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U.U.D.M. Project Report 2011:12
Examensarbete i matematik, 30 hpHandledare och examinator: Johan TyskJuni 2011
Department of MathematicsUppsala University
A Comparison of Local Volatility and Implied Volatility
Hui Ye
1
PrefacePublished in 1973, the Black-Scholes model has undoubtfully become one of the most frequently
used models in the financial derivative pricing field during the past few decades.As the option
pricing benchmark, the Black-Scholes model has shown an extensive application range, however,
some assumptions originally made in order to derive this model have been found unlikely to hold
in the reality. One of such controversial assumptions is that the volatility of underlying asset
remains constant throughout the entire option life. This has been criticized by more and more
researchers and practitioners as it is not in line with the research results on volatilities and the
observations accumulated from the real market data. Because of this shortcoming of Black-Scholes
model, people are eager to find such a model that incorporates the variability of the implied
volatility in the estabilishment of the model and still has similar volatility numerical values as
opposed to those implied by the Black-Scholes model. Our paper here takes a step in that direction
by investigating the replicating ability of a local volatility model. This study is featured by focusing
on the relationships between the implied volatility inferred by the Black-Scholes model, the local
volatility specified by the local volatility model and the volatility given by the Dupire's formula for
implied volatility.
2
Abstract
This paper mainly studies the relationships between the implied volatilities inferred by the Black-
Scholes model and the volatilities derived by the local volatility model. By studying the difference
between other volatilities and the implied volatilities, we can search for models that have similar
volatilities to those of Black-Scholes models, and yet still process more realistic and plausible price
processes that do not depend on a constant volatility, unlike the ones of the Black-Scholes models.
Our search for such models are illustrated by the local volatility model.
3
Acknowledgement
My gratitude and appreciation goes out to both of my supervisors. I would like thank Prof. Johan
Tysk for all the valuable advice during this study. Also I would like to thank Senior Lecturer Erik
Eström for the inspiration of the topic of this study. I would not be able to complete this study and
finish this paper without their constructive advice, kind help and sincere critiques. I personally
benefited a lot from this learning experience. I am really grateful for their help and this studying
opportunity. I am looking forward to working with them again someday.
4
ContentsChapter 1 Introduction-----------------------------------------------------------------------------------------5
1.1 Motivation---------------------------------------------------------------------------------------------------5
1.2 Objectives---------------------------------------------------------------------------------------------------5
1.3 Chapter Review---------------------------------------------------------------------------------------------6
Chapter 2. Background-----------------------------------------------------------------------------------------8
2.1 The Local Volatility Models-------------------------------------------------------------------------------8
2.2 The Dupire Model(Method)-----------------------------------------------------------------------------10
Chapter 3 Implied Volatility Models------------------------------------------------------------------------17
3.1 The Local Volatility Model------------------------------------------------------------------------------17
3.1.1 Option Pricing-------------------------------------------------------------------------------------------17
3.1.2 Implied Volatilities and The Local Volatilities------------------------------------------------------22
3.1.3 Implied Volatilities and The Dupire Volatilities-----------------------------------------------------41
3.1.4 Summary of Three Types of Volatilities-------------------------------------------------------------55
Chapter 4 Conclusions and Future Studies-----------------------------------------------------------------58
Notation---------------------------------------------------------------------------------------------------------60
AppendixA-----------------------------------------------------------------------------------------------------61
Appendix B-----------------------------------------------------------------------------------------------------89
Bibliography--------------------------------------------------------------------------------------------------130
5
Chapter 1. Introduction
1.1 Motivation
In general, volatility is a measure for variation of price of a certain financial instrument over time
in finance. There are many types of volatilities categorized by different standards. For example,
historical volatility is a type of volatility derived from time series based on the past market prices; a
constant volatility is an assumption of the nature of volatility that we usually make in deriving the
Black-Scholes formula for option prices. An implied volatility, however, is a type of volatility
derived from the market-quoted data of a market traded derivative, such as an option.
One of the most frequently used models, the Black-Scholes model which assumes a constant
volatility is used to derive the corresponding implied volatility for each quoted market price for
options. Indeed, the Black-Scholes model has been a great contribution to option pricing area
Nevertheless, there are still some facts that contradict the key assumptions in Black-Scholes model,
especially the constant volatility assumption. The evidence to this contradiction is a long-observed
pattern of implied volatilities, in which at-the money options tend to have lower implied volatilities
than in- or out-of-the-money options. This pattern is called "the volatility smile"(sometimes
referred to as "volatility skew") which was starting to show in American markets after the huge
stock market crash in 1987.
One explanation for this phenomenon is that in reality the volatility of an underlying asset is not
really a constant value throughout the lifespan of the derivative. That is why the volatility curve
plotted by the using of the values of implied volatility inferred by Black-Scholes model does not
appear to be horizontal, but displays a "volatility smile" in the plots. This, however, motivates us to
wonder whether such a model can be found, that gives a series of values of volatility close enough
to the volatility values in the volatility smile, i.e. , the implied volatility; or more specifically what
the difference between the volatility given by this alternative model and the corresponding implied
volatility inferred by Black-Scholes model is, if any.
Having this thought in mind, we can also apply this scheme of searching for suitable models to
testing among different types of models. Our demonstration in this paper uses the local volatility
model.
1.2 Objectives
In this paper, the alternative model type for the Black-Scholes model we use is the local volatility
model.
6
Our objective here is to set up the pricing model for options using the stock price processes and
other conditions specified by the local volatility model, solve the option values for this model,
calculate the corresponding implied volatilities for this model, thus to achieve our goal of
comparing these two volatilities, the implied volatilities and the local volatilities.
Besides the local volatility given by the local volatility model, we also want to compare the implied
volatilities to another local volatility, the dupire volatility. The Dupire volatility is a way of
calculating volatility under the Dupire model, which treats the strike price and the maturityKtime instead of the stock price and current time point as variables in the option valueT S tfunction . We will introduce this Dupire model and Dupire volatility in detail in),;,( tSTKVChapter 2. This additional analysis would give us some additional points of views to this local
volatility model here.
1.3 Chapter Review
Chapter 1, Introduction, mainly talks about the theoretical and practical reasons that motivate us to
write about this topic on implied volatility models in this paper, and sets straight the objectives of
our research as well.
Chapter 2, Background, introduces us to two types of models that will be focused on in the later
chapters of this paper. They are the the local volatility models and the Dupire local volatility model.
This chapter familiarize us to the basic knowledges of these models, and we will discuss these
models in detail including pricing option values and evaluating volatility values in Chapter 3.
Chapter 3, The Implied Volatility Models, concentrates on the difference between the implied
volatilities that are inferred by Black-Scholes model and the volatility factors that are specified by
the local volatility models, with all parameters of these two models staying the same. The former is
obtained by solving the volatility implied by the Black-Scholes formula for options reversely with
known option values. The latter in our paper here is the local volatility specified by the local
volatility model's price processes with known stock prices and time. And also, we are curiously that
what the difference between the implied volatility inferred by the Black-Scholes model evaluated
by a reverse calculation and the Dupire volatility computed by the Dupire method of calculating
volatility(which is called Dupire volatility) is, since both of the option values used in these two
models are essentially based on the Black-Scholes model. Hence, in short, between implied
volatility and local volatility(of the local volatility model), the implied volatility and the Dupire
volatility, we do two sets of cross-references by evaluating the distances between them to find their
inner connections. This is also the main idea of searching for a more realistic alternative asset price
processes for the Black-Scholes model(each different asset price processes correspond to a specific
7
option ricing model), which has the implied volatility close to that of an asset price process which
follows a Black-Scholes model. We finish this chapter by analysing our numerical results and plots.
Chapter 4, Conclusions, which sums up the conclusions for our research and the results in this paper.
8
Chapter 2. Background
In this chapter, we briefly introduce the models we use in this paper.
2.1 The Local Volatility Models
In the 1970s, when Black-Scholes formula was initially derived, most people were convinced that
the volatility of a certain asset given the current circumstance was a constant number. Then, later
on,after the economic crash in 1987, people were starting to doubt the constant volatility
assumption. Especially after more and more evidence of volatility smile was collected, people tend
to believe that the implied volatilities can not remain constant during the whole time. They
probably have some dependent relationships with some other factors in the option pricing model as
well. One of such guesses is that, the implied volatility could be depending on the stock price
and time . And if we study a model of price processes with a volatility that depends on the)(tS tstock price and time , we can try to explore the inner connection between the implied)(tS tvolatility , and the local volatility . The volatility in such models depends on theimpσ )),(( ttSσ
stock price and time . This is why we call these types of models the local volatility models,)(tS twhose volatilities are determined locally.
Hence, we take one example out of this category, and consider a case where the volatility is
decreasing with respect to the stock prices.
Given the local volatility model under an EMM(equivalent martingale measure, we use the same
acronym in the following) as following,Q
, (2.1)dtrdWStSdS t ⋅+⋅⋅= ),(σ
where we assume,
, (2.2)t
t SS 1)( =σ
. (2.3)0=rBy (2.2) and (2.3), the original model (2.1) is degenerated into the following form:
. (2.4)ttt dWSdS ⋅=
Denote the option value function as .),( tSV t
Hence, it follows from Ito formula and equation (2.4) that,
22
2
)(21 dSSVdS
SVdt
tVdV
∂∂
+∂∂
+∂∂
=
9
22
2
)(21 dWSSVdS
SVdt
tV
∂∂
+∂∂
+∂∂
=
(2.5)dSSVdtS
SV
tV
∂∂
+∂∂
+∂∂
= )21( 2
2
Thenwe consider delta-hedged portfolio,
. (2.6)SSVV∂∂
+−=π
Gven the martingale measure , thus under arbitrage-free condition, we will arrive at theQ
condition that,. (2.7)dtrd ⋅⋅= ππ
We re-write (2.6) in differential form that,
. (2.8)dSSVdVd∂∂
+−=π
Compare (2.8) with (2.7), then insert (2.5), the corresponding partial differential equation (PDE)
for model (2.4) takes the form,
. (2.9)021
2
2
=∂∂
+∂∂ S
SV
tV
If we let , and represent the option value, the first-order partial),( tSV t ),( tSV tt ),( tSV tSS
derivative with respect to variable , the second-order partial derivative with respect to variablet, respectively, (2.9) can be expressed in the following way,tS
. (2.10)0),(21),( =+ tSSVtSV tsstt
Model (2.4) is known as one of the local volatility models, whose form can be included into the
SDEMRDModel category inside the matlab database.
Creating the Local Volatility Model from Mean-Reverting Drift (SDEMRD) Models
The SDEMRD class derives directly from the SDEDDO class. It provides an interface in which the
drift-rate function is expressed in mean-reverting drift form:
, (2.11)tt
ttt dWtVXtDdtXtLtSdX )(),(])()[( )( ⋅+⋅−= α
where,
Xt is an NVARS-by-1 state vector of process variables;
S is an NVARS-by-NVARSmatrix of mean reversion speeds;
L is an NVARS-by-1 vector of mean reversion levels;
10
D is an NVARS-by-NVARS diagonal matrix, where each element along the main diagonal is the
corresponding element of the state vector raised to the corresponding power of α;
V is an NVARS-by-NBROWNS instantaneous volatility rate matrix;
dWt is an NBROWNS-by-1 Brownianmotion vector.
SDEMRD objects provide a parametric alternative to the linear drift form by reparameterizing the
general linear drift such that:
. (2.12))()(),()()( tStBtLtStA −==
Hence, we can create in matlab the model in
. (2.4)ttt dWSdS ⋅=
by inputing the following command in Matlab. SDEMRD objects display the familiar Speed and
Level parameters instead of A and B.
Table 2.1: The Local Volatility Model in Matlab
2.2 The Dupire Model(Method)
Frankly, this Dupire model is more of a method for calculating local volatilities than a pricing
model itself.
First of all, let us discuss some basic developments on the implied volatilities so far.
>> obj = sdemrd(0, 0, 0.5, 1) % (Speed, Level, Alpha, Sigma)
obj =
Class SDEMRD: SDE with Mean-Reverting Drift-------------------------------------------Dimensions: State = 1, Brownian = 1
-------------------------------------------StartTime: 0StartState: 1Correlation: 1
Drift: drift rate function F(t,X(t))Diffusion: diffusion rate function G(t,X(t))Simulation: simulation method/function simByEuler
Alpha: 0.5Sigma: 1Level: 0Speed: 0
11
One of the basic assumption in Black-Scholes is that the volatility of the underlying asset stays
constant during the entire time of option's lifespan. Hence, we can know from the Black-Scholes
formula for option prices, that, option prices has the following form
).,,;,( TKtSVV σ=
If we quote from the market date, the option price and underlying asset price of0VV = 0SS =an option with strike price and maturity at time point , we can obtain an0KK = 0TT = 0tt =equation from Black-Scholes formula for ,σ
. (2.13)),,;,( 00000 TKtSVV σ=
From Black-Scholes formula, we can calculate the Greeks, in particular, vega,
.0)()( 21 >==∂∂
= −− τφτφσ
ν ττ dKedSeV rq
Hence can be uniquely determined by equation (2.13).0σσ =
Since the volatility of the underlying asset is constant by assumption of the Black-Scholesσ
model. Then, theoretically the implied volatility derived from (2.13) should be a constant,0σσ =
i.e., independent of the strike price and maturity chosen here. However, in reality, this is0K 0T
contradicted by the existences of volatility smile and volatility skew. In fact, the implied volatility
inferred from option prices with different strike prices and expiration dates is a function ofσ
, [1].TK , ),( TKσσ =
The dependence on strike prices can be shown by the following figure 2.1 and figure. 2.2, given a
fixed maturity time and a fixed initial price at time point . The curve in0TT = 0SS = 0tt =figure 2.1 is called the volatility smile, the curve in figure 2.2 is called the volatility skew.
Similarly, the dependence on the maturity time can be illustrated by curve in figure 2.3, givenT
the stock prices and the strike prices stay unchanged. This shows the term structure ofS K
volatility.
12
Figure 2.1: Volatility Smile
Figure 2.2: Volatility Skew
Figure 2.3: The Term Structure of Volatility
13
To explore the characteristics of implied volatility in a more mathematical way, let us discuss the
model analytically.
Under risk-neutral measure, the underlying asset price process is
, (2.14)tdWtSdtqrSdS ),()( σ+−=
where is the risk-free interest rate, is the dividend yield, is the asset prices,r q S TttW ≤≤0}{
is a Brownian motion(Wiener process), is the asset's volatility that depends on asset pricesσ
and time .S tThus, by using the same approach as in section 2.1, we obtain the PDE for this option under Black-
Scholes model,
. (2.15)0)(),(21
2
222 =−
∂∂
−+∂∂
+∂∂ rV
SVSqr
SVStS
tV
σ
Adding the terminal and boundary conditions to equation (2.15), we can estabilish the following
value problem for option price, in particular, an European call option price.
Definition 2.1 is called the fundamental solution of the Black-Scholes equation, if),;,( TtSG ξ
it satisfies the following terminal value problem to the Black-Scholes equation:
⎪⎩
⎪⎨⎧
−=
=−∂∂
−+∂∂
+∂∂
=
)17.2(),(),(
)16.2(,0)(2 2
22
2
ξδ
σ
STSV
rVSVSqr
SVS
tvLv
where is the Dirac function.. □)(,0,0,0 xTtS δξ <<∞<<∞<<
Problem 2.2 Let be a call option price, satisfying the following terminal),,;,( TKtSVV σ=
value problem:
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
∞<≤−=
≤≤∞≤≤
=−∂∂
−+∂∂
+∂∂
+ )19.2()0(.)(),()0,0(
)18.2(,0)(),(21
2
222
SKSTSVTtS
rVSVSqr
SVStS
tV
σ
Suppose at*SS = )0(, 1** Tttt <≤=
is given as the boundary condition,),0(),(),,;,( 21** TTTKTKFTKtSV ≤≤∞<<=σ
find . □),0(),,( 21 TtTStS ≤≤∞<≤=σσ
Before we are ready to discuss the Dupire method, let us familiarize ourselves with some
theoretical background [1] beforehand.
14
Theorem 2.3 If the fundamental solution is regarded as a function of , then),;,( ηξtSG ηξ ,
it is the fundamental solution of the adjoint equation of the Black-Scholes equation. That is, let
,),;,(),( ηξηξ tSGv =
then satisfies),( ηξv
⎪⎩
⎪⎨
⎧
−=
=−∂∂
−−∂∂
+∂∂
−=∗
)21.2(),(),(
)20.2(,0)()()(2
22
22
Stv
rvvqrvvvL
ξδξ
ξξ
ξξ
ση
where □.,0,0 1ηξ <∞<<∞<< tS
Corollary 2.4 Theorem 2.1 indicates, if the fundamental solution of equation (2.18) is
, then),;,(* tSG ηξ
□).,;,(),;,( * tSGtSG ηξηξ =
The proof of above theorem 2.1 and corollary 2.2 can be referred to Lishang Jiang(1994)[1].
Then, let us move on to discuss the Dupire method in detail.
We denote an European call option price as
, define the second derivative of the option prices with respect to strike prices),;,( TKtSVV =
. (2.22)),;,(2
2
TKtSGKV
=∂∂
By equation (2.22) and (2.23), satisfies the system thatG
⎪⎩
⎪⎨⎧
−=
=−∂∂
−+∂∂
+∂∂
)24.2(,)(),(
)23.2(,0)(),(21
2
222
KSTSG
rGSGSqr
SGStS
tG
δ
σ
where is the Dirac function. We know that , thus (2.24) can be written)( KS −δ )()( xx δδ =−
as,
. (2.25))()(),( SKKSTSG −=−= δδ
Then by Definition 2.1, we know that is the fundamental solution to equation),;,( TKtSG(2.18). By Theorem 2.3, is the fundamental solution, as a function of (),;,( TKtSG TK , tS,are paramters), similar to (2.23) and(2.24), thus satisfies the following system,
15
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
∞<≤−=<∞<≤
=−∂∂
−−∂∂
+∂∂
−
)27.2()0(.)(),;,()26.2(),0(
,0)()()),((21 22
2
2
KSKTKtSGTtK
rGKGK
qrGKTKKT
G
δ
σ
We substitute (2.22) into (2.26), (2.27), then integrate both sides twice with respect to K in interval
. Since we know that,],[ ∞K
i) given a certain , if , for a call option, the following items will all tend to 0, i.e.,S ∞→K
,0)(,,,, 2222 →∂∂
∂∂
∂∂ GK
KKGKGK
KVKV σσ
ii) ηηδηηηδξξ
dSKdSdKK
)()()( −−=− ∫∫ ∫∞∞ ∞
,)(
)()(0
+
+∞
−=
−−= ∫KS
dSK ηηδη
iii) ,),;,( 2
2
KVdVdTtSG
K K ∂∂
−=∂∂
=∫ ∫∞ ∞
ξξ
ξξ
iv) ),,;,(),;,( TKtSVdTtSVK
−=∂∂
∫∞
ξξξ
v) ,),;,( 2
2
VKVKdVdTtxG
KK+
∂∂
−=∂∂
= ∫∫∞∞
ξξ
ξξξξ
vi) .),()),(( 2
22222
2
2
KVKTKdGTd
K ∂∂
=∂∂
∫ ∫∞ ∞
σηηηση
ξξ
Thus, we can transform the system of (2.33), (2.34) based on into the following),;,( TKtSG
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
∞<≤−==
<∞<≤
=−∂∂
−−∂∂
+∂∂
−
+ )29.2()0(.)(),;,()28.2(),0(
,0)(),(21
2
222
KKStTKtSVTtK
qVKVKqr
KVTKK
TV
σ
From equation (2.28), we obtain the explicit expression for implied volatility
. (2.30)
2
22
21
)(),(
KVK
qVKVKqr
TV
TK
∂∂
+∂∂−+
∂∂
=σ
16
This idea of Dupire's of calculating volatility seems to be simple and nice in theory.
However, when it becomes to the reality, when traders want to apply this into real market, the first
obstacle we must overcome is calculating the derivatives of option price, i.e., .2
2
,,KV
KV
TV
∂∂
∂∂
∂∂
And in fact, there is no simple analytical way to do it but to resort to some numerical approach, for
example, finite difference method, etc. Nevertheless, as we are about to see in chapter 3 section 2,
the numerical approach is not good enough for calculating this Dupire volatility, as a slight amount
change in option value would lead to some significant change in the value of derivatives, thus the
volatility value. We can almost say that using (2.30) to calculate implied volatility is ill-posed
[1](we will go to details about this in Chapter 3).
17
Chapter 3. Implied Volatility Models
In this chapter, we compare two different types of volatilities, the local volatility and Dupire
volatility, with implied volatilities under the structure of local volatility model.
3.1 The Local Volatility Model
As we establish in Section 2.1 that, given an asset's price process under an EMM with the riskQfree interest rate that0=r
, (3.1)dWSdS ⋅=
we will have the option pricing problem for an European call option as
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
∞→→−→
−=
=
≤≤=+
+
)5.3(.1),(,),(
)4.3(,)(),()3.3(,0),0(
)2.3()0(0),(21),(
SastSVKStSV
KSTSVtV
TttSSVtSV
S
sst
3.1.1 Option Pricing
Since there is no simple analytical solution for the system (3.2)-(3.5), we then have to resort to the
numerical way to solve the option terminal value problem for this system.
We use software Matlab in this paper to solve numerical problems.
After a closer examination, we realize that we have a terminal boundary value problem here instead
of an initial one, hence in order to use the built-in initial boundary value solver function in Matlab,
we have to substitute some variables in the problem to shift the terminal boundary problem to an
initial boundary problem in order fit this problem into the solving range of the built-in function.
If we denote the time-to-maturity as , then it becomes obvious that if any one of thesetT -=τ
three variables( ) is fixed, the other two will either move in the same direction or in theTt,,τopposite ones. Thus, for every given , we have difference between and is fixed, writtenτ T tin the differential form, i.e.,
. (3.6)dtdT -=
Also, in our model here, we have the asset price not dependent on time as shown in (3.1), thus we
are ready to substitute the time point variable with the maturity time parameter , and regardt Tthe maturity time as a variable as well as treating time-to-maturity as a parameter fromT τ
now on .
18
Thus, system (3.2)-(3.5) can be transformed into the following system,
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
∞→→−→−==
=
≤≤==+−
+
)10.3(.1),(,),()9.3(,)(),()8.3(,0),0(
)7.3()0(0),(21),(
0
0
SasTSVKSTSVKStTSV
TV
tTtTSSVTSV
S
TssT
Then, we can apply the built-in function pdepe in Matlab to solve the above problem.
pdepe is a function that solves initial-boundary value problems for parabolic-elliptic Partial
Differential Equations (PDEs) in one-dimension.
pdepe solves PDEs of the form:
. (3.11)⎜⎜⎝
⎛⎟⎠⎞
∂∂
+⎜⎜⎝
⎛⎟⎠⎞
∂∂
∂∂
=∂∂
∂∂ −
xuutxs
xuutxfx
xx
tu
xuutxc mm ,,,),,,(),,,(
The PDE holds for and . The interval must be finite. can be 0,fttt ≤≤0 bxa ≤≤ ],[ ba m1, or 2, corresponding to slab, cylindrical, or spherical symmetry, respectively. If , then0>m amust be non-negative.
In (3.11), is a flux term and is a source term. The coupling of the),,,(xuutxf∂∂ ),,,(
xuutxs∂∂
partial derivatives with respect to time is restricted to multiplication by a diagonal matrix
. The diagonal elements of this matrix are either identically zero or),,,(xuutxc∂∂ ),,,(
xuutxc∂∂
positive. An element that is identically zero corresponds to an elliptic equation and otherwise to a
parabolic equation, and there must be at least one parabolic equation. An element of thatccorresponds to a parabolic equation can vanish at isolated values of if those values of arex xmesh points. Discontinuities in and/or due to material interfaces are permitted provided thatc s
a mesh point is placed at each interface[2].
For and all , the solution components satisfy initial conditions of the form0tt = x
. (3.12))(),( 00 xutxu =
For all and either or , the solution components satisfy boundary conditions oft ax = bx =the form
. (3.13)0),,,(),(),,( =∂∂
+xuutxftxqutxp
Particularly, in our PDE (3.2) here, if we denote in (3.2), as , as , asS x T t ),( TSV
, then (3.7)-(3.10) become),( txu
19
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
∞→→→−==
=
≤≤=⋅=
+
)17.3(.1),(,),()16.3(,)(),()15.3(,0),0(
)14.3()0(21
0
0
xastxuxtxuKxttxu
tu
tttuxu
x
Txxt
In fact, from a mere observation in the real market, we know that underlying stock price 100=Sis quite high for an option with strike price . Then we can replace the infinity requirement10=K
of limits in equation (3.17) by setting stock price to , given a strike price . Then,100=S 10=K
(3.10) and (3.17) become
, (3.18)100,1),(,-),( === SwhereTSVKSTSV S
. (3.19)100,1),(,-),( === xwheretxuKxtxu x
Then (3.14)-(3.17) take the new forms of (3.20)-(3.23),
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
===−==
=
≤≤=⋅=
+
)23.3(.100,1),(,-),()22.3(,)(),()21.3(,0),0(
)20.3()0(21
0
0
xwheretxuKxtxuKxttxu
tu
tttuxu
x
Txxt
Now, let us rearrange (3.20) in the following form
. (3.24)⎟⎠⎞
⎜⎝⎛ −⋅⋅
∂∂
=∂∂ − )(
2100 uuxx
xx
tu
x
Comparing (3.24) to (3.11), i.e.
, (3.11)⎜⎜⎝
⎛⎟⎠⎞
∂∂
+⎜⎜⎝
⎛⎟⎠⎞
∂∂
∂∂
=∂∂
∂∂ −
xuutxs
xuutxfx
xx
tu
xuutxc mm ,,,),,,(),,,(
we find out that., (3.25)0=m
, (3.26)1),,,( =∂∂xuutxc
, (3.27))(21),,,( uux
xuutxf x −⋅=∂∂
. (3.28)0),,,( =∂∂xuutxs
Next step, let us identify (3.21)-(3.23) to initial and boundary conditions of the form (3.12)-(3.13).
(3.22) itself already complies with the form of (3.12), i.e., the initial condition. Hence, identifying
(3.21) and (3.23) to the boundary conditions of the form (3.13), i.e.,
20
, (3.13)0),,,(),(),,( =∂∂
+xuutxftxqutxp
is equivalent to finding pairs of values of function and function , which satisfies),,( utxp ),( txq
the form in (3.13), given the flux function by (3.27).)(21),,,( uux
xuutxf x −⋅=∂∂
If we substitute (3.27) into (3.13), we have
. (3.29)0)(21),(),,( =−⋅⋅+ uuxtxqutxp x
And the boundary conditions (3.21) and (3.23) are
⎪⎩
⎪⎨⎧
===
=
)31.3(.100,1),(,-),(
)30.3(,0),0(
xwheretxuKxtxu
tu
x
We insert (3.30) into (3.29) at , then0=x
. (3.32)0)0(21),0(),,0( =−⋅⋅+ xuxtqutp
For (3.32) to hold, one option is to put and to 0, i.e.,),,0( utp ),0( tq
⎩⎨⎧
==
)34.3(.0),0()33.3(,0),,0(
tqutp
Similarly, we insert (3.31) into (3.29) at , then100=x
. (3.35)0|)(21),100(),,100( 100=−⋅⋅+ =xx uuxtqutp
We simplify (3.35), obtain
.0
)(21),100(),,100(
|))(1(21),100(),,100(
|)(21),100(),,100(
100
100
=
⋅+=
−−⋅⋅+=
−⋅⋅+
=
=
Ktqutp
Kxxtqutp
uuxtqutp
x
xx
This is to say,
. (3.36)021),100(),,100( =⋅+ Ktqutp
For (3.36) to hold, we can simply choose a pair of values of and ,),,100( utp ),100( tq
⎪⎩
⎪⎨⎧
=
=
)38.3(.1),100(
)37.3(,21-),,100(
tq
Kutp
In conclusion, our boundary conditions now take the form of
21
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
=
=
==
)42.3(.1),100(
)41.3(,21-),,100(
)40.3(,0),0()39.3(,0),,0(
tq
Kutp
tqutp
After specifying all the conditions and function forms, we are ready to gather together all the
thoughts stated above to write them into a program file pdex_u.m (which is included inAppendixA,
Table A.1)in Matlab. We set the values for each one of the variables and parameters, the initial
value of maturity , strike price , risk free interest rate , dividend yield00 == tT 10=K 0=r, using 201 mesh points in the option price range from 0 to 100 and 51 mesh points in the0=q
maturity range from 0 to 5, to simulate the numerical option value at each price level. The option
value curves, option values plotted against asset prices, under different time-to-maturity periods τ
are shown in Figure 3.1(the more complete series of curves of option value at different levels of
time-to-maturity is included in Appendix A). The option price surface with respect to the time-to-
maturity and asset price is shown in Figure 3.2.τ
Figure 3.1: The OptionValue Curves of EuropeanCalls and the Payoff Diagram at Maturity
22
Figure 3.2: The OptionValue Surface for EuropeanCall Options
As shown in figure 3.1 and figure 3.2, without any unexpected outcome , the option value curve
and surface under this local volatility model have no substantial difference to those of a vanilla
European call option under a generic Black-Scholes model. The longer the period of time-to-
maturity, the more valuable the call options; the higher the stock/asset price, the closer to payoff
the option values at maturity.
In the following subsection, we try to find out the internal connection between the implied
volatilities and the local volatilities.
3.1.2. Implied Volatilities and The Local Volatilities
With the preparations in Section 3.1.1, we can now move on to calculate the implied volatilities of
this local volatility model for each mesh point in the two-dimensional space consisted of asset
prices and time. As we mentioned in Section 2.2, the volatility is determined by the Black-Scholes
formula for option prices uniquely, given the other inputs, such as asset price , interest rate ,S r
dividend , maturity , time , strike , option price and so on. This is to say, we canq T t K V
derive a unique implied volatility from Black-Scholes model, in other words,impσ
exists and is unique.),,;,( VtKTSimp σσ =
23
The existence of the implied volatility can be observed from the corresponding relationship
between the option price and the implied volatility . This is true by the formulation of theV impσ
Black-Scholes formula for option pricing. The problem of the uniqueness of the implied volatility
can be solved by the monotonicity of the option price with respect to the maturity timeimpσ V
. For a call option value , where are variables. areT ),,;,( tKTSVVCall σ= TS , tK ,,σ
parameters. We know that the one of the Greeks in Black-Scholes formula for call options, vega,
[3]. Hence, given any value set of , we will find a unique0- <∂∂
=∂∂
=tV
TV
ν ),;,,( tKTSV
value for , which is called the implied volatility, denoted as . For example, , forσ impσ 0σσ =imp
an input set of .),;,,(),;,,( 0000 tKTSVtKTSV =
Therefore, we can regard as a function of , where are parameters, is alsoσ TS , tK , V
quoted from market price, i.e. . While at the same time, the local volatility),,;,( VtKTSσσ =
denoted as can be easily observed from the price processes of this local volatility model, thatlocσ
, for each mesh point in the price axis. Thus, the distance between two correspondingSloc1
=σ
volatilities can be easily calculated. The program for implied volatilities' calculation pdex_imp.m is
included inAppendixA TableA.2.
The plots of implied volatilities and local volatilities are shown in the following.
24
Figure 3.3: The Implied Volatility Curves(Plotted against the Stock Price )S
Figure 3.3 is the implied volatility curve plotted against the stock prices at three different time
points. As we can see in figure 3.3, the implied volatility of the option is quite large (In fact, when
the stock price is close to 0, the implied volatility tends to infinity. We will discuss this in detail at
this end of Section 3.1.2) at those points where the stock prices are close to 0, and as the stockS
price goes up, the implied volatilities gradually fall back to a relatively low and stable level.S
The implied volatility decreases at a decreasing speed as the stock price increase. From an
economic point view, if the stock prices drop to a level close to 0, then the options based on the
same stock will be extremely risky, thus the indicator of riskiness will be extremely large, i.e.
as . On the contrary, the higher the stock price , the less risky the call,∞→impσ 0→S S
option value . However, the decreasing of the riskiness of the underlying asset that the option isV
based on, is not enough to reduce all of the risks that the option is facing, some of which are some
systematic risks, such as the macroeconomic status and so on. Hence, as the stock price increases,
eventually, the implied volatilities will tend to a stable non-negative level in general. However, in
our stock price process here, we assume the risk-free interest rate is 0. This means that investors are
25
not rewarded at an interest for taking the systematic risk, this means that the systematic risk0=r
is 0. Hence, in our special case here(the risk-free interest rate is 0), when the stock price tends to
infinity, the implied volatility tends to 0. The term structure of the implied volatility is shown in
figure 3.4.
Figure 3.4: The Implied Volatility Curves(Plotted against the Time-to-maturity )τ
Figure 3.4 shows the term structure of the implied volatilities at different stock price levels. As the
maturity time comes closer, the option value will become more volatile, hence the implied
volatility will become higher.And as the stock price increases, the curve of implied volatilities
plotted against the time-tom-maturity will shift downward as a whole.
26
Figure 3.5: The Local Volatility CurveSloc1
=σ
Figure 3.5 shows the local volatility curve that given by which only depends on theSloc1
=σ
stock price . And, we know that as , as well asS ∞→=Sloc1
σ 0→S 01→=
Slocσ
as .∞→S
From the illustration of above figure 3.3-3.5, we find out that the implied volatility and theimpσ
local volatility almost have the same tendency of change. Then, we are more curious to findlocσ
out exactly how far away they are from each other.
The Distance between Implied Volatilities and Local Volatilities
Since we have calculated the value of implied volatilities and are aware of that the localimpσ
27
volatility has the form , then we can find out the distance between and bySloc1
=σ impσ locσ
distance function .S
d implocimp1
−=−= σσσ
We put our theory here into practice by program file pdex_dis_imp_loc.m written in matlab(this
program is include in Appendix A). All the parameters and indicators that need to be specified are
gathered in the following table 3.1.
Table 3.1: The Initial Variable Set-up for Program pdex_dis_imp_loc.m
The plots of this section is shown in the following figure 3.6 and figure 3.7.
Price(stock/asset price) 201 mesh points, from 0 to 100.Strike(option strike price) 10Rate(risk-free interest rate) 0Time(time-to-maturity) 51 mesh points, from 0 to 5Value(option value) 51×201 values, calculated in Section 3.1.1
Limit(the upper bound for volatility searchinginterval)
10 times
Yield(dividend yield) 0Tolerance(calculation accuracy) 10-16
Class(option type) call option
28
Figure 3.6: The Comparison of Implied Volatility and Local Volatility ( )10=K
Figure 3.6 is a demonstration of how the distance between the implied volatility and the local
volatility changes as the stock price increases.At first, the the local volatility curve is above the
implied volatility curve, then as the stock price increases, the local volatility decreases more rapidly,
then at a certain stock price level, they intersect, and after that the implied volatility curve lies
above the the local volatility curve. The change of distance between them is shown by the distance
curve marked in black in figure 3.6. Before the stock price reaches the strike price, as the stock
prices increases, the distance decreases rapidly; then at the point when the stock price is equal to the
strike price, the distance reaches 0; after the stock price exceeds the strike price, the distance
gradually increases to a certain relatively low level and stays that way as the stock price continues
to increase.
29
Figure 3.7: TheAbsolute Difference between Implied Volatility and the Local Volatility Curve
(PlottedAgainst , )imploc σσ − S 10=K
When plotting distance curves between the implied volatilities and the local volatilities solely, we
obtain the three curves shown in figure 3.7, each of which represents a different time-to-maturity
level. We can hardly tell them apart without magnifying them since they are lying very close to
each other in figure 3.7. Nevertheless, we can almost say affirmatively that the distance between
these two volatilities are essentially 0 at the point where the stock price is equal the optionSstrike price ; while at those points where the price doesn't reach the strike price level from theKbelow, the distance between them are relatively far away. On the other hand, as the stock price goes
up from above the strike price, the distance between them then will be maintained at a quite stable
level. The more accurate and actual computation results can be read from numerical results
included inAppendix B.
30
Figure 3.8: TheAbsolute Difference between Implied Volatility and the Local Volatility Curve
(PlottedAgainst )imploc σσ − T
In order to show how the distance between the implied volatility and the local volatility changes as
time goes by, we plot figure 3.8. Notice in the distance function that
,S
d implocimp1
−=−= σσσ
the only time-sensitive factor in it is the implied volatility . Hence, at the parts of curvesimpσ
where the maturity time is far away, the distance curves in figure 3.8 reveal some similarT
nature to implied volatility curves in figure 3.4, as they both tend to stay at a relative stable level,
almost parallel to the time-to-maturity axis. Another interesting fact can be observed in figure 3.8 as
well is that: when the stock price is below the strike price , the distance curve shiftsS K
downward as the stock price goes up; when the stock price is above the strike price ,S S K
the distance curve shifts upward as the stock price continues to increase. The distance curveS
hits the bottom when the stock price is equal to the strike price . This observation again, isS K
consistent with our conclusions in figure 3.6 and figure 3.7.
31
The Limits of Implied Volatilities and Local Volatilities at S=0.
One vague statement that we have not really explained in this section is that we say the implied
volatility is very large for those points at which the stock prices are close to 0.SAlthough the plots of the implied volatility curves have indicated that the implied volatilities would
more than likely to go to infinity when the stock price tends to 0. However, we still need more
concrete evidence to prove our speculation here.
We know from initial condition of option pricing that when the stock price falls back to 0, the
option value is also 0, meaning that the ownership of this asset is worthless. Then it makes no sense
to talk about the implied volatility of the option value. Thus, we choose a small neighbourhood of 0
on the stock price axis with its left side end open. For example, we choose . ByS ]10,0( 10−=Susage of the option pricing scheme described in Section 3.1.1, we calculate the option values within
this small interval. We modify our previous program for option pricing by equally choosing 201
mesh points on interval and adding a single point to the collection of]10,0[ 10−=S 100=Smesh points. In this way, we can both achieve the pricing for option prices at small stock price
points and keep our boundary conditions unchanged. This altered program for option pricing is
named as pdex_u_small_s.m, which is included inAppendixA. Then, we use the same scheme for
implied volatility calculations as before. The program file of implied volatility computation for
small , pdex_imp_small_s.m is included inAppendixA as well. The more detailed initialSparameter set-up for program pdex_imp_small_s.m is displayed in the following table 3.2.
Table 3.2: The Initial Variable Set-up for Program pdex_imp__small_s.m
The plots of this program are shown in figure 3.9-3.12. Figure 3.9 depicts the implied volatility
curves plotted against the stock prices at different time-to-maturities. However, according to figure
Price(stock/asset price) 202 mesh points, 100 points from 0 to 10-10 ,and 1.
Strike(option strike price) 10Rate(risk-free interest rate) 0Time(time-to-maturity) 51 mesh points, from 0 to 5Value(option value) 51×202 values
Limit(the upper bound for volatility searchinginterval)
109 times
Yield(dividend yield) 0Tolerance(calculation accuracy) 10-18
Class(option type) call option
32
3.9, the implied volatility at is not a very large number, although the slopeimpσ 1010.10 −⋅=S
is quite steep in the neighbourhood of . Notice that the scale of the volatility axis and the0=S
scale of the stock price axis are obviously different, the latter is enormously larger comparing to the
former. So, if we put both axis to the same measure scale, then the implied volatility curve will be
extremely steep in this interval [0,10-10]. From the above argument, we realize that the steepness of
the implied volatility curve is not necessary an accurate way to determine whether the implied
volatility tends to infinity at .0=S
As an alternative of graphical analysis, let us consider the derivative of implied volatility with
respect to the stock price . Notice that our partition of the interval [0,10-10] is enough small,Simp
∂∂σ
hence we can use the implied volatility value at each mesh point and the step size of the partition to
approximate the derivatives at each point. For example, we select some results of impliedSimp
∂∂σ
volatility from program's data(the numerical results of the implied volatilities calculated by
pdex_imp_small_s.m, which are included inAppendix B).
Table 3. 3: The Slopes of Implied Volatility Curves
As we can see in table 3.3, that the derivative is a really large negative number whenSimp
∂∂σ
τ0impσ
1impσ 100 - SSS =∆0
10
SSimpimpimp
∆
−=
∂
∂ σσσ
2.8 3.71288296174587 3.6618307127546212105.0- −⋅
1110.021-82.4861021044979-
⋅≈
3.8 3.23962470398632 3.1956475324642212105.0- −⋅
1110-0.874.21168795434304-
⋅≈
4.8 2.92041541613200 2.8811834011648012105.0- −⋅
1110.780-4.40027846402993-
⋅≈
33
the stock price is . And we know that the implied volatility is a positiveS 1010−impσ
number at . Hence, if the value of the slope of the implied volatility curve stays1010−=SSimp
∂∂σ
at the current amount, the implied volatility will eventually go to plus infinity as the stock price
continues to decrease from below . This is to say, if holds, then1010−=S 02
2
≥∂∂Simpσ
. Now, all we have to do make sure that is true. We plot the slopes of+∞=→ impSσ
0lim 02
2
≥∂∂Simpσ
implied volatilities into curves in figure 3.10, with respect to the stock prices . And indeed, as itS
is shown in figure 3.10, the slopes of the "slope curves" are truly non-positive, i.e., . In02
2
≥∂∂Simpσ
fact, we can also read from the numerical results of slopes of implied volatility curves to arrive at
the same conclusion(see table 3.4). The slopes of implied volatility curves are decreasing as the
stock price decreases, i.e., . Then, we can say surely that the implied volatility02
2
≥∂∂Simpσ
impσ
tends to infinity as the stock price goes to 0(The more complete numerical results of slopesS
are included inAppendix B).
In short, this is to say, because
i) for ;0|0>=SSimpσ )10( 10
0−=οS
ii) , for ;)10(1| 100
οσ
⋅−=∂
∂=SS
imp
S)10( 10
0−=οS
iii) for , ;02
2
≥∂∂Simpσ
00 SS ≤≤ )10( 100
−=οS
that we have
for , and thus .∞=→ impSσ
0lim 00 SS ≤< ∞=
→ impSσ
0lim
34
Table 3. 4: The Slopes of Implied Volatility Curves
Figure 3.9: The Implied Volatility Curves(Against Stock/Asset Price )S
\Sτ 12105.0 −⋅ 12100.1 −⋅ 12105.1 −⋅ 1210.02 −⋅
2.8 -102104497982.486 -60234023483.1758 -42968495088.3899 -33463371536.9570
3.8 -87954343044.2116 -51889446183.0994 -37017182609.4161 -28829357814.5602
4.8 -78464029934.4002 -46292532890.8126 -33025342546.6038 -25721008566.0331
35
Figure 3.10: The Slope Curves of Implied Volatility Curves(Against Stock/Asset Price )S
Now that we know for a fact that the implied volatility tends to infinity as stock price goes to 0, and
also that the local volatility tends to infinity as the stock price goes to 0, since . Then,Sloc1
=σ
we are even more curious about the their speeds of converging to infinity, in order to make better
judgements when approximating the implied volatility by local volatility at small stock prices.
First of all, we draw figures for the local volatility curve and the slope curve, which are shown in
figure 3.11 and figure 3.12.
36
Figure 3.11: The Local Volatility Curve (Against Stock/Asset Price ).50-1 SS==σ S
Figure 3.12: The Slope Curve of Local Volatility Curve .51-
3 21-1
21- S
SS ⋅==σ
37
We know from , that And the shape of curves in figureSloc1
=σ .0limlim0
=∞=∞→→ locSlocSσσ ,
3.11 and 3.122 confirms that. Notice that the order of magnitude of the vertical axis in figure 3.11
and 3.12 are completely different from that of figure 3.9 and 3.10. So, we can say affirmatively that
the speed of the local volatility's converge to infinity is much more faster than that of the implied
volatility's. But the question is that how much faster the former is. We know in general, that the
implied volatility curve is below the local volatility curve on interval [0,10-10], i.e.,
. We also know that, the corresponding slope curves of the implied5.00 −=<<< Slocimp σσ
volatilities are above those of the local volatilities due to the fact that their signs are negative, i.e.,
. These motivate us to find out the exact magnitude of implied05.0 5.1 <∂
∂=
∂∂
=⋅− −
SSS imploc σσ
volatility and its 1st order derivative , or at least the range of it. One way to do it isimpσSimp
∂∂σ
by fitting and into some certain potential functions' forms. More specifically,impσSimp
∂∂σ
we find the value ranges of and , andα β upperlower ααα −<−<−<0
that satisfy and0<<< upperlower βββ upperlower SS impαα σ <<<0
.05.05.0 <∂
∂⋅−<
∂∂
<∂
∂⋅−
SS
SSS upperlower
impββ σ
We establish our algorithm for searching such ranges by the following way. First of all, from a
simple observation of previous plots and some tests on the matlab program, we choose a rough
lower bound of searching region of , -0.5(motivated by the degree of S in local volatility), aα
upper bound -0.01(motivated by some simple testing on matlab). Also, we choose a rough lower
bound of searching region of , -1.5(motivated by the degree of S in slope of local volatility), aβ
upper bound -0.01(motivated by some simple testing matlab). Set the searching step size to 0.01,
then we can begin our search for such suitable values of and . The search programα β
pdex_imp_slope_small_s_fit.m is included inAppendixA.And the results of our searching is
38
.90.0,00.1,12.0,03.0
−=−=
−=−=
upper
lower
upper
lower
ββ
αα
Thus, we can estimate the implied volatility and its 1st order derivatives as
and , for .21.03.000 −− <<< SS impσ 05.05.0 09.00.01 <⋅−<∂
∂<⋅− −− S
SS impσ
]10,0( 10−∈S
The plots for demonstrating such estimations are shown in figure 3.13 and figure 3.14.
The result of this estimation is to say that if we were to express the implied volatility in aimpσ
potential form, that , where . Since the local volatilityασ Simp = 03.02.10 −<<− α
is given, hence , i.e., .5.0−= Slocσ 0limlimlim0 5.0
12.0
005.0
03.0
0=≤≤= −
−
→→−
−
→ SS
SS
Sloc
imp
SS σσ
0lim0
=→
loc
imp
S σσ
Similarly, if we present the 1st order derivative of the implied volatility with respect to stock/asset
price in a potential form, that , where . Also, knowingSimp
∂∂σ ασ Simp = 90.00.01 −<<− β
the 1st order derivative of a local volatility , we then can compute the limit5.15.0 −⋅−= Slocσ
, i.e., .0.50-.50-limlim
.50-
.50-lim0 5.1
9.0
005.1
0.1
0=
⋅⋅
≤
∂∂∂
∂
≤⋅⋅
= −
−
→→−
−
→ SS
S
SSS
Sloc
imp
SS σ
σ
0lim0
=
∂∂∂
∂
→
S
Sloc
imp
S σ
σ
In addition, we can write that . Again, let us look back on the distance between the∞=→
imp
locS σ
σ0
lim
implied volatility and the local volatility. Facts are , ,5.0−= Slocσ 5.15.0 −⋅−=∂∂ SSlocσ
, and for .21.03.000 −− <<< SS impσ 05.05.0 09.00.01 <⋅−<∂
∂<⋅− −− S
SS impσ
]10,0( 10−∈S
Hence the distance function satisfieslocimpd σσ −=
⎪⎩
⎪⎨
⎧
−⋅−=⋅−−⋅−>∂
∂−∂=
∂∂
>−>=−=
−−−−
−−
),(5.0)5.0(5.0
,0-
9.05.19.05.1
12.05.0
SSSSSS
d
SSd
imploc
imploclocimp
σσ
σσσσ
where, .]10,0( 10−∈S
39
And we know that and−∞→−⋅=−⋅− −−− )1(.50-)(5.0 6.05.19.05.1 SSSS
as , which means as∞→−⋅=− −−− )1( 38.05.012.05.0 SSSS 0→S ∞→−= locimpd σσ
.0→S
This is to say that the distance between these two volatilities is infinitely far away. So, when the
stock price is really close to 0, it would be not suitable to approximate the implied volatility by the
local volatility due to the fact that their distance function tends to infinity at 0.
40
Figure 3.13: The Fitting to Potential Function of The Implied Volatility Curves
Figure 3.14: The Fitting to Potential Function of The Implied Volatility Slope Curves
41
3.1.3. Implied Volatilities and The Dupire Volatilities
In this section, we concentrate on finding the inner connections between the implied volatilities and
the Dupire volatilities. The approach and ideas to solve the implied volatilities for the local
volatility model are the same as shown in Section 3.1.2 stated above. So, our focus here is mainly
concentrated on finding the Dupire volatilities.
The Dupire method is an approach of calculating volatilities by using market-quoted inputs, such
as , , , and so on. This local volatility is called the Dupire volatility.TV∂∂
KV∂∂ V 2
2
KV
∂∂
From formula (2.30) in Chapter 2, we know that
. (3.43)
2
22
21
)(),(
KVK
qVKVKqr
TV
TK
∂∂
+∂∂−+
∂∂
=σ
And also , thus0== qr
. (3.44)
2
22
21
),(
KVK
TV
TK
∂∂
∂∂
=σ
Up till now, the option value we have discussed is a function of asset price and maturity timeS, i.e., , where strike price is regarded as a parameter, relatively fixed, comparing toT ),( TSV Kand .S T
According to formula (3.44), in order to calculate the Dupire volatility, we have to obtain the values
for the derivative and first.2
2
KV
∂∂
TV∂∂
To calculate the derivatives, our plan here is to choose a spectrum of strike prices , and for eachK, we use the same scheme of pricing options as before while the strike price is treated as if itK K
is a constant. Thenwe collect all the option values calculated in this way and rearrange them not
only by the time-to-maturity indexes and the stock price indexes, but also by the strike price
indexes.
Hence, the option prices calculated in this way will be a 3-dimensional space composed of time,
price and strike. As abstract as it is, this would enable us to implement the finite difference method
to calculate the two deorivatives mentioned in (3.44).
Then, let us discuss the practical way of implementing the finite difference method.
42
Finite difference method is a numerical method that approximates the solutions to differential
equations by replacing derivative expressions with approximately equivalent difference quotients.
We use the forward difference algorithm here. Suppose function 's 1st and 2nd order)(xfderivatives at exist, then we can use the following method to approximate andax = )(af ′
.)(af ′′
; (3.45)0,)()()( →−+
=′ hash
xfhfaf
. (3.46)0,)()(2)2()( 2 →++−+
=′′ hash
xfhxfhfaf
If we apply this algorithm to our call option here, using the same notation, that,
for a given asset price and a maturity time ,S T
, (3.47)2
);,();,(2)2;,(),(h
KtxuhKtxuhKtxuTKCKK++−+
=
, (3.48)h
KtxuKhtxuTKCT);,();,(),( −+
=
where the variable represents asset price , variable stands for the maturity time , andx S t Tis a parameter in function .K u
First of all, let us compute the option values. The parameters' set-up in this subsection is organized
in table 3.5.
Table 3.5: The Initial Parameter Set-up for Program pdex_dupire.m
The numerical results of option values are calculated by program pdex_dupire_option.m(the m file
is included inAppendixA). The plot of option value surface is shown in figure 3.15 and the option
value curve plotted against the strike price is shown in figure 3.16.KFigure 3.15 and figure 3.16 are in accordance with our intuitions, that the option value isV
Price(stock/asset price) 101 grid points, from 0 to 100.Strike(option strike price) 21 mesh points, from 9.18① to 11.Rate(risk-free interest rate) 0Time(time-to-maturity) 21 mesh points, from 0 to 2Value(option value) 21×101×21 values
Limit(the upper bound for volatility searchinginterval)
100 times per annum
Yield(dividend yield) 0Tolerance(calculation accuracy) 10-6
Class(option type) call option
① The starting point of the mesh point setseems like an unconventional choice. However, this enables us to placethe middle point of 21 mesh points on a point with value K=10, i.e., K21=11, K11=10 K1=9.18. The value of→middle point of strike price spectrum is in line with the strike price we use in calculating the implied volatility.
43
positively correlated with the stock price and negatively correlated with the strike price .S KThe latter is not quite obvious shown in figure 3.15, but can be observed from figure 3.16.Although
the strike price's impact on the option value is not as much as the stock price , still a higher strikeSprice level will lead to a decrease in option value to a certain extent.
Figure 3.15: The OptionValue Surface
44
Figure 3.16: The OptionValue Curves(Against the Strike Price )K
Then, we can use the numerical results of the option prices and the algorithms for derivatives
calculations to compute the Dupire volatilities. The program code is pdex_dupire.m (included in
AppendixA). The plots from this program are shown in figure 3.17 and figure 3.18.
The dependence of the Dupire volatility on the time-to-maturity is reflected in figure 3.17.Asτ
the maturity time approaches, the Dupire volatility increases at an increasing speed. This pattern,
however, is in consistence with the implied volatility curves in figure 3.4. They all display a
property of the volatility curves that (if the derivatives exist).0,0 2
2
<∂∂
<∂∂
TTσσ
Besides the above, curves in figure 3.17 also show that the Dupire volatility is sensitive to strike
price K as well. A small increment of 0.1 in strike price K leads to a huge upward shift of about 1 in
the volatility curve.
45
Figure 3.17 The Dupire Volatility Curve(PlottedAgainst Time-to-maturity )τ
All other things equal, panel a of figure 3.18 is plotted with a strike , while panel b is.99=Kplotted with a strike . However, this small difference in choosing strike prices leads to a10=Khuge difference in the shapes of Dupire volatility curves, as shown in panel a and panel b of figure
3.18.
Figure 3.18, panel a, describes the Dupire volatility curve at , . Figure 3.18,9.9=K 9.1=τ
panel b, depicts the Dupire volatility curve at , . Their shapes are quite different10=K 9.1=τ
from the the implied volatility's and the local volatility's. The Dupire volatility curves change the
convexity at least once throughout the the range within which the stock price changes.As for the
implied volatility and local volatility curves, their convexity don't change on the interval for stock
prices, this can be observed from curves in figure 3.6. Besides the differenceswith other volatility
curves, the Dupire volatility curves themselves are different from one another. Graphically, we can
see that the convexity changes at least twice in panel a while it only changes at least once in panel b.
Moreover, it's quite obvious that the curve is more peaked in panel b since the difference between
the extreme point and the lowest point in the Dupire volatility curve is much larger.Without ruling
out the impact resulted from choosing the finite difference method in calculation, we can still tell
that the Dupire volatility is extremely sensitive to strike prices K.
46
Figure 3.18 a : The Dupire Volatility Curve(K=9.9)
Figure 3.18 b : The Dupire Volatility Curve(K=10)
47
Although, it appears that the Dupire volatility will drop to 0 rather rapidly as the stock price
increases.But in fact, the Dupire is not as nice as it seems in the diagram. Recall the algorithms of
derivatives and the formula for calculating the Dupire volatility,
, (3.47)2
);,();,(2)2;,(),(h
KtxuhKtxuhKtxuTKCKK++−+
=
. (3.48)hKtxuKhtxuTKCT);,();,(),( −+
=
. (3.44)
2
22
21
),(
KCK
TC
TK
∂∂
∂∂
=σ
computed by (3.48) is a positive real number. This is guaranteed by the monotonicity),( TKCTof option value with respect to the time-to-maturity . However, we can not say the same forV Tthe 2nd order derivative because (3.47) is not necessarily a positive number since),( TKCKK
function is not necessarily a convex function with respect to in the interval we),,( Ktxu Kchoose. Hence, the numerical solution of (3.44) could be a complex number with a non-zero
imaginary part. In such scenarios, the Dupire volatility can not be computed by the formulation
(3.44) proposed by Dupire.
However, the numerical existence problem of the Dupire volatility while using the finite difference
method is only part of the difficulties we might encounter in Dupire volatility calculation here.
We notice that, in figure 3.18, after the stock price reaches a certain high level, the Dupire volatility
begins to display some irregularity and inconsistency in the movement of the volatility curve as the
stock price continues to increase.And eventually, the Dupire volatility vanishes when the stockSprice reaches 80(take figure 3.18 for an example). In retrospect, during our previous discussion in
section 3.1.2, the implied volatility curve in figure 3.3 shows a relatively stable property when the
stock price is at a relatively high level, and this property continues to remain so even when theSstock price continues to increase.And naturally, we would wonder why is that these twoSdifferent volatilities have so extraordinarily different behavior at high stock prices. To solve this
mystery, we need to go back to the Dupire formula for volatility calculations, equation (3.44).
Dupire formula states that, the volatility of an option can be calculated by the following,
. (3.44)
2
22
21
),(
KCK
TC
TK
∂∂
∂∂
=σ
48
The derivatives (for computing the Dupire volatility) required here, are the essential2
2
KC
TC
∂∂
∂∂
,
inputs for determining the Dupire volatility's value.
Let us first examine the 1st order derivative of option price with respect to the maturity-time, .TC∂∂
We know from experience and intuition that is a positive number, our interpretation is thatTC∂∂
longer time-to-maturity produces a high option value. In fact, from an observation of the numerical
lincluded inAppendix B), we find out that, when the price of underlying asset/stock is high enough,
the option value curves lie parallel to the payoff diagram at maturity. This is shown by figure 3.19.
And, another fun fact is that, at large stock prices, the increment between two option values whose
maturity-times are adjacent to each other is a constant. This is to say, , wherekKTSTC
=∂∂ ),,(
is a constant and is quite large comparing to the strike price . Besides our numerical0>k S K
results included inAppendix B, we can conclude the same results by the curve in figure 3.20.
49
Figure 3.19 : The OptionValue Curve( )1.0:0.2:0=τ
Figure 3.20 : The DerivativeTCCT ∂∂
=
50
Figure 3.21 : The DerivativeKCCK ∂∂
=
As for the derivative , we consider its corresponding 1st order derivative .2
2
KCCKK ∂
∂=
KCCK ∂∂
=
We can read from the numerical results of program pdex_dupire_option.m that, the 1st oder
derivative tends to be piecewise constant when the stock price is relatively quite highKCCK ∂∂
=
comparing to the strike price (say from 81 to 100). The graph to illustrate this is shown in figureK
3.20. Hence, the 2nd order derivative is 0 in the region that . This2
2
KCCKK ∂
∂= ]100,81[∈S
indicates that, the Dupire volatility
can not be evaluated by the finite difference numerical method for
2
22
21
),(
KCK
TC
TK
∂∂
∂∂
=σ
[81,100].∈S
This explains why the Dupire volatility can not be computed for large stock prices when using the
51
finite difference numerical method.
And notice that in figure 3.18 that, there is a small jump around , after examining the]80,70[∈S
numerical results around those points, we find out that this is caused by the incompatibility between
and . To be more specific, incompatibility is referred to that at thoseTCCT ∂∂
= 2
2
KCCKK ∂
∂=
jump points, the value of still exists and is positive under the evaluation by finiteTCCT ∂∂
=
difference method, while the value of has already become 0. In conclusion, this2
2
KCCKK ∂
∂=
explains why the Dupire volatility curve behaves in this way in figure 3.18.
Distance Between Implied Volatilities and Dupire Volatilities
By distance function , we are able to compute the distance between the implieddupireimpd σσ −=
volatility and the Dupire volatility.
Then, we canmove on to calculate the distance between implied volatilities and Dupire volatilities.
We include the program file pdex_dis_dupire.m in AppendixA. The plots for illustrating the
distance between the implied volatilities and the Dupire volatilities are shown in the following
figure 3.22 and 3.22. Panel a and panel b in figure 3.22, are the distance curves plotted against the
stock price at , and at , . Figure 3.23 is a plot of theS 9.1=τ 9.9=K 9.1=τ 10=Kdistance value of mesh points within the interval , plotted against the strike price]49,40[∈S
(The plots for the other groups are included inAppendixA).KFor the curve in figure 3.22, panel a, on one hand, the distance curve has some similar patterns as
opposed to the Dupire volatility curve in figure 3.18 a, i.e., the distance curve is truncated for large
stock price. The distance between these two volatilities can not be measured by distance function
using finite difference method due to the fact that the Dupire volatility can not be calculated from
the finite difference method for large stock prices. The reason of this is stated in the illustration of
figure 3.18 a. On the other hand, the distance curve in figure 3.22 has also inherited some properties
from the implied volatility curve demonstrated in figure 3.3, that as the stock price goes up the
distance between these two volatilities decreases as long as the stock price does not exceed the
certain upper boundary for which the Dupire volatility is truncated by using the finite difference
numerical method.
As for the curve in figure 3.22, panel b, resembles some similar features of the curve in figure 3.18
b. The distance between the Dupire volatility and the implied volatility reaches its maximum point
52
around , and hits its minimum around at which the difference between the47=S 70=SDupire volatility and the implied volatility is almost 0.And the curve in panel b is more peaked than
the curve in panel a. Because of the sensitivity of the Dupire volatility curve to strike price K, hence
the absolute difference curve between the Dupire volatility and the implied volatility is highly
sensitive to the strike price K as well.
Because of these properties, we have to impose more constraints on the usage of approximating the
implied volatility by Dupire volatility, since we not only have to make sure the distance between
these two volatilities is controllable but also in numerical existence under the finite difference
method we use here. Besides these, in fact, there are still more obstacles in applying this
approximation scheme. This is because the Dupire volatility calculated by the finite difference
numerical method does not always show a nice consistency between adjacent mesh points. We can
observe this problem from the numerical results included inAppendix B.
Figure 3.22 a: TheAbsolute Difference between Implied Volatility and The Dupire Volatility
(PlottedAgainst Stock Price )dupireimp σσ − S
53
Figure 3.22 b: TheAbsolute Difference between Implied Volatility and The Dupire
Volatility(PlottedAgainst Stock Price )S
54
Figure 3.23 : TheAbsolute Difference between Implied Volatility and The Dupire Volatility at
MeshPoints(PlottedAgainst )K
Figure 3.23 shows the distance between the volatilities of each mesh point. The circled points are
sparce in figure 3.23, due to the same reason that we stated in the illustration of figure 3.18 that the
Dupire volatilities' numerical existence is not guaranteed by our algorithms of finite difference
method. We can not evaluate numerically for some points with our current algorithm.
To sum up our investigation into the inner connections between the implied volatility and the
Dupire volatility, we draw the following conclusions. If we can make the option value to be convex
with respect to the strike price on a certain finite interval, then our scheme of volatilityK
calculation by finite difference method is practical. Hence, we can compare the volatility calculated
by the Dupire method to the one computed by using Black-Scholes formula rebersely. Even if we
can obtain a convex function of option value with respect to the strike price by imposingK
conditions on the local volatility model or choosing a suitable interval, the distance between the
implied volatility still will not be of some specific regularity. As we can see in figure 3.22, the
lower bound of the distance seems to be unstable, since every time we change the upper bound of
stock prices, the lower bound will be very likely to change accordingly.And as for the upper bound
55
of the distance, we can determine it by comparing the value of the left endpoint of the curve and the
value of the peak point of the curve. But still, this upper bound of distance is not fixed either. And
according our conclusions at the end of Section 3.1.2 that the implied volatility tends to infinity as
, unless we can prove the Dupire volatility has the same speed of convergence to infinity, it0→S
would be really hard to approximate the implied volatility by the Dupire volatility. More
importantly, the distance curve does not converge to anything or have a certain pattern of changing.
It is described in LiShangJiang (1994) [1] that the Dupire volatility is ill-posed. Based on all of the
research in section 3.1.3, the Dupire volatility does seem to a method not very appropriate to
simulate the implied volatility under our local volatility model here.
3.1.4. Summary of Three Types of Volatilities
After previous subsections of discussions with three types of volatilities, the local volatility, the
implied volatility, the Dupire volatility, we now summarize their features comprehensively.
Apparently, the local volatility we discuss here, is determined only by the stock priceSloc1
=σ
. Given a certain time-to-maturity and a strike price , the implied volatility, however, isS τ K
determined by the stock price process (provided that all other parameters, such as risk-freeS
interest rate and dividend yield is known). Or, in other words, with all other parameters inr q
the option pricing model given, the stock price process and the time-to-maturity co-S τ
determine the implied volatility . As for the Dupire volatility, it is determined),( τσσ Simpimp =
by the maturity time and the strike price , while the stock price and the time pointT K S t
and all other parameters influence the Dupire volatility merely as parameters.And it's worth
mentioning that the Dupire volatility is very sensitive to the strike price .K
We draw these three types of volatilities in the same diagram. Figure 3.24, panel a, is plotted with a
strike price , and panel b is drawn with a strike price ..99=K 10=K
Throughout the entire interval for stock price , we conclude the following.S
For the local volatility,
i) when the price of stock/underlying asset is below the option's strike price ,(the option isS K
out-of-the-money) the local volatility is larger than the implied volatility; conversely, when the
price of stock/underlying asset is above the option's strike price ,(the option is in-the-S K
56
money) the local volatility is smaller than the implied volatility.
ii) as the stock price tends to 0, the local volatility tends to infinity; as the stock price tends to
infinity, the local volatility eventually tends to 0.
iii) as the stock price increases, the local volatility decreases at a decreasing speed.
For the implied volatility,
i) as the stock price tends to 0, the implied volatility tends to infinity at a much slower speed than
the local volatility; as the stock price tends to infinity, the implied volatility tends to a stable non-
negative level, particularly, when the risk-free interest rate is set to be 0, then the impliedr
volatility tends to 0 as the stock price tends to infinity.
ii) as the stock price increases, the implied volatility decreases at a decreasing speed.
For the Dupire volatility,
i) the Dupire volatility curve changes its convexity at least once on the stock price interval.
ii) the Dupire volatility is highly sensitive to the value of strike price .K
iii) judging by the numerical approach, the finite difference method we use, as the stock price goes
up to a relatively high level, the Dupire volatility can not be numerically evaluated after it reaches 0
at certain point in the stock price axis.
iv) based on our investigation in this paper, the Dupire volatility is bounded within the range of
variation of the stock price .S
57
Figure 3.24. a The Volatility Curves( )dupireimploc σσσ ,,
Figure 3.24. b The Volatility Curves( )dupireimploc σσσ ,,
58
Chapter 4. Conclusions and Future Studies
In this paper, we mainly consider a local volatility model with the local volatility . We5.0−= Slocσ
evaluate the model numerically after deriving this option pricing model by using its specified
underlying asset's price processes.
We focus the study on three types of volatilities, the local volatility specified by the model itself
, the implied volatility inferred by the Black-Scholes model , and the so-called Dupirelocσ impσ
volatility given by the Dupire formula for implied volatilities . By showing the absolutedupireσ
difference between the implied volatility and the local volatility, and between the implied volatility
and the Dupire volatility, we illustrate the inner connections between these volatilities. From our
research here, we conclude that the difference between the local volatility and the implied volatility
is bounded for large values of stock price ; however, the difference between them tends toSinfinity since the implied volatility tends to infinity at a much slower speed than the local volatility
as the stock price tends to 0. At the same time, we also investigate the5.0−= Slocσ Srelationships between the implied volatility and the volatility given by the Dupire formula. For
large values of stock price , the Dupire volatility tends to 0 as well as the implied volatility does;Sfor small small values of stock price , the results shown by this numerical study is inconclusive.SBesides, the Dupire volatility is also time-dependent and highly sensitive to the change of the
values of strike price .KWe know that, our research here is based on an option pricing model(featured by its stock price
processes) that has a implied volatility similar to the one inferred by the Black-Scholes model, yet
still processes a price process that does not depend on the assumption of constant volatility, which
is more plausible and realistic. Hence, intuitively, we would expect these three volatilities possess
some similarities. And our results of this paper verify this speculation, all three types of volatilities
tend to 0 for large values of stock price ; the local volatility and the implied volatility tend toSinfinity for small values of stock price although they have difference speeds of convergenceStowards infinity.
However, procedures stated and demonstrated above may have some limitations as well. For
example, if we choose another local volatility model other than this one we use in this paper, one
question may arise naturally is that whether it is possible to solve the option values for such a model,
if the mechanics of this model does not allow the option value to be solved by any built-in function
of programming software. An obvious alternative is to resort to some other numerical methods to
treat the PDE we might encounter discretely, such as the finite difference method and so on.
Nevertheless, the accuracy of such approximations might be questionable. To avoid such "vains",
59
one useful shortcut is that we can observe the situation in the real market, make guesses such as the
one in Heston model[4] that the stock's volatility might be negatively correlated to the stock price,
to find suitable models guided by experiential knowledge and intuitions.
For future studies, we can also studies other local volatility models other than this simple one we
use here in this paper. Moreover, we can always search for another numerical option pricing
approach other than the one we use here. Besides the alternative approaches and models for implied
volatility study. Another future study direction can be looked into is the investigation of the
tendency of change of the Dupire volatility as the stock price tends to 0, whether it is really
bounded on the stock price axis, whether it tends to infinity as the implied volatility and the local
volatility do, still needs to be verified. Furtherore, other numerical approaches that can fix the
truncation problem of the Dupire volatility at large values of stock price can be studied as well.SThis concludes our conclusions of this paper and directions for future study references to this topic.
60
NotationsSDE Stochastic Differential EquationPDE Partial Differential EquationEMM Equivalent Martingale MeasureK Strike PriceS Stock/Asset Pricet Time pointT Maturity Timeτ Time-to-maturity
TttW ≤≤0}{ Brownian Motion, Wiener Process
V Option ValueG The 2nd Order Derivative of Option Price with
Respect to the StrikeC Call Option Value
impσ The Implied Volatility
locσ The Local Volatility
locσ The Dupire Volatility
r The Risk-free Interest Rate
tνThe Stochastic Volatility
ρ The Correlation between Brownian Motions
61
AppendixA
Program codes and plots
1. pdex_u.m
Program code
TableA.1pdex_u.mfunction value = pdex_u(m, T, x, tau)
%Function calculates the difference between two types of volatilities.
i = 51; %The number of meshpoints in the range of the maturity time T.
t0 = 0; %The initial value of maturity time T.
tT = 5; %The endpoint value of maturity time T.
j = 201; %The number of meshpoints in the range of the asset prices S.
K = 10; %The strike price K.
x = linspace(0,100,j); %The meshpoints in the range of asset prices S.
T = linspace(t0,tT,i); %The meshpoints in the range of maturity time T.
t = linspace(t0,tT,i); %The actual variable notation we use in pdepe
tau = linspace(0,tT-t0,i); %Time-to-maturity tau. tau=T-t0
m = 0; %Parameter in built-in function pdepe, indicates the PDE is slab.
sol = pdepe(m,@pdex1pde,@pdex1ic,@pdex1bc,x,t);
%Using the built-in function pdepe to solve the PDE for option value V.
u = sol(:,:,1);
%Solve the PDE, denote the first solution as the option value.
value =u;
% -------------------------------------------------------------------------
% The curve of distance plotting against the asset price
figure;
plot(x,value);
colormap hsv;
xlabel('Stock Prices');
ylabel('Option Value');
title('European Call Options');
figure;
plot(x,value(51,:),'Color','red'); hold on;
plot(x,value(11,:),'Color','blue'); hold on;
plot(x,value(1,:),'Color','black'); hold on;
62
Plots
xlabel('Stock Prices');
ylabel('Option Value');
title('European Call Options');
% -------------------------------------------------------------------------
% The surface consists of the distance between two kinds of volatilities
figure;
surf(x,tau,value);
colormap hsv;
title('Distance between two kinds of volatilities');
xlabel('Stock Prices');
ylabel('Time-to-maturity \tau');
zlabel('Implied Volatilities');
title('The European Call Option Value Surface');
% -------------------------------------------------------------------------
function [c,f,s] = pdex1pde(x,t,u,DuDx)
c = 1;
f = 1/2*(x*DuDx-u);
s = 0;
end
% -------------------------------------------------------------------------
function u0 = pdex1ic(x)
u0 = max((x-K),0);
end
% -------------------------------------------------------------------------
function [pl,ql,pr,qr] = pdex1bc(xl,ul,xr,ur,t)
K = 10;
pl = ul;
ql = 0;
pr = -1/2*K;
qr = 1;
end
end
63
Figure A.1
2. pdex_imp.m
Program code
TableA.2
pdex_imp.mfunction value = pdex_imp(m, T, x, tau)
%Function calculates the difference between two types of volatilities.
i = 51; %The number of meshpoints in the range of the maturity time T.
t0 = 0; %The initial value of maturity time T.
tT = 5; %The endpoint value of maturity time T.
j = 201; %The number of meshpoints in the range of the asset prices S.
K = 10; %The strike price K.
x = linspace(0,100,j); %The meshpoints in the range of asset prices S.
T = linspace(t0,tT,i); %The meshpoints in the range of maturity time T.
t = linspace(t0,tT,i);
tau = linspace(0,tT-t0,i); %Time-to-maturity tau. tau=T-t0
imp_v = ones(i,j);%Implied volatilities
m = 0; %Parameter in built-in function pdepe, indicates the PDE is slab.
64
sol = pdepe(m,@pdex1pde,@pdex1ic,@pdex1bc,x,t);
%Using the built-in function pdepe to solve the PDE for option value V.
u = sol(:,:,1);
%Solve the PDE, denote the first solution as the option value.
Limit = 100;
Yield = 0;
Tolerance = 1e-18;
Class={'Call'};
for i=1:51
for j=1:201
imp_v(i,j) = blsimpv(x(j), K, 0, tau(i), u(i,j), Limit, Yield,
Tolerance, Class);
end
end
value =imp_v;
% -------------------------------------------------------------------------
% The curve of implied volatilities
figure;
plot(x,value(29,:),'.','Color','black','LineWidth',2);hold on;
plot(x,value(39,:),'Color','blue','LineWidth',2);hold on;
plot(x,value(49,:),'Color','magenta','LineWidth',2);
xlabel('Stock Prices');
ylabel('Implied Volatilities');
title('European Call Options');
% -------------------------------------------------------------------------
% The Term Structure of Implied Volatilities for European Call Options
figure;
plot(tau,value(:,2),'Color','black','LineWidth',2);hold on;
plot(tau,value(:,12),'Color','cyan','LineWidth',2);hold on;
plot(tau,value(:,22),'Color','red','LineWidth',2);hold on;
plot(tau,value(:,32),'Color','blue','LineWidth',2);hold on;
plot(tau,value(:,42),'Color','black','LineWidth',2);hold on;
plot(tau,value(:,52),'.','Color','magenta','MarkerSize',6);
colormap hsv;
xlabel('Time-to-maturity \tau');
ylabel('Implied Volatilities');
title('The Term Structure of Implied Volatilities for European Call
Options');
% -------------------------------------------------------------------------
% The implied volatility surface
figure;
surf(x,tau,value);
65
colormap hsv;
xlabel('Stock Prices');
ylabel('Time-to-maturity \tau');
zlabel('Implied Volatilities');
title('The Implied Volatility Surface of European Call Options');
% -------------------------------------------------------------------------
function [c,f,s] = pdex1pde(x,t,u,DuDx)
c = 1;
f = 1/2*(x*DuDx-u);
s = 0;
end
% -------------------------------------------------------------------------
function u0 = pdex1ic(x)
u0 = max((x-K),0);
end
% -------------------------------------------------------------------------
function [pl,ql,pr,qr] = pdex1bc(xl,ul,xr,ur,t)
K = 10;
pl = ul;
ql = 0;
pr = -1/2*K;
qr = 1;
end
end
66
Plots
Figure A.2
3. pdex_dis_imp_loc.m
Program code
TableA.3
pdex_dis_imp_loc.mfunction value = pdex_dis_imp_loc(m, T, x, tau)
%Function calculates the difference between two types of volatilities.
i = 51; %The number of meshpoints in the range of the maturity time T.
t0 = 0; %The initial value of maturity time T.
tT = 5; %The endpoint value of maturity time T.
j = 201; %The number of meshpoints in the range of the asset prices S.
K = 10; %The strike price K.
x = linspace(0,100,j); %The meshpoints in the range of asset prices S.
T = linspace(t0,tT,i); %The meshpoints in the range of maturity time T.
t = linspace(t0,tT,i);
tau = linspace(0,tT-t0,i); %Time-to-maturity tau. tau=T-t0
67
sig = ones(i,j);% The local volatilities
imp_v = ones(i,j);%Implied volatilities
dis = ones(i,j);% The distance between local & implied volatilities
m = 0; %Parameter in built-in function pdepe, indicates the PDE is slab.
sol = pdepe(m,@pdex1pde,@pdex1ic,@pdex1bc,x,t);
%Using the built-in function pdepe to solve the PDE for option value V.
u = sol(:,:,1);
%Solve the PDE, denote the first solution as the option value.
Limit = 10;
Yield = 0;
Tolerance = 1e-16;
Class={'Call'};
for i=1:51
for j=1:201
sig(i,j)=sqrt(1/x(j));
imp_v(i,j) = blsimpv(x(j), K, 0, tau(i), u(i,j), Limit, Yield,
Tolerance, Class);
dis(i,j)=abs(imp_v(i,j)-sig(i,j));
end
end
value =dis;
% -------------------------------------------------------------------------
% The curve of distance plotting against the asset price
figure;
plot(x,sig);
xlabel('Stock Prices');
ylabel('Local Volatility');
title('European Call Options')
% -------------------------------------------------------------------------
% The curve of distance plotting against the asset price
figure;
plot(x,value(31,:),'.','Color','red','LineWidth',2);hold on;
plot(x,value(41,:),'Color','blue','LineWidth',2);hold on;
plot(x,value(51,:),'Color','black','LineWidth',2);hold on;
xlabel('Stock Prices');
ylabel('Distance between Two Types of Volatilities');
title('European Call Options');
% -------------------------------------------------------------------------
% The curve of distance plotting against the time-to-maturity
figure;
plot(tau,value(:,2),'Color','red','LineWidth',2);hold on;
68
plot(tau,value(:,3),'Color','blue','LineWidth',2);hold on;
plot(tau,value(:,12),'Color','cyan','LineWidth',2);hold on;
plot(tau,value(:,21),'Color','green','LineWidth',2);hold on;
plot(tau,value(:,42),'Color','black','LineWidth',2);hold on;
xlabel('Time-to-maturity');
ylabel('Distance between Two Types of Volatilities');
title('European Call Options');
% -------------------------------------------------------------------------
% The surface consists of the distance between two kinds of volatilities
figure;
surf(x,tau,value);
title('Distance between Two Types of Volatilities');
xlabel('Stock Prices');
ylabel('Time-to-maturity');
zlabel('Distance between Two Types of Volatilities');
title('The Distance Surface of European Call Options');
% -------------------------------------------------------------------------
function [c,f,s] = pdex1pde(x,t,u,DuDx)
c = 1;
f = 1/2*(x*DuDx-u);
s = 0;
end
% -------------------------------------------------------------------------
function u0 = pdex1ic(x)
u0 = max((x-K),0);
end
% -------------------------------------------------------------------------
function [pl,ql,pr,qr] = pdex1bc(xl,ul,xr,ur,t)
K = 10;
pl = ul;
ql = 0;
pr = -1/2*K;
qr = 1;
end
end
69
Plots
Figure A.3: The Distance Surface of EuropeanCall Options
4. pdex_u_small_s.m
Program code
TableA.4
pdex_u_small_s.m
function value = pdex_u_small_s(m, T, x, tau)
%Function calculates the difference between two types of volatilities.
i = 51; %The number of meshpoints in the range of the maturity time T.
t0 = 0; %The initial value of maturity time T.
tT = 5; %The endpoint value of maturity time T.
j = 202; %The number of meshpoints in the range of the asset prices S.
K = 10; %The strike price K.
x = ones(1,j);
xx = linspace(0,10^-100,j-1);
for j=1:201
x(1,j)=xx(j);
end
x(1,202)=100;
70
%The meshpoints in the range of asset prices S.
T = linspace(t0,tT,i); %The meshpoints in the range of maturity time T.
t = linspace(t0,tT,i); %The actual variable notation we use in pdepe
tau = linspace(0,tT-t0,i); %Time-to-maturity tau. tau=T-t0
m = 0; %Parameter in built-in function pdepe, indicates the PDE is slab.
sol = pdepe(m,@pdex1pde,@pdex1ic,@pdex1bc,x,t);
%Using the built-in function pdepe to solve the PDE for option value V.
u = sol(:,:,1);
%Solve the PDE, denote the first solution as the option value.
u_small_s = ones(i,j-1);
for i=1:51
for j=1:201
u_small_s(i,j)= u(i,j);
end
end
value =u_small_s;
% -------------------------------------------------------------------------
% The curve of distance plotting against the asset price
figure;
plot(xx,value);
colormap hsv;
xlabel('Stock Prices');
ylabel('Option Value');
title('European Call Options');
figure;
plot(xx,value(51,:),'Color','red'); hold on;
plot(xx,value(11,:),'Color','blue'); hold on;
plot(xx,value(1,:),'Color','black'); hold on;
xlabel('Stock Prices');
ylabel('Option Value');
title('European Call Options');
% -------------------------------------------------------------------------
% The surface consists of the distance between two kinds of volatilities
figure;
surf(x,tau,u);
colormap hsv;
title('Distance between two kinds of volatilities');
xlabel('Stock Prices');
ylabel('Time-to-maturity \tau');
zlabel('Implied Volatilities');
71
5. pdex_imp_small_s.m
Program code
TableA.5
title('The European Call Option Value Surface');
% -------------------------------------------------------------------------
function [c,f,s] = pdex1pde(x,t,u,DuDx)
c = 1;
f = 1/2*(x*DuDx-u);
s = 0;
end
% -------------------------------------------------------------------------
function u0 = pdex1ic(x)
u0 = max((x-K),0);
end
% -------------------------------------------------------------------------
function [pl,ql,pr,qr] = pdex1bc(xl,ul,xr,ur,t)
K = 10;
pl = ul;
ql = 0;
pr = -1/2*K;
qr = 1;
end
end
pdex_imp_small_s.m
function value = pdex_imp_small_s(m, T, x, tau)
%Function calculates the difference between two types of volatilities.
i = 51; %The number of meshpoints in the range of the maturity time T.
t0 = 0; %The initial value of maturity time T.
tT = 5; %The endpoint value of maturity time T.
j = 202; %The number of meshpoints in the range of the asset prices S.
K = 10; %The strike price K. K=1,S=20 imp_1 K=10,S=100,imp_2
x = ones(1,j);
xx = linspace(0,10^-10,j-1);
for j=1:201
x(1,j)=xx(j);
end
72
x(1,202)=100;
%The meshpoints in the range of asset prices S.
T = linspace(t0,tT,i); %The meshpoints in the range of maturity time T.
t = linspace(t0,tT,i);
tau = linspace(0,tT-t0,i); %Time-to-maturity tau. tau=T-t0
imp_v = ones(i,j);%Implied volatilities
m = 0; %Parameter in built-in function pdepe, indicates the PDE is slab.
sol = pdepe(m,@pdex1pde,@pdex1ic,@pdex1bc,x,t);
%Using the built-in function pdepe to solve the PDE for option value V.
u = sol(:,:,1);
%Solve the PDE, denote the first solution as the option value.
Limit = 1000000000;
Yield = 0;
Tolerance = 1e-18;
Class={'Call'};
for i=1:51
for j=1:202
imp_v(i,j) = blsimpv(x(j), K, 0, tau(i), u(i,j), Limit, Yield,
Tolerance, Class);
end
end
imp_v_small_s = ones(i,j-1);
for i=1:51
for j=1:201
imp_v_small_s(i,j)= imp_v(i,j);
end
end
value =imp_v_small_s;
% -------------------------------------------------------------------------
% The curve of implied volatilities
figure;
plot(xx,value(29,:),'Color','black','LineWidth',2);hold on;
plot(xx,value(39,:),'Color','blue','LineWidth',2);hold on;
plot(xx,value(49,:),'Color','magenta','LineWidth',2);
xlabel('Stock Prices');
ylabel('Implied Volatilities');
title('European Call Options, S=0:100*10^-^1^2:0.5*10^-^1^2');
% -------------------------------------------------------------------------
slope = ones(i-1,j-2);
73
xxx = ones(1,j-2);
for i=1:50
for j=1:199
slope(i,j)=(imp_v_small_s(i+1,j+1)-imp_v_small_s(i+1,j+2))/(xx(j)-
xx(j+1));
xxx(1,j)=xx(j+1);
end
slope(i,200)=slope(i,199);
end
xxx(1,200)=xx(201);
% -------------------------------------------------------------------------
% The curve of implied volatility curves' slope
figure;
plot(xxx,slope(29,:),'Color','black','LineWidth',2);hold on;
plot(xxx,slope(39,:),'Color','blue','LineWidth',2);hold on;
plot(xxx,slope(49,:),'Color','magenta','LineWidth',2);
xlabel('Stock Prices');
ylabel('The Slope of Implied Volatility Curves');
title('European Call Options, S=0:100*10^-^1^2:0.5*10^-^1^2');
% -------------------------------------------------------------------------
loc = ones(i,j-1);
for i=1:51
for j=1:201
loc(i,j)=sqrt(1/(xx(j)));
end
end
% -------------------------------------------------------------------------
% The curve of local volatilities
figure;
plot(xx,loc,'Color','blue','LineWidth',2);
xlabel('Stock Prices');
ylabel('Local Volatilities');
title('European Call Options, S=0:100*10^-^1^2:0.5*10^-^1^2');
% -------------------------------------------------------------------------
slope_loc = ones(i-1,j-2);
for i=1:50
for j=1:200
slope_loc(i,j) =-(1/2)*(xxx(1,j)^(-3/2));
end
end
% -------------------------------------------------------------------------
% The curve of local volatility curves' slope
figure;
plot(xxx,slope_loc,'.','Color','black','LineWidth',2);
74
6. pdex_imp_slope_small_s_fit.m
Program code
TableA.6
xlabel('Stock Prices');
ylabel('The Slope of Local Volatility Curves');
title('European Call Options, S=0:100*10^-^1^2:0.5*10^-^1^2');
% -------------------------------------------------------------------------
function [c,f,s] = pdex1pde(x,t,u,DuDx)
c = 1;
f = 1/2*(x*DuDx-u);
s = 0;
end
% -------------------------------------------------------------------------
function u0 = pdex1ic(x)
u0 = max((x-K),0);
end
% -------------------------------------------------------------------------
function [pl,ql,pr,qr] = pdex1bc(xl,ul,xr,ur,t)
K = 10;
pl = ul;
ql = 0;
pr = -1/2*K;
qr = 1;
end
end
pdex_imp_slope_small_s_fit.mload local_volatility_imp_small_s.mat
imp= ans;
xx = linspace(0,10^-10,201);
load local_volatility_imp_small_s_slope.mat
xxx = linspace(xx(2),10^-10,200);
x=ones(1,8);
% -------------------------------------------------------------------------
%lower bound for implied volatility curve,tau=0.1
x(1)=-0.5;
for j=1:201
while imp(2,j)<xx(j)^x(1)
x(1)=x(1)+0.01;
end
75
end
% -------------------------------------------------------------------------
%upper bound for implied volatility curve,tau=0.1
x(2)=-0.01;
for j=1:201
while imp(2,j)>xx(j)^x(2)
x(2)=x(2)-0.01;
end
end
% -------------------------------------------------------------------------
%lower bound for implied volatility curve,tau=5
x(3)=-0.5;
for j=1:201
while imp(51,j)<xx(j)^x(3)
x(3)=x(3)+0.01;
end
end
% -------------------------------------------------------------------------
%upper bound for implied volatility curve,tau=5
x(4)=-0.01;
for j=1:201
while imp(51,j)>xx(j)^x(4)
x(4)=x(4)-0.01;
end
end
% -------------------------------------------------------------------------
%lower bound for implied volatility curve,tau=0.1
x(5)=-1.5;
for j=1:200
while slope(1,j)>-0.5*(xxx(j))^(x(5));
x(5)=x(5)+0.01;
end
end
% -------------------------------------------------------------------------
%upper bound for implied volatility curve,tau=0.1
x(6)=-0.01;
for j=1:200
while slope(1,j)<-0.5*(xxx(j))^(x(6));
x(6)=x(6)-0.01;
end
end
% -------------------------------------------------------------------------
%lower bound for implied volatility curve,tau=5
x(7)=-1.5;
76
for j=1:200
while slope(50,j)>-0.5*(xxx(j))^(x(7));
x(7)=x(7)+0.01;
end
end
% -------------------------------------------------------------------------
%upper bound for implied volatility curve,tau=5
x(8)=-0.01;
for j=1:200
while slope(50,j)<-0.5*(xxx(j))^(x(8));
x(8)=x(8)-0.01;
end
end
% -------------------------------------------------------------------------
value=x;
% -------------------------------------------------------------------------
% The curves of implied volatilities
figure;
plot(xx,imp(2,:),'.','Color','cyan','LineWidth',2);hold on;
plot(xx,imp(51,:),'.','Color','black','LineWidth',2);hold on;
xlabel('Stock Prices');
ylabel('Implied Volatilities');
title('European Call Options, S=0:100*10^-^1^2:0.5*10^-^1^2');
% -------------------------------------------------------------------------
loc=ones(1,201);
% -------------------------------------------------------------------------
for j=1:201
loc(j)=(xx(j))^(x(1));
end
plot(xx,loc,'Color','green','LineWidth',4);hold on;
for j=1:201
loc(j)=(xx(j))^((x(1)+x(2))/2);
end
plot(xx,loc,'Color','magenta','LineWidth',2);hold on;
for j=1:201
loc(j)=(xx(j))^(x(2));
end
plot(xx,loc,'Color','blue','LineWidth',4);hold on;
for j=1:201
loc(j)=(xx(j))^(x(3));
end
plot(xx,loc,'Color','blue','LineWidth',2);hold on;
for j=1:201
loc(j)=(xx(j))^((x(3)+x(4))/2);
77
7. pdex_dupire_option.m
Program code
TableA.7
end
plot(xx,loc,'Color','red','LineWidth',2);hold on;
for j=1:201
loc(j)=(xx(j))^(x(4));
end
plot(xx,loc,'Color','green','LineWidth',2);hold on;
% -------------------------------------------------------------------------
% The curve of slopes
figure;
plot(xxx,slope(1,:),'Color','black','LineWidth',4);hold on;
plot(xxx,slope(50,:),'Color','blue','LineWidth',4);hold on;
xlabel('Stock Prices');
ylabel('The Slope of Implied Volatility Curves');
title('European Call Options, S=0:100*10^-^1^2:0.5*10^-^1^2');
% -------------------------------------------------------------------------
locc=ones(1,200);
for j=1:200
locc(j)=-0.5*(xxx(j))^(x(5));
end
plot(xxx,locc,'Color','red','LineWidth',2);hold on;
for j=1:200
locc(j)=-0.5*(xxx(j))^(x(6));
end
plot(xxx,locc,'Color','magenta','LineWidth',2);hold on;
% -------------------------------------------------------------------------
for j=1:200
locc(j)=-0.5*(xxx(j))^(x(7));
end
plot(xxx,locc,'Color','green','LineWidth',2);hold on;
for j=1:200
locc(j)=-0.5*(xxx(j))^(x(8));
end
plot(xxx,locc,'Color','cyan','LineWidth',2);hold on;
% -------------------------------------------------------------------------
pdex_dupire_option.mfunction value = pdex_dupire_option(m, K, x, tau)
%Function calculates the difference between two types of volatilities.
78
i = 21; %The number of meshpoints in the range of the maturity time T.
t0 = 0; %The initial value of maturity time T.
tT = 2; %The endpoint value of maturity time T.
j = 101; %The number of meshpoints in the range of the asset prices S.
h = 21; %The number of meshpoints in the range of the strike prices K.
x = linspace(0,100,j); %The meshpoints in the range of asset prices S.
T = linspace(t0,tT,i); %The meshpoints in the range of maturity time T.
t = linspace(t0,tT,i);
tau = linspace(0,tT-t0,i); %Time-to-maturity tau. tau=T-t0
K = linspace(9.18,11,h); %The meshpoints in the range of the strike prices K.
sig_2 = ones(i,j,h);%The square of implied volatility
imp_v = ones(i,j,h);%The implied volatilities.
sig_dupire = ones(i,j,h);%The Dupire volatility
dis = ones(i,j,h);%The distance between Dupire&implied volatilities.
u = ones(i,j,h);
m = 0; %Parameter in built-in function pdepe, indicates the PDE is slab.
for h=1:21
sol = pdepe(m,@pdex1pde,@pdex1ic,@pdex1bc,x,t);
%Using the built-in function pdepe to solve the PDE for option value V.
u(:,:,h) = sol(:,:,1);
%Solve the PDE, denote the first solution as the option value.
end
value = u;
% -------------------------------------------------------------------------
%The Option Value Surface
figure;
uu = ones(21,101);
for h=1:21
for j=1:101
uu(h,j)=u(21,j,h);
end
end
surf(x,K,uu);
xlabel('Stock/Asset Prices');
ylabel('Strike Prices');
zlabel('Option Value');
title('The Option Value Surface,\tau=2');
% -------------------------------------------------------------------------
%The Option Value Curve(Agaist the Strike K)
79
8.pdex_dupire.m
Program code
TableA.8
figure;
uu = ones(21,101);
for h=1:21
for j=1:101
uu(h,j)=u(21,j,h);
end
end
plot(K,uu(:,1),'Color','red','LineWidth',2);hold on;
plot(K,uu(:,51),'Color','blue','LineWidth',2);hold on;
plot(K,uu(:,101),'Color','black','LineWidth',2);hold on;
xlabel('Strike Prices K');
ylabel('Option Value V');
title('The Option Value Curve,\tau=2');
% -------------------------------------------------------------------------
function [c,f,s] = pdex1pde(x,t,u,DuDx)
c = 1;
f = 1/2*(x*DuDx-u);
s = 0;
end
% -------------------------------------------------------------------------
function u0 = pdex1ic(x)
u0 = max((x-K(h)),0);
end
function [pl,ql,pr,qr] = pdex1bc(xl,ul,xr,ur,t)
pl = ul;
ql = 0;
pr = -1/2*K(h);
qr = 1;
end
end
pdex_dupire.mfunction value = pdex_dupire(m, K, x, tau)
%Function calculates the difference between two types of volatilities.
i = 21; %The number of meshpoints in the range of the maturity time T.
t0 = 0; %The initial value of maturity time T.
tT = 2; %The endpoint value of maturity time T.
j = 101; %The number of meshpoints in the range of the asset prices S.
80
h = 21; %The number of meshpoints in the range of the strike prices K.
x = linspace(0,100,j); %The meshpoints in the range of asset prices S.
T = linspace(t0,tT,i); %The meshpoints in the range of maturity time T.
t = linspace(t0,tT,i);
tau = linspace(0,tT-t0,i); %Time-to-maturity tau. tau=T-t0
K = linspace(9.18,11,h); %The meshpoints in the range of the strike prices K.
u = ones(i,j,h); %The option values
sig_2 = ones(i,j,h);%The square of implied volatility
imp_v = ones(i,j,h);%The implied volatilities.
sig_dupire = ones(i,j,h);%The Dupire volatility
dis = ones(i,j,h);%The distance between Dupire&implied volatilities.
m = 0; %Parameter in built-in function pdepe, indicates the PDE is slab.
Limit = 100;
Yield = 0;
Tolerance = 1e-18;
Class={'Call'};
C_T = ones(i,j,h);
C_KK = ones(i,j,h);
for h=1:21
sol = pdepe(m,@pdex1pde,@pdex1ic,@pdex1bc,x,t);
%Using the built-in function pdepe to solve the PDE for option value V.
u(:,:,h) = sol(:,:,1);
%Solve the PDE, denote the first solution as the option value.
end
for j=1:101
for i=1:20
for h=1:19
C_T(i,j,h) = (u(i+1,j,h)-u(i,j,h))/(t(i+1)-t(i));
C_KK(i,j,h)=(u(i,j,h+2)+u(i,j,h)-2*u(i,j,h+1))/((K(h+1)-K(h))^2);
sig_2(i,j,h)=2*C_T(i,j,h)/(K(h)^2*C_KK(i,j,h));
if sig_2(i,j,h)>=0
sig_dupire(i,j,h)=sqrt(sig_2(i,j,h));
else
sig_dupire(i,j,h)=NaN;
end
end
end
81
end
dupire = ones(i-1,j-1,h-2);
for i=1:20
for j=1:100
for h=1:19
dupire(i,j,h)=sig_dupire(i,j+1,h);
end
end
end
value = dupire;
vol= ones(j-1,h-2);
xx = linspace(x(2),100,100);
KK = linspace(9.18,K(h-2),h-2);
TT= linspace(T(1),T(i-1),i-1);
% -------------------------------------------------------------------------
for j=1:100
for h=1:19
vol(j,h)=dupire(20,j,h);
end
end
figure;
plot(xx,vol(:,9),'.','Color','red'); hold on;
xlabel('Stock Prices S');
ylabel('Dupire Volatility');
title('European Calls,Dupire Volatility Curve, \tau=1.9, K=9.9');%K=9.908
figure;
plot(xx,vol(:,10),'.','Color','red'); hold on;
xlabel('Stock Prices S');
ylabel('Dupire Volatility');
title('European Calls,Dupire Volatility Curve, \tau=1.9, K=10');%K=9.999
% -------------------------------------------------------------------------
vol_t=ones(20,19);
for h=1:19
for i=1:20
vol_t(i,h)=dupire(i,20,h);
end
end
tt=linspace(T(1),T(20),20);
figure;
plot(tt,vol_t(:,9),'.','Color','red'); hold on;
plot(tt,vol_t(:,10),'.','Color','blue'); hold on;
82
9.pdex_dis_imp_dupire.mProgram code
TableA.9
xlabel('Time-to-maturity \tau');
ylabel('Dupire Volatility');
title('European Calls,Dupire Volatility Curve, S=20');
% -------------------------------------------------------------------------
function [c,f,s] = pdex1pde(x,t,u,DuDx)
c = 1;
f = 1/2*(x*DuDx-u);
s = 0;
end
% -------------------------------------------------------------------------
function u0 = pdex1ic(x)
u0 = max((x-K(h)),0);
end
function [pl,ql,pr,qr] = pdex1bc(xl,ul,xr,ur,t)
pl = ul;
ql = 0;
pr = -1/2*K(h);
qr = 1;
end
end
pdex_dis_imp_dupire.mfunction value = pdex_dis_imp_dupire(m, T, x, tau)
%Function calculates the difference between two types of volatilities.
i = 21; %The number of meshpoints in the range of the maturity time T.
t0 = 0; %The initial value of maturity time T.
tT = 2; %The endpoint value of maturity time T.
j = 101; %The number of meshpoints in the range of the asset prices S.
h = 21; %The number of meshpoints in the range of the strike prices K.
x = linspace(0,100,j); %The meshpoints in the range of asset prices S.
T = linspace(t0,tT,i); %The meshpoints in the range of maturity time T.
t = linspace(t0,tT,i);
tau = linspace(0,tT-t0,i); %Time-to-maturity tau. tau=T-t0
K = linspace(9.18,11,h); %The meshpoints in the range of the strike prices K.
u = ones(i,j,h); %The option values V.
sig_2 = ones(i,j,h);%The square of implied volatility
83
imp_v = ones(i,j,h);%The implied volatilities.
sig_dupire = ones(i,j,h);%The Dupire volatility
dis = ones(i-1,j,h-2);%The distance between Dupire&implied volatilities.
m = 0; %Parameter in built-in function pdepe, indicates the PDE is slab.
Limit = 100;
Yield = 0;
Tolerance = 1e-18;
Class={'Call'};
C_T = ones(i,j,h);
C_KK = ones(i,j,h);
for h=1:21
sol = pdepe(m,@pdex1pde,@pdex1ic,@pdex1bc,x,t);
%Using the built-in function pdepe to solve the PDE for option value V.
u(:,:,h) = sol(:,:,1);
%Solve the PDE, denote the first solution as the option value.
for i=1:21
for j=1:101
imp_v(i,j,h) = blsimpv(x(j), K(h), 0, tau(i), u(i,j,h), Limit,
Yield, Tolerance, Class);
end
end
end
for h=1:19
for j=1:101
for i=1:20
C_T(i,j,h) = (u(i+1,j,h)-u(i,j,h))/(t(i+1)-t(i));
C_KK(i,j,h)=(u(i,j,h+2)+u(i,j,h)-2*u(i,j,h+1))/((K(h+1)-K(h))^2);
sig_2(i,j,h)=2*C_T(i,j,h)/(K(h)^2*C_KK(i,j,h));
if sig_2(i,j,h)>=0
sig_dupire(i,j,h)=sqrt(sig_2(i,j,h));
else
sig_dupire(i,j,h)=NaN;
end
dis(i,j,h)=abs(imp_v(i,j,h)-sig_dupire(i,j,h));
end
end
end
84
10.Dupire_plots_dis.mProgram code
TableA.10
distance= ones(i-1,j-1,h-2);
TT= linspace(T(1),T(i-1),i-1);
xx = linspace(x(2),100,100);
KK = linspace(9.18,K(h-2),h-2);
for i=1:20
for j=1:100
for h=1:19
distance(i,j,h)=dis(i,j+1,h);
end
end
end
value = distance;
% -------------------------------------------------------------------------
function [c,f,s] = pdex1pde(x,t,u,DuDx)
c = 1;
f = 1/2*(x*DuDx-u);
s = 0;
end
% -------------------------------------------------------------------------
function u0 = pdex1ic(x)
u0 = max((x-K(h)),0);
end
function [pl,ql,pr,qr] = pdex1bc(xl,ul,xr,ur,t)
pl = ul;
ql = 0;
pr = -1/2*K(h);
qr = 1;
end
end
dupire_plots_dis.m
for type=1:2
if type==1
load local_volatility_dis_dupire.mat;
distance;
%The distance between the implied volatility and the dupire
85
%volatility
% -------------------------------------------------------------------------
i = 21; %The number of meshpoints in the range of the maturity time T.
t0 = 0; %The initial value of maturity time T.
tT = 2; %The endpoint value of maturity time T.
j = 101; %The number of meshpoints in the range of the asset prices S.
h = 21; %The number of meshpoints in the range of the strike prices K.
x = linspace(0,100,j); %The meshpoints in the range of asset prices S.
T = linspace(t0,tT,i); %The meshpoints in the range of maturity time T.
% -------------------------------------------------------------------------
TT= linspace(T(1),T(i-1),i-1);
xx = linspace(x(2),100,100);
% -------------------------------------------------------------------------
distance_1 = ones(20,19);
%For plotting distance surface,S=29.
distance_2 = ones(20,19);
%For plotting distance surface,S=31.
distance_3 = ones(20,19);
%For plotting distance surface,S=33.
distance_4 = ones(20,19);
%For plotting distance surface,S=35.
distance_5 = ones(20,19);
%For plotting distance surface,S=35.
distance_6 = ones(20,19);
%For plotting distance surface,S=35.
ttau = linspace(0,1.9,20);
%Time-to-maturity,time-to-maturity=linspace(0.1,2,20).
KK = linspace(9.1,10.9,19);
%Strike prices, from K(2) to K(20),strike=linspace(9.1,10.9,19).
% -------------------------------------------------------------------------
for i=1:20
for h=1:19
distance_1(i,h)=distance(i,55,h);%S=55
distance_2(i,h)=distance(i,60,h);%S=60
distance_3(i,h)=distance(i,65,h);%S=65
distance_4(i,h)=distance(i,70,h);%S=70
distance_5(i,h)=distance(i,75,h);%S=75
distance_6(i,h)=distance(i,80,h);%S=80
end
end
% In order to demonstrate the relationships between
%strike prices K and distancetance between the volatilities, we select some
%effective data from the computing results by picking column 1-20 in time
%meshpoints, column 11-70 in asset price meshpoints(eliminating those
86
%out-of-money points), column 2-20 in strike prices meshpoints, i.e.,
%time-to-maturity=linspace(0.1,2,20),
%price=linspace(10,69,60),strike=linspace(9.1,10.9,19).
% -------------------------------------------------------------------------
figure;
plot(KK,distance_1,'o','Color','red'); hold on;
plot(KK,distance_2,'o','Color','red'); hold on;
plot(KK,distance_3,'o','Color','red'); hold on;
xlabel('Strike Prices K');ylabel('Distance');title('European Calls,
S=55:65:1, \tau=0:1.9:0.1');
figure;
plot(KK,distance_4,'o','Color','blue'); hold on;
plot(KK,distance_5,'o','Color','blue'); hold on;
plot(KK,distance_6,'o','Color','blue'); hold on;
xlabel('Strike Prices K');ylabel('Distance');title('European Calls,
S=70:80:1, \tau=0:1.9:0.1');
% -------------------------------------------------------------------------
figure;
plot(xx,distance(20,:,10),'.','Color','blue'); hold on;
xlabel('The Stock Prices S');
ylabel('The Distance');
title('The Distance Curve \tau=1.9, K=9.9');
% -------------------------------------------------------------------------
end
if type==2
load local_volatility_dis_dupire1.mat;
distance=ans;
%The distance between the implied volatility and the dupire
%volatility
% -------------------------------------------------------------------------
i = 21; %The number of meshpoints in the range of the maturity time T.
t0 = 0; %The initial value of maturity time T.
tT = 2; %The endpoint value of maturity time T.
j = 101; %The number of meshpoints in the range of the asset prices S.
h = 21; %The number of meshpoints in the range of the strike prices K.
x = linspace(0,100,j); %The meshpoints in the range of asset prices S.
T = linspace(t0,tT,i); %The meshpoints in the range of maturity time T.
% -------------------------------------------------------------------------
TT= linspace(T(1),T(i-1),i-1);
xx = linspace(x(2),100,100);
% -------------------------------------------------------------------------
distance_1 = ones(20,19);
%For plotting distance surface,S=29.
87
distance_2 = ones(20,19);
%For plotting distance surface,S=31.
distance_3 = ones(20,19);
%For plotting distance surface,S=33.
distance_4 = ones(20,19);
%For plotting distance surface,S=35.
distance_5 = ones(20,19);
%For plotting distance surface,S=35.
distance_6 = ones(20,19);
%For plotting distance surface,S=35.
ttau = linspace(0,1.9,20);
%Time-to-maturity,time-to-maturity=linspace(0.1,2,20).
KK = linspace(9.1,10.9,19);
%Strike prices, from K(2) to K(20),strike=linspace(9.1,10.9,19).
% -------------------------------------------------------------------------
for i=1:20
for h=1:19
distance_1(i,h)=distance(i,55,h);%S=55
distance_2(i,h)=distance(i,60,h);%S=60
distance_3(i,h)=distance(i,65,h);%S=65
distance_4(i,h)=distance(i,70,h);%S=70
distance_5(i,h)=distance(i,75,h);%S=75
distance_6(i,h)=distance(i,80,h);%S=80
end
end
% In order to demonstrate the relationships between
%strike prices K and distancetance between the volatilities, we select some
%effective data from the computing results by picking column 1-20 in time
%meshpoints, column 11-70 in asset price meshpoints(eliminating those
%out-of-money points), column 2-20 in strike prices meshpoints, i.e.,
%time-to-maturity=linspace(0.1,2,20),
%price=linspace(10,69,60),strike=linspace(9.1,10.9,19).
% -------------------------------------------------------------------------
figure;
plot(KK,distance_1,'o','Color','red'); hold on;
plot(KK,distance_2,'o','Color','red'); hold on;
plot(KK,distance_3,'o','Color','red'); hold on;
xlabel('Strike Prices K');ylabel('Distance');title('European Calls,
S=55:65:1, \tau=0:1.9:0.1');
figure;
plot(KK,distance_4,'o','Color','blue'); hold on;
88
Plots
Figure A.4
plot(KK,distance_5,'o','Color','blue'); hold on;
plot(KK,distance_6,'o','Color','blue'); hold on;
xlabel('Strike Prices K');ylabel('Distance');title('European Calls,
S=70:80:1, \tau=0:1.9:0.1');
% -------------------------------------------------------------------------
figure;
plot(xx,distance(20,:,10),'.','Color','blue'); hold on;
xlabel('The Stock Prices S');
ylabel('The Distance');
title('The Distance Curve \tau=1.9, K=10');
% -------------------------------------------------------------------------
end
end
89
Appendix B
Numerical results
1. pdex_dis_european.m
Table B.1 Distance between Implied Volatilities and Local Volatilities
tau=0.1:1:10; K=10; S=0.5:100:200; r=0,q=0.
S\tau 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.5 0.38121710463
0.56202628141
0.63764349171
0.68138362441
0.70930400844
0.72911719886
0.74291556128
0.75364093769
0.76192500543
0.76835603124
1 0.1568472335
0.29727630276
0.35479060181
0.38721567973 4.07E-01 0.421240
722490.43071234359
0.43781149542
0.44313212615
0.44716781166
1.5 0.079851112732
0.19668822929
0.2435547783
0.26928166874 2.85E-01 0.295276
198120.30227281675
0.30734066609
0.3110442074
0.31380231805
2 0.043809557089
0.14377621278
0.18298753735
0.20390049192 2.16E-01 0.224242
335570.22953945373
0.23325742891
0.23591591075
0.23786393361
2.5 0.02474594886
0.11143558702
0.1446000573
0.16173964357 1.72E-01 0.177780
360960.18181887993
0.18457521156
0.18650862371
0.18790345719
3 0.014203857135
0.0898260792
0.11794522421
0.13198601408 1.40E-01 0.144623
02340.14769458832
0.14974048042
0.15115022207
0.15215057439
3.5 0.0084491081817
0.074479836628
0.098240347451
0.10966933365 1.16E-01 0.119555
947560.12187045214
0.12337902905
0.1243997755
0.12511008085
4 0.0055858472887
0.063066055416
0.082975577613
0.092172906451 9.70E-02 0.099811
4427840.10152720649
0.10262216219
0.1033476191
0.10383974336
4.5 0.0045533637491
0.054246253429
0.070705784348
0.077984865619 8.17E-02 0.083773
8806830.085013609228
0.085785923341
0.086283607658
0.086609017504
5 0.0047126469886
0.047194877022
0.060540937843
0.066169042355 6.89E-02 0.070431
24010.071291874146
0.071810535036
0.072130824443
0.072327792519
5.5 0.0056538076325
0.041372734601
0.05190326799
0.05611261 5.81E-02 0.059114
3919010.059673824716
0.059992805149
0.060174579524
0.060272535422
6 0.0070967849401
0.036409200015
0.044400245524
0.047395403057 4.87E-02 0.049361
9884590.049683789077
0.049846465072
0.049920628511
0.049942812564
6.5 0.0088342403076
0.032035861076
0.03775363071
0.039718598468 4.05E-02 0.040845
369120.040982073006
0.041023930092
0.041015623378
0.040980723531
7 0.01069303823
0.028046981074
0.031758792998
0.032864376975
0.033225244523
0.033324246804
0.033319580777
0.03326987766
0.03319990549
0.033123053761
7.5 0.012501325247
0.024276244109
0.026261945981
0.026672000066
0.026701507477
0.026619142471
0.02650945314
0.026392418869
0.026278157234
0.026171507635
8 0.014052756095
0.02058576494
0.021148046921
0.02102232082
0.020805855795
0.020593371306
0.020408605068
0.020244541551
0.02010037908
0.019973545563
8.5 0.015063536815
0.016865705996
0.016334160232
0.015826438994
0.015439353051
0.015140736457
0.014905557566
0.014711667948
0.014549148614
0.014409806611
9 0.015129590141
0.013041900052
0.011764958072
0.011016796266
0.010525756632
0.010177103033
0.0099124032809
0.0097028458776
0.0095310413388
0.0093855427985
9.5 0.013730267484
0.0090831345279
0.00740791223
0.0065417578271
0.0060045460935
0.0056349789657
0.0053587646869
0.0051445833034
0.0049707869967
0.0048243885744
10 0.010406040359
0.0049945820073
0.0032425736833
0.0023627687265
0.0018257235286
0.001459545036
0.0011868861975
0.00097685756307
0.00080694483241
0.00066398776611
10.5 0.005160389942
0.00081050255125
0.00074786909528
0.0015526631874
0.002051516054
0.0023946005087
0.0026511013317
0.0028498912811
0.0030111923524
0.0031471124541
11 0.0013059134515
0.003422241281
0.0045723422008
0.0052321319763
0.0056617628258
0.0059653676034
0.006195482706
0.0063771527103
0.0065259062484
0.0066518516811
90
11.5 0.0083133524762
0.0076521224059
0.0082398686139
0.0087000494039
0.0090337945192
0.0092848025767
0.0094802628419
0.0096398184484
0.0097727311917
0.0098863242765
12 0.01540850842
0.011833934932
0.011758341859
0.011975683396
0.012192629057
0.012380042065
0.012534511146
0.012667475864
0.012781791537
0.012881106418
12.5 0.022340759103
0.015928928666
0.015132151877
0.01507486044
0.015159820894
0.015274164483
0.015382940499
0.015485458329
0.015578761692
0.015662291003
13 0.028990507838
0.019906798096
0.018363816975
0.018011322582
0.01795345945
0.01798704686
0.018046501523
0.018115557493
0.018185686826
0.018252226043
13.5 0.035313118715
0.023746903873
0.021455587684
0.020796979258
0.020588740211
0.020535916997
0.020543130669
0.020576512578
0.020621633129
0.020670147192
14 0.041303919531
0.027437723908
0.024410298008
0.023442098458
0.02307863793
0.022935706712
0.022888355026
0.022884468891
0.022903157154
0.022932735512
14.5 0.046977771882
0.030975367047
0.027231831216
0.025955655362
0.025434416284
0.025199369031
0.025095698235
0.02505341487
0.025044670193
0.025054551154
15 0.052357625122
0.034361798768
0.029925332871
0.028345716354
0.027666001814
0.027338190131
0.027176979634
0.027095552705
0.027058752796
0.027048355536
15.5 0.057468438526
0.037603089227
0.032497202861
0.030619778034
0.02978227942
0.029362075098
0.029142570478
0.029021593306
0.028956431687
0.028925365949
16 0.06233421219
0.040707846011
0.034954911837
0.032785032742
0.031791333255
0.031279794296
0.031001624261
0.030840991838
0.030747417323
0.030695471679
16.5 0.066976740184
0.043685927164
0.037306699812
0.034848550969
0.033700640722
0.033099189668
0.032762280519
0.032562141424
0.032440303343
0.032367421914
17 0.071415265637
0.046547475603
0.0395612159
0.036817380303
0.035517223359
0.034827345203
0.034431840259
0.034192536166
0.034042733123
0.033948988042
17.5 0.075666552972
0.049302265676
0.041727154201
0.038698568419
0.037247757471
0.03647072641
0.036016913411
0.035738909582
0.035561539522
0.035447102274
18 0.079745143719
0.051959316108
0.043812931147
0.040499124418
0.038898647128
0.038035293137
0.037523540525
0.037207351955
0.037002862946
0.036867975407
18.5 0.083663649072
0.054526706727
0.045826433953
0.042225937761
0.040476062846
0.039526589252
0.038957291722
0.038603409198
0.038372251515
0.038217196949
19 0.087434215469
0.057011535637
0.047774852134
0.043885676083
0.041985950732
0.040949811999
0.040323345954
0.039932165472
0.039674746106
0.039499820557
19.5 0.091062448777
0.05941996276
0.049664588793
0.045484681701
0.043434018692
0.042309863306
0.041626553283
0.041198311677
0.040914952366
0.040720437202
20 0.094562727534
0.061757304776
0.051501238828
0.047028882183
0.044825707651
0.043611385185
0.042871482336
0.042406201744
0.042097101424
0.041883237931
20.5 0.097779148496
0.064028133239
0.053289616548
0.04852372445
0.046166156076
0.044858781467
0.044062454724
0.043559898396
0.043225100799
0.042992067653
21 0.10093642391
0.066236422444
0.055033816098
0.049974136011
0.047460165238
0.046056228459
0.045203568095
0.044663209776
0.044302576835
0.044050471111
21.5 0.10305275848
0.068385506871
0.056737289279
0.051384512401
0.048712170879
0.047207677394
0.046298709579
0.045719718077
0.045332909804
0.045061732002
22 0.21320071636
0.070478602084
0.058402930072
0.052758726704
0.049926224817
0.048316851668
0.047351561625
0.046732801224
0.046319262696
0.046028906104
22.5 0.21081851068
0.072518153653
0.060033158968
0.054100156339
0.051105987844
0.049387241651
0.048365602408
0.047705648547
0.047264604528
0.046954849144
23 0.20851441406
0.074506370792
0.061629999657
0.055411720577
0.052254733619
0.050422099457
0.049344103011
0.048641271487
0.048171728923
0.04784224004
23.5 0.20628424925
0.076442776523
0.063195151912
0.056695925885
0.053375362192
0.051424435469
0.050290123418
0.049542510406
0.049043268617
0.048693600079
24 0.20412414523
0.078332350451
0.064730065224
0.057954911958
0.054470420906
0.052397017712
0.05120650898
0.050412038581
0.049881706546
0.049511308499
24.5 0.20203050891
0.080175842065
0.066235881994
0.059190497931
0.055542130522
0.053342374657
0.052095888596
0.051252364462
0.05068938413
0.050297614902
25 0.2 0.081665618899
0.067713849378
0.060404233472
0.056592414445
0.054262801387
0.052960675352
0.052065833123
0.051468507385
0.051054648916
25.5 0.1980295086
0.083062450688
0.069164544621
0.061597419351
0.057622929811
0.055160368978
0.05380306995
0.052854627699
0.052221151443
0.051784427521
26 0.19611613514
0.085190237614
0.070588864638
0.062771213136
0.058635097848
0.056036936075
0.054625066841
0.053620771397
0.052949264065
0.052488860464
91
26.5 0.19425717247
0.19425717247
0.071986435886
0.063926484615
0.059630129607
0.05689416345
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86.5 0.10752066611
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91.5 0.1045416747
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100 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1
S\tau 2.8 3.8 4.8 2.8 3.8 4.8 2.8 3.8 4.8
0.5 0.7970325885
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1 0.46282569744
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16 0.030972427271
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0.070693449213
0.070409892393
21 0.043456710001
0.043592833362
0.043748999225
0.066733047232
0.066138501622
0.066021773891
0.073642095562
0.070746687704
0.070465262829
21.5 0.044386977409
0.044513951261
0.044664981072
0.066863293505
0.066242332122
0.066121163429
0.073836654874
0.070799731695
0.070521144415
22 0.045274281133
0.045392308372
0.045538359121
0.06699204058
0.066344025782
0.066218419364
0.073994014592
0.070852875144
0.070577718799
22.5 0.046121321077
0.046230608445
0.046371824852
0.067119360691
0.066443666749
0.066313628352
0.073833779299
0.070906273465
0.070635244718
23 0.04693057424
0.047031331758
0.04716784711
0.067245322895
0.066541336012
0.066406873973
0.074222847592
0.070959917134
0.070694027358
23.5 0.047704317496
0.047796757781
0.047928694682
0.067369991702
0.066637111577
0.066498236836
0.074520462211
0.071013381636
0.070754452294
24 0.048444647691
0.048528985162
0.048656456202
0.067493429115
0.066731068441
0.066587794669
0.07433241794
0.071068364456
0.070817050865
24.5 0.049153499411
0.049229949373
0.049353057718
0.067615692091
0.066823278856
0.066675622411
0.07489299361
0.071123396518
0.070882464253
25 0.049832660728
0.049901438345
0.050020278269
0.06773683877
0.06691381221
0.066761792291
0.073325784302
0.071181633726
0.070951565154
25.5 0.050483787184
0.050545106343
0.050659763709
0.067856916685
0.067002735305
0.066846373919
0.073051447299
0.071239071682
0.071025406178
26 0.051108414239
0.051162486315
0.051273039018
0.067975979579
0.067090112396
0.066929434374
0.072176217389
0.071301149908
0.071105382548
26.5 0.051707968368
0.051755000905
0.05186151929
0.068094068491
0.067176005187
0.067011038262
0.075347758329
0.071366117092
0.071193188956
27 0.052283776974
0.052323972294
0.052426519547
0.068211231302
0.067260473037
0.067091247804
0.071478477472
0.071436441766
0.071291064253
27.5 0.052837077263
0.052870631014
0.052969263542
0.068327505479
0.067343572903
0.067170122905
0.072975846373
0.071513843508
0.071401683167
28 0.053369024192
0.053396123857
0.053490891655
0.068442933921
0.067425359571
0.067247721241
0.0732542702
0.071605788105
0.071528358541
28.5 0.053880697614
0.053901520994
0.053992467994
0.068557555509
0.067505885712
0.067324098291
0.10232890202
0.071702504331
0.071675192652
29 0.054373108696
0.054387822376
0.054474986785
0.0686713922
0.067585201718
0.067399307441
0.073808061064
0.071827538701
0.071846953945
29.5 0.054847205699
0.054855963517
0.054939378153
0.068784488574
0.067663356032
0.067473400038
0.077940810257
0.071974327726
0.072049298247
30 0.055303879197
0.055306820723
0.055386513326
0.06889688007
0.067740395316
0.067546425457
0.10153461651
0.072151358172
0.072288530015
30.5 0.055743966782
0.055741215824
0.055817209359
0.069008583601
0.067816363929
0.067618431117
0.075601441211
0.07237247855
0.072571458586
98
2. pdex_dupire_option.m
Table B.2 The Option Prices(Dupire)
31 0.056168257321
0.056159920459
0.056232233414
0.069119631144
0.067891304966
0.067689462627
0.10101525446
0.072647791442
0.072905307897
31.5 0.05657749481
0.056563659982
0.056632306639
0.069230027803
0.067965259395
0.067759563788
0.10075854437
0.072977284618
0.073296905316
32 0.05697238186
0.056953116995
0.0570181077
0.069339840422
0.06803826654
0.067828776641
0.078907327269
0.073390069657
0.07375240366
32.5 0.057353582869
0.05732893458
0.057390275996
0.069449047301
0.068110363663
0.067897141567
0.075723867936
0.073888660394
0.074276446336
33 0.057721726885
0.057691719237
0.057749414584
0.069557711285
0.068181587697
0.06796469731
0.075998458814
0.074476375812
0.074871910211
33.5 0.058077410219
0.058042043569
0.058096092854
0.069665831699
0.068251972963
0.06803148099
K=9.9
S\tau 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0 0 0 0 0 0 0 0 0 0 0
1 0 1.91E-14 1.16E-11 4.51E-10 5.69E-09 3.76E-08 1.66E-07 5.62E-07 1.56E-06 3.72E-06 7.91E-06
2 0 3.34E-12 1.05E-09 2.73E-08 2.53E-07 1.31E-06 4.76E-06 1.36E-05 3.23E-05 6.75E-050.00012713157451
3 0 2.73E-10 4.44E-08 7.67E-07 5.24E-06 2.15E-05 6.43E-050.00015506859276
0.00032059671056
0.00059068170724
0.00099584894721
4 0 1.37E-08 1.14E-06 1.31E-05 6.68E-050.00021812284945
0.00054199907023
0.0011195159533
0.0020276522572
0.0033319099577
0.0050823861226
5 0 4.67E-07 1.99E-05 0.0001529940418
0.00058462302286
0.0015362392532
0.0032066480711
0.0057394902939
0.0092186656366
0.013675218611
0.019099755736
6 0 1.13E-050.00024602192531
0.00127843345
0.003724382041
0.0079833820041
0.01420233193
0.022346168737
0.032273776654
0.043794600344
0.056707864116
7 00.00019519023949
0.0022104430681
0.0079067555402
0.017892985955
0.031870633401
0.049188119641
0.069161427185
0.091179873385
0.11474412381
0.13946397533
8 0 0.0024228679091
0.01459628447
0.036921319922
0.066552088053
0.10079280743
0.1377580672
0.1762236819
0.21541159448
0.25483849606
0.29419521557
9 0 0.021201244573
0.071080129065
0.13214181253
0.19599456598
0.2593537999
0.32102819311
0.38061737414
0.438087612
0.49352514727
0.54704515082
10 0 0.12718251664
0.25539614581
0.36803720922
0.46751818768
0.55710878826
0.63909929461
0.71508657869
0.7861952856
0.85322620406
0.91679131498
11 0.2 0.50254658508
0.68054072381
0.8148529031
0.92722378072
1.0258019866
1.114702802
1.1963371659
1.2721940973
1.3433349919
1.4105439168
12 1.2 1.270918019
1.3790047998
1.4822973701
1.5769386601
1.6640002525
1.7447679443
1.8203265546
1.8915331414
1.9590200488
2.0232886323
13 2.2 2.2131523359
2.2564469337
2.3146117901
2.3777518806
2.4417128517
2.504837372
2.5664322316
2.6262854401
2.684366814
2.7407090605
14 3.2 3.2020504235
3.2155427474
3.2418413202
3.2769971539
3.3174670412
3.3609453015
3.4060050077
3.4517705118
3.4977150331
3.5435047105
15 4.2 4.2002800583
4.2038304393
4.2139479811
4.230940928
4.2536140062
4.2805643801
4.310620708
4.3428741778
4.3766577964
4.4114885486
99
16 5.2 5.2000345827
5.2008614544
5.2043014327
5.2116385076
5.2231300041
5.2384204904
5.2569487148
5.2781301403
5.3014400373
5.3264431079
17 6.2 6.2000039579
6.2001796243
6.2012407488
6.2041304536
6.2094875721
6.2175348948
6.2282078995
6.2412896199
6.2565004065
6.2735560509
18 7.2 7.2000004282
7.2000351994
7.2003379194
7.2013927363
7.2037197328
7.2076904905
7.2134874775
7.2211441007
7.2305936167
7.2417139098
19 8.2 8.2000000445
8.2000065599
8.2000876153
8.2004490089
8.2014006567
8.2032534083
8.2062441326
8.2105182613
8.2161381489
8.2231023717
20 9.2 9.2000000045
9.2000011748
9.2000217855
9.2001392154
9.2005087933
9.2013321126
9.2028066765
9.2050944859
9.2083091978
9.2125159653
21 10.2 10.2 10.200000204
10.200005229
10.200041737
10.200179037
10.200529587
10.201228011
10.202407585
10.204183346
10.206643087
22 11.2 11.2 11.200000035
11.200001219
11.200012161
11.200061268
11.200205032
11.200524264
11.201112385
11.202062876
11.203459329
23 12.2 12.2 12.200000006
12.200000277
12.20000346
12.200020465
12.200077521
12.200218896
12.200503439
12.200997913
12.201769738
24 13.2 13.2 13.200000001
13.200000062
13.200000965
13.200006695
13.2000287
13.200089584
13.200223589
13.200474283
13.200890572
25 14.2 14.2 14.2 14.200000014
14.200000265
14.200002152
14.200010431
14.200036013
14.20009762
14.20022179
14.200441364
26 15.2 15.2 15.2 15.200000003
15.200000072
15.200000682
15.200003731
15.20001425
15.200041971
15.200102193
15.200215674
27 16.2 16.2 16.2 16.200000001
16.200000019
16.200000213
16.200001316
16.200005561
16.200017799
16.20004646
16.200104032
28 17.2 17.2 17.2 17.2 17.200000005
17.200000066
17.200000459
17.200002144
17.200007458
17.200020868
17.200049588
29 18.2 18.2 18.2 18.2 18.200000001
18.20000002
18.200000158
18.200000818
18.200003092
18.200009273
18.200023383
30 19.2 19.2 19.2 19.2 19.2 19.200000006
19.200000054
19.20000031
19.20000127
19.200004081
19.200010919
31 20.2 20.2 20.2 20.2 20.2 20.200000002
20.200000018
20.200000116
20.200000518
20.200001781
20.200005054
32 21.2 21.2 21.2 21.2 21.2 21.200000001
21.200000006
21.200000043
21.20000021
21.200000772
21.200002321
33 22.2 22.2 22.2 22.2 22.2 22.2 22.200000002
22.200000016
22.200000084
22.200000332
22.200001059
34 23.2 23.2 23.2 23.2 23.2 23.2 23.200000001
23.200000006
23.200000034
23.200000142
23.20000048
35 24.2 24.2 24.2 24.2 24.2 24.2 24.2 24.200000002
24.200000014
24.200000061
24.200000216
36 25.2 25.2 25.2 25.2 25.2 25.2 25.2 25.200000001
25.200000005
25.200000026
25.200000097
37 26.2 26.2 26.2 26.2 26.2 26.2 26.2 26.2 26.200000002
26.200000011
26.200000043
38 27.2 27.2 27.2 27.2 27.2 27.2 27.2 27.2 27.200000001
27.200000005
27.200000019
39 28.2 28.2 28.2 28.2 28.2 28.2 28.2 28.2 28.2 28.200000002
28.200000009
40 29.2 29.2 29.2 29.2 29.2 29.2 29.2 29.2 29.2 29.200000001
29.200000004
41 30.2 30.2 30.2 30.2 30.2 30.2 30.2 30.2 30.2 30.2 30.200000002
42 31.2 31.2 31.2 31.2 31.2 31.2 31.2 31.2 31.2 31.2 31.200000001
100
43 32.2 32.2 32.2 32.2 32.2 32.2 32.2 32.2 32.2 32.2 32.2
44 33.2 33.2 33.2 33.2 33.2 33.2 33.2 33.2 33.2 33.2 33.2
45 34.2 34.2 34.2 34.2 34.2 34.2 34.2 34.2 34.2 34.2 34.2
46 35.2 35.2 35.2 35.2 35.2 35.2 35.2 35.2 35.2 35.2 35.2
47 36.2 36.2 36.2 36.2 36.2 36.2 36.2 36.2 36.2 36.2 36.2
48 37.2 37.2 37.2 37.2 37.2 37.2 37.2 37.2 37.2 37.2 37.2
49 38.2 38.2 38.2 38.2 38.2 38.2 38.2 38.2 38.2 38.2 38.2
50 39.2 39.2 39.2 39.2 39.2 39.2 39.2 39.2 39.2 39.2 39.2
51 40.2 40.2 40.2 40.2 40.2 40.2 40.2 40.2 40.2 40.2 40.2
52 41.2 41.2 41.2 41.2 41.2 41.2 41.2 41.2 41.2 41.2 41.2
53 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2 42.2
54 43.2 43.2 43.2 43.2 43.2 43.2 43.2 43.2 43.2 43.2 43.2
55 44.2 44.2 44.2 44.2 44.2 44.2 44.2 44.2 44.2 44.2 44.2
56 45.2 45.2 45.2 45.2 45.2 45.2 45.2 45.2 45.2 45.2 45.2
57 46.2 46.2 46.2 46.2 46.2 46.2 46.2 46.2 46.2 46.2 46.2
58 47.2 47.2 47.2 47.2 47.2 47.2 47.2 47.2 47.2 47.2 47.2
59 48.2 48.2 48.2 48.2 48.2 48.2 48.2 48.2 48.2 48.2 48.2
60 49.2 49.2 49.2 49.2 49.2 49.2 49.2 49.2 49.2 49.2 49.2
61 50.2 50.2 50.2 50.2 50.2 50.2 50.2 50.2 50.2 50.2 50.2
62 51.2 51.2 51.2 51.2 51.2 51.2 51.2 51.2 51.2 51.2 51.2
63 52.2 52.2 52.2 52.2 52.2 52.2 52.2 52.2 52.2 52.2 52.2
64 53.2 53.2 53.2 53.2 53.2 53.2 53.2 53.2 53.2 53.2 53.2
65 54.2 54.2 54.2 54.2 54.2 54.2 54.2 54.2 54.2 54.2 54.2
66 55.2 55.2 55.2 55.2 55.2 55.2 55.2 55.2 55.2 55.2 55.2
67 56.2 56.2 56.2 56.2 56.2 56.2 56.2 56.2 56.2 56.2 56.2
68 57.2 57.2 57.2 57.2 57.2 57.2 57.2 57.2 57.2 57.2 57.2
69 58.2 58.2 58.2 58.2 58.2 58.2 58.2 58.2 58.2 58.2 58.2
101
70 59.2 59.2 59.2 59.2 59.2 59.2 59.2 59.2 59.2 59.2 59.2
71 60.2 60.2 60.2 60.2 60.2 60.2 60.2 60.2 60.2 60.2 60.2
72 61.2 61.2 61.2 61.2 61.2 61.2 61.2 61.2 61.2 61.2 61.2
73 62.2 62.2 62.2 62.2 62.2 62.2 62.2 62.2 62.2 62.2 62.2
74 63.2 63.2 63.2 63.2 63.2 63.2 63.2 63.2 63.2 63.2 63.2
75 64.2 64.2 64.2 64.2 64.2 64.2 64.2 64.2 64.2 64.2 64.2
76 65.2 65.2 65.2 65.2 65.2 65.2 65.2 65.2 65.2 65.2 65.2
77 66.2 66.2 66.2 66.2 66.2 66.2 66.2 66.2 66.2 66.2 66.2
78 67.2 67.2 67.2 67.2 67.2 67.2 67.2 67.2 67.2 67.2 67.2
79 68.2 68.2 68.2 68.2 68.2 68.2 68.2 68.2 68.2 68.2 68.2
80 69.2 69.2 69.2 69.2 69.2 69.2 69.2 69.2 69.2 69.2 69.2
81 70.2 70.2 70.2 70.2 70.2 70.2 70.2 70.2 70.2 70.2 70.2
82 71.2 71.2 71.2 71.2 71.2 71.2 71.2 71.2 71.2 71.2 71.2
83 72.2 72.2 72.2 72.2 72.2 72.2 72.2 72.2 72.2 72.2 72.2
84 73.2 73.2 73.2 73.2 73.2 73.2 73.2 73.2 73.2 73.2 73.2
85 74.2 74.2 74.2 74.2 74.2 74.2 74.2 74.2 74.2 74.2 74.2
86 75.2 75.2 75.2 75.2 75.2 75.2 75.2 75.2 75.2 75.2 75.2
87 76.2 76.2 76.2 76.2 76.2 76.2 76.2 76.2 76.2 76.2 76.2
88 77.2 77.2 77.2 77.2 77.2 77.2 77.2 77.2 77.2 77.2 77.2
89 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2
90 79.2 79.2 79.2 79.2 79.2 79.2 79.2 79.2 79.2 79.2 79.2
91 80.2 80.2 80.2 80.2 80.2 80.2 80.2 80.2 80.2 80.2 80.2
92 81.2 81.2 81.2 81.2 81.2 81.2 81.2 81.2 81.2 81.2 81.2
93 82.2 82.2 82.2 82.2 82.2 82.2 82.2 82.2 82.2 82.2 82.2
94 83.2 83.2 83.2 83.2 83.2 83.2 83.2 83.2 83.2 83.2 83.2
95 84.2 84.2 84.2 84.2 84.2 84.2 84.2 84.2 84.2 84.2 84.2
96 85.2 85.2 85.2 85.2 85.2 85.2 85.2 85.2 85.2 85.2 85.2
102
97 86.2 86.2 86.2 86.2 86.2 86.2 86.2 86.2 86.2 86.2 86.2
98 87.2 87.2 87.2 87.2 87.2 87.2 87.2 87.2 87.2 87.2 87.2
99 88.2 88.2 88.2 88.2 88.2 88.2 88.2 88.2 88.2 88.2 88.2
100 89.2 89.2 89.2 89.2 89.2 89.2 89.2 89.2 89.2 89.2 89.2
S\tau 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
0 0 0 0 0 0 0 0 0 0 0
1 1.53E-05 2.75E-05 4.64E-05 7.41E-050.00011341175203
0.00016697550108
0.00023782522063
0.00032910653414
0.00044401435868
0.00058576931504
20.00022067999625
0.00035840307213
0.00055120139488
0.00081008693919
0.0011457390057
0.001568229717
0.0020868021025
0.0027097940127
0.003444434106
0.0042969095229
3 0.0015652088983
0.00232488863
0.0032977621876
0.0045025225042
0.0059534442069
0.0076607407391
0.0096310373156
0.011867604641
0.014371152736
0.017139996515
4 0.0073133693226
0.010045431675
0.013286442016
0.01703414809
0.021278814468
0.026005274828
0.031194706009
0.03682532307
0.042874400968
0.049318134385
5 0.025456259835
0.032693663426
0.040748602748
0.049553884084
0.059043489635
0.069153917451
0.079825036475
0.091000508178
0.10262822289
0.11466010092
6 0.070821922558
0.08596101798
0.10196671941
0.1187013653
0.13604750413
0.15390438334
0.17218507854
0.19081564588
0.20973150689
0.22887783337
7 0.1650343899
0.19121380706
0.21781945044
0.244711038
0.27177798073
0.29893261882
0.32610538537
0.35324261563
0.3803011638
0.40724838531
8 0.33327949703
0.37196102974
0.41016857603
0.44786096342
0.48501096665
0.52160413185
0.55763640856
0.59311159408
0.6280385518
0.66242930213
9 0.59878015632
0.6488643985
0.69742761247
0.74458271417
0.79042856811
0.835056123
0.8785485402
0.92098086942
0.96242126459
1.0029302018
10 0.97737133948
1.0353450975
1.0910074945
1.1446013934
1.1963357375
1.246387318
1.2949061453
1.3420193722
1.3878375375
1.432456734
11 1.4743978845
1.5353516444
1.5937491291
1.6498755595
1.7039728958
1.7562427195
1.8068548214
1.8559517387
1.9036571594
1.9500779387
12 2.0847498321
2.1437333738
2.2005035423
2.2552793736
2.308250494
2.3595793557
2.4094053611
2.4578479574
2.5050117684
2.5509881865
13 2.7953881903
2.8484986869
2.9001440864
2.9504172772
2.999401267
3.047175449
3.0938144107
3.1393870433
3.1839567533
3.2275805384
14 3.5889229355
3.6338343919
3.678168249
3.7218850034
3.7649586179
3.8073768611
3.8491388833
3.8902521136
3.9307292634
3.9705858084
15 4.4470076526
4.4829445508
4.5191075823
4.5553567004
4.5915836472
4.6277049374
4.6636572088
4.6993942632
4.7348815406
4.7700950423
16 5.3527833394
5.3801667899
5.4083592574
5.4371739808
5.4664602629
5.4960951561
5.5259776824
5.5560266707
5.5861743215
5.6163666786
17 6.2921923489
6.3121705847
6.3332792344
6.3553365626
6.3781888092
6.4017048966
6.4257721849
6.4502952711
6.4751909441
6.5003889063
18 7.254359612
7.2683800178
7.2836227218
7.2999449248
7.3172181824
7.335327155
7.3541682318
7.3736493369
7.3936876814
7.4142100109
19 8.2313682922
8.2408703412
8.2515239824
8.2632385601
8.2759247159
8.2894962838
8.3038714703
8.3189734649
8.3347310098
8.3510781727
20 9.2177404412
9.2239806698
9.2312102531
9.2393880442
9.2484651028
9.2583881461
9.2691021847
9.2805515674
9.2926824661
9.3054424593
21 10.209846408
10.213829142
10.218605226
10.224171708
10.230513553
10.237607175
10.245423382
10.253928492
10.26308757
10.27286409
103
22 11.205369943
11.207846592
11.210925443
11.214627948
11.21896314
11.223930318
11.229521484
11.23572229
11.242515045
11.249878617
23 12.202880984
12.204384885
12.206325569
12.208736578
12.211641147
12.215053697
12.218981403
12.223424839
12.228380201
12.23383944
24 13.201522162
13.202415682
13.203613966
13.205153583
13.207063892
13.209367458
13.21208073
13.215214376
13.218774537
13.222763121
25 14.200792817
14.201313162
14.202039117
14.20300468
14.204239682
14.20576946
14.207614843
14.209792224
14.212313954
14.215188679
26 15.200407476
15.200704971
15.20113713
15.20173263
15.202518419
15.203519013
15.204756054
15.206248202
15.20801091
15.210056714
27 16.200206849
16.20037408
16.200627212
16.200988825
16.201481432
16.202126695
16.202944793
16.203954223
16.205171219
16.206609951
28 17.200103807
17.200196363
17.200342433
17.20055888
17.200863456
17.201274106
17.20180834
17.202482997
17.203313542
17.204314154
29 18.200051547
18.200102049
18.200185186
18.200313024
18.200498933
18.200757064
18.201101821
18.201547638
18.202108308
18.202796983
30 19.200025349
19.200052548
19.200099271
19.200173848
19.200285972
19.200446367
19.200666393
19.200957864
19.201332495
19.201801837
31 20.200012355
20.200026832
20.200052787
20.200095798
20.200162673
20.200261268
20.200400237
20.200588895
20.200836814
20.201153725
32 21.200005974
21.200013597
21.200027863
21.200052409
21.200091885
21.200151884
21.200238806
21.200359771
21.200522351
21.200734472
33 22.200002867
22.200006843
22.200014609
22.200028482
22.200051563
22.200087733
22.200141609
22.200218484
22.200324191
22.200465003
34 23.200001367
23.200003423
23.200007613
23.200015386
23.200028761
23.200050377
23.200083487
23.200131938
23.200200114
23.200292861
35 24.200000648
24.200001703
24.200003946
24.200008266
24.200015955
24.200028769
24.200048956
24.200079255
24.200122892
24.200183532
36 25.200000306
25.200000843
25.200002036
25.200004419
25.200008806
25.200016345
25.200028563
25.200047373
25.200075105
25.200114477
37 26.200000144
26.200000416
26.200001046
26.200002352
26.200004838
26.200009244
26.200016588
26.200028186
26.200045692
26.200071088
38 27.200000067
27.200000204
27.200000535
27.200001247
27.200002647
27.200005206
27.200009592
27.200016698
27.20002768
27.20004396
39 28.200000031
28.2000001
28.200000273
28.200000659
28.200001443
28.20000292
28.200005525
28.200009853
28.200016702
28.200027078
40 29.200000015
29.200000049
29.200000139
29.200000347
29.200000784
29.200001632
29.200003171
29.200005793
29.200010041
29.200016618
41 30.200000007
30.200000024
30.200000071
30.200000182
30.200000425
30.20000091
30.200001814
30.200003394
30.200006016
30.200010163
42 31.200000003
31.200000012
31.200000036
31.200000096
31.20000023
31.200000506
31.200001035
31.200001983
31.200003593
31.200006196
43 32.200000001
32.200000006
32.200000018
32.20000005
32.200000124
32.20000028
32.200000589
32.200001155
32.20000214
32.200003766
44 33.200000001
33.200000003
33.200000009
33.200000026
33.200000067
33.200000155
33.200000334
33.200000671
33.200001271
33.200002283
45 34.2 34.200000001
34.200000005
34.200000014
34.200000036
34.200000086
34.200000189
34.200000389
34.200000753
34.200001381
46 35.2 35.200000001
35.200000002
35.200000007
35.200000019
35.200000047
35.200000107
35.200000225
35.200000445
35.200000833
47 36.2 36.2 36.200000001
36.200000004
36.20000001
36.200000026
36.20000006
36.20000013
36.200000263
36.200000501
48 37.2 37.2 37.200000001
37.200000002
37.200000006
37.200000014
37.200000034
37.200000075
37.200000155
37.200000301
104
49 38.2 38.2 38.2 38.200000001
38.200000003
38.200000008
38.200000019
38.200000043
38.200000091
38.200000181
50 39.2 39.2 39.2 39.200000001
39.200000002
39.200000004
39.200000011
39.200000025
39.200000054
39.200000108
51 40.2 40.2 40.2 40.2 40.200000001
40.200000002
40.200000006
40.200000014
40.200000031
40.200000065
52 41.2 41.2 41.2 41.2 41.2 41.200000001
41.200000003
41.200000008
41.200000018
41.200000039
53 42.2 42.2 42.2 42.2 42.2 42.200000001
42.200000002
42.200000005
42.200000011
42.200000023
54 43.2 43.2 43.2 43.2 43.2 43.2 43.200000001
43.200000003
43.200000006
43.200000014
55 44.2 44.2 44.2 44.2 44.2 44.2 44.200000001
44.200000002
44.200000004
44.200000008
56 45.2 45.2 45.2 45.2 45.2 45.2 45.2 45.200000001
45.200000002
45.200000005
57 46.2 46.2 46.2 46.2 46.2 46.2 46.2 46.200000001
46.200000001
46.200000003
58 47.2 47.2 47.2 47.2 47.2 47.2 47.2 47.2 47.200000001
47.200000002
59 48.2 48.2 48.2 48.2 48.2 48.2 48.2 48.2 48.2 48.200000001
60 49.2 49.2 49.2 49.2 49.2 49.2 49.2 49.2 49.2 49.200000001
61 50.2 50.2 50.2 50.2 50.2 50.2 50.2 50.2 50.2 50.2
62 51.2 51.2 51.2 51.2 51.2 51.2 51.2 51.2 51.2 51.2
63 52.2 52.2 52.2 52.2 52.2 52.2 52.2 52.2 52.2 52.2
64 53.2 53.2 53.2 53.2 53.2 53.2 53.2 53.2 53.2 53.2
65 54.2 54.2 54.2 54.2 54.2 54.2 54.2 54.2 54.2 54.2
66 55.2 55.2 55.2 55.2 55.2 55.2 55.2 55.2 55.2 55.2
67 56.2 56.2 56.2 56.2 56.2 56.2 56.2 56.2 56.2 56.2
68 57.2 57.2 57.2 57.2 57.2 57.2 57.2 57.2 57.2 57.2
69 58.2 58.2 58.2 58.2 58.2 58.2 58.2 58.2 58.2 58.2
70 59.2 59.2 59.2 59.2 59.2 59.2 59.2 59.2 59.2 59.2
71 60.2 60.2 60.2 60.2 60.2 60.2 60.2 60.2 60.2 60.2
72 61.2 61.2 61.2 61.2 61.2 61.2 61.2 61.2 61.2 61.2
73 62.2 62.2 62.2 62.2 62.2 62.2 62.2 62.2 62.2 62.2
74 63.2 63.2 63.2 63.2 63.2 63.2 63.2 63.2 63.2 63.2
75 64.2 64.2 64.2 64.2 64.2 64.2 64.2 64.2 64.2 64.2
105
76 65.2 65.2 65.2 65.2 65.2 65.2 65.2 65.2 65.2 65.2
77 66.2 66.2 66.2 66.2 66.2 66.2 66.2 66.2 66.2 66.2
78 67.2 67.2 67.2 67.2 67.2 67.2 67.2 67.2 67.2 67.2
79 68.2 68.2 68.2 68.2 68.2 68.2 68.2 68.2 68.2 68.2
80 69.2 69.2 69.2 69.2 69.2 69.2 69.2 69.2 69.2 69.2
81 70.2 70.2 70.2 70.2 70.2 70.2 70.2 70.2 70.2 70.2
82 71.2 71.2 71.2 71.2 71.2 71.2 71.2 71.2 71.2 71.2
83 72.2 72.2 72.2 72.2 72.2 72.2 72.2 72.2 72.2 72.2
84 73.2 73.2 73.2 73.2 73.2 73.2 73.2 73.2 73.2 73.2
85 74.2 74.2 74.2 74.2 74.2 74.2 74.2 74.2 74.2 74.2
86 75.2 75.2 75.2 75.2 75.2 75.2 75.2 75.2 75.2 75.2
87 76.2 76.2 76.2 76.2 76.2 76.2 76.2 76.2 76.2 76.2
88 77.2 77.2 77.2 77.2 77.2 77.2 77.2 77.2 77.2 77.2
89 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2 78.2
90 79.2 79.2 79.2 79.2 79.2 79.2 79.2 79.2 79.2 79.2
91 80.2 80.2 80.2 80.2 80.2 80.2 80.2 80.2 80.2 80.2
92 81.2 81.2 81.2 81.2 81.2 81.2 81.2 81.2 81.2 81.2
93 82.2 82.2 82.2 82.2 82.2 82.2 82.2 82.2 82.2 82.2
94 83.2 83.2 83.2 83.2 83.2 83.2 83.2 83.2 83.2 83.2
95 84.2 84.2 84.2 84.2 84.2 84.2 84.2 84.2 84.2 84.2
96 85.2 85.2 85.2 85.2 85.2 85.2 85.2 85.2 85.2 85.2
97 86.2 86.2 86.2 86.2 86.2 86.2 86.2 86.2 86.2 86.2
98 87.2 87.2 87.2 87.2 87.2 87.2 87.2 87.2 87.2 87.2
99 88.2 88.2 88.2 88.2 88.2 88.2 88.2 88.2 88.2 88.2
100 89.2 89.2 89.2 89.2 89.2 89.2 89.2 89.2 89.2 89.2
106
3. pdex_dupire.m
Table B.3 The Dupire Volatilities
4. pdex_dis_dupire.m
Table B.4 Distance between Implied Volatilities and the Dupire Volatilities
S=1:100 .9.1.99 == τ,K
S=1:10 S=11:20 S=21:30 S=31:40 S=41:50 S=51:60 S=61:70 S=71:80 S=81:90 S=91:100
dupireσ0.1103875
1351
0.0997995
08443
0.1008571
9287
0.1054613
1299
0.1063929
1169
0.1005420
3662
0.0927723
37099
0.0851126
23971NaN NaN
dupireσ0.1076708
6022
0.0997230
90679
0.1011916
7192
0.1058786
9067
0.1060496
9887
0.0997735
64282
0.0920217
31358
0.0731887
71445NaN NaN
dupireσ0.1056060
4521
0.0997211
77438
0.1015750
5219
0.1062409
3047
0.1056361
0557
0.0989937
26871
0.0913724
05838
0.0814369
46953NaN NaN
dupireσ0.1041218
578
0.0997553
88519
0.1020028
7325
0.1065384
4185
0.1051580
4199
0.0982061
27539
0.0904617
37785
0.0677596
35682NaN NaN
dupireσ0.1030239
5962
0.0998121
17284
0.1024683
1984
0.1067633
7199
0.1046217
8906
0.0974167
90179
0.0899351
55032
0.0714249
27393NaN NaN
dupireσ0.1021527
2677
0.0998936
68878
0.1029625
3212
0.1069097
8776
0.1040338
4167
0.0966223
86341
0.0892097
0572Inf NaN NaN
dupireσ0.1014207
1648
0.1000051
7119
0.1034749
6498
0.1069737
5558
0.1034008
074
0.0958330
65625
0.0882457
96623
0.0451730
90455NaN NaN
dupireσ0.1007983
6181
0.1001518
7943
0.1039938
0133
0.1069533
181
0.1027290
1956
0.0950573
49174
0.0863168
13945Inf NaN NaN
dupireσ0.1003138
6197
0.1003395
3131
0.1045064
1173
0.1068483
8764
0.1020247
7141
0.0942440
41036
0.0868018
02914Inf NaN NaN
dupireσ0.0999892
99656
0.1005734
5679
0.1049998
4282
0.1066605
6525
0.1012934
5784
0.0934875
21274
0.0834390
478030 NaN NaN
tau=1.9;K=10.
x Distance x Distance x Distance x Distance x Distance
1 0 8 0 15 0.038940004132 22 0.068873
053982 29 0.10268879475
2 0 9 0 16 0.0096295917963 23 0.075617
101167 30 0.10557478798
3 0 10 0.0055731734841 17 0.011647
847367 24 0.081526241573 31 0.107758
81617
4 0 11 0.033925944179 18 0.028040
893949 25 0.086754277933 32 0.108238
5101
5 0 12 0.64329959398 19 0.041173
752426 26 0.091414513072 33 0.138504
78093
107
5. pdex_imp_small_s.m
Table B.5 Implied Volatilities Volatilities for Small Stock Prices
6 0 13 0.16942858353 20 0.051991
380509 27 0.095590702894 34 NaN
7 0 14 0.08371187034 21 0.061090
850366 28 0.099342677508
impσ 1.0:1:.10=τ
S(10-12) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.5 17.067374456
12.374808848
10.261732387
8.9885985531
8.1130805179
7.4628010286
6.9547831538
6.5434512861
6.2012658136
5.9107854141
1 16.807915572
12.18978099
10.109905584
8.8566500881
7.9947402955
7.3545311423
6.8543562506
6.4493577404
6.1124270276
5.82639813
1.5 16.654951304
12.08068344
10.020377406
8.7788393657
7.9249514453
7.2906788408
6.7951276005
6.3938628779
6.060030152
5.7766257294
2 16.545877707
12.002883154
9.9565293611
8.7233457874
7.8751775815
7.2451379284
6.7528835684
6.3542812586
6.0226576181
5.7411246589
2.5 16.460958653
11.942308056
9.9068155815
8.6801358652
7.8364204939
7.209676281
6.7199886499
6.3234591426
5.9935553914
5.7134794798
3 16.391368347
11.892664854
9.8660723467
8.6447221475
7.8046556713
7.1806119612
6.6930277045
6.2981968286
5.9697025155
5.6908206547
3.5 16.332384569
11.850586173
9.8315366027
8.6147033947
7.777729559
7.1559747029
6.6701732129
6.2767820665
5.949482378
5.671612589
4 16.281181594
11.814056928
9.8015548616
8.588642614
7.7543533712
7.1345853953
6.6503314813
6.2581901385
5.9319274897
5.6549362737
4.5 16.235932943
11.781774575
9.7750582899
8.5656109033
7.7336940002
7.1156818209
6.6327955033
6.2417586221
5.9164124196
5.6401976148
5 16.195389228
11.75284809
9.7513157264
8.5449728167
7.7151815258
7.0987425504
6.6170816156
6.2270343513
5.9025093054
5.6269901858
5.5 16.158657868
11.726640897
9.7298047959
8.5262743574
7.6984087626
7.0833950405
6.6028442555
6.2136935511
5.8899124495
5.6150236033
6 16.125078854
11.702682275
9.7101392194
8.5091798063
7.6830746052
7.0693638011
6.5898278831
6.2014967921
5.8783957896
5.6040831291
6.5 16.094150304
11.680614273
9.6920252539
8.4934338658
7.6689500762
7.056439333
6.5778381697
6.1902619962
5.8677874123
5.5940054608
7 16.065481626
11.660158307
9.6752342837
8.4788378405
7.6558569635
7.0444585829
6.5667238684
6.1798474509
5.8579535156
5.584663498
7.5 16.038762836
11.641093281
9.6595848691
8.4652340351
7.6436538279
7.0332921417
6.5563649398
6.1701406833
5.848787905
5.5759563647
8 16.01374378
11.623240771
9.6449305857
8.4524951865
7.6322265243
7.0228355728
6.5466645087
6.1610509242
5.8402048775
5.56780265
8.5 15.990219649
11.606454692
9.6311515484
8.4405170969
7.6214815954
7.013003367
6.5375432565
6.1525038543
5.8321342619
5.5601356968
9 15.968020631
11.590613922
9.6181483551
8.4292133712
7.6113415506
7.0037246238
6.5289354154
6.144437851
5.8245178747
5.5529002353
9.5 15.947004359
11.57561691
9.6058376668
8.4185115747
7.6017414208
6.9949398993
6.5207858447
6.1368012483
5.8173069336
5.5460499256
10 15.927050293
11.561377673
9.5941489195
8.4083503759
7.5926261953
6.9865988605
6.513047855
6.1295502995
5.8104601329
5.5395455306
10.5 15.908055465
11.547822766
9.5830218366
8.3986773845
7.5839488832
6.9786585124
6.5056815634
6.1226476348
5.8039421884
5.5333535325
11 15.889931225
11.534888951
9.5724045197
8.3894474919
7.5756690245
6.9710818353
6.4986526298
6.116061078
5.7977227194
5.5274450719
11.5 15.872600695
11.522521389
9.562251963
8.3806215802
7.5677515323
6.9638367257
6.4919312744
6.109762726
5.7917753797
5.5217951215
108
12 15.855996774
11.510672215
9.5525248828
8.3721655049
7.5601657807
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3.3729323903
2.9467235518
2.6590764343
3.3281068247
2.9080900942
2.6245981012
11 3.4808665784
3.0397361357
2.7420773646
3.3720420021
2.9459561886
2.6583916222
3.3275950072
2.9076489609
2.6242044013
11.5 3.4774448643
3.0367877122
2.7394464688
3.3711613895
2.9451972491
2.6577143268
3.3270864019
2.9072105958
2.6238131718
12 3.4741664188
3.0339627257
2.7369257066
3.3702903382
2.9444465488
2.6570443834
3.3265809684
2.9067749641
2.6234243815
12.5 3.4710196322
3.0312511737
2.7345061539
3.3694286409
2.943703909
2.6563816328
3.326078667
2.9063420315
2.6230379998
13 3.4679942542
3.0286442256
2.7321799319
3.3685760971
2.9429691569
2.6557259207
3.3255794587
2.9059117646
2.6226539969
13.5 3.4650811899
3.0261340456
2.7299400498
3.3677325127
2.9422421252
2.6550770977
3.3250833054
2.9054841303
2.6222723434
14 3.462272332
3.0237136489
2.7277802758
3.3668976995
2.941522652
2.6544350194
3.3245901695
2.9050590964
2.6218930104
14.5 3.4595604226
3.0213767827
2.7256950309
3.3660714755
2.9408105801
2.6537995457
3.3241000142
2.9046366312
2.6215159695
15 3.4569389386
3.0191178268
2.723679301
3.365253664
2.9401057576
2.6531705408
3.3236128034
2.9042167034
2.6211411931
15.5 3.4544019954
3.0169317116
2.7217285628
3.3644440941
2.939408037
2.6525478732
3.3231285016
2.9037992825
2.6207686538
16 3.4519442663
3.0148138476
2.7198387216
3.3636425996
2.9387172752
2.6519314152
3.3226470738
2.9033843384
2.6203983248
115
16.5 3.4495609138
3.0127600672
2.718006059
3.3628490197
2.9380333336
2.6513210433
3.322168486
2.9029718418
2.6200301798
17 3.4472475317
3.0107665741
2.7162271879
3.3620631982
2.9373560777
2.6507166371
3.3216927045
2.9025617635
2.6196641929
17.5 3.4450000951
3.0088299006
2.7144990143
3.3612849835
2.9366853766
2.6501180802
3.3212196961
2.9021540752
2.6193003388
18 3.4428149173
3.0069468704
2.7128187044
3.3605142284
2.9360211038
2.6495252595
3.3207494286
2.9017487488
2.6189385925
18.5 3.4406886135
3.0051145672
2.7111836562
3.3597507899
2.9353631359
2.648938065
3.3202818701
2.901345757
2.6185789295
19 3.4386180682
3.0033303068
2.7095914747
3.3589945293
2.9347113534
2.6483563899
3.3198169891
2.9009450726
2.6182213256
19.5 3.4366004073
3.0015916131
2.7080399507
3.3582453114
2.9340656399
2.6477801305
3.3193547549
2.9005466693
2.6178657573
20 3.4346329743
2.9998961971
2.7065270419
3.3575030054
2.9334258824
2.647209186
3.3188951374
2.9001505209
2.6175122013
20.5 3.4327133082
2.9982419386
2.7050508566
3.3567674835
2.9327919712
2.6466434583
3.3184381067
2.8997566019
2.6171606347
21 3.4308391251
2.9966268697
2.703609639
3.3560386218
2.9321637992
2.6460828521
3.3179836336
2.8993648869
2.6168110351
21.5 3.4290083014
2.9950491609
2.7022017568
3.3553162997
2.9315412627
2.6455272746
3.3175316894
2.8989753513
2.6164633803
22 3.4272188595
2.9935071082
2.7008256895
3.3546003997
2.9309242604
2.6449766356
3.3170822458
2.8985879708
2.6161176486
22.5 3.4254689541
2.9919991218
2.6994800188
3.3538908078
2.9303126939
2.6444308472
3.3166352751
2.8982027213
2.6157738187
23 3.4237568613
2.9905237163
2.6981634192
3.3531874126
2.9297064674
2.643889824
3.31619075
2.8978195794
2.6154318695
23.5 3.4220809675
2.9890795016
2.6968746506
3.3524901059
2.9291054876
2.6433534828
3.3157486435
2.8974385219
2.6150917804
24 3.4204397608
2.9876651749
2.6956125504
3.3517987823
2.9285096637
2.6428217423
3.3153089293
2.897059526
2.6147535311
24.5 3.4188318222
2.9862795137
2.6943760279
3.351113339
2.927918907
2.6422945237
3.3148715813
2.8966825693
2.6144171016
25 3.4172558183
2.984921369
2.693164058
3.3504336759
2.9273331314
2.6417717499
3.3144365741
2.8963076298
2.6140824723
25.5 3.4157104944
2.9835896599
2.6919756762
3.3497596954
2.9267522528
2.641253346
3.3140038823
2.8959346859
2.6137496237
26 3.4141946687
2.9822833681
2.6908099738
3.3490913025
2.9261761893
2.6407392388
3.3135734814
2.8955637161
2.6134185369
26.5 3.4127072264
2.9810015331
2.689666094
3.3484284045
2.9256048609
2.6402293571
3.3131453469
2.8951946995
2.6130891931
27 3.411247115
2.979743248
2.6885432276
3.3477709108
2.9250381897
2.6397236313
3.3127194549
2.8948276155
2.6127615739
27.5 3.4098133397
2.9785076557
2.6874406097
3.3471187333
2.9244760997
2.6392219935
3.3122957817
2.8944624437
2.6124356611
28 3.4084049589
2.977293945
2.6863575166
3.3464717859
2.9239185167
2.6387243776
3.3118743043
2.8940991641
2.612111437
28.5 3.4070210812
2.9761013476
2.6852932629
3.3458299847
2.9233653685
2.6382307191
3.3114549996
2.8937377571
2.6117888839
29 3.405660861
2.9749291352
2.6842471986
3.3451932476
2.9228165842
2.6377409549
3.3110378452
2.8933782031
2.6114679845
29.5 3.4043234961
2.9737766165
2.6832187069
3.3445614947
2.9222720951
2.6372550234
3.310622819
2.8930204832
2.6111487219
30 3.4030082245
2.972643135
2.6822072021
3.3439346478
2.9217318337
2.6367728646
3.3102098992
2.8926645786
2.6108310791
30.5 3.4017143218
2.9715280665
2.6812121272
3.3433126306
2.9211957344
2.6362944199
3.3097990642
2.8923104707
2.6105150398
31 3.4004410986
2.970430817
2.6802329522
3.3426953685
2.9206637328
2.6358196319
3.309390293
2.8919581414
2.6102005876
116
Table B.6 Slopes of Implied Volatility Curves for Small Stock Prices
31.5 3.3991878987
2.9693508211
2.6792691726
3.3420827888
2.9201357663
2.6353484445
3.3089835647
2.8916075726
2.6098877066
32 3.3979540965
2.9682875397
2.6783203075
3.3414748203
2.9196117735
2.6348808032
3.3085788588
2.8912587468
2.609576381
32.5 3.3967390955
2.9672404589
2.6773858983
3.3408713935
2.9190916944
2.6344166544
3.3081761551
2.8909116464
2.6092665952
33 3.3955423261
2.966209088
2.6764655071
3.3402724404
2.9185754707
2.6339559459
3.3077754337
2.8905662545
2.6089583339
33.5 3.3943632447
2.9651929584
2.6755587159
3.3396778944
2.9180630448
2.6334986266
Simp
∂∂σ 1.0:1:.10=τ
S(10-12) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.5-5.1891776734E+11
-3.7005571602E+11
-3.0365360519E+11
-2.6389692989E+11
-2.3668044473E+11
-2.1653977265E+11
-2.008538064E+11
-1.8818709149E+11
-1.7767757202E+11
-1.6877456831E+11
1-3.0592853667E+11
-2.1819510081E+11
-1.7905635691E+11
-1.5562144493E+11
-1.3957770057E+11
-1.2770460292E+11
-1.1845730022E+11
-1.1098972492E+11
-1.0479375114E+11
-99544801152
1.5-2.1814719272E+11
-1.5560057232E+11
-1.2769608959E+11
-1.1098715654E+11
-99547727451
-91081824878
-84488064142
-79163238721
-74745067703
-71002140968
2-1.6983810811E+11
-1.2115019628E+11
-99427559128
-86419844277
-77514175180
-70923294871
-65789836985
-61644231863
-58204453405
-55290358241
2.5-1.3918061196E+11
-99286402507
-81486469644
-70827435477
-63529645284
-58128639510
-53921890733
-50524628104
-47705751931
-45317650112
3-1.1796755617E+11
-84157362651
-69071487939
-60037505613
-53852224503
-49274516618
-45708983254
-42829524107
-40440274981
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3.5-1.0240594972E+11
-73058489273
-59963482306
-52121561486
-46752375764
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-39683463169
-37183856060
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4-90497302476
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4.5-81087430244
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-33878541010
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-29448541587
-27806228353
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5-73462720576
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-43021860936
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-30695019725
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5.5-67158028491
-47917243416
-39331152969
-34189102294
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-26032744710
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6-61857098769
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-36227931163
-31491880845
-28249058067
-25848936291
-23979426888
-22469591719
-21216754672
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6.5-57337355840
-40911932535
-33581940359
-29192050597
-26186225371
-23961500249
-22228602694
-20829090686
-19667793366
-18683925628
117
7-53437580211
-38130051516
-31298829157
-27207610829
-24406271275
-22332882299
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-18331221250
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7.5-50038112671
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-20913137794
-19400862220
-18179518144
-17166054960
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8-47048262625
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8.5-44398035379
-31681541218
-26006386606
-22607451432
-20280089660
-18557486324
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-16132006748
-15232774359
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9-42032542776
-29994024416
-24621376423
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-19200259550
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-16299141442
-15273205252
-14421882247
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9.5-39908133398
-28478472904
-23377494636
-20322397586
-18230450859
-16682077479
-15475979336
-14501897665
-13693601329
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10-37989654989
-27109814406
-22254165877
-19345982882
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-15880696310
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10.5-36248480446
-25867630440
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-14057867158
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11-34661059834
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11.5-33207841803
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-16912150696
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12-31872458884
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12.5-30641104278
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13-29502049589
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13.5-28445267645
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14-27462134615
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-8954943237
14.5-26545192640
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-15552537962
-13520700071
-12129362509
-11099517396
-10297302232
-9649393346.9
-9111744272
-8656224523.2
15-25687959079
-18333205230
-15050517190
-13084316527
-11737920725
-10741338079
-9965031521.6
-9338046497.6
-8817759553
-8376949092.7
15.5-24884772043
-17760139874
-14580141192
-12675438844
-11371151022
-10405733572
-9653702079.1
-9046321698
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-8115273726.4
16-24130664395
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-12291539515
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-10090629583
-9361389745.3
-8772416032.3
-8283670061.8
-7869581336.2
16.5-23421260286
-16715927102
-13723037919
-11930391869
-10702829485
-9794198702.9
-9086399374.4
-8514741165.2
-8040363265.1
-7638447294
118
17-22752689653
-16238897528
-13331483115
-11590026578
-10397514030
-9514824713.7
-8827231721.8
-8271892317.2
-7811055466.5
-7420611605.3
17.5-22121517143
-15788548318
-12961826379
-11268695495
-10109271968
-9251072912.5
-8582555919.5
-8042622473.9
-7594569229.3
-7214955777.9
18-21524682697
-15362696563
-12612276160
-10964841425
-9836706558.6
-9001665307
-8351186466.2
-7825820827.5
-7389855721.9
-7020483485.5
18.5-20959451636
-14959391448
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-10677072732
-9578569554.3
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-8132063893.3
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-7195977613.2
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19-20423372495
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-10404141886
-9333741966
-8541432449.9
-7924238437.7
-7425754046.8
-7012094628.5
-6661620068.3
19.5-19914241263
-14213602489
-11669061956
-10144927269
-9101217740
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-7240798875.8
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20-19430070909
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-11385483505
-9898417670.1
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-8126320946.7
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20.5-18969065297
-13539181994
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-8669538402.8
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-6897428719.5
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21-18529596789
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-8468820400.7
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-7190031172.3
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-6362467506
-6044488908.4
21.5-18110186922
-12926330367
-10612412093
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-8277260938.7
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22-17709489691
-12640410666
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22.5-17326277027
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23-16959426149
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23.5-16607908510
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24-16270780107
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24.5-15947172962
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25-15636287615
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25.5-15337386488
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26-15049788008
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26.5-14772861389
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-8657588107.8
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-5733385339.4
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130
Bibliography
[1]Lishang Jiang(1994)Mathematical Modeling and Methods of Option Pricing, 312, Tongji University.[2] Matlab Indexesfunction pdepe.[3]Wikipedia,Greeks(Finance), http://en.wikipedia.org/wiki/Greeks_(finance) .[4] Steven L. Heston (1993)A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond andCurrency Options, Yale University.