Backtesting SVI Pararneterization of Implied Volatilities

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impa
Instituto Nacional de Matemática Pura e Aplicada
Backtesting SVI Pararneterization of
Implied Volatilities
Author: Rafael Balestro Dias da Silva
Advisor: Vinícius Viana Luiz Albani
Rio de Janeiro
July 2016
Acknowledgements
I dedicate the present Work to my parents Ana and Mauro,
girlfriend Luiza for the understanding in relation to my co
Who helped me a lot With the mathematics and to my goo
the presence and great debates. Also, I Would like to than
Raphael, Luis, Pedro and Zeca - and to my friends at Vinc
special thank to Professor Jorge Zubelli, Ph.D and to my a‹
the immeasurable support during the construction of this
ii
Who made it possible, to my beloved
ns
(f.
Á
tant absence, to my brother Vitor
friends Gabriela and Leonardo for
to my friends at Ancar Ivanhoe
i Partners - Renato and Tiago. A
:lvisor Vinicius Albaní, Ph.D for all
master thesis.
Abstract
Since the stock market crash in 1987, many generalizations of Black-Scholes model have been
introduced, in order to incorporate the so-called volatilíty smile effect. In this master thesis
We study the numerical aspects of the stochastic Volatility inspired (SVI) parameterization of
implied volatility smile. lntroduced at the end of 1990°s, this model - Which is basically a
parameterized functional form - presents a nice smile adherence, Which made it popular Within
market practitioners mainly by its low computational cost. We present numerical experiments
using real data, evaluating how reliable Would the fitted parameters be on calculating volatilities
for future dates.
Key Words: Financial Options, Financial Derivatives, Volatility Smile, Black-Scholes Formula,
SVI fit, Stochastic Differential Equation, Variance l\/lodelling.
iii
Contents
Index
Introduction
1 Local Volatility Surface and Dupire”s Equation
1.1 Derivation of Dupire Equation . . . . . . . . . . . . .
1.2 Local Volatility and Implied Volatility . . . . . . . .
2 SVI Parameterization: An Introduction
2.1 Roger Lee°s Moment Formula . . . . . . . . . . . . .
2.2 Static Arbitrage . . . . . . . . . . . . . . . . . . . . .
2.2.1 Calendar Spread Arbitrage . . . . . . . . . . .
2.2.2 Butterfly Arbitrage . . . . . . . . . . . . . . .
2.3 Different SVI Formulations for Implied Volatility Slices
2.3.1 The Raw SVI Parameterization . . . . . . . .
2.3.2 The natural SVI parameterization . . . . . . .
3 Parameters l\/Iodelling and Preliminary Results
3.1 The Algc›rithm . . . . . . . . . . . . . . . . . . . . .
3.2 The Implementation . . . . . . . . . . . . . . . . . .
3.3 Illustrative Results . . . . . . . . . . . . . . . . . . .
4 Backtesting SVI
4.1 Backtesting l\/Iethodology . . . . . . . . . . . . . . . .
4.2 DAX Index spot options . . . . . . . . . . . . . . . .
4.2.1 Backtest Results for DAX Index Spot Options
4.3 SPX Index spot options . . . . . . . . . . . . . . . .
4.3.1 Backtest Results for SPX Index Spot Options
4.4 PETR4 Equity Options . . . . . . . . . . . . . . . . .
4.4.1 Backtest Results for PETR4 Equity Options .
4.5 Oonfidence Intervals . . . . . . . . . . . . . . . . . .
4.5.1 Numerical Example . . . . . . . . . . . . . . .
4.6 An Alternative Approach . . . . . . . . . . . . . . . .
5 Concluding Remarks
Appendix
V
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o o o o o Q o Q Q Q Q Q Q O O
Q Q Q o o Q n n Q Q Q o o a n
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Bibliography
OONTENTS
51
List of Figures
2.1
2.2
2.3
2.4
2.5
2.6
3.1
3.2
3.3
r'\r'\t'\r'\r'\r'\r”r'\r'\r'\r'\r“r'\r'\r'\r'\r'\r'\
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4.21
4.22
4.23
4.24
4.25
4.26
4.27
<;>1›-|>›-lšC.»JO0hD
Generic Curve Shape of a Raw SVI Parameterization . . . . . . . . . . . . . . . 1
Íncreasing parameter a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1ncreasing parameter b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1 reasing parameter p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Í reasing parameter m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1 reasing parameter U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
E(J
E(J
E(J
SPX Implied Volatility Curve at October, 13th 2015 . . . . . . . . . . . . . . . . 19
SPX Implied Volatility Curve at October, 14th 2015 . . . . . . . . . . . . . . . . 19
SPX Implied Volatility Curve at October, 15th 2015 . . . . . . . . . . . . . . . . 20
DAX Index Call Options” SVI Fitted Curve (expiring: March/ 16) . . . . . . . . 23
gíy) function for DAX Index Calls Options SVI Fit . . . . . . . . . . . . . . . . 23
DAX Index Put Options” SVI Fitted Curve (expiring: March/ 16) . . . . . . . . 23
gíy) function for DAX Index Puts Options SVI Fit . . . . . . . . . . . . . . . . 24
D>+1 Market vs SVI Implied Vols. Comparison for DAX Call Options . . . . . . 25
D>+1 Market Prices vs Model Prices Comparison for DAX Call Options . . . . . 25
D>+1 Market vs SVI Implied Vols. Comparison for DAX Put Options . . . . . . 25
D>+1 Market Prices vs Model Prices Comparison for DAX Put Options . . . . . 26
D>+3 Market vs SVI Implied Vols. Comparison for DAX Call Options . . . . . . 26
L>+3 Market Prices vs Model Prices Comparison for DAX Call Options . . . . . 26
D'+3 Market vs SVI Implied Vols. Comparison for DAX Put Options . . . . . . 27
_J+3 Market Prices vs Model Prices Comparison for DAX Put Options . . . . . 27
L'+5 Market vs SVI Implied Vols. Comparison for DAX Call Options . . . . . . 27
D>+5 Market Prices vs Model Prices Comparison for DAX Call Options . . . . . 28
_, Market vs SVI Implied Vols. Comparison for DAX Put Options . . . . . . 28
Market Prices vs Model Prices Comparison for DAX Put Options . . . . . 28
SPX 1ndex Call Options” SVI Fitted Curve (expiring: March/ 16) . . . . . . . . 29
g(y) function for SPX 1ndex Calls Options SVI Fit . . . . . . . . . . . . . . . . 29
SPX Index Put Options” SVI Fitted Curve (expiring: March/ 16) . . . . . . . . . 30
gíy) function for SPX 1ndex Puts Options SV1 Fit . . . . . . . . . . . . . . . . . 30
D'+1 Market vs SVI Irnplied Vols. Comparison for SPX Call Options . . . . . . 31
D>+1 Market Prices vs À/Iodel Prices Comparison for SPX Call Options . . . . . 31
D'+1 Market vs SVI Irnplied Vols. Comparison for SPX Put Options . . . . . . 32
D*-'fl Market Prices vs À/Iodel Prices Comparison for SPX Put Options . . . . . 32
D>+3 Market vs SVI Irnplied Vols. Comparison for SPX Call Options . . . . . . 32
D›+3 Market Prices vs À/Iodel Prices Comparison for SPX Call Options . . . . . 33
D>+3 Market vs SVI Irnplied Vols. Comparison for SPX Put Options . . . . . . 33
J
J
J
_J-J-cn
D'-'v5
vii
viii LIST OF FIGURES
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4
›-|\›-|\›-|\›-|\›-|\›-|\›-|\›-|\
FI
Q)
II
z)
II
z)
Cí
Q)
9
0
OO\lÓ3CI!›-|šO~DN>›-\
9
0
1
2
3
D>+3 Market Prices vs Model Prices Comparison for SPX Put Options .
D'+5 Market vs SVI Implied Vols. Comparison for SPX Call Options . .
DH'-5 Market Prices vs Model Prices Comparison for SPX Call Options .
D>+5 Market vs SVI Implied Vols. Comparison for SPX Put Options . .
D'+5 Market Prices vs Model Prices Comparison for SPX Put Options .
PETR4 Equity Call Options” SVI Fitted Curve (expiring: March/ 16) . . 35
g(y) function for PETR4 Equity Cal1s Options SVI Fit . . . . . . . . . .
PETR4 Equity Put Options” SVI Fitted Curve (expiring: March/ 16) . .
DH-1 Mark
D>+1 Mark
D'+1 Mark
D*-|-1 Mark
L>+3 Mark
D'-I-3 Mark
D>+3 Mark
L>+3 Mark
D'-I-5 Mark
J
J
JJL'-I-5 Mark
D>+5 Mark
et
et
et
et
et
et
et
et
et
'+5 Market
et
et
D>+1 DAX Calls Prices Confidence Interval . .
D'+1 DAX Call Model Prices with its Confidence Intervals . . . . . . . .
gíy) function for PETR4 Equity Puts Optionsvs SVI Implied Vols. Comparison
S1Prices vs Model Prices Compari
vs SVI Implied Vols. Comparison
Comparison for PETR4 Put OptionsPrices vs Model Prices
SVI Fit . . . . . . . . . .
for PETR4 Call Options .
on for PETR4 Call Options
for PETR4 Put Options .
vs SVI Implied Vols. Comparison for PETR4 Call Options .
Prices vs Model Prices Comparison for PETR4 Call Options
vs SVI Implied Vols. Comparison for PETR4 Put Options .
Prices vs Model Prices Comparison for PETR4 Put Options
vs SVI Implied Vols. Comparison for PETR4 Call Options .
Prices vs Model Prices Comparison for PETR4 Call Options
vs SVI Implied Vols. Comparison for PETR4 Put Options .
Prices vs Model Prices Comparison for PETR4 Put Options
SPX European Calls - Interpolated Total Variance Surface . . . . . . . .
SPX European Calls - Interpolated Implied Volatilities Surface . . . . . .
SVI Backtest with Interpolated Volatility Surface . . . . . . . . . . . . .
List of Tables
3.1
›-|\›-|\›-I\›-l\›-|\›-|\ Cä<;fl›-|>O0hD›-\
Calibrated Parameters for the Example Curves . . .
Calibrated Parameters for DAX Index Spot Options
Quantitative Indexes for DAX Index Spot Options .
Calibrated Parameters for SPX Index Options . . .
Quantitative Indexes for SPX Index Options . . . .
Calibrated Parameters for PETR4 Index Options .
Quantitative Indexes for PETR4 Index Spot Options
ix
LIST OF TABLES
Introduction
The price of securities such as bonds, stocks or ƒutures cannot be predicted. Because it is
subjected to different sources of risk and shocks, derivative contracts were introduced to protect
market practítioners and companies from such randomness.
As an example, an European call (or European put) option is a contract that gives to its
holder the right to buy (or sell) a given amount of an asset at a specific maturity time for a
fixed strike price. In contrast, American options can be exercised at any time until its maturity.
These so-called Vanilla options are the most liquid and intuitively reflect the market expectation
of the asset price future behaviour.
More complex derivatives are key tools in hedging and trading strategies, however, they
may be not sufficiently liquid to avoid arbitrage opportunities, i.e., they must be appropriately
priced with mathematical models, taking into account the available market information.
In the classical Black-Scholes model [2l, the dynamics of the instantaneous logarithmic of
returns of an asset is given by an Itô process with constant coefficients, the asset”s drift and the
volatility. Under no-arbitrage arguments, European call option price is given by the well-known
Black-Scholes formula.
In practice, volatility is not constant as we can see when we use Black-Scholes formula to
identify it. It changes with strike and maturity. These volatilities are the so-called implied
volatilities. The volatility smile can be seen when we plot such implied volatilities with the
option”s time to maturity fixed and vary the strike price. This name comes from its shape,
since volatility usually presents lower levels for strike prices close to the underlying asset price,
or at-the-money strikes, and higher levels as we go far from the at-the-money.
After the stock market crash in 1987, many generalizations of Black-Scholes model were
proposed, where the volatility (or diffusion) coefficient is assumed stochastic. More precísely,
let the asset price evolution be given by:
dst : Lltstdt + mstdwt, É 2 O,
where ,at is the drift, ut the variance (the square of volatility) and {l/Vi, t 2 0} is a Wiener
process. One possible way to model the variance ut as a stochastic process is the following [23]:
d'Ut : O¿(›St, Ut, JF T6(St, Ut,
oz and Ú are continuous and well-behaved functions, T is a constant and {Zt, t 2 0} is another
Wiener process, correlated with Wt. In a first observation, this assumption clearly adds ran-
domness sources into the model. Also, note that setting T : 0 and ,at as a constant, we would
have the standard time dependent Black-Scholes variance.
The stochastic volatility models describe in a more precise way the evolution of the under-
lying asset prices within a period of time. This is of particularly importance, when we want to
evaluate path-dependent derivatives. However, the simulation of theses models have a higher
computational cost.
1
2 Introduction
A market where every contingent claim can be replicated through a portfolio of negotiable
assets is called a complete market [16]. This is equivalent to the existence of a unique equiva-
lent martingale measure, which gives, for example, unique European call prices. A necessary
condition for this property to hold is the number of sources of uncertainty be less or equal to
the number of underlying assets in a portfolio. So, a portfolio of assets modeled by stochastic
volatility models fails to satisfy such condition, since the number of sources of uncertainties is
in general higher than the number of assets.
Local volatility models introduced in [6] and [5], can be seen as a particular class of stochastic
volatility models (see [11]), where the diffusion coefficient, or local volatility surface, of the asset
price dynamics is a deterministic function of the underlying asset”s current price and time. This
model is interesting because it keeps the market completeness and the possibility of hedging
positions through the derivatives. Ã/Ioreover, in [6] it is proved that the local volatility is the
unique diffusion coefficient consistent with quoted European option prices. However, local
volatility calibration is an ill-posed inverse problem, and some regularization technique must
be applied. See [4, 7, 8, 21].
Even though there is a map between local and implied volatilities, in practice, its eval-
uation involves a complicated tridimensional interpolation method, which is not necessarily
well-behaved in terms of smoothness and continuity, since the maturity dates are discrete and
sparse.
Therefore, the quest for a model which is simple and easy to calibrate, that fits the market
smile and satisfies non-arbitrage conditions brought the community to study some simplifica-
tions of stochastic volatility models. Among them, there are the stochastic volatility inspired
(SVI) models [10, 13], which consists in techniques to parametrize the implied volatility smile.
It was firstly developed at Merryl Lynch in 1990 and publicly disseminated in [10].
Since it is relatively easy to implement and has interesting properties such as absence of
static arbitrage, SVI models have been largely studied by the community in the past two
decades. In addition, it has been shown in [12] that SVI is not arbitrary, since the large
maturity limit of Heston implied volatility holds SVI non-arbitrage properties.
This master thesis aims to explore the problem of modeling implied volatility smiles in order
to guarantee some iinportant properties such as absence of calendar-spread arbitrage and static
arbitrage, good fit to bid-ask implied volatilities and good behavior under extreme strikes and
maturities. To do so, we will apply known SVI parameterization techniques over real data from
liquidly traded European options and compare this fixed parameters with the realized implied
volatilities for the next trading days.
Structure of the thesis In Chapter 1, we present the derivation of Dupire”s partial dif-
ferential equation and the inverse problem of local volatility surface calibration from quoted
European option data through the implied diffusion process for the underlying asset”s price
observed from the option prices for different strikes. In Chapter 2, we make a review of the-
oretical results concerning SVI models. Chapter 3 is concerned with implied volatility surface
modeling for real data. Some backtest techniques for market data are described in Chapter 4.
In Chapter 5 we draw some concluding remarks.
Chapter 1
Local Volatility Surface and Dupire”s
Equation
It is well-known that the the probability density of the underlying asset price atthe option”s
maturity time can be inferred from European vanilla option prices [3, 19]. In [6, 5], it was
shown that there exists a unique state-dependent diffusion coefficient consistent with these
distributions. Such parameter, denoted by oL(S, t), is known as local volatility surface.
There are at least two interesting interpretations of local volatility surface, it can be seen as
an average of all possible instantaneous volatilities in a stochastic process, this was the initial
intuition of practítioners. Another one, it determines how the implied volatilities evolve, since,
by no-arbitrage arguments, there is a one-to-one map between both quantities. Below, we
explain both concepts.
1. 1 Derivation of Dupire Equation
Suppose the stock price follows the diffusion process below:
clS
Ê” : /lidt JF UL(Si›7Íš Soldl/Vi» É É 0»
1:
where {Wt, t 2 0} is a standard Brownian Motion, ,at : rt - Dt is the risk-neutral drift, given
by the risk-free rate rt minus the dividend yield Dt, and oL(St, t) is the local volatility surface.
The undiscounted risk-neutral price C : C' exp (LT rtdt) of an European option with strike
K and maturity T is given by:
õ(S(), K, I JE [HlãX<[0, ST _ I/ (ST _ K)§0(ST, T, SQ)ClST,
K
with <p(ST, T ; S0) the probability density of the stock price St at time t : T (conditional to
S0). It evolves according to the Fokker-Planck equation:
18222 Ô Õgo
§E(ULST<PJ - ÊWSTWJ I Ô?-
Now, assuming that go satisfies sufficient conditions such that we can differentiate under the
integral sign, it follows that:
3
4 CHAPTER 1. LOCAL VOLATILITY SURFACE AND DUPIRE”S EQUATION
/\./
ÔC OO
8? _ _ fK ÊÚ(STvTv S0)dSTv
érõ
Í :</>(K›TšS0J›
and in T: N
ÕC °° Ô
Ô? - [K (ST _ T, Sg)CIlST,
By the Fokker-Planck equation, the later equation is equivalent to
W 1o2 2 2 ô/K <sT - K> (¿@<f›-LSTt›> - ¶<isTt›>) ist.
Using some algebra tricks and integrating by parts twice, we find Dupire”s equation:
aí 2k¢o%í ~ oõ
(,)T:0-LT¶¬L/sL(T) . (I.I.1)
Now, we can isolate the volatility term to get the so-called Dupire”s formula:
ÔC ~ ÕC
_ - T C - K-2 or ”( ” ( off)
2 ÕK2
which is well-defined due to the static no-arbitrage condition
o2oTK, > 0
when K > 0.
This is the most important conclusions in Dupire”s article In theory, the term in the right-
hand of (1.1.2) can be easily evaluated with a complete set of European options prices, moreover,
this represents the unique expression for local volatility surface. In what follows, we shall use
equation (1.1.2) as the definition of local volati1ity surface. Note that, in practice, Dupire”s
formula is highly unstable, since we must interpo1ate noisy and scarce data and differentiate it,
in other words, local vc›latility surface calibration is an ill-posed inverse problem. See [4, 7].
1.2 Local Volatility and Implied Volatility
As one could guess, it is possible to express local volatility in terms of the implied volatility, con-
ditional to K, T and S0, derived from the Black-Scholes formula, and denoted by ot;5(K, T ; S0).
In this section, we follow [11] closely.
Firstly, define the variables:
T
FT I S0 exp pitdt)
0
1.2. LOCAL VOLATILITY AND IMPLIED VOLATILITY 5
is the forward price of the stock at t : 0,
K
92108 E
<,.z(st,, y, T) z õ§t.S(sO, Fte?/, T)T
is the Black-Scholes total implied variance. In terms of the above variables, the Black-Scholes
formula for the future value of the option price becomes:
is the log-strike, and
C”Bs(FTzy›wJ = FT[^¡(d1)_@y/V(d2)lM-.r>-@~N<-ipiÉs
Defining u(T) : LT (rt - Dt) dt, u(y,T) : C(FTey,T) and uL(y,T) : oL(FTe@/,T)2 (the
local variance), the PDE in (1.1.1) can be rewritten as:
Ôu VL Ô211 Õu
_ I T . 1.2.28,. 2 (ôyt t,y)+tz< >«› < >
Differentiating the Black-Scholes formula in (1.2.1), we have:
82035 : _1_i+L2 8035
ÕLU2 8 2a) 2a›2 Õw ”
82035 : 1 _ ä ÕGBS
Õydw 2 to Õw ”
Õ2CBS ÔCB5 ÕCB5
TW : Ty + 28? (1.2.3)
Now, we want to describe equation (1.2.2) in terms of the implied variance UBS. Since, for every
y and T, u(y, T) = CBs(FT, y, oa), differentiating the equation above by y and T, it follows that:
ÕU _ ÕGBS ÔCBS Õwíy- ôy + Ôw ay, (1.2.4)
ou ô2oBS ó>2oBS ou o2oBS ou 2 ôott ouõ_e:õ_l2+2z›i¶íi+ às (ai) WT? M5)
o oo o8-; = fã + ;J(T)@. (1.2.õ)
The expression in (1.2.6) can be rewritten as:
and by (1.2.2), the right-hand side of the above equation is equal to:
Õu u E9211 do
Õ? _”<T)U I 2L (59% Õy>”
6 CHAPTER 1. LOCAL VOLATILITY SURFACE AND DUPIRE”S EQUATION
So, it follows that,
oct-,›So¿ _ it ou oz»
Õw ÔT_2 Ôy2 Ôy”
Ô2 Ô
Substituting the terms Õ-Q; and íw by the expressions in (1.2.4) and (1.2.5) in the PDE (1.2.2),
IJ IJ
we have:
LCBSÔi:V_LLCBS 2_Ôi¬|_ 1 y Ôw¬|_ _L_i+L2 ai 2+yi
Ôw ÔT 2 Õw Õy 2 ao Ôy 8 2o; 2w2 Ôy Ôy2 ”
and simplifying:
ôi_V 1_äôi+1 1 1+y2 Ôi2+Lyi
ÔT_ L wôy 4 4 to w2 Õy 2Õy2 ”
So, we just have to invert the above equation to isolate the local variance:
81
ÕTV : 1.2.7L 1yoa+1 1 1+¿2 o¿2+f‹3>2_a ( ”
ao Õy 4 4 ao w2 Ôy 2 Ô;/2
The intuitive conclusion of this result is that the local variance is expressed as the forward
Õ
Black-Scholes implied volatilities distorted by the skew i.
Ô
Even though this work doesn”t aim to calibrate the loffal volatility surface, but the implied
volatility one, this relation composes an important background to the overall understanding
of the problem of pricing options using different approaches to the variance other than the
constant one assumed in the Black-Scholes model.
While Dupire”s result provided the discussion with a stronger theoretical background, the
models which aim to interpolate and extrapolate the implied volatility surface from given market
data is more straightforward and simpler to implement, with market smile adherence. Thus,
the implied variance can also be expressed as a function of local volatilities.
Chapter 2
SVI Parameterization: An
Introduction
Given the computational and theoretical difficulties related to the simulation of stochastic
volatility models, it makes sense to search for simpler techniques that are easy to implement,
arbitrage-free and that fit the market implied smile. One possible approach is the arbitrage-free
interpolation for implied volatility surfaces, such as [9] and [14]. However, these models are
not well suited to extrapolate the implied smile, and there is no closed-formula for volatilities.
So, we choose the SVI model introduced in [10], which has a closed-form representation for the
irnplied volatility function and guarantees the absence of static arbitrage.
In this chapter we shall present the definitions of different static arbitrage conditions and
how to prevent them in the formulation of the SVI model. More precísely, we shall state
sufficient conditions for the existence of a non-negative martingale defined in the probability
space where European call options prices are defined as the expected value of their payoffs,
under the risk-neutral measure. We shall show also that the model is free of calendar-spread
arbitrage, which means that the total variance must increase with the maturity. Visually, this
means that if we plot several slices of volatility smiles (one for each maturity time) in the same
log-strike >< implied volatility chart, the curves must not cross each other.
Another important characteristic that the SVI model has is the consistency with respect to
Roger Lee”s moment formula [18], which set bounds for the tail slopes of implied volatility at
extreme maturities.
2.1 Roger Lee”s l\/Ioment Formula
In [18], the author presented the moment formula of implied volatilities, which implies some
properties of the implied volatility tail slopes at large and small-strikes, setting bounds for these
slopes, which can be interpreted as arbitrage bounds. We shall review these results.
Let us recall the notation
K
y :1Og< v and : O-BS('S[0iy7T)T7
T
where the latter is the total implied variance of Black-Scholes model for a given maturity T.
Let 5 be defined by B[y[/T :
Roger Lee”s moment formula shows that, when calculating
ÊR : limsupfi and BL : limsupfi,
y-›oo y->-oo7
8 CHAPTER 2. SVI PARAMETERIZATION: AN INTRODUCTION
it follows that /31.-t,5L G [0, 2]. Also, it states the following:
Theorem 1 (Roger Lee”s Moment Formula for Implied Volatility).
1 s 1
%+íR-ã=sup{p:lE[S;¬+p]<oo},
and
1 5 1 _
E1-É;-§:sup{q:lfÍ.[STq]<oo}.
In [18], the theorem above is proved when the asset ST has an arbitrary distribution.
Theorem 1 shows that the best constant for the tails slopes depend only on the number of
finite moments of the distribution of the underlying asset.
Despite we shall not give the complete proof of Theorem 1 since it is not the focus of the
present work, we shall demonstrate two important lemmas that help us to understand why SVI
parameterization is consistent with Roger Lee”s moment formula.
Define 50 = exp (- f0T utdt) as a discount factor and write the Black-Scholes option price
for the strike K(y) as C(K(y)) : CBg(y,w(y)), where K(y) : FT exp(y) is the strike at the
log-moneyness y. We have the following lemmas:
Lemma 1. There exists y* > 0 such that for every y > y*,
w(y) > \/2[@J[/T»
Proof. Since CBS is strict monotone, the lemma follows from this:
ÚBS(y›W(yJJ < ÚBs(yz \/ 2[y[/TJ»
If a: > x*. On the left hand side of the above equation we have:
um Ô(K(y)) z um s01E(sT _ K)+ z 0,
y->oo K->oo
which is guaranteed by the Dominated Convergence Theorem, since E(ST - K)+ < oo. By
L”Hôpital rule, the right-hand limit can be written as:
um o Su ×/21y1T>- õ F [‹1›<@> um @›<p<y>‹1>< @>] - rf F /2.B 1 _ 0 T _ _ _ 0 T
y->oo y->oo
which completes the proof. Ú
Lemma 2. For any É > 2 there exists y* such that for all y < y*,
w(@/) > \/ölyl/T
Hence, for 5 : 2 the conclusion holds if and only if lP”(ST : 0) < 1 /2.
Proof. For Õ > 2 there exists y* such that for all y < y*,
l1””(51¬ < FT @Xp(@/)) < @(-\/f-(fi)[y[) _ @Xp(-3J)<Í)(-\/f+(fi)[”J[),
where f is the characteristic function of the underlying asset”s price at t : T.
2.2. STATIC ARBITRAGE 9
The stated above is true since, as y -> oo the second part converges to lP”(ST : 0), and
the first part approaches 1 (or 1 /2 in case S : 2 for the special case stated in the lemma).
Therefore, due to the strict monotonicity of the Black-Scholes price of an European put option
PBSI
PT.›s(yzw(y)) = fi@1E(K(@/J - STJ* S fi‹›K(y)JP°(ST < FT @><i>(i/)) < PBs(@/z \//fly]/TJ»
for all y < y*. E
The stated above shows that the large and small tails slopes are not larger than 2 and, as a
consequence, the implied Black-Scholes variance is linear in the log-strike y as T oo. These
are characteristics that the SVI parameterization must hold.
2.2 Static Arbitrage
We want to find an implied volatility surface that is free of static arbitrage, which means that
there exists a non-negative martingale model such that:
Õ(KzT) = E@l(5T _ KW»
that is, the undiscounted European call option price at maturity can be written as the expected
value of its final payoff in the risk-neutral measure Q for some process St.
Definition 1. A uolatility surface ao is free of static arbitrage and only
(i) is free from calendar spread arbitrage;
(ii) is free of butterfly arbitrage at each single time slice.
Note that, Item (iJ of Definition 1 guarantees the monotonicity of European call option
prices with respect to theirs maturities, while Item (ii) guarantees the existence of a probability
density. For more information about static arbitrage and implied volatilities, please see [20].
From now on this chapter shall review some theoretical aspects exposed in [13].
2.2.1 Calendar Spread Arbitrage
Consider two European call options of strike K and maturities T1 and T2 with T1 > T2. Since
T1 > T2 we would intuitively expect that the price of the option with the larger maturity
is higher than the other one. That happens because the more time we have ahead, the more
uncertainty we suppose We will be exposed, so we would expect a higher total variance, therefore
a higher price. That is why, whenever this condition is not respected, the calendar spread
arbitrage is characterized.
In order to have an accurate model to price options, one would suppose that this condition
must be respected, that is, the implemented model must guarantee monotonicity for the call
options with same strike and different maturities. That leads us to the following definition:
Definition 2. A uolatility surface of is free of calendar spread arbitrage
Õw
_ t > Oat (yâ ) _ 7
forallyš R andt>0.
10 CHAPTER 2. SVI PARAMETERIZATION: AN INTRODUCTION
This definition is motivated by the following lemma:
Lemma 3. Considering a constant proportional to the underlying asset prices dividends
applicable), the volatility map ao is free of calendar spread arbitrage if and only
Ê_Í(v,t) 2 0,
forally€Randt>0.
Proof. Let (Xt)t¿0 be a martingale and L 2 0, 0 É t1 < t2. Then, the following relation is true:
EKXT2 _ L)+l 2 EKXT1 _ L)+l
/\/ /'\./
Now, let C1 and C2 be options with strikes K1 and K2, and maturities t1 and t2, respectively,
and with the same log-strike or log-moneyness:
K1 K2 <>í I í ieF1 F2 Xp v
t t
Then, the process {Xt}t¿0 defined by % for t 2 0 is a martingale and we have the following
relation:
/-\/ /-\/
ä _ @X1í>(_y)1El(Xzâz _ @X1>(v))+l 2 @X1@(_y)El(Xzz1 _ @><p(v))+l _
if the dividends are proportional. E
This relation means that, for constant moneyness, option prices are non-decreasing in time to
maturity. If this relation stands for Black-Scholes prices CB5(y, w(y, t)) then, w(y, t) is strictly
increasing.
2.2.2 Butterfly Arbitrage
In this subsection we shall define butterfly arbitrage over a time slice of the volatility surface.
Thus, we shall use w(y) in order to simplify the notation, since the time will be hold fixed.
Now, recalling the Black-Scholes formula:
CBs(v, w(v)) = 5(/\f(d+(v)) _ @Xr›(v)/\f(d-(v)))» f01' 81111 G R
with ./\/1 the Gaussian cumulative distribution function and d1(y) : -y/ \/oJ(y) i \/w(y) /2.
Now, let g : R -› R be a function defined by:
g(y)z=(1-ÊÍT) -%y”2(Ê-fã) -1%”. (2.2.1)
Definition 3. A volatility map y -> w(y) given for some maturity is free of butterfly arbitrage
if the corresponding asset”s probability density is non-negative. Please refer to the 1 for the
realtion between w(y) and the underlying asset probability density.
sfíš
\./
N)
Lemma 4. A volatility map y -> w(y) given for some maturity is free of butterfly arbitrage if
and only g(y)2 0 for all y É R and lim d+(y) : -oo.
y-›oo
2.3. DIFFERENT SVI FORMULATIONS FOR II\/IPLIED VOLATILITY SLICES 11
Proof. As mentioned in Chapter 1, the probability density function <p can be inferred from a
set of option prices by:
32001) Õ20Bs(vz w(v))W) I W I .zw >
for any y 6 R and K : Ft exp(y).
Now, differentiating the Black-Scholes formula, for any y G R we have:
( ) d-( )2</>(v) _ @Xp .
It is important to notice that this density function may not always have its integral equals
1. In particular, call prices must go to 0 as y -> oo, which is equivalent to the second part of
Lemma 4. E
2.3 Different SVI Formulations for Implied Volatility Slices
In this section we shall present different functional forms which have been proposed to fit the
implied volatility curve for a given maturity with real data.
Firstly, we shall see the raw SVI parameterization, which was introduced in [10], then we
will present an alternative (and equivalent) parameterization so-called natural SVI parameteri-
zation. We shall not discuss the SVI-JW (Jump-Wing) parameterization since we are interested
in time independent methods for backtesting in different dates.
2.3.1 The Raw SVI Parameterization
Let XR _ {a, b, p, m, o} be a set of parameters. The raw SVI parameterization of total implied
variance is:
«nz X1.) z cz + 1» [pv - m> + do - mr + U2] . <2.õ.1>
where b 2 0, [p] < 1, a G R, o > 0, and a + bo\/1- ,o2 2 0 to guarantee that w(y;XR) 2 0 for
all y 6 R. It is easy to see that for a given set XR, the function y -> w(y, XR) is strictly convex.
Now, we will investigate the intuitive meaning of the formulation parameters XR through
the derivatives of equation to in (2.3.1):dia_ Z 1
Õa 7
ÔwÉ z z›<z.1-m>+¢<zz-m>2+‹›2.
Õ
dia m-y
sw I z z bp»\/(v_m) +0
Õw o
Õ? Z ¢<y-m>2+‹12'
12
Figure 2.1 shows a generic raw SVI curve with the set of parameters XR : (0.05, 0.15, 0.40, 0.30 0 45
CHAPTER 2. SVI PARAMETERIZATION: AN INTRODUCTION
Figures 2.2-2.6 illustrate how variations on each parameter may change the shape of the volatil-
ity curve
E3
.Z
1:I*."1:1E1l
rnpe
I I I I I I I
IÍL3
III I'“¬_'l I'.|'I
III I"-.'I
11.15 .ta
"¬~«.1\\\
|]_`| I I I I I I I-2 -1.5 -1 -11.5 11 11.5 1 1.5 2
Ger'1er1`1: 5"."I Cuwe furlm plied "."1:1I=E1liIit1_.f Exam |:1Ie
"-.
"-.
_ . / _
Lug-F 1:1r¬w'ar1:I I'-1'I1:1r1e'1-mas
Figure 2.1: Generic Curve Shape of a Raw SVI Parameterization
2.3. DIFFERENT SVI FORMULATIONS FOR II\/IPLIED VOLATILITY SLICES
-:-1-:L1
EE
a1:I*.f1:1al
11111
-:-'11-:|.'-
EE
a1:I*.f1:1at
11111
Incraaaing param atara
Ú-5 1 1 1 1 1 1 1
íäifllãanaria E:-:ampla
[ Gar'1ar1`1:E1‹arn1:1Ia1.-.-ithhi1;|hara [
11.45 - -
III.-4
EL35
l1I.3
III I"'¬_'l I'.|'I
IÍL2 - fr; -
_I__r"
E
11.15- / _
|]_`| I I I I I I I
-2 -1.5 -1 -IÍI.5 IJ III.5 1 1.5 2
Lag-F award I'-11 an a'1-11 aaa
Figure 2.2: Increasing parameter a
Incraaaing 1:1ararr1 atar |:1
Ú-5 1 1 1 1 1 1 1
[í5*."IGar1ar11:E:-:ampla |
Ganafla E:1‹:arn1:1Ia1-zith higharh
III.-45 - -
III.-*I - -
IIL35 - -
IIL3 - -
III I"-_'I I'_|'I
IÍL2 - -
lÍI.I5- -
¬¬-11___h¬______
111,1 1 1 1 1 1 1 1
-2 -1.5 -1 -III.5 III III.5 1 1.5 2
L1:11;1-F 1:11*-.»1'ar1:I I'-sl an a'1-11 aaa
Figure 2.3: Increasing parameter b
CHAPTER 2. SVI PARAMETERIZATION: AN INTRODUCTION
Ir1erea.air11;1 |:1arar1'1 et er 1;
Ú-'45 1 1 1 1 1 1 1
í5'1'IGer1er1`eE:a5n'1|:1Ie
Ger1en'e E1a5r1'11:1Ie'1-.-'ith higher 1;
III.=1 - -
11.55 - -
E3 11.5 //_
/ _
mpedvefi
1:1 5.1 I'.l'I
[1.2 \% /r
131.15 _ / _
_-'
1z|_1 1 1 1 1 1 1 1
-2 -1.5 -1 -III.5 III III.5 1 1.5 2
L1:11;1-F1:1r¬w'ar1:II'-111:1r1e'1-r1eaa
Figure 2.4: Increasing parameter ,o
Ir1ereaair11;| 1:1ararr1 eterm
Ú-35 1 1 1 1 1 1 1
í5'1'IGer1en'e E:‹:arr1|:1Ie
Generic E1a5r1'11:1Ie'1-.-'ith higherm
11.5 - -
`“~.
E5
Ê: r-¬_1 -'.r|
.Z
I:I1"1:1al
rr1|:1e [1.2 - -
5.15- BRM _
1z|_1 1 1 1 1 1 1 1
-2 -1.5 -1 -III.5 III III.5 1 1.5 2
Leg-F 1:1rw'ar1:I I'-111:1r1e'1-r1eaa
Figure 2.5: Increasing parameter m
As you can see in Figures 2.2, 2.3, 2.4, 2.5 and 2.6,
2.3. DIFFERENT SVI FORMULATIONS FOR II\/IPLIED VOLATILITY SLICES 15
í5"1'II5er1en`eE:-1:e1r1'11:1Ie
Ger1er1eE:-=:=51r1'11:1Ie'-1.-'ithhi1;1her 1:1 [
Ir11:rea.air11;| param et er 1:1
11.5 - -
`“~.
\
E3
Ê: ru -'.r|
rr1|:1e1:I1f1:1a1 1:1 `r~_~
xx/
11 1 1 1 1 1 1 1 1
-2 -1.5 -1 -III.5 III III.5 1 1.5 2
L1:11;1-F1:1r¬w'ar1:II'-111:1r1e'1-r1eaa
Figure 2.6: Increasing parameter o
o when a increases, the level of total variance also increases, that is, there is a vertical
translation of the curve;
o increasing b, the slopes of the put and call wings also increase, which represents an increase
of total variance for out-of-money strikes;
o when ,o increases, there is a reduction of the call wing total variance;
o changing m, there is a translation of the curve in the x axis;
o increasing co, the curvature reduces in the neighborhood of the at-the-money strike.
2.3.2 The natural SVI parameterization
The natural SVI parameterization
w(v;><.1z-.z)_ 4+ É [1+Çfi(v -11)+ ¢(C(v -11) +/1)” + (1 _1>2)}z (2-3-2)
is equivalent to the raw SVI by the following change of variables:
/ /1 _ 2
(a,b,,o,m,o) : [A+¶(1-,o2),Çl,,o,,Li p, 'O . (2.3.3)
( 2 2 C C
Note that, now there are two “level variables””, A which has a straight unitary derivate, and
w which doesn”t have, while in the raw formulation the only “level variable”” a has unitary
derivate.
CHAPTER 2. SVI PARAMETERIZATION: AN INTRODUCTION
Chapter 3
Parameters Modelling and Preliminary
Results
The goal of this chapter is to find a stable and reliable way to calibrate the raw SVI parameters
in (2.3.1) for a given set of implied volatilities observed in the market. It is well known that the
least square technique for this problem is affected by the large quantity of local minima. Other
problem regarding this modeling is that optimizing over a 5-dimensional set of parameters
usually returns non-stable results.
Therefore, the technique which will be applied is the one presented in [22] with a small mod-
ification developed for us in order to make the choice of m and o more precise. This approach
is particularly interesting because it simplifies the problem dividing it into one linear program-
ming and one two-dimension equation optimization through a simple dimensional reduction.
Moreover, the results are impressively good and the computational cost is low, if compared to
other techniques such as stochastic volatility models.
3. 1 The Algorithm
Let wT0t2¿ : wT be the total variance. Now, applying the following change of variable on the
log-strikes:
y-m
íczí,
o
we turn the SVI parameterization into:
a1T2t2¿(x) = aT + boT(,o;1: + \/ ;e2 + 1).
The equation above shows that, for fixed values of m and o, the SVI curve is completely
specified by a, ob and ,o. Hence, if we define the parameters:
c : boT,
d : ,oboT,
A : aT,
then we rewrite SVI formulation such that it is linear in the variables A, c and d:
wT2t2t(¿1:) : A + da: + cx/ m2 + 1
17
18 CHAPTER 3. PARAMETERS MODELLING AND PRELIMINARY RESULTS
It is important to note that the region where the parameters A, c and d are defined must
respect the original parameters boundaries, so it is derived from these constraints. Define the
parallelepiped D:
D:{(A,c,d)€R3 : AGR, c20, d€R},
which is the region in R3 where these new variables are defined. Therefore, for a given set of n
market observation pairs (att, óoÊ,”30”`§,z¿(,¿)), we define the cost function as:
n 2
f($11›“T"ÃÊÍz1<1>)(A” CI dl _ 2 (A T dia T C\/ T 1 _ wT0Iifhl(i)) '
1:1
So, it is only necessary to find the triplet (A*, c*, d*) for which Vf = 0 and then solve the
problem P below with the corresponding triplet (a*, b*, ,o*):
TL
(P) 13315 2(wm.‹›.T.1›*.z›* (vz) _ wm‹z1~1‹@1<1>)2-
” 1:1
The objective function in the optimization problem above is so-called sum of the squared errors.
Its square root is so-called RMSE.
3.2 The Implementation
The algorithm presented above is strongly dependent on the initial guesses of m and o, i.e.,
it is a heuristic method. Therefore, in order to reduce such dependence, the algorithm must
be tested with different initial values of m and o, in order to choose appropriately the initial
guesses. More precísely, the minimization is initialized with all combinations of m and o ranging
from 0.01 to 0.99, with the step size 0.01. We choose the pair that presents the lower RMSE
values. This adds some computational cost to the problem but not sufficient to become a
problem, since each calibration takes around 2-3 minutes to be performed.
Also, a loop structure with 10 iterations has been applied for every pair of initials (m, o).
The MATLAB@ code of the function created to calibrate these parameters can be found in
Appendix 5.
Note that, the bid-ask spread is not always well-behaved, mainly when we go far from the
at-the-money strikes. To overcome this issue, we exclude the outliers, since they do not add
relevant information to the analysis and usually are related to low trading volume options.
3.3 Illustrative Results
Before implementing a backtest strategy for the fitted parameters, we illustrate the smile ad-
herence of the SVI parameterization. The data is composed by SPX index options traded at
13th, 14th and 15th of October, 2015, with the shortest maturity time available - 21th of Oc-
tober. The resulting SVI curves compared with the market implied volatilities can be found in
Figures 3.1, 3.2 and 3.3.
3.3. ILL USTRATIVE RESULTS
-:-'.1-:L1
EE
e1:I1.11:1a1
11111
-:-'.1-:L1
EE
1:I1"1:1al
m|:1e
5F'1{151e;1c|:1ir1r11;|ea1I1:11:1ti1:1r15e51iI:1ra1e1:I eur1Ie'»15.mar1:etmi11 '1-'1:1I5-1II11et1:1I:1er,15th
Ú-15 1 1 1 1 1 1
mark et 1:1 1:15.
í51.11 Fit
-II"
11.111-
11.11'- -
':-
11.1E--
-5'
_ "'- _III zln I'.|'I
.'¬-.
_ IV' _f=~ I
¬* 1
5 1,5
ñ.
'-.-' _ _
-51 -“11.15 _ _ 1--1 _
'-L* _
5:31- -11 J'
z J 3. .z'_¬;. 1
.-"'-.
11.12
"'-' '-.-'-._.--: ..-'
1z|_1I] 1 1 1 1 1 1
- .III-1 -III.III2 III III.III2 III.III=1 III.IIIEi IILIII5 III.1
L1:11;1-F1:1r¬a-'ar1:II'-111:1r'1e'1-r1e55
Figure 3.1: SPX Implied Volatility Curve at October, 13th 2015
5F'1{151e1c|:1ir1r11;|ea1I1:11:1ti1:1r15e51iI:1ra1e1:I eur1Ie'»15.mar1:etmi11 '1-'1:1I5-11II1et1:1I:1er,1=1th
Ú-15 1 1 1 1 1 1 1
market 1:11:15. |
í5111r11
11.11'
:-
-:ja
11.1E- -
__. '-.-
"-.
11.15- -
'12
-:"
_ _.¬. .I"'¬. _._v:. ¬._.'
.___ .I-:I
-:::-
.-"'
:=~ E
5.15 51-1 __ -:" .-"-.-.-'
.-"-.
.z"'-.
-:"':-
11.12- 515 -
"-. -.-"
-"' ¬:":-
.;'_"-. .-'-. .-'-. -:§':-.-' '-._.-' '-._.-' -
1111 1 1 1 1 1
-III.III1 III III.III1 III.III2 III.III5 III.III-1
L1:11;|-F1:1r¬w'ar1:II'-.-11:1r1e'-,-r1e55
Figure 3.2: SPX Implied Volatility Curve at October, 14th 2015
11.115 11.115 11.111' 11.1111
20 CHAPTER 3. PARAMETERS MODELLING AND PRELIMINARY RESULTS
5F'1t 151 e:-t1:1ir1`r'11;|e51I 1:11:1ti1:1r15 e51it1r5te1:I eu11.1e'115.m5rketmi1:1 '111:1I5-1III1et1:1t1er,15th
I I I I I I
market 1:11:15.
51.11 Fit
11.2 - -
1'
11.111 - 25 «ta -
':-
e5 3:' E 1 'I-F' 1E-E
= 5
_ II-
1:I1"1:1al
2:' E
m|:1e
_ IE- _
-L»
.P _.-'_
-"':___- .
11.12- _~“` -
-131 . I11. 1 - 5;-__., 5. V -
¬.__..' _:.¬._
..I...' .'... ¬_. 5... .
13 113 1 1 1 1 1 1
- .IIIEI -III.III-1 -III.III2 III III.III2 I1.III-1 III.I1E III.III5
L1:11;1-F1:1r¬w'ar1:II'-111:1r1e'1-r1e55
Figure 3.3: SPX Implied Volatility Curve at October, 15th 2015
a b m ,o o RMSE
SPX calls at October, 13th 0.0872 1.3634 0.0302 0.2365 0.0200 1.18x10_2
SPX calls at October, 14th 0.0537 2.1805 0.0399 0.3435 0.0299 0.90;1:10`3
SPX calls at October, 15th 0.0463 2.6030 0.0299 0.2067 0.0200 1.60x10`2
Table 3.1: Calibrated Parameters for the Example Curves
Table 3.1 presents the parameters values calibrated from the SPX data and used to find the
SVI curves plotted in Figures 3.1-3.3. As one can see, the curves fit the market data well and
the RMSE is low.
Chapter 4
Backtesting SVI
After presenting the SVI parameterization, now, we shall backtest it. More precísely, we shall
verify how accurate it is as a prediction method.
4.1 Backtesting Methodology
We use the following market data:
o Underlying asset end of the day price;
0 Risk-free interest rate (annualized);
o Evaluate the forward price corresponding to the underlying asset (based on the end of
the day price and the risk-free interest rate);
o Time to maturity (in years);
o European call and put bid and ask prices for every strike with negotiations on the date.
Using the data above, it is possible to calculate the Black-Scholes implied volatility corre-
sponding to the bid and ask prices, and calibrate the raw SVI (2.3.1) parameters through the
algorithm described in 3. Once the parameters are identified, we apply two different tests that
verify non-arbitrage properties. The first one is to test butterfly arbitrage, where the function
defined in (2.2.1) must be non-negative in R. The second one is to test Roger Lee”s extreme
slope (1) conditions, where we take into consideration the largest and smallest log-strikes in
the data.
Now, we can calculate implied volatilities for any log-strike using the SVI function. So,
we can price any option on the considered asset/maturity pair at any time we want. In what
follows, we test how accurate the calibrated SVI curve is to predict option prices at future dates.
If D denotes the negotiation date considered in calibration, we evaluate option prices at the
dates D+1, D¬'-3 and D+5, comparing them with market prices. We plot the corresponding
implied volatilities of the model prices with market bid and ask prices. The prediction is
considered accurate if the model implied volatilities fall between the market bid and ask ones.
These results shall be presented in the subsequent subsections.
The following functions are used to measure the distance between model prices and data,
i.e., how accurate the model is:
21
22 CHAPTER 4. BACKTESTING SVI
o Data misfit for volatilities:
”°' 2E <0.7SVI _ O.7:l\4arket)
S:1 n 2 7
\ Z <0.7:l\4arket)
' IS
where n is the total number of log-strikes in the data, os]/I is the SVI implied volatility
and oM“"”ke'” is the implied volatility of the mean of market bid and ask prices.
o Normalized É2-error of prices:
Ê (CIYSVI _ CIš\darket)2
' IS
\ É <CIš\darket)2 7
SI
where CSW is the Black-Scholes option price calculated with the SVI volatility for each
log-strike.
ZM=
o Average error over spot:
[ <CISVI _ GMarket):|
77189 7
where S0 is the underlying asset spot price.
4.2 DAX Index spot options
DAX Index European type options are traded daily at Eurex Exchange in EUR currency and
have a high volume of trading, so it matches our requirements for the test. We took all the
necessary data for call and put options on D :22-Jan-2016, and then, we fitted the curve and
tested the non-arbitrage properties. The fitted parameters can be found in Table 4.1.
Figures 4.1 and 4.3 show the calibrated SVI curve for DAX Index options expiring in 15-
Mar-2016. Note that, the fitted curve for put options in Figures 4.3 is not free of butterfly
arbitrage, since the function g(y) defined in (2.2.1), and plotted in Figure 4.4 has negative
values. It can also be observed in Figure 4.3, since one can notice that the curve presents non-
desirable concavity. The same does not happen with call prices, as we can see in Figures 4.1
and 4.2, where the volatility curve is convex and the function g(y) is non-negative.
a I) m ,o o
DAX Index Spot Call Options 0.1083 48.6448 0.1600 0.9807 0.0100
DAX Index Spot Put Options 0.2552 -0.2282 0.01 4.0272 0.0303
Table 4.1: Calibrated Parameters for DAX Index Spot Options
4.2. DAX INDEX SPOT OPTIONS
II21.ë-.1I{In1:Ie:=›:1Ca1I5 5111 Fitte1:I1Cur-.-'e an III1-22-15
5 1 1 1 I
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5511551155151
Q- _
S- _
1'_ _
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51- -
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3- _
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_ 1 _ 1 1 1
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Figure 4.2: g(y) function for DAX Index Calls Options SVI Fit
II21.11G{In1:Ie><F'ut5 51.11 Fittedfiur'-.-'e un |II1-22-15
Ú 34 1 1 1 1 1 1
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Figure 4.3: DAX Index Put Options” SVI Fitted Curve (expiring: March/ 16)
24 CHAPTER 4. BACKTESTING SVI
Di'-.Ii Puts G['¡r]
Ú 5 I I I I
Ú- _
ÊH
U5 -
1 _
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L ug-Forward Munevn ess
Figure 4.4: g(y) function for DAX Index Puts Options SVI Fit
4.2.1 Backtest Results for DAX Index Spot Options
Figures 4.6, 4.10, 4.14, 4.8, 4.12 and 4.16 show the comparison between market prices and the
predicted prices at the dates D + 1, D + 3 and D + 5. The predictions are evaluated With the
SVI volatilities calibrated at D.
Figures 4.5, 4.9, 4.13, 4.7, 4.11 and 4.15 compares market implied volatilities of bid and ask
prices and the volatilities of prices predicted by the SVImodel.
Even though the SVI volatilities curve do not fall perfectly inside the bid-ask spread, the
predicted prices presented good adherence to the market ones.
Calls D-'fl D+3 D+5
Data misfiú fm» Vols. 1.071 ~ 1.531 ~ 2.3.1«ír1
Normalized Zz-error of prices 2.31.1 _ 6.()2.1 _ 9.76.1O_2
Average error over spot 1.01.1 _ 320.1 _ 4.1O.1O_3
Puts E E D+5
Data mísfiú før VQIS. 3.251 - 3.021 - 11.76.10-2
Normalized K2-error of prices 3.40.1 _ 3.841 _ 4.87.1O_2
Average error over spot 820.1 _ 5.45.1 _ 6.93.1O_4C)C)CJ_|:C)CJC) ›I>L\Dl\DI-\oo›-II-I C)C)CJ_|:C)CJC) ›I>1\awC,Ooow›-I
Table 4.2: Quantitative Indexes for DAX Index Spot Options
4.2. DAX INDEX SPOT OPTIONS
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Figure 4.5: D+1 l\/Iarket vs SVI Implied Vols. Comparison for DAX Call Options
11F'ri1:55
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Figure 4.6: D+1 l\/Iarket Prices vs l\/Iodel Prices Comparison for DAX Call Optlons
E3._
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Figure 4.7: D+1 l\/Iarket vs SVI Implied Vols. Comparison for DAX Put Optlons
EI.3Ei
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E3._
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13'
Figure 4.9: D+3 l\/Iarket vs SVI Implied Vols. Comparison for DAX Call Options
11Pn`ee5
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Figure 4.10: D+3 l\/Iarket Prices vs l\/Iodel Prices Comparison for DAX Call Optlons
550
D+ 1 D.1i11{ Ind exüplinne F'r1`1::e5 Elaclcl est
CHAPTER 4 BACKTESTING SVI
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4.2. DAX INDEX SPOT OPTIONS
E3._
e1:l'1"1:1eI
Figure 4.11: D¬'-3 Market vs SVI Implied Vols. Comparison for DAX Put Options
Figure 4.12: D+3 l\/Iarket Prices vs l\/Iodel Prices Comparison for DAX Put Options
Figure 4.13: D+5 l\/Iarket vs SVI Implied Vols. Comparison for DAX Call Options
1-1111
D+3 De-11 |1'1dekOpti1:11'1el11'1|:1. '1f1:1| Beekteet
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CHAPTER 4. BACKTESTING SVI
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Figure 4.14: D-1-5 l\/Iarket Prices vs l\/Iodel Prices Comparison for DAX Call Options
ílllü 5111 Filte1:lCur¬~.1e
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L1:11;1-Fer-1.-er1jM1:11'1e-111¬eee
Figure 4.15: D+5 l\/Iarket vs SVI Implied Vols. Comparison for DAX Put Options
D+5 IIÉI.-111-llnde>cC.`1|:1lio1'1e Prieee Beekleel
45Ú 1 1 1 1 1 1
11 Medel Pneee
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L1:1g|-Fer-1.-er1jM1:111e-111¬eee
Figure 4.16: D+5 l\/Iarket Prices vs Model Prices Comparison for DAX Put Options
4.3. SPX INDEX SPOT OPTIONS 29
4.3 SPX Index spot options
SPX Index Spot Options are negotiated at Chicago Board Options Eaíchanged and is the most
liquid equity options in the world. The underlying currency is USD (American Dollar).
Figures 4.17 and 4.19 show fitted SVI curves for the SPX Index options expiring in 18-Mar-
2016 and traded at 22-.Ian-2016. As one can see through Figures 4.18 and 4.20, the SVI curves
for call and put options are free of butterfly arbitrage.
a b m ,o o
SPX Index Spot Call Options 0.1998 0.1273 0.0644 -5.5263 0.0022
SPX Index Spot Put Options 0.2381 1.2779 0.0603 0.3069 0.0200
Table 4.3: Calibrated Parameters for SPX Index Options
5F'}{lnclekCelle 5'1"l F1l:I:e1:lCu1*-.-'e 511 III1-22-IE
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Figure 4.17: SPX Index Call Options” SVI Fitted Curve (expiring: March/ 16)
5F'Í1{C3l|3G[I1.-']
15
15 l _
1--1 _
12 _
15 _
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L5g|-F51'-1.-er1jM511e-zmeee
Figure 4.18: g(y) function for SPX Index Calls Options SVI Fit
30 CHAPTER 4. BACKTESTING SVI
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Figure 4.19: SPX Index Put Options” SVI Fitted Curve (expiring: l\/Iarch/16)
25 SPX PIIIZS G['1.1']
20 -
15 \ -
É 10 -
5 _
5_ ííí' _
_ 1 1 1 1 1 1 1
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Figure 4.20: g(y) function for SPX Index Puts Options SVI Fit
4.3.1 Backtest Results for SPX Index Spot Options
Figures 4.22, 4.26, 4.30, 4.24, 4.28 and 4.32 show the comparison between market prices and
the predicted prices at the dates D + 1, D + 3 and D + 5. The predictions are evaluated with
the SVI volatilities calibrated at D.
Figures 4.21, 4.25, 4.29, 4.23, 4.27 and 4.31 compares market implied volatilities of bid and
ask prices and the volatilities of prices predicted by the SVI model.
As we can see in the figures and in Table 4.4 the prices given by the SVI model are close to
the market ones, which means that, this model can be used to predict prices.
4.3. SPX INDEX SPOT OPTIONS
Calls D+1 D+3 D+5
Data misfit for vols. 1.23.11 _1 1.24.1 _1 1.53.10_2
Normalized K2-error of prices 2.17.1 _ 2.11.1 _ 1.04.10_1
Average error over spot 2.70.1 _ 2.50.1 _ 1.40.10_3
Puts E E D+5
Data misfit for vols. 7.93.1 _ 8.97.1 _ 6.05.10_2
Normalized K2-error of prices 4.65.1 _ 9.42.1 _ 8.84.10_2
Average error over spot 2.50.1 _ 4.40.1 _ 1.80.10_3Cl)CÇJ<_J_Â;LL)LL)L.)wwwlfic.o1- ÇL>Ç_>CL>_Á;CL>Ç_>Ç_>oowwbô0.11:
Table 4.4: Quantitative Indexes for SPX Index Options
_lII105"."lFilte1:l1I2111--.fe
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Figure 4.21: D+1 l\/Iarket vs SVI Implied Vols. Comparison for SPX Call Options
D 1 SPI l111:lek1í25ll C1|:1l15ne F'n1:e5 Beekteet
BÚ 51 1 1
15 1115rkel 51:15.
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Figure 4.22: D-'fl l\/Iarket Prices vs l\/Iodel Prices Comparison for SPX Call Options
stes
._
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CHAPTER 4. BACKTESTING SVI
EL44 -
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Figure 4.23: D+1 Market Vs SVI Implied Vols. Comparison for SPX Put Options
cmF'riees._
On
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D+1 SPI Index Put Options Prices Bsektest
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ie
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Figure 4.24: D-'fl Market Prices Vs Model Prices Comparison for SPX Put Options
E3._
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++*
_ _ _
ä * ä
|]_1Ê I I I I I I
- .U4 -EI.III2 IJ I].IJ2 IILIJ4 IILIIIE Ilüfl EI.1
Leg-I' ur'-.-«srd munewiess
Figure 4.25: D¬'-3 Market Vs SVI Implied Vols. Comparison for SPX Call Options
4.3. SPX INDEX SPOT OPTIONS
nPrices
._
Opo
Figure 4.26: D-I-3 Market Prices Vs Model Prices Comparison for SPX Call Options
E3._
mp`e|:I'Ifoet
Figure 4.27: D+3 Market vs SVI Implied Vols. Comparison for SPX Put Options
nPrices
._
Opo
Figure 4.28: D+3 Market Prices Vs Model Prices Comparison for SPX Put Options
EIEI
SEI
Í"EI
EEI
5EI
-Él-4:1
SEI
2EI
1D
D+3 SPI Inde>:OsJI Options Prices Eiacktest
im] I I I I I I
EI.5
EL45
EI.4
1:! |'.II.'I lJ'I
EL3
|z|_z I I I I I I I
-EIS -EI.1 -EI.El5 EI ELEI5 EI.1 EI.15 EI.2
35Ú I I I I I I I
qr 5"."I Fit
_ _* _
SIJEI
25EI
I'\I.'I II 4:1
_ |'.|'I 4:1
IIJEI
5EI
-3tI-<Í`.=-
-It-=l`.`=-
O1+.-_.+__ G
<:> market obs.
_ ir Siri Fit
üü
e Pc
_ _ O-1. -=L'> *$-_ +++T:i.;}<_> _
+ee*Í í Ê Ê É
Log-f on.-àsrd rnonepness
D+3 5P}{InI:Ie><PutOptionsImp. 'Ifol Bscktest
-8.04 -EI.EI2 EI EI.EI2 ELEI4 EI.EIEi ELEIB EI.i
-III-
4°
_ *_ _
-Jr
*-
-3*
_ -F _
-Itr
+
+
e
e
* s
íülll Siri Fittedüur'-.fe
se D+3 Eticllrnp. trois
D+3.¿'-skIrnp.I.foIs
$-
_ 1* *_ _
Log-For'-«sro Monepn ess
pi
D+3 SPI Index Put Options Prices Becktest
~f.> msritet obs
2ser*
Ê
'É'
O
-IP
_ O _
Ê_ Ê _
' .sí '
Log-For'-Nerd Monepn ess
I I I I I I I
-8.15 -Ili -EI.El5 I] EI.I]5 Ili U.15 EI.2
CHAPTER 4. BACKTESTING SVI
íülll Siri Fitteciüur'-.re
rir D+5 Eiiciimp. trois
D+5.¿'-sicImp.IroIs
EI.2S - -
EI.2E¡ - -
_ - *_ \ _
_* ¬-_
+ ¬.
stes
F' I"¬¬.'| -P-
'¬.
mped'ro
= *_
*es.
_ *F _oca +_*
*I-*I-
*#-
'if'
_ 4° _EI.2 *r
-315
|z|_1Ê I I I I I
- .EIB -EI.III-4 -III.II|2 III ELIII2 III.III-fi IILIIIE
Log-Ion.-.-srdmoneiizness
Figure 4.29: D¬'-5 Market Vs SVI Implied Vols. Comparison for SPX Call Options
D+5 SPI Inciexfisii Options Prices Bscictest
ÍÊÚ *_ I I I I I
<:> msritet obs.
P se Siri Fit
IEIEI- -
BEI- + -
+-
Q O
BEI- -nPrices
._
Opo
<+I- -Iii-4U- ._ _ -
-c-*É
Égfiä
's
EU- Éärs _ _
É s ts
I I I I I
-REIS -EI.III-4 -III.II|2 III ELIII2 III.III-fi IILIIIE
Log-Ion.-.-srdmoneiizness
Figure 4.30: Dr-5 Market Prices vs Model Prices Comparison for SPX Call Options
D+5SPi€In|:|exPutOptionsImp.*roIEIscictest
os4 ,
_____oosrInuzúcuwz
EL42 _ se D+5EIid|mp.'lroIs
D+5.fi-.skImp.¬roIs
o4_ Pxfixtä* _
oss_ _
E3
Í: cú cn
._
eciiroet
4:1 Lú -Ii
mn F' Lú I"'I-i.'l
EL3 - -4» -
EI.2EI - *Ê *Ê -
-It-
U.2Ei - sie -
DÊÊH -HI pis É oi? um
Figure 4.31: D+5 Market Vs SVI Implied Vols. Comparison for SPX Put Options
4.4. PETR4 EQUITY OPTIONS 35
i
‹í> msritet obs.
Stri Fit
«$-
2I]EI -
IF-
Iso _
s
\'_'I
:Ê -c
Ç '-If
E
15.
P Ino _
4:-
-=1`;>
-2+
$-
5EI É -
*I-Ii*
*Ê
Ê*ers rir I* E
I I I I-II Is oi -nos o o os oi
Log-For'-«sro iirionepn ess
Figure 4.32: Dr-5 Market Prices vs Model Prices Comparison for SPX Put Options
4.4 PETR4 Equity Options
Petrobras PN equity options are the most liquid equity options in Brazil. They are negotiated
at BMFʧB0fuespa Stock E5126/tange in BRL (Brazilian Reais). The volume of daily trading for
these options are not as big as the ones in the examples above, Which can give us less accurate
results.
Figures 4.33 and 4.35 shovv fitted SVI curves for PETR4 Equity options expiring in 21-Mar-
2016 and traded at 5-Feb-2016. As one can see in Figures 4.34 and 4.36, both SVI curves are
free of butterfly arbitrage.
0. Õ m ,0 U
PETR4 Equity Call Options 0.1998 0.1273 0.0644 -5.5263 0.0022
PETR4 Equity Put Options 0.2381 1.2779 0.0603 0.3069 0.0200
Table 4.5: Calibrated Parameters for PETR4 Index Options
PETFI4 Equitpúeiis Stri Fitted Cur-.fe on U2-IEI-IE
I 2 I I I I I I ¬
Stri Fit Imp. trois
Eiidimp. trois
ir'-.sk|mp. trois
1 1 _
1 _
Ú?
É
ʧ III EI -
'É
É
E.
É
'¬¬_ _.
EI Ei "`-_ “_ -
THE _,_I-P"_'_-___'-I-_'-,T-
¬"\.`¬`_` _'_'____¡_,__-I-'__
"'~¬._¬`_ ___'_'__I-I-_
EI? -
I I I I I I I
II|.3 -IJ.2 -Ili IJ III.i 0.2 0.3 III.-4
Log-For'-I-erdiwioneymess
Figure 4.33: PETR4 Equity Call Options” SVI Fitted Curve (expiring: March/16)
CHAPTER 4. BACKTESTING SVI
18 i i s Pit csiis em i i
Is_ I _
I~=I_ _
12- _
Io_ _
ÉII:o
S_ _
E-
._ /\
2 _ _
E I I I I I
- .EI-4 EI III2 III IILIII2 EI.III-4 IILIIIE III IIIE
L op-For'-.-Isrd i\I'Ione'IIn ess
Figure 4.34: g(y) function for PETR4 Equity Calls Options SVI Fit
PETFI-4 Equity Puts Stri Fitted Cur-re on III2-IIII-IE
Stri Fitimp trois
Eiidimp trois
I4 I I I I I I
I 3 _ í.-'I'-skimp. trois
12- -
*IO2 II- _
É
D
Ib-
'El
É 1_ .-ff' _
CL
É .z
.-'
os_"-~__¬¬ _
¬-¬¬_¬_H 'J-
¬¬¬-H ¬. .-ff,-¬-'¬-I.
¬¬-I.-I¬_`_ __.--I¬_`_ __.-
EI S _"“--_ --""_ _`_i`í_'_'__ _
I;|_;I' I I I I I I I I
-II|.4 -IIL3 -III.2 -EI.i III III.i III.2 IIL3 III.-4 III.5
LoI;I-For'-I.-risriji-Iione'IIness
Figure 4.35: PETR4 Equity Put Options” SVI Fitted Curve (expiring: March/16)
PETFI-=I I=-IIIs cw]
3 I I I I I I I I
/*z
25- -
2_ _
ÉII:o
i5_ _
1- _
ELE t I I I I I I I\I- .4 -III.3 -0.2 -0.1 U III.i III.2 11.3 IJ.==i III.5
Log-For\II'erdi'-Iione'IIness
Figure 4.36: g(y) function for PETR4 Equity Puts Options SVI Fit
4.4. PETR4 EQUITY OPTIONS 37
4.4.1 Backtest Results for PETR4 Equity Options
Figures 4.38, 4.42, 4.46, 4.40, 4.44 and 4.48 show the comparison between market prices and
the predicted prices at the dates D + 1, D + 3 and D + 5. The predictions are evaluated with
the SVI volatilities calibrated at D.
Figures 4.37, 4.41, 4.45, 4.39, 4.43 and 4.47 compares market implied volatilities of bid and
ask prices and the volatilities of prices predicted by the SVI model.
As we can see in Table 4.6, the predictions are less accurate than in the previous examples,
probably due to the smaller amount of data and the smaller trading volume. However the
predicted prices presented skewness similar to the market ones, which is a nice feature.
¬'-Calls D¬'-1 D¬'-3 D 5
Data misfit for vols. 1.23.10_1 1.24.1i _ 1.53.10_2
Normalized Zz-error of prices 2.17.10_1 2.11.1 _ 1.04.10_1
Average error over spot 2.70.10_3 2.50.1 _ 1.40.10_3
Puts D+1 E D+5
Data misfit for vols. 7.93.10_2 8.97.1 _ 6.05.10_2
Normalized É2-error of prices 4.65.10_2 9.421 _ 8.84.10_2
Average error over spot 2.50.10_3 4.40.1 _ 1.80.10_3C)L_)C›_*:C)Ç›L_› cowwbôc.oI~I-I
Table 4.6: Quantitative Indexes for PETR4 Index Spot Options
EI 1 PETFI4 Eouit'-_: Ceiis Options Imp. troi Secictest
íišifl Stri Fittedüurve
re D+i EIII:iimp. trois
D+1 .-'1¬Isi=:Imp. trois
I
EIEI5- -
-:-'II-:|.'-
-E
[Ig K /
.-"
Ê .f
;oss_ / _
'É
É
Ê' EIS -
se
ors ,,, + _
or *I _
USE -
+
I I I I I I I
III 4 IJ 3 IJ 2 -III.i IJ IJ 1 IJ 2 III 3 III 4
L oo-For'-«sro i\I'Ione'IIn ess
Figure 4.37: D+1 Market vs SVI Implied Vols. Comparison for PETR4 Call Options
CHAPTER 4. BACKTESTING SVI
EI+1 PETFI-4 Equity Coil Options Prices Eiscittest
I I I I I I I
tt* ‹í> msritet obs.
syi ,II
_ Q _
1.4
1.2
1- _
*-
O
nprices
F' CIO
._
Opo 1:1 is-.I _ Ir<>
EL4 - -
S
<`HI-
-=íi>+ -II-
-IF
os _ _
Q 'Ô
I I I I I I I
-“Dri -III.3 -III.2 -III.i III III.1 III.2 IIL3 III.-4
Log-ionI-'srI:imoneyrIess
Figure 4.38: D-i-1 Market Prices vs Model Prices Comparison for PETR4 Call Options
D+1 PETFI4 Equity Put Options Imp. troi Becictest
I.-4 I I
_oo sri Firiúúcuwe [
fr- D+1 Bidimp. trois
EI+i .ft-skimp. trois1.3-
I.2- -
/'
mpeotrostes
= I
o.s_ ,,,+* _
'III-
EI.S - -
-IIE
-IIE
|z|_;I' I I I Ii* I I T I I
-III.3 -III.2 -III.i III III.1 III.2 III.S III.4 III.5 IILE
Loq-For'-ysroiyioneymess
Figure 4.39: D+1 Market vs SVI Implied Vols. Comparison for PETR4 Put Options
EI+i PETFI4 Equity Put Options Prio es Becktest
3 i i i i i i i i _
<> msritet oi:Is.
«I Stri Fit
2.5 _ IS _
_ ä _2 S
nPrices íI.I¬
S
__ I
Opo -O-*I-
I _ S _
o.s _ I* _
I
s
I
[IU I I I I I I I I
- .S -III.2 -Ili I] IJ.1 III.2 IIL3 IJ.-4 IJ.5 III.Ei
Loq-For'-yeroiyioneymess
Figure 4.40: D+1 Market Prices vs Model Prices Comparison for PETR4 Put Options
4.4. PETR4 EQUITY OPTIONS
ES._
eotroet
Figure 4.41: D¬'-3 Market vs SVI Implied Vols. Comparison for PETR4 Call Options
Figure 4.42: D+3 Market Prices vs Model Prices Comparison for PETR4 Call Options
Figure 4.43: D+3 Market vs SVI Implied Vols. Comparison for PETR4 Put Options
rop
EI.EI5 -
EI.S5
Z ¬-IJ U1
EI.S5
EI.5
onprices._
UD
SS._
dtroert
mpe
EI.EI -
EI.S
EL?
ELE
I
EI+S PETFI4 Equity Osiis Options Imp. troi Eiscktest
rir D+S EIiI:iimp. trois
D+3.¿'-sicimp.trois
\\~\\______/Ef
*II-
III- qr. 5* _
O
-ir -III-
_oo syi Firrúúcowz L
+*+'-
I.S
-Sus
Loq-For'-«sro iyioneyn ess
EI+3 PETFI4 Equity Csii Options Prices Bscictest
I I I I I I I I
-III.4 -III.S -EI.2 -III.1 EI III.i EI.2 EI.S III.4
IS-
1.4-
1.2-
1-
III CO
ELE -
EI.4 -
EI.2 -
i i i i i i i i
<> msritet obs.
Stri Fit
É
O
O
III-
O
O
'III''O giz*
*¡"`<:›
O O-IE OO O-IIII-“fr
I.S
5
I I I I I I I
III.4 -III.S -EI.2 -III.1 EI III.i EI.2
Log-I oryrsro moneyness
D+3 PETFI4 E quity P ut Options im p. troi Becittest
-:- cú-
1.4-
1.2-
I
1:1 CCI
EI.S
EI.4
o ._2ü
às iII+3 Eiiciimp. trois
D+3.4.skimp.trois
I
_oo syi Firiúúcuwz [
/
"13'
'IF. 44 1* *L *_
I-*T _
_* _
EI4
.2
I I I I I I
-0.1 IJ III.i EI.2 IJ.S I].4 III.5
Loq-For'-«sro iyioneyn ess
CHAPTER 4. BACKTESTING SVI
EI+S PETFI4 Equity Put Options Prices Bscktest
2 5 I I I I I I
‹í> msritet obs.
Stri Fit
2 _ _
1.5- Ê -
O
nPrices
._: *C-
O
Opo I_ 4. _
O
É
É
o.s_ _
$ S
I I I I I I
-DELE -III.1 III III.i IIL2 IIL3 III.4 IIL5
Loq-For'-ysroiyioneimess
Figure 4.44: D¬'-3 Market Prices vs Model Prices Comparison for PETR4 Put Options
EI+5 PETFI4 Equity Csiis Options Imp. troi Bscktest
i.S
rir D+5 Eiiciimp. trois1.2 _ D+5.¿¬-.sicimp.trois
_oo sri Firrúúcowz L
1.1-
1- _
*~»~_____/ E3
mpeotrost
1:1 'ss
ELF- -
O * 1* + _ +
EI.S- _* Ip -
O
os _ _ _
I;|_ I I I I I I I
ÍIL4 -IIL3 -IIL2 -III.i III III.1 IIL2 IIL3 III.4
Loq-For'-ysroiyioneimess
Figure 4.45: D+5 Market vs SVI Implied Vols. Comparison for PETR4 Call Options
EI+5 PETFI4 Equity Coil Options Prices Becictest
I I I I I I I
*L <> msritet oi:Is.
-z;;I qr. 5"I"I Fit
1 _ _
1.2
III
_ 4* _
nprices
5:' CCI
._
Opo 1:1 izn _ _* _
O
EL4 _ -
+
III
'S II»
ELE- <;› 4° _
<:› II»
c *I *Ir
O .I-_; Q
[IU I I I I I I I
- .4 -IJ.3 -11.2 -III.i IJ Ili I].2 IIL3 III.4
Log-Ion.-.-srcimoneyness
Figure 4.46: D-E5 Market Prices vs Model Prices Comparison for PETR4 Call Options
4.4. PETR4 EQUITY OPTIONS
EI+5 PETFI4 Equity Put Options Imp. troi Bscictest
2 I I
_oo sri Firiúúcuwe L
41- D+5Bidimp.troIs
I-3 - EI+5.-'1¬-.sI<Imp.trois
1.E- _
1.4- _
E3._
mp`eI:itrost
E
1 _ _
/Í/;è
o s `__- --'__`T¡_T1 1 * M 1
4 ¬.. *
_ 4
EI.E _ sie -
I;|_ I I I I I I
ÍIL2 -III.1 III III.i EL2 EL3 EI.4 EL5
Loq-For'-ysroiyioneimess
Figure 4.47: D4-5 Market vs SVI Implied Vols. Comparison for PETR4 Put Options
EI+5 PETFI4 Equity Put Options Prices Becittest
2-5 I I I I I I
‹í> msritet obs.
Stri Fit
2- _
I.s_ _
S
nPrices
._
Opo I _ 3 _
S
É
EI.5 _ _
_ S
* S
gn I I I I I I
- .2 -III.1 III ELI EL2 ELE EI.4 ELE
Loq-For'-«sro iyioneyn ess
Figure 4.48: D+5 Market Prices vs Model Prices Comparison for PETR4 Put Options
42 CHAPTER 4. BACKTESTING SVI
4.5 Confidence Intervals
Although we are considering volatility as a deterministic function of the log-strike, we state
confidence intervals, based on the bid-ask spread of option prices as follows:
o From D0 data, we evaluate the modulus of the difference between observed bid-ask implied
volatilities, take the maximum over strikes and denote this quantity by Õ.
o Denoting the SVI curve at each log-strike lc by 25;/¡(/c), we consider 25V¡(/c) i Õ/2 upper
and lower bounds.
0 We calculate Black-Scholes prices using such bounds on the volatility to define confidence
intervals for the option prices.
4.5. 1 Numerical Example
Now, we evaluate such confidence interval using the D+1 backtest data for DAX Index Calls
4.2.1.
Õ = 0.0423
Defining siüoniicience interysi iorE1I'1'-.IE Csiis Prices on III+1
IÊÚÚ I I I I -
III Iyiooeitlp Eiounosry Prices
III Model Lo¬II.rerBounoeryPrices
Iyisrirzet Prices
IEIDEI .
CCI Z II
-III-
-II-BIG
-*HIP
*II-515
-III-III- III-
*III-
*III
SEIEI *iii -
*iii-
EO
ED
\'_'I
EI:I_
E
E
“EL *Ik-
O 1.-2*
Irflr*
4EIEI * _*ils
*-12-
II=**
* III»III
I'*~I.`I Z Z
III «III IIIII IIIIII
III-III -IIIIII
E I I I I
- .15 Eli EID5 EI ELU5 ELI
L oq-For'-«sro iyioneyn ess
Figure 4.49: D+1 DAX Calls Prices Confidence Interval
We can also determine a confidence level for our interval by performing a Kupiec test [17] by
determining the number of violations, i.e., the number of market prices outside the confidence
interval. On the example above,the number of violations was 0 as we can see in Figure 4.50.
4.6. AN ALTERNATIVE APPROACH 43
Model Pnc es snci Interysis in Perspectiye
EEUEI , ,
EEIIJEI _ _
Ú? -_-_
É Isoo_ _ _E _
_, _
ã _
III? _
IIIJ
É _
E Iooo " _
CD
'El
Cv
E
I'.|'I Z
.=I=-D I*-I.'I 1:! E-
i
'='i_ -I
IIHIic
ic
'|II
It
:-1xxIII
x
L"ic
|_›‹:
1:1 III 1:! I'_|'I 1:1 1:1 E
__
Loq-For'-«sro iyioneyn ess
Figure 4.50: D-i-1 DAX Call Model Prices with its Confidence Intervals
4.6 An Alternative Approach
As an attempt to refine the technique proposed in Section 4.1, to forecast implied volatilities
for the day D + j with maturity time T, we evaluate, by an interpolation method, implied
volatilities for the day D with maturity time T - gi.
So, we use the two-dimensional interpolation method proposed in [15] and proceed as follows:
0 Calculate implied volatilities smiles for several different expiring dates 15.; a set of indexes
for discrete time-to-maturity quantities) with data taken from the market;
o For each maturity t G It,¿,t.¿+1I and strike K we calculate Uf,,,p(K,t)t by interpolating
linearly between UÊmp(K, t¿)t.¿ and 0Êmp(K, tz-+1)t,-+1, where such values were obtained with
SVI.
0 So, we evaluate the surface of implied volatilities (K, t) H aÊ.mp(K, t).
As an example, we calibrated the surface for SPX European Options with data taken from
@Bl00mberg in À/Iay,13th. The total variance and implied volatility surfaces can be seen in
Figures 4.51 and 4.52:
44 CHAPTER 4. BACKTESTING SVI
kiEI
eitreeiiiences
EE.I'..I'.
z __..- . -
oos __-~" f_ .-.,.. siso
Ú.E|E i pmfl
oos ..~~" “-
u_u 2EI5EI
EI 2EIEiEI _
Tim e-t o-I'-Iiet ury SIIIIIES
'EI-'
'_
1:1 Oil
OJ
E
ID
Ê.
:_
ID
Iš
Figure 4.51: SPX European Calls - Interpolated Total Variance Surface
_-I.
.' Í -" '
_Ê ois _.-“"Ê__.-É ' -.Â
É _}' :_. :
É:_ oI4 _.~' '
'CI
EDss ois-_
E
'E oI2 _
E
Cire oii _ ro-
11':
Iš oi _| _ _
zooo _._.-._ .~ “U5no2EI5EI 4
21E|EI
215III Ú U2
EÊÚÚ Ú Time-to-iyietury
Strikes
Figure 4.52: SPX European Calls - Interpolated Implied Volatilities Surface
As an example, we took market data on May, 18th for SPX calls expiring in May, 27th (j =
9). On the figure 4.53 you can find the SVI curve fitted with ínterpolated implied volatilities
for D+j on May, 13th (a,Êmp(K, 9)) as well as the bid and ask implied volatilities from market.
4.6. AN ALTERNATIVE APPROACH
ES._
mp`eI:itr'oet
0.1?
0.1E
0.15
0.14
III _ Lú
III _ I"'I-I.'I
1:! _ _
SPIIE Cells Im piied troistiiities Sscktesteci 'with Stri from Osiihrsted Surface
-I.
-I.
O
____ I I
_D+] S tri Fitt e ci C urye
_ _ ¬ 4 Bio Imp. trois on T-I
`_~_ .¿'-.skimp.trois onT-I I
¬-I. -_
-I.
III + 4
_ _* _
0.1 _ _
0.09 _ _
0.0E _ _
O
I I I I I I
-ELEIS -ELEIES -ELEI2 -ELEIIS -III.EIi -III.IIIEIS III EI.EIEIS
LoI;I-For'-ysroi-Iione'III¬ess
Figure 4.53: SVI Backtest with Interpolated Volatility Surface
Since the Data Misfit for the volatilities was 6.47:1110_2, the single example above shows us
no clear advantage in relation to the original experiment of the present work. However, we
believe it°s worth looking more carefully at it on posterior works.
CHAPTER 4. BACKTESTING SVI
Chapter 5
Concluding Remarks
In this work we intended to backtest SVI parameterization of Black-Scholes implied volatilities
in order to see how accurate such model is to predict option prices.
After showing that SVI satisfies important non-arbitrage properties (see [18], I20I, [11] and
1131), we presented an algorithm based on techniques from [22] to calibrate the SVI parameter-
ized volatilities from market data. The algorithm included some modifications to increase the
robustness of reconstructions, in order to be less dependent on the choice of the initial guess
of the parameters m and 0. This provided us the necessary tools to start the backtest and
perform the numerical examples presented in Chapter 4.
Finding reliable option data to test our algorithms was the first difficulty we faced. We had
to gather all the necessary information about bid and ask prices of European options, choosing
the ones with high trading volume. Thats why we have chosen DAX (EUREX), SPX (CBOE)
and PETR4 (BMF&Bovespa) vanilla options.
A visual way to evaluate how accurate are the calibrated SVI curves is to observe its
position in relation to market implied volatilities. Under this perspective, SVI parameters
did not produced accurate predictions for all strikes, specially those far from the at-the-money,
however, it predicted important features presented by the market data, such as skewness.
So, this model can be advantageous for practítioners since it is computationally cheaper than
stochastic volatility models.
Moreover, using SVI functional forms to price options for future scenarios might be an
interesting tool in portfolio optimization techniques, such as the Black-Litterman inspired one,
where the user inserts returns expectations into the model. See Going further on this
research, we would like to suggest the same kind of test with smaller time frames in order to
find out if it is possible to better understand how the volatilities react considering marginal
disturbs on other variables.
47
CHAPTER 5. CONCLUDING REMARKS
Appendix
l\/[ATLAB Code for Quasi-Explicit SVI Calibration
function Ia, b, rho, m, sigma, rmse] = SVI_fit(bid_vo1s,ask_vo1s,T,1og_strikes)
%defining variables
mid_vo1s = (ask_vo1s + bid_vo1s)/2;
mid_tota1_vo1 = mid_vo1s*T;
= 1ength(mid_vo1s);
parameters_matrix = zeros(9801,6);
T1
ui0;
for m_initia1 = 0.01 : 0.01 : 0.99
for sigma_initia1 = 0.01 : 0.01 : 0.99
% changing variable
1og_strikes_chg = (1og_strikes - m_initia1)/sigma_initial;
mid_tota1_vo1 = mid_vo1s*T;
% Solving the linear system to reach zero for the cost function.
A:
Z:
Par
In sum(1og_strikes_chg) sum(sqrt(1og_strikes_chg.^2+1));
sum(1og_strikes_chg) sum(1og_strikes_chg.^2)
1og_strikes_chg'*sqrt(1og_strikes_chg.^2 +1);
sum(sqrt(1og_strikes_chg.^2 +1)) 1og_strikes_chg'*sqrt(1og_strikes_chg.^2 +1)
sum(log_strikes_chg.^2 +1)];
[sum(mid_tota1_vol); mid_tota1_vo1'*1og_strikes_chg;
mid_tota1_vo1'*sqrt((log_strikes_chg.^2 +1))];
ameters = A\z;
a_ti1de = Parameters(1);
dz
C:
aí
bi
rho
Z
0°0°
Z
/ s
Parameters(2);
Parameters(3);
a_ti1de/T;
c/(sigma_initia1*T);
= d/(b*sigma_initia1*T);
ow, considering a, b and rho we will minimize the problem choosing m and
igma
49
50 Appendix
i = 0;
m_loop = m_initial;
sigma_loop = sigma_initial;
while i <l0
X = S3/m('X'zIl 21);
obj_function = @(X) sum((a + b*(rho*(log_strikes - x(1)) +
sqrt((log_strikes - x(1)).^2 + x(2)^2)) - mid_vols).^2);
x = fminsearch(obj_function,[m_loop , sigma_loop]);
m_loop=x(1);
sigma_loop=x(2);
i=i+1;
end
m=m_loop;
sigma=sigma_loop;
fitted_values = a + b*(rho*(log_strikes - m) +
sqrt((log_strikes - m).^2 + sigma^2));
lqs = sum((mid_vols - fitted_values).^2);
u=u+l;
parameters_matrix(u,1)=a;
parametersnnatrix(u,2)=b;
parameters_matrix(u,3)=rho;
parameters_matrix(u,4)=m;
parameters_matrix(u,5)=sigma;
parameters_matrix(u,6)=lqs;
end
end
% defining m and sigma looking for the least sguare errors sum
min_error = min(parameters_matrix(:,6));
n = length(parametersnnatrix);
for r = l:n;
if parametersJnatrix(r,6) == min_error
a = parameters4natrix(r,1);
b = parametersnnatrix(r,2);
rho = parametersJnatrix(r,3);
m = parameters¿natrix(r,4);
sigma = parameters_matrix(r,5);
rmse = sqrt(parameters4natrix(r,6));
end
end
% graphic construction for visual analysis
liminf = min(log_strikes);
limsup = max(log_strikes);
x_axis= linspace(liminf,limsup,10000);
fitted_curve = a + b*(rho*(x_axis - m) + sqrt((x_axis - m).^2 + sigma^2));
plot(log_strikes,bid_vols,'c*');
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Appendix 51
hold on
plot(log_strikes,ask_vols,'g*');
plot(x_axis,fitted_curve,'r');
xlabel('Log-forward moneyness');
ylabel('Tmplied Volatilities');
legend('SVT Fit Imp. Vols','Bid