Fitting Local Volatility: Analytic and Numerical Approaches in Black-Scholes and Local Variance Gamma Models

205 Pages·2020·12.09 MB·English

by  Andrey Itkin

Fitting Local
Volatility
Analytic and Numerical Approaches
in Black-Scholes and
Local Variance Gamma Models
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Fitting Local
Volatility
Analytic and Numerical Approaches
in Black-Scholes and
Local Variance Gamma Models
Andrey Itkin
New York University, USA
World Scientific
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British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
FITTING LOCAL VOLATILITY
Analytic and Numerical Approaches in Black-Scholes and Local Variance Gamma Models
Copyright © 2020 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
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ISBN 978-981-121-276-5
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January3,2020 20:32 FittingLocalVolatility–9inx6in b3761-main pagev
Foreword
– By Alex Lipton
Co-Founder and CTO, Silamoney, Partner,
Numeraire Financial, Connection Science Fellow,
Massachusetts Institute of Technology, Cambridge, USA
Prof. Andrey Itkin, a well-known and highly respected financial engineer,
gives a tour de force performance in this short, insightful book. To under-
stand his contribution properly, we need to step back in history.
Derivatives, including options, forwards, and futures, have been around
at least since early modern times. While forwards and futures are linear
instruments,whicharerelativelysimpletohandle,optionsarenonlinearin
nature and are more difficult to deal with. They come in all kind of flavors
and include vanilla calls and puts with hockey-stick payoffs, European op-
tions with more general payoffs, American, Bermudan and Asian options,
as well as barrier and other exotic options. The corresponding underliers
include stocks, bonds, currencies, commodities, etc.
The original scientific approaches to options valuation were developed
byBachelier(1900),Bronzin(1908),Boness(1964),Samuelson(1965),and
others. Yet, explosive development of the field is due to the seminal work
of Black–Scholes–Merton (BSM), Black and Scholes (1973), and Merton
(1973), who developed a novel theory of rational option pricing, which
gained universal acceptance and eventually became an indispensable tool
of modern financial engineering. The main advantage of their theory com-
pared to its predecessors is its ability to tie together pricing and hedging.
By necessity, BSM theory is based on several idealized assumptions, which
donotholdintherealworld.Theseassumptionsareasfollows:(A)markets
are efficient and frictionless, i.e., there are no transaction costs and taxes;
(B) there are no restrictions on short sales, frequency of trading, or the
amountofsharesboughtandsold;(C)thestockpricesaregovernedbylog-
normal stochastic processes with time-independent parameters. Provided
v
January3,2020 20:32 FittingLocalVolatility–9inx6in b3761-main pagevi
vi Foreword
that these assumptions hold, one can construct a perfect hedge for a con-
tingentclaimandpriceitunderarisk-neutralmeasure.IntheBSMframe-
work every stock is characterized by a single number σ, which is called its
volatility,whileitsphysicaldriftµisunimportant,andcanberiskneutral-
ized, so that µ → r−q, where r is the risk-free interest rate, and q is the
dividend rate. Thus, in order to price an option, we can assume that the
correspondingstockpriceisgovernedbythefollowingstochasticdifferential
equation (SDE):
dS(t)
=(r−q)dt+σdW(t), (1)
S(t)
S(0)=S ,
0
where W is a standard Wiener process.
ForaEuropeanoption,thecorrespondingBSMpartialdifferentialequa-
tion (PDE) and the terminal condition have the form
1
V (t,S)+ σ2S2V (t,S)+(r−q)SV (t,S)−rV (t,S)=0, (2)
t 2 SS S
V (T,S)=v(S),
where T is the option maturity, and v(S) is its payoff. For calls and puts,
we have
v(S)=(ω(S−K)) , (3)
+
where
(cid:26)
x,x≥0,
(x) = (4)
+ 0 x<0,
and ω is the call/put indicator, ω = 1 for calls, and ω = −1 for puts. It is
easy to show that BSM prices of vanilla calls and puts with strike K and
maturity T have the form
(cid:20) (cid:18) ln(F (t,T)/K)+ 1σ2τ(cid:19)
V (t,S,T,K;r,d,σ)=ωe−rτ F (t,T)N ω √ 2 (5)
σ τ
(cid:18) ln(F (t,T)/K)− 1σ2τ(cid:19)(cid:21)
−KN ω √ 2 ,
σ τ
whereN(.)isthecumulativenormaldistribution,τ =T−t,andF (t,T)=
exp((r−q)τ)S(t)istheforwardprice.Pricesofmorecomplicatedoptions
can be computed either analytically or numerically.
However, in real markets, idealized BSM assumptions do not hold.
Market prices are such that different vanilla options on a given underlier
January3,2020 20:32 FittingLocalVolatility–9inx6in b3761-main pagevii
Foreword vii
must be priced using different implied volatilities σ (t,S,T,K) in Eq. (3),
I
so that every underlier has its implied volatility surface, rather than a sin-
gle volatility. For a fixed maturity T, this fact results in the volatility skew
or smile; for a fixed strike K it manifests itself as the term structure of
volatility. As a result, considerable modeling insight is required for pricing
vanillaand,especially,exoticoptionsconsistentlywiththemarketandwith
each other.
Local volatility models assume that the stock price is governed by
Eq. (1), with the constant σ replaced by the so-called local volatility
σ (t,S).ThesemodelsextendthestandardBSMframeworkinarelatively
L
mild fashion, so that they are complete and allow for a perfect hedge. In
general, local volatility models cannot generate proper implied volatility
dynamics and tend to misprice exotic options, while at the same time pric-
ing vanillas perfectly. The variance gamma model of Madan and Seneta
(1990) can be generalized to the local variance gamma model of Carr and
Nadtochiy (2017).
Jump diffusion models, first proposed by Merton (1976), augment the
regular diffusion with Poissonian jumps. These models are necessarily
incompleteanddon’tallowforaperfecthedge,sincesmallandlargemove-
ments of the underlier cannot be handled simultaneously. For many un-
derliers, local volatility models augmented with jumps tend to agree with
market prices well.
Stochastic volatility models are useful in many situations. The most
popularmodelofthiskindisduetoHeston(1993).Themodelassumesthat
volatility is driven by a mean-reverting square-root process and generates
closed-form expressions for call and put prices. For certain underliers, for
example, liquid currency pairs, stochastic volatility models are adequate,
though never perfect.
Finally, universal volatility models proposed by Lipton (2002) combine
the best features of local, jump diffusion and stochastic volatility models;
as a result, they produce reliable prices and hedges not only for calls and
puts but also for many exotic options.
A large portion of Itkin’s book is devoted to the calibration of local
volatility models. The instrument of choice for calibrating these models
to the market is the celebrated forward equation of option pricing due
to Dupire (1994). An alternative but equivalent formulation is given by
Derman and Kani (1994). The Dupire equation allows one to price calls
and puts with different strikes and maturities at once. Depending on the
context,itcanheusedtoexpressσ (t,S)intermsofσ (t,S),or,conversely,
I L
January3,2020 20:32 FittingLocalVolatility–9inx6in b3761-main pageviii
viii Foreword
toexpressσ (t,S)intermsofσ (t,S).Specifically,writtenintermsofcall
L I
prices C(t,S,T,K), the Dupire equation reads
1
C (T,K)− σ2 (T,K)K2C (T,K)
T 2 L KK
+(r−q)KC (T,K)+qC(T,K)=0,
K
C(0,K)=(S−K) . (6)
+
Foragivenσ ,Eq.(6)canbesolvedforwardtogeneratethecorresponding
L
C(T,K), and then σ via Eq. (5). Alternatively, and more importantly,
I
for a given price surface C(T,K), Eq. (6) can be used to calibrate the
corresponding local volatility model and get
C (T,K)+(r−q)KC (T,K)+qC(T,K)
σ2 (T,K)= T K . (7)
L 1K2C (T,K)
2 KK
Once σ is known, one can use an appropriately modified BSM pricing
L
problem to find prices of more complicated options.
Equation(7)isbeautifulintheory,butratherdifficulttouseinpractice
for a variety of reasons, first and foremost due to the fact that in real
markets C(T,K) is known for a rather sparse set of discrete pairs Ξ =
{(T ,K )}, i = 1,...,N. This book is dedicated to developing novel and
i i
technicallyadvancedmethodsforsolvingthecalibrationprobleminearnest.
InPartI,theauthordiscussesthegeneralconceptoflocalvolatilityand
introduces the all-important no-arbitrage conditions. He develops various
forms of no-arbitrage interpolation which are used in the rest of the book.
The advantage of these interpolations is that, in addition to preserving
no-arbitrage conditions by construction, they allow closed-form solutions
for various special cases of the general volatility problem. In addition, the
authordiscussesarecentworkofCarrandPelts(2015)onnoarbitragefor
the implied volatility surface, and puts it into the general framework.
In Part II, the calibration problem for a discrete set of maturities
and strikes is analyzed in great detail. The author extends the work of
Andreasen and Huge (2011), and, more closely, Lipton and Sepp (2011), to
produceacontinuouspiecewiselinearlocalvolatilitysurfaceσ (t,S)inthe
L
spirit of Itkin and Lipton (2018). He also uses modern regression methods
to perform parameter calibration, starting with the Stochastic Volatility
Inspired (SVI) local volatility parameterization, Gatheral (2006), and pro-
ceeding to his own parameterization, Itkin (2015), which he developed a
decade ago while working as a market maker. The author convincingly
demonstrates the advantages of this parameterization by comparing the fit
with representative market data as well as with SVI.
January3,2020 20:32 FittingLocalVolatility–9inx6in b3761-main pageix
Foreword ix
Finally,inPartIII,theauthorexpandsthelocalvariancegammamodel
ofCarrandNadtochiy(2017)inthespiritofCarrandItkin(2018a,2018b).
He adds a drift and makes the stochastic driver geometric rather than
arithmeticbyusingaclevertimechange.Afterthat,theauthorshowshow
to calibrate the model to the market in an efficient manner by exploiting
various new no-arbitrage interpolations introduced in Part I.
Thisbookexpandsthepreviousworkdonebytheauthor,somesoloand
someinco-authorshipwithPeterCarrandAlexanderLipton,byproviding
additional material, which makes the exposition clear and uniform. One of
the key advantages of the book is its coverage of the most recent findings
in the general area of volatility calibration. The other one is that it is a
treasuretroveofnumericalrecipesmakingthevolatilitysurfacecalibration
both fast and accurate.
Itkin’s book can be used as a foundation for an advanced master or
PhD course in financial engineering and mathematical finance. It can also
beusedbypractitionersandacademicswhowanttolearnthemostmodern
and efficient approaches to building local and implied volatility surfaces in
a fast and accurate way. I recommend it wholeheartedly.
References
[1] Andreasen,J.andHuge,B.(2011).Volatilityinterpolation,Risk Magazine,
pp. 76–79.
[2] Bachelier, L. (1900). Theorie de la speculation, Annales de l’Ecole Normale
Superieure, 17, pp. 21–86.
[3] Black, F. and Scholes, M. (1973). The pricing of options and corporate lia-
bilities, Journal of Political Economy, 81, pp. 637–659.
[4] Boness, A.J. (1964). Elements of a theory of a stock option value, Journal
of Political Economy, 72, pp. 163–675.
[5] Bronzin, V. (1908). Theorie der Pra¨miengesch¨afte; Franz Deuticke.
[6] Carr, P. and Itkin, A. (2018a). An expanded local variance gamma model,
Available at https://arxiv.org/pdf/1802.09611.pdf.
[7] Carr, P. and Itkin, A. (2018b). Geometric local variance gamma model,
Available at https://arxiv.org/abs/1809.07727.
[8] Carr, P. and Nadtochiy, S. (2017). Local Variance Gamma and explicit cal-
ibration to option prices, Mathematical Finance, 27(1), pp. 151–193.
[9] Carr, P. and Pelts, G. (2015). Duality, deltas, and derivatives pricing, in
SteveShreve’s65thBirthdayConference,http://www.math.cmu.edu/CCF/
CCFevents/shreve/abstracts/P.Carr.pdf.
[10] Derman, E. and Kani, I. (1994a). Riding on a smile, RISK, pp. 32–39.
[11] Dupire, B. (1994). Pricing with a smile, Risk, 7, pp. 18–20.
[12] Gatheral, J. (2006). The volatility surface: A practitionals guide. Wiley,
Hoboken.