black-Scholes Model is the tradable assets, as the
variance is assumed to be constant, as mentioned earlier. This contributes to the market being
complete, because the underlying stock can be traded continuously, which hedges the option.
However, this is not the case with stochastic volatility models, where a common feature (of several
of the stochastic volatility models) is that they are determined by two risk factors, the market risk
and the volatility risk (Boswijk, 2001, page 1). This means that the randomness/risk of the model
can be traced both to the tradable assets as well as the volatility of the asset’s return (Moodley,
2005, page 7-8). This introduces a new risk measure, however because the volatility cannot be
traded in the real world, and thereby hedged, the market is incomplete and this complicates the
pricing process.
Much like in the derivation of the Black-Scholes Model the random variables must be hedged in
order to create a riskless portfolio. This is done by constructing a portfolio П, which consists of the
option that are being priced, a -∆ quantity of the stock and a -∆1 quantity of another asset (where the
value depends on the volatility). This gives the following portfolio:
25
Π = 𝑓 − Δ𝑆 − Δ1 𝑓1
The infinitesimal change of the portfolio can then be defined as:
𝑑Π = {
− Δ1 {
𝜕𝑓 1 2 𝜕 2 𝑓
𝜕 2𝑓
1 2 𝜕 2𝑓
+ 𝑣𝑆
+
𝑣𝑆𝑏𝜌
+
𝑏
} 𝑑𝑡
𝜕𝑡 2
𝜕𝑆 2
𝜕𝑣𝜕𝑆 2 𝜕𝑣 2
𝜕𝑓1 1 2 𝜕 2 𝑓1
𝜕 2 𝑓1 1 2 𝜕 2 𝑓1
+ 𝑣𝑆
+
𝑣𝑆𝑏𝜌
+ 𝑏
} 𝑑𝑡
𝜕𝑡 2
𝜕𝑆 2
𝜕𝑣𝜕𝑆 2 𝜕𝑣 2
+{
𝜕𝑓
𝜕𝑓1
𝜕𝑓
𝜕𝑓1
− Δ1
− Δ} 𝑑𝑆 + { − Δ1
} 𝑑𝑣
𝜕𝑆
𝜕𝑆
𝜕𝑣
𝜕𝑣
In order to make the portfolio riskless, the option must be hedged. This is done by first letting the
dv-term be eliminated by setting:
𝜕𝑓
𝜕𝑓
𝜕𝑓1
− Δ1
= 0 → Δ1 = 𝜕𝑣
𝜕𝑓1
𝜕𝑣
𝜕𝑣
𝜕𝑣
and then eliminating the dS-term by setting:
𝜕𝑓
𝜕𝑓
𝜕𝑓1
𝜕𝑓
𝜕𝑓1
𝜕𝑓 𝜕𝑣 𝜕𝑓1
− Δ1
−Δ=0→Δ=
− Δ1
→Δ =
−
𝜕𝑆
𝜕𝑆
𝜕𝑆
𝜕𝑆
𝜕𝑆 𝜕𝑓1 𝜕𝑆
𝜕𝑣
What is left is now the riskless portfolio which is defined as:
𝜕𝑓 1 2 𝜕 2 𝑓
𝜕 2𝑓
1 2 𝜕 2𝑓
𝑑Π = { + 𝑣𝑆
+ 𝑣𝑆𝑏𝜌
+ 𝑏
} 𝑑𝑡
𝜕𝑡 2
𝜕𝑆 2
𝜕𝑆𝜕𝑣 2 𝜕𝑣 2
− Δ1 {
𝜕𝑓1 1 2 𝜕 2 𝑓1
𝜕 2 𝑓1 1 2 𝜕 2 𝑓1
+ 𝑣𝑆
+
𝑣𝑆𝑏𝜌
+ 𝑏
} 𝑑𝑡
𝜕𝑡 2
𝜕𝑆 2
𝜕𝑣𝜕𝑆 2 𝜕𝑣 2
Following the arguments from section 2.2, the return of the riskless portfolio value must now equal
the risk free interest rate. This is shown as below:
𝑑Π = 𝑟Π𝑑𝑡 = 𝑟(𝑓 − Δ𝑆 − Δ1 𝑓1 )𝑑𝑡
26
Inserting the equations in the above formula and rearranging, gives the following (see Gatheral,
2006, page 4-7 for a more detailed derivation):
𝜕𝑓 1 2 𝜕 2 𝑓
𝜕𝑓
𝜕 2𝑓
1 2 𝜕 2𝑓
𝜕𝑓
(𝑎
+ 𝑣𝑆
+
𝑟𝑆
−
𝑟𝑓
+
𝑣𝑆𝑏𝜌
+
𝑏
+
−
𝜆𝑏)
=0
𝜕𝑡 2
𝜕𝑆 2
𝜕𝑆
𝜕𝑆𝜕𝑣 2 𝜕𝑣 2
𝜕𝑣
Black-Scholes
Correlation
Volatility Volatility premium
, which is the general solution for stochastic volatility models. The solution has been shortened by
rewriting the arbitrary function f(St, vt, t) as the term (𝑎 − 𝜆𝑏), where the terms a and b refers to the
drift and volatility from the variance process that was assumed in equation 3.1.1.2. For clarity the
Black-Scholes solution has been highlighted in the equation, as well as the correlation, volatility
and volatility premium part. However, as a risk neutral world is assumed, the volatility premium is
non-existing and therefore equal to zero.
3.2
The Heston Stochastic Volatility Model
The Heston Model was introduced in 1993 by Steven L. Heston and is today one of the most widely
used stochastic volatility models on the market. The model was the one of the first models to depict
an alternative (semi)closed-form solution to the Black-Scholes Model for the pricing of a European
call option.
In the following paragraph the process of the Heston model will be introduced and the partial
differential equation for the Heston model will be outlined. Later the basic for the calculation of the
option prices will be set by the introduction of the characteristic functions and the Fourier
Transformation, which will then be implemented on the Heston model.
3.2.1 The Process of the Heston model
The Heston model was the first known model of its kind to depict a (semi)closed solution for option
pricing after the Black-Scholes Model. The model also differs from other stochastic volatility
models, as the development of the underlying asset is assumed to be correlated to the volatility
process. This is in compliance with the empirical studies shown in section 2.3.1. This along with the
27
easiness that is related to the implementation of the model has contributed greatly to the
attractiveness of the model throughout the years.
The Heston model is compiled of two partial differential equations that will be discussed in the
following. The Heston model is suitable for pricing stock options, as it also builds on the
generalized Wiener Process as introduced earlier in this thesis in section 2.1.
In the Heston model the stock price is assumed to develop in accordance to the following diffusion:
𝑑𝑆𝑡 = 𝜇𝑡 𝑆𝑡 𝑑𝑡 + √𝑣𝑡 𝑆𝑡 𝑑𝑍1
Equation 3.2.1.1
The stock price is dependent on a drift and a variance, which is also similar to the Black-Scholes
Model with the exception that the volatility now depends on the time (and it no longer a constant).
It is here noticeable that the instantaneous variance, vt, is based on a square root process similar to
the so-called CIR-process, first developed by Cox, Ingersoll and Ross in 1985. They introduced a
model that described the evolution of interest rates, and the instantaneous interest rate was said to
follow a CIR-process, which was a square root process (for more information see the direct source).
Following from this, the variance in the Heston model must therefore satisfy the following
stochastic differential equation (Heston, 1993, page 328):
𝑑𝑣𝑡 = 𝜅(𝜃 − 𝑣𝑡 )𝑑𝑡 + 𝜂√𝑣𝑡 𝑑𝑍2
Equation 3.2.1.2
The terms κ, θ and vt above describe the mean-reverting volatility of the process, as with the general
stochastic volatility model mentioned in the previous section. The mean speed of reversion, 𝜅,
determines the relative speed of the volatility or the weight that the long-run variance and current
variance are given. The average level of the stock, 𝜃, is the long-run variance that the drift pulls the
volatility towards. The vt term is the current variance, while 𝜂 is the volatility of the volatility (the
last will be elaborated on later).
The variance will furthermore always remain positive as long as:
2𝜃𝜅 − 𝜂2 > 0
Equation 3.2.1.3
In accordance to the general stochastic volatility model presented in section 3.1.1, the terms Z1 and
Z2 are Wiener processes that must be correlated with each other. This is shown in the following:
28
(𝑑𝑍1 𝑑𝑍2 ) = 𝜌𝑑𝑡
Equation 3.2.1.4
In the above equation the term ρ is the correlation coefficient between the return of the underlying
stock and the changes in the variance. This relationship explains the before mentioned phenomenon
of the leverage effect. This correlation has proven to be a great advantage to the Heston model as
this is also present in empirical studies that have been performed over the years. The correlation,
which is often negative, will ensure that the volatility for example will rise if the value of the
underlying asset falls dramatically. In addition the variance is also mean-reverting, which is also
evident in the market. The mean-reverting process is the term 𝜅(𝜃 − 𝑣).
In the correlation shown above in equation 3.2.1.4 the terms Z1 and W are both are Wiener
processes that follow a standard normal distribution with mean zero and variance one (N(0,1)). The
process of Z2 can be described as a function of the process Z1 and an independent Brownian Motion
W, as written below:
𝑍2 = 𝜌𝑍1 + √1 − 𝜌2 𝑊
Equation 3.2.1.5
This implies that the processes Z1 and Z2 will be fully dependent on each other, whenever the
correlation coefficient is either 1 or -1. However, if the correlation coefficient is instead equal to
zero, the process of Z2 will instead be dependent on the process of W – making Z1 and Z2
independent of each other.
One of the advantages of the Heston model is that it can be applied to a various number of
distributions – while the Black-Scholes model assumes a normal distribution, where the only
adjustable parameters are the mean and variance. The Heston model is able to “handle” different
distributions because the correlation factor, ρ, has an effect on the heaviness of the tails of a
distribution and the volatility of volatility factor, η, has an effect on the kurtosis on the distribution.
This means that the parameters of the Heston model can be adjusted in order to better fit the market
data in question. However, this also means that the correlation coefficient have a direct effect on the
volatility smile. A positive correlation (where 𝜌 > 0) will make the distribution of the returns
positively skewed, resulting in a thin left tail and a fat right tail. This is due to the fact that the left
tail of the probability density function is associated with low variance and therefore will not be
spread, whereas the right tail will (Heston, 1993, page 336-338). This is also coherent with the
29
rationale behind the theory – a positive correlation between the underlying stock and the variance
means that an increase in the asset price will, ceteris paribus, result in a higher variance which will
affect the asset price in a positive way. In practice most indexes though have a negative correlation,
and the smile is skewed. Therefore the opposite occurs.
The Heston model furthermore differs from other stochastic volatility models in the simplicity of its
implementation. The solution to the Heston model is based on characteristic functions, which
enables the implementation to be faster than with other stochastic volatility models, which for
example may use Monte Carlo simulation. The advantage of the characteristics functions lies in the
technique that only requires a numeric solution of integrals, which makes the Heston solution a
(semi)closed form solution (the semi part due to the complexity of the numeric solution). The
theory underlying characteristic functions will be explained more deeply in section 3.2.3.
A disadvantage of the Heston model is that it poorly prices options with a short maturity. The
underlying reason for this is that the Heston stochastic volatility model is based on a Geometric
Brownian Motion, and therefore the model cannot predict extreme events on the option market.
Furthermore, the stochastic volatility that is to be estimated is not observable in the market. This
makes the estimation process difficult, and the model is in addition very sensitive to its parameters.
This makes the calibration of the model parameters vital for the model to be able to price the
options accurate. This also means that the more realistic the model is required to be, the more
complex the calibration of the parameters should be (Mikhailov et al, 2008).
3.2.2 PDE for the Heston Model
The general stochastic volatility formula presented in section 3.1.1 is fairly easy to transfer to the
Heston model. The value of any asset must according to the Heston model satisfy the following
partial differential equation (PDE) where the term 𝑎(𝑆𝑡 , 𝑣𝑡 , 𝑡) = 𝜅(𝜃 − 𝑣𝑡 ) (the mean-reverting
process) and 𝑏(𝑆𝑡 , 𝑣𝑡 , 𝑡) = 𝜂√𝑣𝑡 . When inserting the values of a and b from the Heston model into
the solution for general stochastic volatility models, see equation 3.1.1.2, the following solution the
Heston model is presented:
𝜕𝑓
1
𝜕2 𝑓
𝜕𝑓
𝜕2 𝑓
1
𝜕2 𝑓
𝜕𝑓
+ 2 𝑣𝑆 2 𝜕𝑆2 + 𝑟 𝜕𝑆 𝑆 − 𝑟𝑓 + 𝑣𝑆𝜂𝜌 𝜕𝑆𝜕𝑣 + 2 𝑣𝜂2 𝜕2 𝑣 + (𝜅(𝜃 − 𝑣) − 𝜆𝜂𝜌) 𝜕𝑣 = 0
𝜕𝑡
Eq. 3.2.2.1
30
However, as mentioned earlier, it is assumed that the option valuation is in the risk neutral world.
The process of the underlying stock under risk neutral assumptions is defined as:
𝑑𝑆𝑡 = 𝑟 ∙ 𝑆𝑡 𝑑𝑡 + √𝑣𝑡 𝑆𝑡 𝑑𝑍1
Equation 3.2.2.2
where the drift has been replaced by the risk free interest rate, r (and r-q when taking dividends into
account). Here it is important to notice the term λ(S, v, t), which is the price of volatility risk. As the
pricing is assumed to be risk neutral, the volatility risk term λ must equal zero. This eliminates the
term of ληρ in equation 3.2.2.1.
3.2.3 Characteristic Functions and the Fourier Transformation
In the world of option pricing there exists numerous ways of estimating the fair value/price of a
stock option. These methods include finite difference method, simulation etc. As an alternative
method of pricing options, the mapping of the characteristic function of the density function has
been recognized for its easy and less complex computation. Empirical studies have shown that the
distribution of the stock option doesn’t follow a Gaussian distribution in the real world. However,
the use of characteristic functions and Fourier transformation doesn’t require the distribution of the
underlying to be known, and therefore there has been a growing interest for the use of this method.
In addition the method also offers a fast computational advantage when compared to many of the
other methods available.
The main idea underlying this method is to take the integral of the option payoff function over the
probability function. The probability function is then retrieved by inverting the Fourier transform.
The characteristic function defines the probability distribution of a random variable, and every
function has its own characteristic function from which the density function can be computed
(Schmelzle, 2010, page 2).
The general definition of a characteristic function with respect to u is given by (Schmelzle, 2010,
page 8):
+∞
𝜙𝑇 (𝑢) ∶= 𝐸[𝑒 𝑖𝑢𝑥𝑇 |𝑥𝑡 = 0] = ∫−∞ 𝑒 𝑖𝑢𝑥𝑇 𝑓𝑇 (𝑥)𝑑𝑥
Equation 3.2.3.1
31
𝑆
where 𝑥𝑇 = 𝑙𝑜𝑔 𝑆𝑇 . This is also known as a Fourier transform, which is basically used to transform
0
a complex function to another function of the same variable. The characteristic function is defined
for arbitrary real numbers u, where i is an imaginary number 𝑖 = √−1. The f(x) is the probability
density function, and the stochastic process appears for −∞ < 𝑢 < ∞. The relationship between the
probability density functions and the characteristic function is said to be “one to one”. This means
that one can derive the one from the other, and the relationship between the two is illustrated below
(Schmelzle, 2010, page 7-11):
1
+∞
𝑓𝑇 (𝑥) = ℱ −1 [𝜙𝑇 (𝑢)] = 2𝜋 ∫−∞ 𝑒 −𝑖𝑢𝑥𝑇 𝜙𝑇 (𝑢)𝑑𝑢
Equation 3.2.3.2
This relationship is also what makes the use of characteristic functions in option pricing popular, as
it is possible to derive the probability density function from the characteristic function, even though
the density function is not known in closed-form11 (Cherubi et al, 2010, page 32). This is possible,
as the probability density function can be expressed in terms of an integral in which the
characteristic function is a part of. This is done by the inversion of the Fourier transform.
One of the main advantages of the Fourier transformation is that under certain conditions the inner
and scalar products will be the same in the Fourier transforms. This is especially an advantage when
the characteristic function is use to describe a distribution. This is expressed as the Planchard
Theorem or Parseval’s Theorem as written below:
∞
∞
∫−∞ 𝑓(𝑥)𝑔̃ (𝑥)𝑑𝑥 = ∫−∞ 𝑓̃(𝑥)𝑔 (𝑥)𝑑𝑥
Equation 3.2.3.3
The Fourier transform has several properties however these will not be expressed in detail in the
thesis. For more information see the original article by Schmelzle, 2010.
11
The only known density functions in closed-form are the normal distribution, the Cauchy distribution and the inverse
Gaussian distribution (Cherubi et al, 2010, page 34)
32
3.2.4 Implementation of the Fourier transform on the Heston model
The Heston model was the first model to introduce the Fourier transformation methods to option
pricing (Kahl and Lord, 2010, page 1). The Fourier transformation can be applied to a given
characteristic function in order to derive the probability density function of the distribution, as
explained earlier.
The characteristic function of the normal distribution is known as the following (Gatheral, 2006,
page 57):
𝜙𝑇 (𝑢) = 𝐸[𝑒 𝑖𝑢𝑥𝑇 ] = 𝑒 −0.5𝑢(𝑢+𝑖)𝜎
2𝑇
= 𝑒 −0.5𝜎
2 𝑖𝑢𝑇−0.5𝜎 2 𝑢2 𝑇
Equation 3.2.4.1
which is also the characteristic function for the Black-Scholes model, as it assumes normally
distributed log-return of the underlying stock.
In order to predict a future price of a stock, which by definition is uncertain, the process depends on
the probability distributions. The value of a typical European call option can be expressed in terms
of the probability of it being exercised at-the-money. This gives the following definition
(Schmelzle, 2010 page 17 and Kahl and Lord, 2010, page 1):
𝑄
𝑄
𝑄
𝑆
𝑡
𝐶(𝑆0 , 𝐾, 𝑇) = 𝐸𝑡 𝑇 [(𝑆𝑇 − 𝐾)+ ]𝑒 −𝑟𝑇 = 𝐸𝑡 𝑇 [(𝑆𝑇 ∙ 1{𝑆𝑡 >𝐾} )] − 𝐾 ∙ 𝐸𝑡 𝑇 [(1{𝑆𝑡 >𝐾} )] = 𝑃(𝑡,𝑇)
∙
𝑆
𝐸𝑡 𝑇 [(1{𝑆𝑡 >𝐾} )] − 𝐾 ∙ 𝑃[(𝑆𝑇 > 𝐾)]
Equation 3.2.4.2
= 𝐹(𝑡, 𝑇) ∙ 𝑆𝑇 [(𝑆𝑇 > 𝐾)] − 𝐾 ∙ 𝑃[(𝑆𝑇 > 𝐾)]
where the latter equation is similar to the Black-Scholes formula. When pricing a European call
option, the value depends on the probability function of the stock price measure being larger than
the exercise price.
The basic theory of option pricing says that the price of a call option at maturity is the expected
positive value of its payoff discounted back to today, as shown below:
𝐶(𝑆, 𝐾, 𝑇, 𝑟, 𝑞) = 𝐸 𝑄 [(𝑆𝑇 − 𝐾)+ ]𝑒 −𝑟𝑇
Equation 3.2.4.3
Where the process of the underlying stock behaves as the following:
33
𝑆𝑇 = 𝑆0 𝑒 (𝑟−𝑞)𝑇 ∙ 𝑒 𝑥𝑇
Equation 3.2.4.4
Inserting this in equation 3.2.4.3 yields the following, where the constant 𝑆0 𝑒 (𝑟−𝑞)𝑇 has been put
outside the parenthesis:
𝐶(𝑆, 𝐾, 𝑇, 𝑟, 𝑞) = 𝐸 [𝑆0 𝑒
= 𝐸 [(𝑒
𝑥𝑇
−
(𝑟−𝑞)𝜏
(𝑒
𝑥𝑇
−
+
𝐾
𝑆0 𝑒 (𝑟−𝑞)𝜏
+
𝐾
𝑆0 𝑒 (𝑟−𝑞)𝜏
) ] 𝑒 −𝑟𝜏
) ] 𝑆0 𝑒 (𝑟−𝑞)𝑇 𝑒 −𝑟𝜏
From Gatheral (2006, page 58) the price of a call from a characteristic function is given by:
1
+∞ 𝑑𝑢
𝐶𝐺𝐴𝑇𝐻𝐸𝑅𝐴𝐿 (𝑆, 𝐾, 𝑇) = 𝑆 − √𝑆𝐾 𝜋 ∫0
1
𝑢2 +
4
𝑖
𝑅𝑒 [𝑒 −𝑖𝑢𝑘 𝜙𝑇 (𝑢 − 2)] Equation 3.2.4.5
Where r = q = 0. However, this is a simplification to the real world, and the formula needs to be
rewritten in terms of r = q ≠ 0, so that both the risk free interest rate and a dividend yield is
incorporated into the formula. This is done by producing the following call price, where the spot
price is assumed to be zero:
+∞
𝐾
1
𝑑𝑢
𝑖
−𝑖𝑢𝑘
𝐶𝐺𝐴𝑇𝐻𝐸𝑅𝐴𝐿 (1,
,
𝑇)
=
1
−
∙
𝐾
∫
𝑅𝑒
[𝑒
𝜙
(𝑢
−
)]
√1
𝑇
𝜋 𝑢2 + 1
2
𝑆0 𝑒 (𝑟−𝑞)𝑇
0
4
Then accounting for dividends in the real spot price is done by the following equation of the call
price:
𝐶(𝑆0 , 𝐾, 𝑇, 𝑟, 𝑞) = 𝑆0 𝑒 −𝑞𝑇 ∙ 𝐶𝐺𝐴𝑇𝐻𝐸𝑅𝐴𝐿 (1,
𝐾
𝑆0 𝑒 (𝑟−𝑞)𝑇
, 𝑇)
For simplicity the term k is defined as:
𝑘 = 𝑙𝑜𝑔 (
𝐾
𝑆0
𝑒 (𝑟−𝑞)𝑇
)
This yields the following function for the price of the call option:
34
+∞
1
𝑑𝑢
𝑖
𝐶(𝑆0 , 𝑘, 𝑇, 𝑟, 𝑞) = 𝑆0 𝑒 −𝑞𝑇 ∙ 1 − √𝐾 ∫
𝑅𝑒 [𝑒 −𝑖𝑢𝑘 𝜙𝑇 (𝑢 − )]
𝜋 𝑢2 + 1
2
0
4
To find the price of the call option, the characteristic function must now be inserted into the above
formula, and solved. The characteristic function of the Heston model is defined as:
𝜙𝑇 (𝑢) = 𝑒 𝐶(𝑢,𝜏)𝜃+𝐷(𝑢,𝜏)𝑣𝑡
Equation 3.2.4.6
With vt = v0 as the initial variance and the following defined as:
2
𝐶(𝑢, 𝜏) = 𝜆 {𝑟 − 𝑇 − 𝜂2 𝑙𝑜𝑔 (
1−𝑔𝑒 −𝑑𝜏
1−𝑔
−𝑑𝜏
)}
Equation 3.2.4.7
1−𝑒
𝐷(𝑢, 𝜏) = 𝑟 − (
)
1 − 𝑔𝑒 −𝑑𝜏
𝛽 ± √𝛽 2 − 4𝛼𝛾 𝛽 ± 𝑑
𝑟±=
=: 2
2𝛾
𝜂
𝑑 = √𝛽 2 − 4𝛼𝛾
𝛼=−
𝑢2
2
−
𝑖𝑢
2
𝛽 = 𝜆 − 𝜌𝜂𝑗 − 𝜌𝜂𝑖𝑢 𝛾 =
𝑟−
𝑔= +
𝑟
+ 𝑖𝑗𝑢, j= 0,1
𝜂2
2
It is here important to emphasize that the initial conditions of C and D must be equal to zero, so that
the initial parameters are not evaluated at infinity – this will cause the minimization function to
choose the initial values as estimates of the parameters.
In order to test the functionality of the Fourier Transformation applied to the Heston model, it is
first applied to the standard Black-Scholes model. This enables the Fourier Transformation method
to be verified, as the Black-Scholes solution to the Fourier Transformation can be verified by the
direct solution of the Black-Scholes pricing equation.
The characteristic functions differ with the process at hand, and the characteristic function for the
Black-Scholes model is defined as:
1
𝜙𝑇 (𝑢) = 𝐸[𝑒 𝑖𝑢𝑥𝑇 ] = 𝑒𝑥𝑝 {− 2 𝑢(𝑢 + 𝑖)𝜎 2 ∙ 𝑇}
Equation 3.2.4.8
35
Chapter 4: Data
The data used in the analysis of the Heston model consists of index option data that has been
extracted from Wharton Research Data Service (WRDS). The option market data more specifically
consists of the data from the S&P 500 index (Standards and Poors), which is a gathering of 500
large-cap common stocks in a value weighted index. The index is traded on the NASDAQ and the
New York Stock Exchange (NYSE), and is by some considered to be an ideal proxy for the total
market in the US as the index covers about 75% of the U.S. Equities12. All data used in this thesis of
the
S&P
500
index
has
been
downloaded
from
the
website
http://wrds-
web.wharton.upenn.edu/wrds/process/wrds.cfm after opening a free account. The data more specific
consists of daily closing prices of the S&P 500 index with matching dividend yield and highest
closing bid- and lowest closing ask-prices. The selected time period to analyse the Heston model
across is from January to March 2010. However, the actual plotting of the volatility surface will not
be done on the entire period, but the attached code is easily modified for this. The chosen data set
consists of one monthly observation day, which has been chosen to be the 3rd Thursday of each
month. This underlying reason for this is that the S&P500 index options expires at the third Friday
of the month
(http://www.randomwalktrading.com/main/index.php?option=com_content&view=article&id=216
&Itemid=338 visited at 08/31/2011).
As an input in the analysis, it is also required to find a measurement for the risk free interest rate, as
it is to be used as the discounting rate as well as the drift in the process of the underlying stock
(because of the assumption made of the risk neutral world). In this thesis the risk free rate used in
the analysis is assumed to be constant – which means that the assumed interest rate structure is
assumed to be flat. This is a simplification of reality, as empirical studies have shown that the risk
free term structure is far from flat – in reality the interest rate yield curve is usually upward sloping.
Often treasury bills and treasury bonds have been used as the risk free rate. The reason for this is
that it is assumed that the chance of a government defaulting on an obligation, which is
12
Source: http://www.standardandpoors.com/servlet/BlobServer?blobheadername3=MDTType&blobcol=urldata&blobtable=MungoBlobs&blobheadervalue2=inline%3B+filename%3DFactsheet_SP_500.pdf&
blobheadername2=ContentDisposition&blobheadervalue1=application%2Fpdf&blobkey=id&blobheadername1=contenttype&blobwhere=1243931099288&blobheadervalue3=UTF-8 visited on 08/15/2011
36
denominated in its own currency it relatively low (close to none-existing) (Hull, 2008, page 74).
Therefore, this is considered to be near the closest an investor can come to making a risk free
investment. However, the treasury rates are considered to be artificially low by practitioners as a
result of tax and regulatory issues. Instead, derivatives traders often use LIBOR rates as proxy for
the risk free interest rate. The LIBOR rate13 is the interest rate at which large banks in London are
willing to make a large wholesale deposit with other banks (Hull, 2008, page 74) – this is
essentially the same as borrowing money to the bank in question. As a result of this, it is required
that the financial institution satisfies certain creditworthiness, and it is required typically that it at
least have an AA-rating. As such, one can say that the LIBOR rate is not totally risk free, as there
will always be a chance, however very little, for the bank to default on the loan.
In this thesis it has be chosen to find an estimate of the risk free interest rate using the BlackScholes Model. As mentioned earlier, the only unknown parameter in the Black-Scholes formula is
the volatility, which cannot be directly observed in the real market. However, when inserting the
implied volatility into the Black-Scholes Model, the market price of the stock option can be found.
As a result of this, the market price can be defined as (Wilmott, 2005, page 184):
𝐶(𝑆0 , 𝐾, 𝑇) = 𝐶𝐵𝑆 (𝑆0 , 𝐾, 𝜎𝑖𝑚𝑝𝑙 (𝑆0 , 𝐾, 𝑇), 𝑇)
Equation 4.1
As the implied volatility is a known parameter through the data collection in this thesis, the risk free
interest rate can be found by “backing out” from the Black-Scholes formula. The dividend yield can
also be found through the same way as the risk free interest rate. However, for simplicity it is
assumed that the dividend yield is constant, and therefore it will serve as an input in the calibration
of the risk free interest rate. The dividend yield extracted from WRDS is the daily dividend yield,
and the dividend yield used in the analysis is calculated as an average dividend yield of all S&P 500
stock options. The risk free interest rate is then found by minimizing the difference of the BlackScholes Model prices with the known market prices, CM, with respect to r, as seen below:
min = min 𝐶𝐵𝑆 (𝑆0 , 𝐾, 𝑇, 𝜎𝑖𝑚𝑝𝑙 , 𝑟, 𝑞) − 𝐶𝑀 (𝐾, 𝑇)
𝑟
13
𝑟
Equation 4.2
The LIBOR rate is only quoted for maturities up to 12 months.
37
As an estimate of the market prices in this minimization process, the average bid-ask price is used,
which have been used in many empirical studies throughout the years. The average bid-ask quote
has been calculated from the following:
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑝𝑟𝑖𝑐𝑒𝑖 =
𝑏𝑖𝑑𝑖 +𝑎𝑠𝑘𝑖
2
Equation 4.3
The initial value14 of the risk free interest rate is set equal to an annualized rate of an American
Treasury bill15. This is done in the hope that the estimation of the risk free rate will be closer to the
true risk free rate than otherwise. However, as it turns out from working with it, changing the initial
estimate of the risk free interest rate does not change the estimation process and the final estimated
value of the interest rate – this means that the minimization process performed is stable.
This approach gives a proxy for the risk free interest rate, however, it is important to emphasise that
the estimate is based on a model (the Black-Scholes Model) which fails to describe the market data,
as some of its assumptions are too constrained. Furthermore, the use of an average dividend yield
can also have an effect on the estimated interest rate, however smaller than the previous one. In
spite of the acknowledgement of the disadvantages of the method applied to find an estimate of the
risk free interest rate, it is assumed that the interest rate backed out from market prices and the
Black-Scholes formula it a “good enough” proxy for the real risk free interest rate. This should of
course also be seen in relation to the fact that interest rate processes is not the main focus in the
thesis. It is therefore assumed that the calibrated risk free interest rate is reliable and can be used in
the further analysis of implied volatility surface.
Chapter 5: Fitting the implied volatility surface
5.1
Model Calibration
In order to fit the Heston model to the implied volatility surface of the S&P500 index options, it is
first necessary to calibrate the parameters of the model. There exist several methods for this,
14
Initial risk free interest rate is set to 0.33 (the 1-year Daily Treasury rate from 01/15/2010)
http://www.treasury.gov/resource-center/data-chart-center/interestrates/Pages/TextView.aspx?data=yieldYear&year=2010 visited on 08/29/2011.
15
38
however empirical studies performed by Bakshi, Cao and Chen (1997) have shown that simply
fitting the implied parameters of the Heston model to the ones observed in the market, is not
sufficient enough in order to make the model fit the market. Another approach that empirically has
been used to estimate the parameters of the Heston model is to use the so-called inverse problem,
which is solved by finding the parameters of the model that produce the right market price. The
purpose of the calibration of the parameters is simply to make the Heston model fit as closely as
possible to the market data, and thereby reducing the error margin between the estimated model
price from the Heston model and the market price observed.
The following parameters in the Heston model need to be calibrated or estimated:
𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟𝑠(𝑝) = 𝜅, 𝜃, 𝑣, 𝜂, 𝜌
The calibration itself is performed by minimizing the following non-linear least-squares
optimization problem, as written below (Mikhailov et al, 2008, page 76):
p
M
2
min SSE(p) = min ∑N
i=1[Ci (K i , Ti ) − Ci (K i , Ti )] , where i = 1,2 … N
p
Equation 5.1.1
p
In equation 5.1.1, the terms 𝐶𝑖𝑝 and 𝐶𝑖𝑚 are the model price and the price observed in the market
respectively. The term p refers to the set of calibrated parameters values, and N is the number of
options used for the actual calibration.
In order to ensure that the process is positive, the following equation must hold that:
2𝜆𝑣̅ > 𝜂, where 𝜆 = 𝜅
Furthermore, the following conditions should be set for the initial parameters:
The speed of mean reversion, 𝜅, should be non-negative
0<𝜅
The long-run volatility, 𝜃, should be non-negative
0<𝜃
The initial variance should be non-negative
0 < 𝑣𝑡
The volatility on volatility should be non-negative
0< 𝜂
The correlation coefficient should be in the interval
−1 < 𝜌 < +1
39
This should minimize the time needed to calibrate the model, as the intervals (that needs to be
searched) are smaller with theses constraints on the initial parameters than without.
5.2
Setting the initial parameters
In order to calibrate the Heston parameters to the empirical data from the S&P 500 index options, it
is necessary to determine a set of initial parameters. The initial values can be chosen in several of
ways, and differ in the literature. This thesis will only use one of these methods and this is chosen to
be an historical estimate of the initial values of the Heston model parameters.
As presented earlier in the thesis, the variance process of the Heston model is defined as the
following:
𝑑𝑣𝑡 = 𝜅(𝜃 − 𝑣𝑡 )𝑑𝑡 + 𝜂√𝑣𝑡 𝑑𝑊𝑡
The historical data chosen to form the basis of the estimate of the initial parameter values is daily
data in the period from January 2010 to March 2010. The variance process is then constructed from
historical prices in a month, so that the time step of the variance process is monthly. This is done in
accordance to Hull (2008, page 282-283)
The variance process is discretized in the following:
𝑣𝑡+1 − 𝑣𝑡 = 𝜅(𝜃 − 𝑣𝑡 )Δ𝑡 + 𝜂 √𝑣𝑡 √Δ𝑡𝑁(0,1),
𝑤ℎ𝑒𝑟𝑒 𝑁(0,1) 𝑖𝑠 𝑡ℎ𝑒 𝑒𝑟𝑟𝑜𝑟 𝑡𝑒𝑟𝑚 𝜀𝑡+1
Similar to the method used in short interest rate models (Bond & Interest Rate, Elisa Nicolato,
handout page 19), the variance process can be found by performing a regression on the change in
the variance process, as written below:
𝑣𝑡+1 − 𝑣𝑡 = 𝜅𝜃Δ𝑡 − 𝑘Δ𝑡√𝑣𝑡 + 𝜉𝑡+1
40
Here the estimate of the first term in the regression (𝜅(𝜃 − 𝑣𝑡 )Δ𝑡) becomes 𝑎̂ = 𝜅𝜃Δ𝑡 and the
second term is estimated as 𝑏̂ = −𝜅Δ𝑡16. By finding the coefficients corresponding to these terms it
is possible to isolate 𝜅̂ and 𝜃̂. This gives the initial values of the two terms for later use in the
calibration.
Following the method of the short interest rate estimation the residuals of the above regression are
saved, and the squared residuals (the error term in previous equation) is to be regressed against the
variance, vt. The squared residuals can be defined as:
2
𝜉𝑡+1
= (𝜂√𝑉𝑡 𝑑𝑊𝑡 )2
Running a regression will yield the following result, where the Brownian motion will be captured in
the error term ut+1. Here it is important to notice that the term d is not discretizing the variance, but
is instead simply the coefficient in front of vt.
2
𝜉𝑡+1
= 𝑐 + 𝑑 ∙ 𝑣𝑡 + 𝑢𝑡+1
2
𝜉𝑡+1
= 𝜂2 Δ𝑡 ∙ 𝑣𝑡
𝑑~𝜂2 Δ𝑡
The term c is not relevant for the initial parameters, and is therefore ignored. Performing the
regression yields an estimate for η2, and by taking the square root of this estimate, the initial
parameter value for η is found. It is here important to emphasize that this has yielded a negative 𝜂2 ,
which is a violation of the basic conditions (taking the square root of a negative number is simply
not possible). However, as the parameter is only meant as an initial estimate of the real parameter, it
is assumed that the estimated value is in absolute terms – making it positive.
The correlation parameter, ρ, is the correlation between the price process and the variance process.
This coefficient is found by using basic statistical calculations. The basic statistical formula for the
correlation coefficient between two variables, X and Y, is defined as:
16
Linear regression as y = a + bx
41
𝜌=
𝑐𝑜𝑣(𝑋,𝑌)
√𝑉𝐴𝑅(𝑋)∙√𝑉𝐴𝑅(𝑌)
Equation 5.1.2
It is therefore necessary to find the covariance between the two time series, the spot price and the
variance. From equation 2.2.4 in section 2.2 the risk neutral process of the underlying stock was
defined. The logarithm of the spot price, xt = ln St, can be found by applying Itô’s Lemma on the
underlying process, which yields the following:
1
𝑑𝑥𝑡 = ((𝑟 − 𝑞) − 𝑣𝑡 ) 𝑑𝑡 + √𝑣𝑡 𝑑𝑍1
2
Here the term Z1 is a Brownian motion under risk neutral assumptions. In order to find the change
in the logarithm of the spot prices, this is transferred to the change in spot prices. The change in the
log spot price process and the variance process are written below:
1
𝑙𝑛𝑆𝑡+1 − 𝑙𝑛𝑆𝑡 = ((𝑟 − 𝑞) − 𝑣𝑡 ) Δ𝑡 + √1 − 𝜌2 𝑑𝐵1 + 𝜌𝑑𝐵2
2
𝑣𝑡+1 − 𝑣𝑡 = 𝜅(𝜃 − 𝑣𝑡 )Δ𝑡 + 𝜂 √𝑣𝑡 𝑑𝐵2
For the process of the underlying stock the change in spot prices should be calculated by using one
observation each month in order to match the frequency of the variance process (which is also one
variance a month). However, in order to make the value of the spot price used in the calculation
more valid, an average of the spot prices for a whole month is calculated, to match the variance of
that same month.
Then, finding the covariance between the two time series yields the following:
𝑐𝑜𝑣(𝑋, 𝑌) = 𝑐𝑜𝑣 (√1 − 𝜌2 𝑑𝐵1 + 𝜌𝑑𝐵2 , √𝑣𝑡 𝑑𝐵2 ) = 𝐸[𝜌𝜂 √𝑣𝑡 Δ𝑡 ∙ 𝑁 2 (0,1)]
𝑐𝑜𝑣(𝑋, 𝑌) = 𝜌𝜂 √𝑣𝑡 Δ𝑡
After finding the covariance between the two time series the following equation can be used to find
an initial estimate of the 𝜌 parameter needed in the calibration of the Heston model parameters.
Rearranging the previous equation, yields the following equation for 𝜌:
42
𝜌=
𝑐𝑜𝑣(𝑋, 𝑌)
√𝑉𝐴𝑅(𝑋) ∙ √𝑉𝐴𝑅(𝑌)
=
𝜌𝜂 √𝑣𝑡 Δ𝑡
√Δ𝑡 ∙ 𝜂√𝑣𝑡 √Δ𝑡
The only parameter missing for the initial values is now the initial value for the variance. This is set
equal to the long-run variance, 𝜃, in the Heston model, and will afterwards change in accordance to
its process.
The initial parameters are set to:
Parameter
𝜿
𝜽
𝒗𝒕
𝜼
𝝆
Initial value
0.7958723
0.03581966
0.03581966
0.1168394
-0.6678461
The results of the calibration are presented in the following table. It is here evident that the
calibrated parameters differ with the market data for which is has been fitted. Overall it can be said
that the calibrated parameters are very similar and nothing sticks out.
Parameter
01/15/2010
02/19/2010
03/19/2010
5.3
𝜿
0.82910108
0.82776839
0.82877590
𝜽
-0.71738058
-0.72926088
-0.80367335
𝒗𝒕
0.49492123
0.50368195
0.41344328
𝜼
0.07578355
0.07855661
0.07488894
𝝆
0.03954365
0.04094678
0.03216978
Simulation of the Heston process
As a performance measurement of the Fourier Transformation it has been chosen to illustrate the
Heston model by the use of Monte Carlo simulation. As simulation is widely recognized for the use
in option pricing, the accuracy of the Fourier Transformation pricing will be compared to the
simulated prices. In the following, the theory behind Monte Carlo simulation will therefore be
introduced.
43
5.3.1 Monte Carlo simulation
There exist many methods for option pricing, including the finite differences method and Monte
Carlo simulation. Each method varies in form and their implementation.
The basic idea underlying Monte Carlo simulation is to value a derivative by generating random
numbers of some uncertainty and probability density. The procedure was first introduced by P.P.
Boyle in 1977, and its relatively easy implementation and simple structure have since its
introduction been a contributing factor to its popularity (Glasserman, 2004).
The Monte Carlo approach is based on the assumption that the value of the derivative is equal to the
expected value of the derivative in the future discounted back to time zero, like the following
equation states:
𝑓(𝑆, 0) = 𝑒 −𝑟𝑇 𝐸[𝑓(𝑆𝑇 , 𝑇)]
Equation 5.3.1.1
The simulation can be divided into 4 steps which are listed in the following:
1. Generate and simulate N number of N(0,1) outcomes under the risk neutral assumption
2. Approximate and calculate N number of terminal values
3. Calculate N number of the final payoff and then calculate the arithmetic average of the
payoff at maturity
4. The price of the option today is then found by discounted this value with the risk free
interest rate
One of the advantages of simulation as an option pricing tool is its accuracy and easy
implementation. This makes it a fine tool for controlling the performance of other models and
methods. However, the accuracy of the simulation process depends very much on the number of
simulations, and the convergence rate which can be described as:
𝜎
√𝑁
where N is the number of simulations. This also means that in order to minimize the simulation
error by half, the number of simulations should be four times higher (Empirical Finance notes,
2010, chapter 4, slide 21). One should also keep in mind that the higher the number of simulations,
44
the longer the time before the process has been simulated. The chosen number of simulations in the
analysis is 10,000.
Unfortunately, the Monte Carlo simulation sometimes contains a bias that can make the simulation
less accurate. However, this can be minimized as shown in the following section.
5.3.2 Milstein Scheme
In order to perform the Monte Carlo simulation the stochastic processes need to be converted from
continuous time to discrete time. However, when performing a Monte Carlo simulation a
discretization error appears from the estimation of the time-discretization of the stochastic
differential equations (Glasserman, P, 2004, page 339-436) - the error arises since the majority of
the models implemented can only be applied approximately. However, as the discretization time
interval is shortened, the discretization error is reduced. The discretization itself can be done in
various ways, and the most popular of them is the so-called Euler Scheme. However, when
generating the variance process according to Euler, negative variances may be produced if the
random numbers generated from normal standard distribution are large and negative. This is
inconsistent with real market data, as the variance cannot go negative. As an alternative method the
so-called Milstein Scheme can be applied. The Milstein Scheme takes method into account a higher
order of the Taylor expansion than the Euler Scheme, and therefore the method performs better
compared to the Euler discretization. More specifically the Milstein Scheme increases the accuracy
by adding a second-order Taylor expansion through the means of Itô’s Lemma.
When applied to the geometric Brownian motion in the Black-Scholes model, the following
equation is the result of the conversion from continuous time to discrete time, ∆t17:
1 ′
𝑏 𝑆𝑖 𝑏𝑆𝑖 (𝑍 2 − 1)Δ𝑡,
𝑤ℎ𝑒𝑟𝑒 𝑍 = 𝑁(0,1),
2
1
= 𝑆𝑖 + (𝑟 − 𝑞)𝑆𝑖 ∆𝑡 + 𝜎𝑆𝑖 √∆𝑡𝑍 + 𝜎 2 𝑆𝑖 (𝑍 2 − 1)Δ𝑡,
2
𝑆𝑖+1 = 𝑆𝑖 + 𝑎 ∙ 𝑆𝑖 ∆𝑡 + 𝑏 ∙ 𝑆𝑖 √∆𝑡𝑍 +
⇒ 𝑆𝑖+1
In the above equation the a and b terms are the drift and volatility of the process corresponding to
the risk free interest rate minus the dividend yield and the variance σ in the Black-Scholes Model.
The conversion from continuous time to discrete time is generally illustrated by the stochastic
17
b'(x) is the first derivative of b(x).
45
process of dS being transformed into a stepwise time measurement of ∆t. The change in the stock
return in the time period, ∆t, is then determined by the above equation.
When applying the Milstein Scheme to the variance process in the Heston model, the following
equation arises:
𝑣𝑖+1 = 𝑣𝑖 + 𝜅(𝜃 − 𝑣𝑖 )Δ𝑡 + 𝜂√𝑣𝑖 √Δ𝑡𝑍 +
𝜂2
Δ𝑡(𝑍 2 − 1) ,
4
𝑤ℎ𝑒𝑟𝑒 𝑍~𝑁(0,1)
Which can be rewritten as the following (Gatheral, 2006, page 22):
2
𝜂
𝜂2
𝑣𝑖+1 = (√𝑣𝑖 + √Δ𝑡𝑍) + 𝜅(𝜃 − 𝑣𝑖 )Δ𝑡 −
Δ𝑡
2
4
It should here be noted, that only the variance process should be discretized by the Milstein
Scheme, as it has been chosen to perform the mixing solution on the processes, as explained in the
following.
5.3.3 Mixing solution
The solution to estimating the option price can also be found through the so-called mixing solution.
The mixing solution was first introduced by Hull & White (1987), who presented a solution with no
correlation between the stock price changes and the volatility changes. However, Romano and
Touzi later understood the importance of this correlation coefficient and thereby extended the
model in 1997 to incorporate a correlation between the stock price changes and volatility changes.
Lewis (2000) then further extended the theorem to handle payoff functions and generalized stock
price volatility coefficients (Lewis, 2002) (with no correlation), whereas the theorem presented by
Romano and Touzi applied to call and put options.
The basic idea underlying the mixing solution is to give an alternative theoretical and computational
method to solve option models (this only applies to certain advanced models, including some
stochastic volatility models). The mixing solution calculates the option values by a weighted sum of
constant volatility prices or a mixture of explicit Black-Scholes prices with a calculated effective
spot and effective volatility (where the latter approach relates to call and put options). The
46
advantage of the mixing solution is therefore that the solution to the model is just to take the
expectation over the volatility process.
The mixing approach can be divided into three basic steps as illustrated below (Lewis, 2005, page
97):
1. Separate the stock price evolution into two processes, independent of each other. One of the
processes should be independent of the volatility process
2. Integrate the variance and mean
3. Insert the integrated variables from step 2 into the Black-Scholes formula (mixing formula)
The solution for the option price process in a risk neutral world can be defined as the following:
𝑆𝑇 = 𝑆0 𝑒
1 𝑇
𝑇
(𝑟−𝑞)𝑇− ∫0 𝑣𝑡 𝑑𝑡 +∫0 √𝑣𝑡 𝑑𝐵𝑡
2
Equation 5.3.3.1
which is similar to the Black-Scholes explicit solution in continuous time. As the process is written
in continuous time, the mixing solution can be applied to it. The Brownian motion can again be
expressed as 𝐵𝑡 = 𝜌𝑊𝑡 + √1 + 𝜌2 𝑍𝑡 , 𝑤ℎ𝑒𝑟𝑒 𝑑𝑊𝑡 𝑑𝑍𝑡 = 0.
If the stock price process is divided into two independent processes, separating the Brownian
motions, the following equation appears:
𝑇
1 𝑇
𝑒𝑓𝑓 (𝑟−𝑞)𝑇− ∫0 (1−𝜌2 )𝑣𝑡 𝑑𝑡+∫0 √(1−𝜌2 )𝑣𝑡 𝑑𝑍𝑡
2
𝑆𝑇 = 𝑆𝑇 𝑒
Where the effective spot in the T is defined as:
𝑒𝑓𝑓
𝑆𝑇
𝑇
1 𝑇 2
𝜌 𝑣𝑡 𝑑𝑡+∫0 𝜌√𝑣𝑡 𝑑𝑊𝑡
= 𝑆0 𝑒 −2 ∫0
It is here noticeable that the effective spot is affected by the values of the variance and the
Brownian motion Wt, and at the same time is independent of the movements in Zt.
The effective variance can be defined as:
𝑒𝑓𝑓
𝑣𝑇
=
1 𝑇
∫ (1 − 𝜌)2 𝑣𝑡 𝑑𝑡
𝑇 0
47
For simplicity the integrals of the mean and the variance are defined as the following:
𝑇
𝐼𝑇1 = ∫ 𝑣𝑡 𝑑𝑡,
𝑇
𝐼𝑇2 = ∫ √𝑣𝑡 𝑑𝑊𝑡
0
0
In order to simulate the process of the effective spot and variance, the differential equations must be
discretized finding the dynamics of the integrals. This can be done by discretizing the variance
process by the Milstein scheme, as discussed in section 5.3.2.
The distribution of the log spot price, xt, can be described as a mean-variance mixture of the normal
distribution as shown below:
𝑒𝑓𝑓
𝑥𝑡 ~(𝑟 − 𝑞)𝑇 + 𝑙𝑜𝑔𝑆𝑇
𝑒𝑓𝑓
+ √𝑇𝑣𝑇 𝑁(0,1)
which is similar to the Black-Scholes Model that assumes a normal distribution. As a result of this,
it is possible to express the price of a call option as the expectance of the Black-Scholes price by
inserting the effective spot and variance in the equation:
𝑒𝑓𝑓
𝑒𝑓𝑓
𝐶(𝑆0 , 𝑣𝑜 , 𝑇) = 𝐸[𝐶𝐵𝑆 (𝑆𝑇 , 𝑣𝑇 , 𝑇)]
Equation 5.3.3.2
The simulation process for the Heston model should then be performed as the Black-Scholes
simulation with the exception of the simulation of the effective spot and variance.
5.4
Comparison of simulated and Fourier prices
In the following paragraph the direct comparison of the simulated call prices and the Fourier prices
from the S&P 500 index options will be conducted. The simulation approach is known for its
accuracy for estimated option prices, as the simulated prices converges to the true market prices as
the number of simulations is increased. Therefore the simulation approach is suitable for evaluating
the performance of the Fourier Transformation method for option pricing. By comparing the two
methods, it is possible to give a rough estimate of the performance of the Fourier transformation.
To test the precision of the Fourier method in pricing options with the Heston Model, it is first
necessary to ensure that the valuation is performed correctly. For simplicity this is done with the
48
Black-Scholes Model for which there exist a closed-form solution for. This means that it is possible
to check the result from the Fourier estimated call prices with the closed-form solution from the
simulated process. However, as the Black-Scholes Model is not the focus of the thesis, the reader is
referred to the appendix, where the code for simulation and Fourier Transformation prices exists.
The simulation is now performed on the Heston model using the mixing solution, as explained
earlier. When comparing the prices from the simulation with the Fourier transform, it is evident that
the two methods are similar for short maturities.
Figure 4 – Plot of prices across log moneyness
Source: Own contribution
Remark: Green lines are prices calculating via the Fourier Transformation, and red lines are simulated through the
mixing solution
49
Even though the two methods appear similar, it is easy to see that they differ in some places. This is
especially the case for longer maturities where the call prices differ. This is evidence that the
Fourier method is not able to completely replace the simulation approach, however, this should be
viewed in the light of the parameters chosen for the Heston model. As mentioned in the calibration
section in 5.1, the calibration of the parameters is very sensitive with respect to the initial
parameters chosen. Depending on the initial values the minimization process will find the closest
minimum to the objective function given the parameters, however, there is no guarantee that the
minimum is a local minimum, and not a global minimum. This is coherent with the importance of
finding the true parameters of the Heston model that best fit the market data. Nonetheless, is has
been assumed that the found minimum is the global. Later it will be illustrated how changing the
parameters in the Heston model can make a big difference on the prices estimated from the model.
Furthermore the fact that the options with a shorter maturity are priced more closely to the
simulated prices can also be a product of the data used for the calibration. As the dataset contains
more option that are in the money, these options will receive a higher weight in the calibration than
the remaining options – which will mean that these options will be priced more accurate.
5.5
The implied volatility surface of the S&P500 index options
In the following section the implied volatility surface of the S&P 500 index options (the market
data) will be compared to the estimated implied volatility surface of the Heston model from the
simulation and the Fourier Transformation. As one of the main implications of the Black-Scholes
model is the constant volatility assumption, it makes no sense to plot the implied volatilities given
in the Black-Scholes model, as it will only give a flat line. However, the two valuation methods can
still be compared with respect to the implied volatility found through the methods for the Heston
model. Here it is also possible to compare the outcome directly to the market fit, as the implied
volatilities is a part of the downloaded dataset. As can be seen in figure 5.5.1, the Fourier method
applied to the Heston model gives a poor fit compared to the actual implied volatility for short
maturities. This is also what was expected. As the time to maturity increases the implied volatility
from the Fourier Transformation fits more closely to the one observed in the market – and actually
has a better fit than the simulated implied volatility (it should here be noted that the simulation can
be improved by increasing the number of simulations). Therefore it seems that for maturities other
than short maturities, the Fourier Transformation method is reliable.
50
Figure 5.5.1 – Plot over the implied volatilities across log moneyness
Source: Own contribution
Remark: The blue line is the imported implied volatility from the market data, the green line is the Fourier calculated
implied volatility and the red line is the simulated implied volatility
Gathering the implied volatilities across moneyness and time to maturity gives an image of the
implied volatility surface – in a 3D plot. Comparing the Heston model with the plotted implied
volatilities observed in the market gives an overview of how the Heston model fits to the market
data. The implied volatility from the market has been backed out of the Black-Scholes model, as it
is known to be the wrong number in the wrong formula that gives the right price. As can be seen
from figure 5.5.2, the Heston model clearly does a better fit that the Black-Scholes model (which is
obvious as it assumes constant volatility, as mentioned). As the plots are constructed as a function
of the log moneyness, the area in the middle refers to at-the-money options (ATM), the area to the
51
right to out-of-the-money (OTM) options and last the area on the left which represents in-themoney (ITM) options.
Figure 5.5.2
Source: Own contribution
Remark: For maturities: 7, 36, 64, 75, 92, 155, 166, 246, 258, 337, 350 days to maturity
Figure 5.5.3
Source: Own contribution
Remark: For maturities: 7, 36, 64, 75, 92, 155, 166, 246, 258, 337, 350 days to maturity
52
It appears as if the implied volatilities from options with a longer time to maturity have a wider
spread than the ones closer to the exercise date. This is more clearly seen on the volatility surface
on the market implied volatility in figure 5.5.3. Furthermore it can be seen that the fit is worse as
the time to maturity is shortened – here the market implied volatilities are nicely organized, where
the implied volatilities from the Heston model are not. This is coherent with the empirical evidence
of the Heston model, as it does not incorporate jumps. The fit of the Heston model is relatively
close to the market implied volatility surface, and therefore it can be concluded that the model can
be fitted nicely. It should here be noted that the deviations can be the result of the calibration
process finding a local minimum instead of a global minimum, which means that there can exists
parameters for the Heston model that fits the implied volatility surface even better.
5.6
The Heston model’ sensitivity to its parameters
The Heston Model is very sensitive with respect to its parameters, and therefore an examination of
the effects of the parameters will be conducted. This will illustrate the effect the parameters have on
the spot price process. In order to ensure a high level of accuracy for the option prices calculated
from the Heston model, it is vital that the calibrated parameters fit the market data as perfectly as
possible. This was also evident from the calibration of the Heston model where the calibrated
parameters differed with the datasets – which means that they were fitted to a particular dataset and
therefore they are not constant. As a result of the sensitivity of the parameters it is relevant to look
at each calibrated parameter and illustrate the effect a change will have on the implied volatilities –
thereby showing the sensitivity of that parameter18.
The default parameters are set to the first calibrated parameters (01/15/2010):
Parameter
Default value
𝜿
𝜽
𝒗𝒕
𝜼
𝝆
0.82910108
0.07578355-
0.03954365
0.49492123
-0.71738058
The mean reversion of speed, 𝜅:
18
The following section is based on Heston, 1993, page 335-339 and lecture notes from Advanced Financial Modeling
53
The mean reversion speed factor determines the speed at which the volatility is pushed towards its
long-run volatility, θ. The mean reversion speed is also a determinant for the volatility clustering in
the returns of the spot. When plotting the implied volatilities and moneyness for the Heston model,
with the only changed parameter being 𝜅, it is clear to see that 𝜅 determines the spread of the
volatility, and that the difference in implied volatilities is greater for option with longer maturities.
This is because the coefficient, ceteris paribus, will have a greater impact the longer the time
horizon (time to maturity).
Figure 5.6.1 – Plot of implied volatilities across log moneyness with kappa changing +/-0.3
Source: Own contribution
The long-run volatility, 𝜃:
As mentioned earlier the long-run volatility is the mean of the variance process, for which the
variance process moves as time go by. The long-run volatility is an increasing function of the price
of the option, so as the volatility increases, the price and the implied volatility of the option also
increases. Changing the long-run volatility has a great impact on the implied volatilities, and for
54
longer time for maturity the effect is greatest. However, as it is evident from figure 5.6.2 the
implied volatilities changes level, but the process remains stable, as the mean reversion speed is
held constant.
Figure 5.6.2 – Plot of implied volatilities across log moneyness with theta changing +/-0.1
Source: Own contribution
The initial volatility, vt:
The initial volatility determines the current level of the volatility, and is by the mean-reversion
speed pushed towards the long-run volatility. Changing the parameter by +/-0.01 has a visible effect
on all the plots in figure 5.6.3. However, for the options with the shorter maturity the effect of
altering the parameter has the greatest effect. This is because the long-run volatility will have the
biggest impact on the current volatility over a period of time, ceteris paribus.
55
Figure 5.6.3 – Plot of implied volatilities across log moneyness with vt changing +/-0.1
Source: Own contribution
Basically there are two parameters that directly have an effect on the volatility smile and thereby the
volatility surface. These parameters are the volatility of volatility (volatility of the variance), η and
the correlation coefficient, ρ.
The volatility of volatility, 𝜂:
As the volatility of volatility parameter has an effect on the kurtosis of the distribution of spot
prices, it can be used to better fit the distribution that the market data follows. This is an advantage
of the Heston model, compared to for example the Black-Scholes Model, whose kurtosis is equal to
zero. Whenever the volatility of volatility is equal to zero, the volatility is deterministic as the
variance doesn’t changes – this makes the distribution normal. When altering the volatility of the
variance by +/-0.1 the OTM options and ITM options are affected. As the correlation coefficient
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between the stock price process and the variance process is negative, a positive increase in the
volatility of volatility will decrease the implied volatility for ITM options whereas the implied
volatilities for the OTM options will increase. The effect on altering the parameter doesn’t seem to
affect ATM options to the same extent.
Figure 5.6.4 – Plot of implied volatilities across log moneyness with eta changing +/-0.1
Source: Own contribution
The correlation coefficient, 𝜌:
The correlation between the stock process and the variance process affects the skewness of the
volatility smile and of the spot returns. As explained earlier, the introduction of the correlation
coefficient in the option pricing model has contributed a great deal to the popularity of the Heston
model, as it is similar to the real world. The Heston model is very sensitive with respect to the
correlation parameter, as it directly has an effect on the skewness of the distribution.
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For stock options the correlation between the price and variance processes are normally negative, as
explained earlier. This means that the volatility is a decreasing function of the spot price, and as the
spot price falls, the volatility increases.
Figure 5.6.5 – Plot of implied volatilities across log moneyness with rho changing +/-0.1
Source: Own contribution
Following the analogy from earlier, this will result in the left tail being “spread out”, as it is
associated with high volatility. At the same time the right side of the distribution, representing the
low volatility, will remain the same, and will therefore not be spread. As it can be seen from figure
5.6.5, the price of an OTM option will increase as the correlation is set up by +0.1, and the opposite
for OTM options.
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All in all it can be concluded that the Heston model is very sensitive with respect to its parameters.
As a result of this it is vital for the accuracy of prices it produces, that the parameters are calibrated
to the market data.
Chapter 6: Conclusion
The main objective of the thesis was to fit the Heston Model to the implied volatility surface of the
S&P 500 index options. To perform this analysis, a part of the basic theory underlying option
pricing was first presented. This included stochastic processes and the Geometric Brownian Motion
which is often used to describe stock option data. After this, the Black-Scholes Model was
introduced in order to establish the need for a stochastic volatility model on the market for pricing
stock options. Several of the assumptions underlying the Black-Scholes Model are violated in the
real world, however the violations differ as some of the assumptions can be relaxed, or the model
can be adjusted (as with dividends and American options). In order to test the empirical
implications of the Black-Scholes Model, an analysis of the empirical stock returns of the S&P 500
index options was then conducted. The analysis was intended to test whether the log returns of the
stock options were normally distributed as well as if the data depicted a constant variance or not.
The analysis showed that the empirical stock returns displayed a high central peak and fat tails,
which are evidence of a non-normal distribution. The log returns of the stock options further
showed a deviation from the normal distribution in the tails. This is evidence of a stochastic
volatility in the market data. Furthermore, when plotting the stock returns from 1980 to 2010 it
showed a non-constant variance, as the returns had been subject to extreme events throughout the
years – beginning with the so-called Black Monday in 1987.
As a result of the analysis of the empirical data, the basis for introducing a stochastic volatility was
made. Some of the most influential stochastic volatility models and the basic jump-diffusion model
were then shortly introduced in order to provide an overview on the existing models that exists on
the market. The chosen model, the Heston Model, was then introduced and its characteristics were
outlined. One of the main reasons for the model’ popularity is that the model incorporates a
correlation between the underlying asset process and the variance process, which is evident in the
market. The Heston Model is also characterised as being mean-reverting. Furthermore, the model
has a (semi)-closed form solution by the means of a numeric solution to an integral (Fourier
59
Transformation), which makes it less computational complex compared to other models. This has
increased the model’ popularity as the time needed to find the option prices is far less than with the
common Monte Carlo simulation. However the Heston model is still considered to have a semiclosed form solution as the solution of the integrals in the Fourier Transformation can be relative
difficult to incorporate. Therefore, the secondary purpose of the thesis is to evaluate the
performance of the Fourier Transformation for option pricing. In order to estimate the performance
of the method, the Heston Model was also simulated by a Monte Carlo simulation, which has been
known to converge to the true market price as a function of the number of simulations chosen.
Although the Heston model contains many advantages compared to other models, it still has the
disadvantage that it poorly fits the implied volatility surface of options with shorter maturities.
In the analysis the Heston Model was calibrated in order to fit the implied volatility surface of the
S&P 500 index options. This calibration process, which minimizes the square difference between
market and model price, is very sensitive with respect to the initial estimate of the parameters, as
there is no guarantee that the used method will find a global minimum and not a local minimum
when solving the calibration process. However, it is assumed that the calibrated model parameters
are global minimums. The calibration was conducted for three months, and the estimates were
shortly compared to each other, which showed that they were very similar. In order to evaluate the
performance of the Heston model and its use of Fourier Transformation for option pricing, the
mixing solution was applied to the Heston model. By simulated an effective variance and
calculating an effective spot price, the solution of the Heston model is then simply to insert the two
values into the standard Black-Scholes Model which will then yield the price of the option. The
simulated prices were then compared to the prices obtained from the Fourier Transformation, and it
showed that the approaches were almost identical for short maturities. However, when comparing
the implied volatilities of the two approaches with the implied volatility backed out from the market
prices it turned out that the Fourier Transformation was actually good at describing and pricing
options with longer maturities. The implied volatility from the Fourier Transformation was actually
closer to the market implied volatility than the simulated was. Therefore it can be said that for
option with longer maturities the Fourier Transformation approach is relevant, as it is relatively
easy to implement and faster compared to the Monte Carlo simulation. The Heston model was then
compared the implied volatility surface of the market. The analysis here showed that the Heston
model deviated from the market data for shorter maturities, but for longer maturities the fit was
good. However, it should also be mentioned that the calibration of the Heston parameters is not
60
guaranteed to produce a global minimum. As a result of this it can be that there are parameters for
the Heston model that more closely can be fitted to the implied volatility surface. As the Heston
parameters are very sensitive for the pricing process, an analysis of the parameters changing were
conducted, all else being held equal. This showed that the Heston model is indeed sensitive with
respect to its parameters, and therefore it is an important aspect to consider when determining the
use of valuation approach. One needs to leverage the computational fast advantages and the
accuracy of the Heston model and its solution in the form of Fourier Transformation with the time
difficulties and continuously calibration of the parameters (that are not stable).
61
List of Literature
Academic Articles:
Bakshi, Gurdip; Cao, Charles; Chen, Zhiwu (1997), Empirical Performance of Alternative Option
Pricing Models, The Journal of Finance, Vol. LII, No. 5
Bergomi, Lorenzo (2005), Smile Dynamics II, Risk, 18
Black, Fischer and Scholes, Myron (1973). "The Pricing of Options and Corporate Liabilities".,
Journal of Political Economy, 81 (3) page 637–654
Carr, Peter and Madan, Dilip B (1999), Option valuation using the fast Fourier transform, Journal
of Computational Finance, page 61-73
Campbell, John Y, Lo, Andrew W and MacKinlay, A Craig (1997), The Econometrics of Financial
Markets, Princeton
Cox, J.C., Ingersoll, J.E. & Ross, S.A. (1985), A Theory of the Term Structure of Interest Rates,
Econometrica: Journal of the Econometric Society, vol. 53, no. 2, pp. 385-407.
Heston, Steven L (1993), A Closed-Form Solution for Options with Stochastic Volatility with
Applications to Bond and Currency Options, The Review of Financial Studies, Vol. 6, No. 2 , pp.
327-343
Hull, John and White, Alan (1987), The Pricing of Options on Assets with Stochastic Volatilities,
The Journal of Finance, Vol. 42, No. 2, pp. 281-300
Kahl, Christian and Lord, Roger (2010), Fourier Inversion Methods in Finance, First version: 31st
August 2009, this version: 26th January 2010
Merton, Robert C. (1973), Theory of Rational Option Pricing, Bell Journal of Economics and
Management Science, 4 (1): 141–183
Merton, Robert C. (1976), Option pricing when the underlying stock returns are
Discontinuous, Journal of Financial Economics, 3, page 125–144.
Mikhailow, Sergei and Nögel, Ulrich (2008), Heston’s Stochastic Volatility Model Implementation,
Calibration and Some Extensions, Wilmott Magasine
Renault, Eric and Touzi, Nizar (1996), Option Hedging and Implied Volatilities in a Stochastic
Volatility Model, Mathematical Finance, Vol. 6, NO. 3, page 279-302
Romano, Marc and Touzi, Nizar (1997), Contingent Claims and Market Completeness in a
Stochastic Volatility Model, Mathematical Finance, Vol. 7, No. 4, page 399–410
62
Schmelzle, Martin (2010), Option Pricing Formulae using Fourier Transform: Theory and
Application
Schoebel, Rainer, and Zhu, Jianwei (1998), Stochastic Volatility With an Ornstein-Uhlenbeck
Process: An Extension, www.uni-tuebingen\uni\wwm\papers.html
Stein, E. and J. Stein (1991), Stock Price Distributions with Stochastic Volatility: An
Analytic Approach, Review of Financial Studies, 4, page 727–752.
Books:
Cherubini, Umberto; Lunga, Giovanni Della; Mulinacci, Sabrina and Rossi, Pietro (2010), Fourier
Transform Methods in Finance (The Wiley Finance Series)
Gatheral, Jim (2006), The Volatility Surface. A Practitioner’s Guide, Wiley Finance
Glasserman, Paul (2004), Monte Carlo Methods in Financial Engineering, Chapter 6, pages
399-343
Hull, John (2008), Options, Futures and other Derivatives, Prentice Hall, 7th edition
Lewis, Alan L (2005), Option Valuation under Stochastic Volatility, Finance Press, 2nd printing
Rebonato R. (1999), Volatility and Correlation, John Wiley, Chichester
Wilmott, Paul (2005), Paul Wilmott Introduces Quantitative Finance, John Wiley, Chichester
Academic reports
Moodley, Nimalin (2005), The Heston Model: A Practical Approach with Matlab Code
Others:
Bond & Interest Rate, Elisa Nicolato, handout
Empirical Finance notes, 2010, chapter 4, slide 21
Advances in Financial Modeling, 2010, notes
Used as inspiration: Kivila, Liis and Ucar, Sibel (2009), Stochastic Volatility Models with
Application to Volatility Derivatives, Thesis, Aarhus School of Business
63
Internet sources: (all visited last time on 08/31/2011)
http://www.ivolatility.com/help/14.html
http://www.randomwalktrading.com/main/index.php?option=com_content&view=article&id=216&
Itemid=338
http://www.standardandpoors.com/indices/sp-500/en/us/?indexId=spusa-500-usduf--p-us-l-http://www.standardandpoors.com/servlet/BlobServer?blobheadername3=MDTType&blobcol=urldata&blobtable=MungoBlobs&blobheadervalue2=inline%3B+filename%3DFact
sheet_SP_500.pdf&blobheadername2=ContentDisposition&blobheadervalue1=application%2Fpdf&blobkey=id&blobheadername1=contenttype&blobwhere=1243931099288&blobheadervalue3=UTF-8
http://www.treasury.gov/resource-center/data-chart-center/interestrates/Pages/TextView.aspx?data=yieldYear&year=2010
http://wrds-web.wharton.upenn.edu/wrds/process/wrds.cfm
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Appendix
See attached CD-rom
65