Zhu J. Applications of Fourier Transform to Smile Modeling: Theory and Implementation
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Contents
1 Option Valuation and the Volatility Smile ......................... 1
1.1 Stochastic Processes for Stocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 BrownianMotion.................................... 1
1.1.2 Stock Price as Geometric Brownian Motion . . . . . . . . . . . . . . 3
1.1.3 It
ˆ
o Process and It
ˆ
o’sLemma........................... 4
1.2 The Black-Scholes Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 Options and Dynamic Hedging . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.2 Risk-NeutralValuation ............................... 7
1.2.3 Self-financing and No Arbitrage . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.4 EquivalentMartingaleMeasures........................ 11
1.3 Volatility Quotations in Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3.1 Implied Volatilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3.2 MarketQuotations ................................... 16
1.3.3 Special Case: FX Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 Characteristic Functions in Option Pricing........................ 21
2.1 Constructing Characteristic Functions (CFs) . . . . . . . . . . . . . . . . . . . . 22
2.1.1 A General Process for Stock Price . . . . . . . . . . . . . . . . . . . . . . 22
2.1.2 Valuation of European-style Options via CFs . . . . . . . . . . . . . 23
2.1.3 Special Case: FX Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 Understanding Characteristic Functions . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2.1 Properties of Characteristic Functions . . . . . . . . . . . . . . . . . . . 29
2.2.2 Economic Interpretation of CFs . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2.3 Examination of Existing Option Models . . . . . . . . . . . . . . . . . 35
2.2.4 Relationship between CF to PDE . . . . . . . . . . . . . . . . . . . . . . . 39
2.2.5 Advantages of CF and Modular Pricing . . . . . . . . . . . . . . . . . . 42
3 Stochastic Volatility Models ..................................... 45
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2 The Heston Model .......................................... 48
3.2.1 Model Setup and Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
xi
xii Contents
3.2.2 PDE Approach to Pricing Formula . . . . . . . . . . . . . . . . . . . . . . 50
3.2.3 Expectation Approach to Pricing Formula. . . . . . . . . . . . . . . . 52
3.2.4 Various Representations of CFs . . . . . . . . . . . . . . . . . . . . . . . . 54
3.3 The Sch
¨
obel-Zhu Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.3.1 Model Setup and Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.3.2 DerivationofCFs.................................... 58
3.3.3 NumericalExamples ................................. 60
3.4 Double Square Root Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.4.1 Model Setup and Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.4.2 NumericalExamples ................................. 69
3.5 Other Stochastic Volatility Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.6 Appendices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4 Numerical Issues of Stochastic Volatility Models ................... 77
4.1 AlternativePricingFormulaswithCFs......................... 78
4.1.1 The Formula
´
a la Black-Scholes . . . . . . . . . . . . . . . . . . . . . . . . 78
4.1.2 The Carr and Madan Formula . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.1.3 TheAttariFormula................................... 80
4.2 Risk Sensitivities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.2.1 DeltaandGamma.................................... 81
4.2.2 VariousVegas....................................... 82
4.2.3 CurvatureandSlope.................................. 84
4.2.4 Volga and Vanna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.3 DirectIntegration(DI) ...................................... 86
4.3.1 TheGaussianIntegration.............................. 86
4.3.2 Multi-Domain Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.3.3 StrikeVectorComputation ............................ 88
4.4 FastFourierTransform(FFT) ................................ 89
4.4.1 AlgorithmsofFFT................................... 89
4.4.2 Restrictions......................................... 91
4.5 DirectIntegrationvs.FFT ................................... 92
4.5.1 Computation Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.5.2 Computation Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.5.3 MatchingMarketData................................ 94
4.5.4 Calculation of Greeks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.5.5 Implementation...................................... 95
4.6 LogarithmofComplexNumber............................... 99
4.6.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.6.2 Three Algorithms Dealing with Branch Cut . . . . . . . . . . . . . . 101
4.6.3 When Main Argument Is Appropriate . . . . . . . . . . . . . . . . . . . 103
4.7 CalibrationtoMarketData...................................104
4.7.1 General Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.7.2 FixingVelocityParameter.............................105
4.7.3 Fixing Spot Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.8 MarkovianProjection .......................................108
Contents xiii
5 Simulating Stochastic Volatility Models ...........................113
5.1 Simulation Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.1.1 Discretization .......................................114
5.1.2 Moment-Matching ...................................115
5.2 Problems in the Heston Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.2.1 NegativeValuesinPaths ..............................116
5.2.2 Log-normal Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.2.3 Transformed Volatility Scheme . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.2.4 QE Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.2.5 The Broadie-Kaya Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.2.6 Some Other Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.3 SimulationExamples .......................................125
5.4 MaximumandMinimum ....................................126
5.5 Multi-Asset Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6 Stochastic Interest Models ......................................135
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.2 The Cox-Ingosoll-Ross Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
6.2.1 TheZero-CorrelationCase ............................138
6.2.2 TheCorrelationCase.................................140
6.3 The Vasicek Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
6.4 The Longstaff Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
6.4.1 TheZero-CorrelationCase ............................145
6.4.2 TheCorrelationCase.................................146
6.5 Correlations with Stock Returns: SI versus SV . . . . . . . . . . . . . . . . . . 148
7 Poisson Jumps .................................................153
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
7.2 SimpleJumps..............................................158
7.3 Lognormal Jumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
7.4 ParetoJumps ..............................................163
7.5 The Kou Model: An Equivalence to Pareto Jumps . . . . . . . . . . . . . . . 165
7.6 AffineJump-Diffusions .....................................168
8L
´
evy Jumps ...................................................173
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
8.2 Stochastic Clock Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
8.2.1 Variance-Gamma Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
8.2.2 Normal Inverse Gaussian Model. . . . . . . . . . . . . . . . . . . . . . . . 181
8.3 Time-Changed L
´
evy Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
8.3.1 Uncorrelated Time-Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
8.3.2 Correlated Time-Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
8.4 The Barndorff-Nielsen and Shephard Model . . . . . . . . . . . . . . . . . . . . 193
8.5 Alpha Log-Stable Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
8.6 Empirical Performance of Various L
´
evyProcesses...............196
xiv Contents
8.7 Monte-CarloSimulation.....................................198
8.7.1 Generating Random Variables. . . . . . . . . . . . . . . . . . . . . . . . . . 198
8.7.2 Simulation of L
´
evy Process ............................201
9 Integrating Various Stochastic Factors............................203
9.1 Stochastic Factors as Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
9.2 Integration Approaches. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
9.2.1 Modular Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
9.2.2 Time-Change Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
9.3 Pricing Kernels for Options and Bonds.........................217
9.4 Criterions for Model Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
10 Exotic Options with Stochastic Volatilities ........................223
10.1 Forward-StartingOptions....................................224
10.2 BarrierOptions ............................................226
10.2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
10.2.2 Two Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
10.2.3 NumericalExamples .................................233
10.3 Lookback Options ..........................................235
10.3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
10.3.2 Pricing Formulas with Stochastic Factors . . . . . . . . . . . . . . . . 238
10.4 AsianOptions .............................................244
10.4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
10.4.2 The Black-Scholes World . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
10.4.3 Asian Options in a Stochastic World . . . . . . . . . . . . . . . . . . . . 249
10.4.4 Approximations for Arithmetic Average Asian Options . . . . 252
10.4.5 A Model for Asian Interest Rate Options . . . . . . . . . . . . . . . . 254
10.5 CorrelationOptions.........................................257
10.5.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
10.5.2 Exchange Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
10.5.3 QuotientOptions ....................................263
10.5.4 Product Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
10.6 OtherExoticOptions .......................................266
10.7 Appendices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
11 Libor Market Model with Stochastic Volatilities ...................273
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
11.2 Standard Libor Market Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
11.2.1 Model Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
11.2.2 Term Structure and Smile of Volatility . . . . . . . . . . . . . . . . . . . 280
11.3 Swap Market Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
11.3.1 Model Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
11.3.2 CorrelationStructure .................................286
11.3.3 ConvexityAdjustmentsforCMS .......................291
11.4 Incorporating Stochastic Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
Contents xv
11.4.1 The Andersen and Brotherton-Ratcliffe Model . . . . . . . . . . . . 295
11.4.2 The Piterbarg Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
11.4.3 The Wu and Zhang Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
11.4.4 The Zhu Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
11.4.5 The Belomestny, Matthew and Schoenmakers Model . . . . . . 312
11.5 Conclusive Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
References..............................................................319
Index..............................................................327
Chapter 1
Option Valuation and the Volatility Smile
In this chapter, we briefly present the basic concepts of option pricing theory. The
readers who are familiar with these topics, can skip this chapter and begin with the
next chapter directly. A Brownian motion is an elemental building-block in mod-
eling the dynamics of stock returns, and correspondingly the geometric Brownian
motion as an exponential function of Brownian motion is the simplest and most pop-
ular process for stock prices, on which the Black-Scholes model is based. Dynamic
hedging in the Black-Scholes model is a self-financing trading strategy that ensures
no arbitrage, and allows us to derive the Black-Scholes equation and formula. It
is shown that no arbitrage implied in the dynamic hedging results in a risk-neutral
process with which all financial derivatives may be valued. From the point of view
of probability measures, a risk-neutral process can be regarded as a process derived
via the change of the historical measure to a measure using the money market ac-
count as numeraire. Finally, we can verify that an equivalent martingale measure
under certain conditions in turn implies no arbitrage. Therefore, no arbitrage, risk-
neutral valuation and equivalent martingale measure are the key concepts in option
pricing theory, and are essentially equivalent to each other, but from different points
of view. Next, we focus on the practical challenge of option pricing: inconsistent
implied volatilities, or the volatility smile, a challenge that arises from the weakness
of the Black-Scholes model, and that we will tackle in this book. Implied volatilities
display different smile patterns, and are extensively used by markets for quotations
of option prices.
1.1 Stochastic Processes for Stocks
1.1.1 Brownian Motion
In financial markets there are a number of prices, rates and indices that display
strong stochastic behavior in the course of time. Nowadays, a Brownian motion is
J. Zhu, Applications of Fourier Transform to Smile Modeling, Springer Finance,
DOI 10.1007/978-3-642-01808-4
1,
c
Springer-Verlag Berlin Heidelberg 2000, 2010
1
2 1 Option Valuation and the Volatility Smile
a basic building-block in modeling financial variables, and extensively used to de-
scribe most periodic changes of random prices, for example, daily returns of stock
prices. Brownian motion was first introduced by the Scottish botanist R. Brown in
1827 to document random movements of small particles in fluid. The French math-
ematician L. Bachelier (1900) applied Brownian motion in his lately honored PhD
thesis “The Theory of Speculation” to analyze the price behavior of stocks and op-
tions. But first in 1905, A. Einstein delivered a formal mathematical analysis of
Brownian motion and derived a corresponding partial differential equation. After-
wards Brownian motion became an important tool in physics. In honor of N. Wiener
who made a great contribution to studying Brownian motion from the perspective
of stochastic theory, a standard Brownian motion is also called a Wiener process.
Let (
Ω
,F ,(F
t
)
t0
,Q)
1
be a complete probability space equipped with a prob-
ability measure Q. F
t
is
σ
-algebra at time t and may be interpreted as a collection
of all information up to time t.
2
A standard Brownian motion is defined as follows.
Definition 1.1.1. Brownian motion: A standard Brownian motion {W(t),t 0} is
a stochastic process satisfying the following conditions,
1. for 0 t
1
t
2
··· t
n
< ∞,W(t
1
),W(t
2
) −W(t
1
),··· ,W(t
n
) −W(t
n−1
) are
stochastically independent;
2. for every t > s 0,W(t) −W(s) is normally distributed with a mean of zero and
a variance of t −s,
W(t) −W(s) ∼ N(0,t −s);
3. for each fixed
ω
∈
Ω
,W(
ω
,t) is continuous in t;
4. W(0)=0 almost surely.
Condition 1 says that non-overlapping increments of W(t) are uncorrelated. Con-
dition 2 states that any increment W (t)−W(s) is distributed according to a Gaussian
law with a mean of zero and a variance of t −s. As a Brownian motion W (t) is de-
fined on F
t
, the probability measure Q is then a Gaussian law. In other words, the
dispersion of a standard Brownian motion is proportional to time length. A further
implication of condition 2 is that a standard Brownian motion is homogeneous in
the sense that the distribution of any increment depends only on time length t −s
and not on time point s. Condition 3 explains that a standard Brownian motion has
continuous paths over time. The last condition just requires that a standard Brown-
ian motion starts at zero almost surely, and implies the expected value of any W(t)
is equal to zero. Summing up, a standard Brownian motion belongs to a class of the
processes with stationary independent increments.
Since a standard Brownian motion is normally distributed and defined on the
entire real line, it may not be applied appropriately to model price levels that are
usually strictly positive. A realistic concept for price movements is a geometric
Brownian motion, an exponential function of a Brownian motion.
1
For more detailed descriptions on probability space,
σ
-algebra and filtration, please refer to other
mathematical textbooks, for example, Øksendal (2003).
2
Throughout this book, for notation simplicity, we will not always express a conditional expecta-
tion by explicitly adding F
t
.
1.1 Stochastic Processes for Stocks 3
1.1.2 Stock Price as Geometric Brownian Motion
If instantaneous returns of an asset S(t) are assumed to follow a fixed quantity plus a
Brownian motion which describes the risky and unpredictable part of S(t), the asset
S(t) itself follows a geometric Brownian motion,
dS(t)
S(t)
=
μ
dt +
σ
dW(t), (1.1)
where the parameters
μ
and
σ
are constant. Due to the randomness of W (t), S(t) is
no longer a deterministic quantity, but governed by some stochastic dynamics. By
intuition, if we are neglecting the term
σ
dW(t) in (1.1), we obtain the following
equation
dS(t)
S(t)
=
μ
dt,
dS(t)
dt
= S(t)
μ
, (1.2)
which is an ordinary differential equation (ODE) and admits a deterministic solution
S(t)=S(0)e
μ
t
.
Here the parameter
μ
can be interpreted as the instantaneous return of S(t).By
adding the random term
σ
dW(t) to the ODE in (1.2), we obtain a stochastic dif-
ferential equation (SDE) as in (1.1). There is no longer a deterministic solution to
(1.1). However, in the sense of expectation we have
E[S(t)] = S(0)e
μ
t
.
This indicates that if an asset S(t) is governed by a geometric Brownian motion, it
has an expected return of
μ
which is also referred to as a drift term. Since the param-
eter
σ
scales W(t) and then characterizes how large the instantaneous stock returns
deviate from its drift, we call this parameter volatility. Additionally, specifying the
stock price to be a geometric Brownian motion implies the following facts:
1. The increments dS(t
1
) and dS(t
2
),t
1
= t
2
, are independent, i.e., uncorrelated,
since the increments dW(t
1
) and dW(t
2
) are independent.
2. The randomness of the instantaneous stock returns is normally distributed and
has the same volatility.
3. As a result of normal distribution, stock returns have zero skewness and no excess
kurtosis, which contradicts the most empirical studies that document the negative
skewness and the excess kurtosis of historical stock returns.
4. The stock prices are always positive.
5. The random paths of the stock prices are continuous and have no jumps.
These implications indicate that a geometric Brownian motion can simply cap-
ture the main characteristics of stock prices, but also imposes some restrictions to
their model behavior. For example, the stock prices exhibit no jumps and have a
constant volatility. More complicated models for stock prices, as shown later, will
4 1 Option Valuation and the Volatility Smile
relax these implications of the geometric Brownian motion by adding more realistic
components.
1.1.3 It
ˆ
o Process and It
ˆ
o’s Lemma
Geometric Brownian motion may be extended to a general stochastic process,
dS(t)=S(t)
μ
dt + S(t)
σ
dW(t)
= a(S(t),t)dt + b(S(t),t)dW(t), (1.3)
where a(S(t),t) and b(S(t),t) could be deterministic functions or stochastic pro-
cesses. If the following two conditions are satisfied,
P
t
0
|a(S(u),u)|du < ∞,t 0
= 1 (1.4)
and
P
t
0
b
2
(S(u),u)du < ∞,t 0
= 1, (1.5)
the process S(t) is called an It
ˆ
o process. P[·]=1 means the condition is fulfilled al-
most surely. In fact, the above two conditions ensure that the solution of the stochas-
tic differential equation of an It
ˆ
o process exists. In other words, we have
S(t)=S
0
+
t
0
a(S(u),u)du+
t
0
b(S(u),u)dW(u) < ∞.
If a(S(t),t) and b(S(t),t) are not functions of time t, namely a(S(t),t)=a(S(t))
and b(S(t),t)=b(S(t)),theIt
ˆ
o process S(t) is also a time-homogenous process. In
this case, conditions (1.4) and (1.5) are simplified to the Lipschitz condition
|a(x) −a(y)|+ |b(x) −
b(y)| D|x −y|, x,y,D ∈ R.
Given an It
ˆ
o process
dX(t)=a(X(t),t)dt + b(X(t),t)dW(t),
a natural question with respect to It
ˆ
o process is, if Y (t)= f (X(t),t) is a function of
the It
ˆ
o process X(t) and time t, what form should the process Y(t) take. In fact, most
financial derivatives are essentially a function of underlying assets and time, there-
fore it is very important and useful to derive the stochastic process of a financial
derivative based on the process of underlying asset. It
ˆ
o’s lemma gives us an instruc-
tive way to derive a stochastic process, and then is a very useful tool to analyze and
evaluate options. The simple version of It
ˆ
o’s lemma is following.
1.2 The Black-Scholes Model 5
Theorem 1.1.2. It
ˆ
o’s Lemma: If X(t) is an It
ˆ
o process, then the function
3
Y(t)=
f (X(t),t) ∈C
2×1
[R×[0, ∞) → R] is an It
ˆ
o process of the following form,
dY(t)=
∂
f
∂
t
dt +
∂
f
∂
X(t)
dX(t)+
1
2
∂
2
f
∂
X
2
(t)
(dX(t))
2
=
∂
f
∂
t
dt +
∂
f
∂
X(t)
dX(t)+
1
2
∂
2
f
∂
X
2
(t)
b
2
(X(t),t)dt
=
∂
f
∂
t
+
1
2
∂
2
f
∂
X
2
(t)
b
2
(X(t),t)
dt +
∂
f
∂
X(t)
dX(t). (1.6)
Applying It
ˆ
o’s lemma, it is easy to obtain the process of log returns x(t)=lnS(t),
dx(t)=
μ
−
1
2
σ
2
dt +
σ
dW(t)
with x(0)=lnS(0).
1.2 The Black-Scholes Model
1.2.1 Options and Dynamic Hedging
The payoff of a European call option at maturity date with strike K is given by
C(T )=max[S(T) −K,0],
which depends directly on the underling stock price S(T). The strong dependence
of a call price on its underlying stock price implies that an option should have a “fair
price” to best represent the co-movements between option and underlying stock. It
then raises a question, what is the exact definition of the fair price for a call? Black-
Scholes’s answer to this question is that a fair price must be an arbitrage-free price
in the sense that a risk-free portfolio comprising of options and underlying stocks
must reward a risk-free return.
Now we consider a portfolio including a call and its underlying. Denote the price
of a call at time t as C(t), C(t) should be a function of S(t) and t. In order to obtain
the stochastic process of C(t), we apply It
ˆ
o’s lemma and obtain the following SDE
for C(S(t),t),
dC(t)=
∂
C
∂
t
+
μ
S
∂
C
∂
S
+
1
2
σ
2
S
2
∂
2
C
∂
S
2
dt +
σ
S
∂
C
∂
S
dW(t). (1.7)
3
The notation C
2×1
stands for the class of all functions that are twice differentiable with respect
to first argument (here X) and once differentiable with respect to second argument (here t).