Construction Of the Implied Volatility Smile - Eurex
Transcript of Construction Of the Implied Volatility Smile - Eurex
Goethe University, Frankfurt/Main
Thesis
Construction Of the ImpliedVolatility Smile
byAlexey Weizmann
May, 2007
Submitted to the Department of Mathematics
JProf. Dr. Christoph Kühn, Supervisor
c©Weizmann 2007
Contents
1 Introduction 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Eurex 32.1 Derivatives on DJ EURO STOXX 50 at Eurex . . . . . . . . . . . . . . . 32.2 Market Making at Eurex . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 Preliminaries 73.1 Mathematical Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 Economic Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3 The Black-Scholes Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.3.1 Model Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3.2 Dynamics of the Underlying . . . . . . . . . . . . . . . . . . . . . 133.3.3 The Black-Scholes Differential Equation . . . . . . . . . . . . . . 143.3.4 Implied Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3.5 The Greeks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4 Vanna-Volga Method 214.1 Option Premium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.2 Implied Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.3 Vanna-Volga Result for EURO STOXX 50 . . . . . . . . . . . . . . . . . 24
5 Investigating ∆∆∆-neutrality 265.1 Portfolio Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.2 The Pricing Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.3 Derivation of the Implied Volatility . . . . . . . . . . . . . . . . . . . . . 305.4 Justification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
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6 Comparison 346.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406.3 Choice of Anker Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
7 Pricing Under Stochastic Volatility 427.1 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427.2 Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437.3 Application on the Pricing Formula . . . . . . . . . . . . . . . . . . . . . 47
8 Evaluation 50
A Tables 51
B Figures 56
C Matlab 60
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List of Figures
2.1 Trading Obligations for PMM and PML . . . . . . . . . . . . . . . . . . 6
4.1 Vanna-Volga Method, τ = 0.1205 . . . . . . . . . . . . . . . . . . . . . . 254.2 Vanna-Volga Method, τ = 0.3999 . . . . . . . . . . . . . . . . . . . . . . 25
6.1 Best Volatility and Premium Estimates, τ = 0.1205 . . . . . . . . . . . . 376.2 Volatility and Premium Residuals, τ = 0.1205 . . . . . . . . . . . . . . . 376.3 Best Volatility and Premium Estimates, τ = 0.3699 . . . . . . . . . . . . 396.4 Volatility and Premium Residuals, τ = 0.3699 . . . . . . . . . . . . . . . 39
B.1 Volatility Term Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 56B.2 Bounds for Implied Volatility Slope . . . . . . . . . . . . . . . . . . . . . 57B.3 Strike Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58B.4 Best Volatility and Premium Estimates, τ = 0.0411 . . . . . . . . . . . . 59B.5 Volatility and Premium Residuals, τ = 0.0411 . . . . . . . . . . . . . . . 59
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List of Tables
3.1 Greeks for European options . . . . . . . . . . . . . . . . . . . . . . . . . 20
6.1 Deviations of Estimates OESX-1206 . . . . . . . . . . . . . . . . . . . . . 366.2 Deviations of Estimates OESX-0307 . . . . . . . . . . . . . . . . . . . . . 38
A.1 Market Futures Prices and Obtained Forward Prices . . . . . . . . . . . . 51A.2 Typical Set of Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 52A.3 Distribution Histogram for Strikes . . . . . . . . . . . . . . . . . . . . . . 54A.4 Deviations of the Recommended Set of Strikes . . . . . . . . . . . . . . . 55
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Even extremely liquid markets, as the market for European style options on equity in-dexes, sometimes fail to provide sufficient data for pricing its options, e.g. particularoptions are not liquid enough.We are to investigate an extension of a well-known and widely spread “market-based”Vanna-Volga method, which not only allows to retrieve reasonable estimates for optionpremiums, but also to determine consistent implied volatilities easily. The theoretical re-sults are then analyzed using the daily settlement prices of Dow Jones EURO STOXX 50call options provided by Deutsche Börse. Introducing a stochastic volatility model wewere also able to deliver an explanation for the formulas, which were previously heuris-tically justified merely by formal expansion of the option premium by Itô.
Acknowledgements
I would like to express my gratitude to JProf. Dr. Christoph Kühn for the time hespend on the discussions and explanations.I deeply appreciate the help of my supervisor at Eurex, Dr. Axel Vischer, his assistantadvices and hints during the time of the research and writing of this thesis.Great thanks to my parents for their backing.
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1 Introduction
1.1 Motivation
We investigate a method for a simple and transparent derivation of the implied volatilitysmile from the market data. A knowledge of the contemporary volatility is crucial fortraders, especially in Foreign Exchange market, since options are priced in terms ofvolatility. But also traders in other markets are interested in an easily reproduciblemethodology to retrieve the volatility smile – for hedging, exploiting arbitrage or tradingvolatility spreads. The procedure also delivers options premiums. This can be used forpricing illiquid options, e.g for deep in-the-money options. The investigated procedurerequires the existence of four liquid options, whose implied volatilities are then readilyavailable. By adjusting the theoretical Black-Scholes price with costs for an over-hedgeone receives the desired market consistent option premium.
An Ornstein-Uhlenbeck stochastic volatility process provides the theoretical frame-work. Being mean-reverting, the volatility process tends to its mean level. Thus, experi-encing a fast mean-reverting volatility, we are able to approximate option premiums bythe Black-Scholes price adjusted by higher order derivatives of the option premium withrespect to the underlying spot price.
The theoretical results are evaluated with options on a highly traded Pan-Europeanindex DJ EURO STOXX 50. The settlement data were provided by Deutsche Börse.
1.2 Outline
A brief overview of Eurex is given in Chapter 2.Mathematical and economical terms referred to in this thesis are dealt with in Chapter 3.In Chapter 4 we use the Vanna-Volga method for deriving option premiums and volatilitysmile.Chapter 5 deals with an extension of the Vanna-Volga method. Obtained results arecompared in Chapter 6.
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Pricing under mean-reverting stochastic volatility together with the resulting pricingformula is explained in Chapter 7. The Appendix contains auxiliary tables and figuresas well as a description of the Matlab program.
The set of data for EURO STOXX 50 provided by Microstrategy∗ contains:
• Underlying close price St
• Call and put settlement prices CMKt P MK
t
• Strike price K• Time to expiration τ = T − t• Implied volatility I
We restrain our observations on options with fixed-strike moneyness m := KS
0.8 < m < 1.2 ,
since they are the most liquid in the market and thus bear the most information. Theother restriction is the consideration of call options only - the same results can beobtained for put options due to the put-call parity (see Definition 5). Put options areused for the estimation of the interest rate.
∗Data portal by Deutsche Börse
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2 Eurex
Eurex is the world leading derivatives exchange. It offers fully electronic trading in alarge number of derivatives. Facilities like Wholesale Trading (OTC), Eurex StrategyWizard or Market Making are available to the market to ensure liquidity and simplicityin trading. We take a look at derivatives on the EURO STOXX 50 traded at Eurexrelevant for the thesis. A brief introduction of the Market Making Program thereafteris connected to the question stated in the Chapter 5.2.
2.1 Derivatives on DJ EURO STOXX 50 at Eurex
Dow Jones EURO STOXX 50 is a blue-chip index containing the top 50 stocks in theEurozone.∗ The stocks, capped at ten percent, are weighted according to their free floatmarket capitalization with prices updated every 15 seconds. DJ EURO STOXX 50 Indexderivatives are the world’s leading euro-denominated equity index derivatives.
European style options on the Index traded at Eurex (OESX) are available withmaturities up to 10 years. The last trading day is the third Friday of each expirationmonth. The final settlement price is calculated as the average of the DJ EURO STOXX50 Index values between 11:50 and 12:00 CET. The contract value is EUR 10. Theminimum price change is 0.1 index point which is equivalent to EUR 1. At least sevenexercise prices are available for each maturity with a term of up to 24 months. For thesematurities the exercise price intervals are 50 index points; The exercise price interval foroptions with maturities larger than 36 month is 100 index points.
Futures on the EURO STOXX 50 (FESX) have excellent liquidity having a minimumprice change of one index point which is equivalent to EUR 10.
VSTOXX, based on the DJ EURO STOXX 50 options, is an implied volatility indextraded on the Deutsche Börse. It is set up as a rolling index with 30 days to expirationand derived by linear interpolation of the two nearest sub-indexes.† Sub-indexes per
∗For current composition of the index visit www.stoxx.com .†For detailed information on the derivation see [11].
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option expiry are computed for the first 24 months, giving 8 sub-indexes in total, whichare updated once a minute.
Volatility futures on VSTOXX (FVSX) can be used for trading calendar and marketspreads, as a hedging tool (e.g. crash risk), speculating (e.g mean-reverting nature ofvolatility).
2.2 Market Making at Eurex
Designated traders, called Market Makers, are granted a license to make tight markets inoptions and several futures contracts. This increases liquidity and transparency. Everyexchange participant may apply to be a Market Maker. Three models, which differ inthe kind of response to quote requests, continuous quotation and products selection, maybe chosen.Specifically:
• Regular Market Making (RMM) is restricted to less liquid options on equities,equity indexes and Exchange Traded Funds (EXTF) and to all options in fixedincome (FX) futures. RMM allows to choose products to quote (if available) withthe obligation to response to quote requests in all exercise prices and all expirations.
• Permanent Market Making (PMM) is available for all equity, equity index, EXTFoptions and options on FX futures. Products in which participants would like toact as PMM can be selected individually. The obligation is to quote for a set ofpre-defined number of expirations 85% of the trading time continuously.
• Advanced Market Making (AMM) is available for any pre-defined package of equityand/or equity index options as well as on FX options with an obligation of contin-uous quotation for a set of exercise prices for a pre-defined number of expirationsand options.
If traders fulfill the obligations they are refunded transactions and exercise fees.We give here a detailed description of PMM in index options only.‡ PMM consists of
three obligation levels – PMM, PMM short (PMS) and PMM long (PML). The obligation
‡For further information on Market Making please visitwww.eurexchange.com > Market Model > Market-Making.
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to quote for an average of 85 percent must be fulfilled for all expirations up to a definedmaximum maturity. Asymmetric quotation is allowed. PMM and PMS are obliged toquote calls and puts in five exercise prices out of a window of seven around the currentunderlying price – one at-the-money, three in-the-money and three out-of-the-moneyexercise prices. Compared to PMM, PMS must quote a larger minimum quote size, butfewer expirations.PML concentrates on long-term expirations – more than 18 and up to 60 months. Theobligations are fulfilled by quoting six exercise prices out of a window of nine aroundthe current underlying price. For that compare Figure 2.1 where the contract monthsup to 10 years are defined as follows:
The three nearest successive calendar months (1-3), the three following quarterlymonths of the March, June, September and December cycle (6-12), the four fol-lowing semi-annual months of the June and December cycle (18-36) and the sevenfollowing annual months of the December cycle (48-120).
A protection tool against system-based risk, called “Market Maker Protection”, is pro-vided by Eurex for Market Makers in PMM and AMM. It averts too many simultaneoustrade executions on quotes by Market Maker by counting the number of traded contractsper product within a defined time interval, chosen by the Market Maker.
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3400 3500 3600 3700 3800 3900 4000 4100 4200 4300 4400 4500 46001
2
3
6
9
12
18
24
30
36
48
60
72
84
96
108
120
Strikes
Con
trac
t Mon
ths
Figure 2.1: Trading obligations for PMM and PML.The solid vertical line denotes the at-the-money strike. The light grey area is the PMMarea with the obligation to quote five out of seven strikes. The dark grey is the PML areawith the obligation to quote six out of nine strikes.
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3 Preliminaries
3.1 Mathematical Terms
We first introduce some theorems and definitions.Let T ∗ > 0 be a finite time horizon. We consider a complete filtered probability
space (Ω,F ,P,F) satisfying the usual conditions,∗ where Ω is the set of states, F is aσ-algebra, P : F → [0, 1] a probability measure and F := (Ft)t∈[0,T ] a filtration generatedby a n-dimensional Brownian motion Wt.
We present a generalized version of Girsanov’s Theorem.Theorem 1 (Girsanov’s Theorem)Let Xt be a process with
Xt −X0 =
∫ t
0
µudu +
∫ t
0
σudWu (3.1)
where µ : Ω × R+ → R and σ : Ω × R+ → R+ are adapted and (F ⊗ R+)-measurable,with ∫ T
0
σ2udu < ∞ P-a.s. and
∫ T
0
|µu|du < ∞ P-a.s.
Let rt be an adapted process with∫ T
0|ru|du < ∞ (P-a.s.) such that
∫ T
0
(µu − ru
σu
)2
du < ∞ .
We set
Zt = exp
(−∫ t
0
µu − ru
σu
dWu −1
2
∫ t
0
(µu − ru
σu
)2
du
).
If E[ZT
]= 1 then we can define a new measure Q such that
dQ
dP= ZT ,
∗Right continuous and saturated for P-null sets.
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then X has the representation under Q
Xt −X0 =
∫ t
0
rudu +
∫ t
0
σudWu ,
where Wt is a Q Brownian motion.
ProofSee [18], Chapter VII, §3b.
A process σt is called square integrable, denoted by σt ∈ L2, if∫ T
0
E[σ2
u
]du < ∞ .
Given a process Xt, processes
t 7→ µ(t,Xt) t 7→ σ(t,Xt) t 7→ r(t,Xt)
also satisfy Girsanov’s theorem.A process of the form (3.1) is referred to as an Itô process. Girsanov’s Theorem allows
a representation of Itô processes with respect to a shifted Brownian motion Wt, whichnaturally defines a new measure, an equivalent martingale measure – measure, givingprobability zero to events, which had probability zero under the initial measure.
The first integral in (3.1) is the Riemann-Stieltjes integral, the second is a stochasticintegral defined to be an L2-limit of an approximating sequence of simple† functions :
limn→∞
∫ b
a
gn(u)dWu =
∫ b
a
g(u)dWu :=n−1∑k=0
g(tk)[Wtk+1
−Wtk
], (3.2)
where∫ b
aE[(gn(u)− g(u))2
]du → 0 and a = t0 < · · · < tn = b.
The stochastic integral in (3.2) is evaluated with forward increments of the Brownianmotion. This has an economic interpretation and is closely related to the point ofarbitrage: interpreting g as the number of assets bought at tk and held till tk+1 and Wt
as the price of a driftless asset at t one profits g(tk)[Wtk+1
−Wtk
]. If one would be able
to anticipate the price evolution, a riskless profit could be possible.Consider a portfolio consisting of n + 1 underlyings (without loss of generality we
†That is, there exist deterministic time points a = t0 < · · · < tn = b, such that σ(t, Xt) is constant oneach subinterval.
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assume the first underlying to be a riskless bond Bt)
V ζt = ζ0
t Bt +n∑
i=1
ζ it ·X i
t .
The vector of adapted processes ζt = (ζ1t , . . . , ζ
nt ), with
∫ t
0|ζ0
u|du < ∞, ∀t ≤ T ∗ andζ it , i = 1 . . . n is square integrable, is called a trading strategy. V ζ
t is referred to as thevalue of the portfolio at time t.
The portfolio V ζ described above is self-financing if its value vary only due to thevariations of the market
V ζt − V ζ
0 =
∫ t
0
ζ0udBu +
n∑i=1
∫ t
0
ζ iu · dX i
u .
Definition 1 (Arbitrage Opportunity)An arbitrage opportunity is a self-financing portfolio ζ such that
V ζ0 = 0 ,
V ζT > 0 , P-a.s.
The subsequent result by Delbaen and Schachermayer links the existence of an equiv-alent martingale measure stated by Girsanov’s Theorem with the absence of arbitrageopportunities.
Theorem 2 (Fundamental Theorem of Asset Pricing)There exists an equivalent martingale measure for the market model if and only if themarket satisfies the NFLVR (“no free lunch with vanishing risk”) condition.
ProofSee [4].
A financial market is called complete if every contingent claim H with maturity T canbe replicated by trading a self-financing strategy ζ, that is the value of the portfolio heldaccording to the trading strategy at time T equals the contingent claim
V ζT = H P-a.s.
Theorem 3 (Complete Market Theorem)A financial market is complete if and only if there exists exactly one equivalent martingalemeasure.
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ProofSee [8].
The following theorem describes the evolution of a continuous semimartingale, a pro-cess Xt that can be written as Mt + At, where Mt is a continuous local martingale andAt is a continuous adapted process of bounded variation. This decomposition, calledthe Doob-Meyer decomposition, is unique for M0 = A0 = 0.‡ Thus an Itô processis a continuous semimartingale, whose finite variation and local martingale parts (asthose on the right-hand side of (3.1) respectively), satisfy that both
∫ t
0µ(u, Xu)du and
〈∫ t
0σ(u, Xu)dWu,
∫ t
0σ(u, Xu)dWu〉 are absolutely continuous. The most general form of
a stochastic integral can be defined with a previsible bounded process as the integrandand a semimartingale as an integrator.
Theorem 4 (Itô’s Lemma)Let f : Rn → R be a twice continuously differentiable function and let X = (X1, . . . , Xn)
be a continuous semimartingale in Rn. Then for all t ≥ 0 holds
f(Xt)− f(X0) =n∑
i=1
∫ t
0
∂f
∂xi(Xu)dX i
u +1
2
n∑i=1
n∑j=1
∫ t
0
∂2f
∂xi∂xj(Xu)d〈X i, Xj〉u
ProofSee [16], IV.32, p. 60.
Itô’s Lemma gives us a tool for handling stochastic processes. Loosely speaking, it isthe stochastic version of the chain rule in ordinary calculus.
The following theorem establishes a link between partial differential equations andstochastic processes and thus is a formula for valuating claims.
Theorem 5 (Feynman-Kač Stochastic Representation Formula)Assume that f is a solution to the boundary value problem
∂f
∂t(t, x) + µ(t, x)
∂f
∂x+
1
2σ2(t, x)
∂2f
∂x2(t, x) = 0 ,
f(T, x) = h(x)
‡A detailed discussion on martingales can be found in [6], Section 2.
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where the process σ(u, Xu)∂f∂x
(u, Xu) is square integrable and X defined as (for s ≥ t)
Xs −Xt =
∫ s
t
µ(u, Xu)du +
∫ s
t
σ(u, Xu)dWu
Xt = x .
Then f has the representation
f(t, x) = E[h(XT )|Ft
],
where Ft is generated by a Brownian motion Wt.
ProofSee [1], Chapter 4, p. 59.
Definition 2 (Novikov’s Condition)A process ϕ satisfies Novikov’s condition, if
E exp
(∫ t
0
1
2ϕ2
udu
)< ∞ .
3.2 Economic TermsDefinition 3 (European Option)A contract, giving its holder the right, not the obligation, to buy one unit of a pre-definedasset, the underlying S, at a predetermined strike price K on the pre-defined date, thematurity date T , is called a European call option. Its payoff is
h(ST ) = (ST −K)+ .
A European put option gives its holder the right to sell one unit of the underlying. Itspayoff is
h(ST ) = (K − ST )+ .
Definition 4 (Forward)A forward contract is an agreement between two parties to buy (sell) one unit of anunderlying for a predefined price, the forward price, on a maturity date.
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Definition 5 (Put-Call Parity)The put-call parity is a relationship linking the option premiums for a call and a putoption with same maturity and strike price
C(t, St; K, T )− P (t, St; K, T ) = St − exp(−rτ)K .
This relationship follows from no-arbitrage arguments and is model-independent.
3.3 The Black-Scholes Model
The Black-Scholes model was introduced 1973 and started a profound study of the theoryof option pricing.
3.3.1 Model Assumptions
The assumptions (see [2]) of the Black-Scholes model reflecting ideal conditions in themarket are summarized below:
1. The market is efficient, that is arbitrage-free, liquid, time-continuous and has afair allocation of information. That implies zero transaction costs.
2. Constant risk-free rate r.
3. The no dividend paying underlying follows a geometric Brownian motion: a processdescribed in (3.4).
4. Short selling are possible.
None of the assumptions is satisfied perfectly. Markets have transaction costs, under-lyings do not follow a geometric Brownian motion and are traded in discrete units orat most in fractions.§ Despite those inconsistencies it is still a benchmark for othermodels and a standard pricing model in the financial world, since it is a function ofobservable variables and is easily implemented having a closed pricing formula. Themain distinguishing feature of the Black-Scholes model is its completeness.
§The distribution of returns appears to be leptokurtic – higher peak around its mean and fat tails,compared to the standard normal distribution.
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3.3.2 Dynamics of the Underlying
A bank account Bt with deterministic continuously compounded interest rate r existsand an investment of B0 = 1 evolves as:
Bt − 1 =
∫ t
0
rBudu (3.3)
equivalent to
Bt = exp(rt) . (3.3′)
The price of the underlying St follows a geometric Brownian motion:
St − S0 =
∫ t
0
µSu du +
∫ t
0
σSu dWu , (3.4)
where µ denotes the instantaneous expected return of the underlying, σ2 is a non-stochastic instantaneous variance of the return and at most a known function of time,Wt is a Brownian motion.This implies a lognormal distribution of the underlying. To see that apply Itô’s lemmaon G := ln St
Gt −G0 =
∫ t
0
(µ− 1
2σ2)du +
∫ t
0
σdWu = (µ− 1
2σ2)t + σWt.
That is ln St ∼ N(ln S0 + (µ− 1
2σ2)t, σ
√t)
and the dynamics of the underlying can bewritten as
St = S0 exp
((µ− 1
2σ2)t + σWt
). (3.4′)
The Black-Scholes option price of European type at time t is then a function of theunderlying St, strike K, maturity τ = T − t, continuously compounded deterministicinterest rate r and constant volatility σ:
V BS
t = η[StΦ(ηd+)− exp (−rτ)KΦ(ηd−)] , (3.5)
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with
η =
+1 for a call,
−1 for a put
Φ(z) =1√2π
∫ z
−∞exp
(−u2
2
)du
d+ =ln St
K+ (r + 1
2σ2)τ
σ√
τd− =
ln St
K+ (r − 1
2σ2)τ
σ√
τ.
The main result (3.5) is obtained by constructing a riskless arbitrage-free portfolio. Atfirst we derive the Black-Scholes differential equation.
3.3.3 The Black-Scholes Differential Equation
We choose the bank account as the numeraire, that is Bt ≡ 1. We are to find theappropriate shift, giving us an equivalent measure Q, under which the discounted priceprocesses are martingales¶
Vt :=Vt
Bt
= EQ[VT |Ft
]. (3.6)
Define a process
ZT = exp
(−λWT −
1
2〈λWT , λWT 〉
)= exp
(−λWT −
1
2λ2T
), (3.7)
where λ := µ−rσ
is called the market price of volatility risk.Since Law(WT −Wt) = Law(X), for X ∼ N (0, T − t) it follows for the characteristicfunction of (WT −Wt) with u ∈ R
E[exp
(iu(WT −Wt)
)]= E
[exp
(iuX
)]= exp
(−u2(T − t)
2
). (3.8)
We set t = 0 for convenience and observe for (3.7)
E[ZT
]= E
[exp
(− λWT −
1
2λ2T
)]= exp
(− 1
2λ2T
)E[exp
(i2λWT
) ](3.9)
= exp
(−1
2λ2T
)exp
(1
2λ2T
)= 1 . (3.10)
¶This property is called the Risk Neutral Valuation, see [1], Prop. 6.9.
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Thus we are able to apply Girsanov’s Theorem and define an equivalent martingalemeasure Q by
Q(F ) = EP[ZT111F
], ∀F ∈ FT . (3.11)
It also follows from Girsanov’s theorem that Wt := Wt + λt is a Q-martingale. And(3.4′) writes in terms of Wt as
ST = St exp
((r − 1
2σ2)τ + σ(WT − Wt)
),
or in terms of a stochastic variable Z ∼ N (0, 1) with density φ(z) = 1√2π
exp(−z2
2
)ST = St exp
((r − 1
2σ2)τ + σ
√τZ
). (3.12)
Applying the Feynman-Kač Stochastic Representation Formula and (3.12) we receive forthe case of a European call option
Vt = exp(−rτ)EQ[h(ST )|Ft
]. (3.13)
In the following we assume St to be Markovian.‖ Let us denote with V = v(t, St) thevalue of the payoff of a contingent claim h := η(ST −K)+. By Itô’s Lemma
Vt − V0 =
∫ t
0
(µSu ∂sv + ∂tv +
1
2σ2S2
u ∂ssv
)du +
∫ t
0
σSu ∂sv dWu , (3.14)
where ∂ denotes the corresponding partial derivative.Furthermore, the stochastic part of the change in the option price is assumed to beperfectly correlated with the underlying changes. This allows to set up a portfolio Π
consisting of a short position in a claim and a long position of ∆ units of the underlying:
Π = −V + ∆S . (3.15)
‖The future behavior of the process St given what has happened up to time t, is the same as thebehavior obtained when starting the process at St; see [20] p. 109.
15
A change in the value of the portfolio over a time interval t
Πt − Π0 = −(Vt − V0) +
∫ t
0
∆udSu . (3.16)
Substituting (3.4) and (3.14) into (3.16) and rearranging
Πt−Π0 =
∫ t
0
(−µSu ∂sv−∂tv−
1
2σ2S2
u ∂ssv+∆uµSu
)du+
∫ t
0
(−σSu ∂sv+∆uσSu
)dWu .
(3.17)
Choosing ∆ = ∂sv (3.17) becomes
Πt − Π0 =
∫ t
0
(− ∂tv −
1
2σ2S2
u ∂ssv
)du . (3.18)
To exclude any arbitrage opportunities the portfolio Π must earn at a risk-free rate r∗∗
Πt − Π0 =
∫ t
0
rΠudu . (3.19)
Setting (3.18) equal to (3.19) and substituting from (3.15) we obtain the Black-Scholesdifferential equation
∂tv + rs ∂sv +1
2σ2s2 ∂ssv − rv = 0 , (3.20)
where v and its derivatives are evaluated at (t, St). The boundary condition for thepartial differential equation above is given by
v(T, ST ) = h(ST ) .
∗∗This follows from simple arguments excluding arbitrage.
16
The solution to (3.20) can now be derived with (3.13).
Vt = exp(−rτ)EQ[h(ST )|Ft
]= exp(−rτ)EQ
[h(ST )|St = s
]= exp(−rτ)
∫ +∞
−∞
[s exp
((r − 1
2σ2)τ + σ
√τz
)−K
]1z0>Kφ(z)dz
with r := (r − 12σ2) and z0 := (ln K
s− rτ)/(σ
√τ) we get
= exp(−rτ)
(∫ +∞
z0
s exp
(rτ + σ
√τz
)φ(z)dz −K
∫ +∞
z0
φ(z)dz
)= exp(−rτ)
(s exp(rτ)√
2π
∫ +∞
z0
exp
(− 1
2(z − σ
√τ)2 +
1
2σ2τ
)dz −KΦ(−z0)
)= exp(−rτ)
(s exp(rτ)
∫ +∞
z0
1√2π
exp
(− 1
2(z − σ
√τ)2
)dz −KΦ(−z0)
)= s
∫ +∞
z0
1√2π
exp
(− 1
2(z − σ
√τ)2
)dz − exp(−rτ)KΦ(−z0)
and recognizing the integrand as the density function of Z ′ ∼ N (σ√
τ , 1) the result (3.5)for a call option follows with Z ′ − σ
√τ ∼ N (0, 1). Put premium follows then with the
put-call parity.
3.3.4 Implied Volatility
The Black-Scholes model is often chosen as a starting point. However, empirical resultshave revealed that the model experiences heavy deviations from the realities of currentoptions markets – the crucial Black-Scholes assumption of constant volatility misprizesa number of options systematically. There are several concepts of volatility to fix thisproblem. Two of them are briefly introduced below.
Historical volatility is based on historical market data over some time period in thepast. It can be computed as the standard deviation of the natural logarithm of close-to-close prices of the underlying:††
ϑ :=1
n− 1
n∑i=1
(log
(xi
xi−1
))2
− 1
n(n− 1)
(n∑
i=1
log
(xi
xi−1
))2
,
††See [9], pp. 239-240.
17
where x1, . . . , xn are the close-to-close prices, equally spaced with distance ∆t, which ismeasured in years. The denominator n− 1 is chosen to form an unbiased estimator andfor an estimator for the historical volatility follows
σh :=
√ϑ
∆t.
A problem coming up is the appropriate period of time over which the estimation shouldbe calculated: a very large set could include many old data, which are of little importancefor the future volatility, since volatility changes over time.
A direct measurement of volatility is thus difficult in practice. Since we assume themarket is efficient, it provides us with proper option premiums. It is also aware of theproper volatility. This feature forms the concept of implied volatility.
Definition 6 (Implied Volatility)Implied volatility I is the volatility, for which the Black-Scholes price equals the marketprice
V BS(t, St; K, T ; I) = V MK. (3.21)
Note, that the put-call parity implies that puts and calls with the same strike haveidentical implied volatilities. Implied volatility can be thought of as a consensus amongthe market participants about the future level of volatility – assuming a fair allocation ofinformation, as well as a same model used by all market participants for pricing options.
A concept closely related to implied volatility is smile effect – volatility obtained frommarket prices is often U-shaped, having its minimum near-the-money, often defined asan interval, for which
0.95 ≤ m ≤ 1.05 .
Deviation of implied volatility from a constant Black-Scholes volatility can be viewed asthe risk premium payable to the holder of the short position, which indirectly impliesvolatility to be fungible.‡‡ Trading volatility is accomplished for example by selling vega– a position achieved by selling an option. This technique makes profit if the underlyingexhibits no movements or falls. Trading a time spread – is a portfolio, consisting oflong and short options with different expiries and, typically, same exercise price. Longtime spreads – buying a long-dated option and selling a short-dated one – become moreworthy with increasing volatility, since a long-dated option has a larger vega.
‡‡A number of products allow brokers to trade pure volatility, for example volatility or variance swaps,volatility indexes or futures on volatility indexes.
18
As indicated by several researchers, volatility tends to be mean-reverting (e.g. see acurrent research on implied volatility indices [5], [9], p. 377 or [13], p. 292). A uniqueimplied volatility given the Black-Scholes price can be found with numerical procedures(such as Newton-Raphson used by Matlab), since
∂CBS
∂σ= Λ > 0 .
This legitimates a market standard to quote prices in terms of implied volatilities.Most of the time implied volatility is larger than historical. Implied volatility increasesin time to maturity and becomes less profound – compare Figure B.1.
3.3.5 The Greeks
Traders are interested in risks connected to a particular option. The sensitivities of anoption can be described by partial derivatives of the option premium with respect to themodel and the parameters.We list the most commonly used of them for vanilla options in the Table 3.1, where Φ(z),η, d+ and d− are defined as in (3.5) and
φ(z) =∂Φ(z)
∂z=
1√2π
exp
(−z2
2
).
Γ and Ξ give the curvature of ∆ and Λ correspondingly. The Greeks containing partialderivatives with respect to volatility, measure sensitivities to misspecifications withinthe model. Other Greeks, as Θ = ∂V/∂t and ρ = ∂V/∂r, are less important – in thecase of Theta we have a deterministic time-decay and the magnitude of Rho is extremelysmall.
Hedging against any of the sensitivities requires another option and the underlyingitself. To eliminate the short-term dependancies on any of the Greeks, hedgers arerequired to set up an appropriate portfolio of the underlying and other derivatives.
Some useful relations and notations.For an option with strike K one has
∆(t; K)CALL −∆(t; K)PUT = 1
0 ≤ ∆(t; K)CALL ≤ 1 .
Especially Foreign Exchange markets speak about plain vanilla options in terms of Delta
19
Greek Representation
Delta ∆=∂V
∂sηΦ(ηd+)
Gamma Γ=∂2V
∂s2
1
Stσ√
τφ(d+)
Vega Λ=∂V
∂σSt
√τφ(d+)
Volga Ξ=∂2V
∂σ2
St
√τd+d−σ
φ(d+)
Vanna Ψ=∂2V
∂s∂σ−d−
σφ(d+)
Speed Υ=∂3V
∂s3−(
d+
σ√
τ+ 1
)φ(d+)
S2t σ√
τ
Dual Delta ∆∗=∂V
∂K−η exp(−rτ)Φ(ηd−)
Table 3.1: Greeks for European options
and quote those in terms of volatility. It abstracts from strike and current underlyingprice, giving a transparent and a user-friendly method. A k∆ option, is an option whose∆ is k/100 for a call and −k/100 for a put. For detailed relationships among the Greekssee [14].
20
4 Vanna-Volga Method
The Vanna-Volga method is commonly used by market participants trading foreign ex-change (FX) options, which arises from the fact, that the FX market has only few activequotes for each maturity:
0∆ straddle is a long call and a long put with the same strike and expiration date– trader bets on raising volatility. The premium of a straddle yields informationabout the expected volatility of the underlying – higher volatility means higherprofit, and as a result a higher premium.
Risk-reversal is a long out-of-the-money call and a short out-of-the-money put with asymmetric ∆. Most common risk-reversals use 25∆ options. Traders see a positiverisk-reversal as an indicator of a bullish market, since calls are more expensive thanputs, and vice-versa.
Vega-weighted butterfly is constructed by a short at-the-money straddle and a long 25∆
strangle.∗ A buyer of a vega-weighted butterfly profits under a stable underlying.A straddle together with a strangle give simple techniques to trade volatility.
The AtM volatility σAtM is then derived as the volatility of the 0∆ straddle and thevolatilities of the risk-reversal (RR) and the vega-weighted butterfly (VWB) are subjectto following relations:†
σRR = σ25∆CALL − σ25∆PUT
σV WB =1
2(σ25∆CALL + σ25∆PUT )− σAtM .
Implied volatility of a risk-reversal incorporates information on the skew of the impliedvolatility curve, whereas that of a strangle – on the kurtosis.
∗Strangle is set up by a long k∆ call and a long k∆ put. The strategy is less expensive than a straddle,being profitable for a higher volatility.
†See [21], p. 35.
21
With the volatilities received in that way, Vanna-Volga allows us to reconstruct thewhole smile for a given maturity. At first one evaluates data received with this methodas proposed by Castagna and Mercurio in [3]. In the research the authors applied Vanna-Volga on EUR/USD exchange rate and obtained good results for strikes with moneyness0.9 < m < 1.1.
We evaluated the method for call options on EURO STOXX 50 for a time period ofone month with two different maturities. In this chapter we only introduce the resultsobtained within Vanna-Volga method. An interpretation and further discussion of (4.4)and (4.5) are given in Chapter 5.2 and Chapter 5.3 correspondingly.
4.1 Option Premium
As already mentioned, moneyness is defined as m :=Kj
St. A daily set of strikes, range
of moneyness, satisfying this requirement Kt := Kj | 0.8 < m < 1.2, Ft, withKi < Kj, for i < j is totally ordered. The Black-Scholes price of a European call optionat time t, with maturity T and strike K is denoted by CBS(t; K); the correspondingsettlement price by CMK(t; K).
We choose some option, the reference option, and use its implied volatility for calcu-lation of the Black-Scholes prices, for Black-Scholes assumes constant volatility. By σ
we denote the implied volatility of the reference option, the reference volatility. At firstwe compute the theoretical values for Vega, Volga and Vanna for Kt using the formulasfrom Table 3.1.
Our aim is to construct a weighted portfolio consisting of three liquid options withstrikes K1, K2, K3. Since those options are frequently traded, their implied volatilitiesσ1, σ2 and σ3 are precise and can be calculated easily. The constructed portfolio shouldbe vega, volga and vanna neutral with respect to an illiquid option with strike K. Thetime weights x1(t; K), x2(t; K), x3(t; K) then make the portfolio instantaneously hedged
22
up to the second order derivatives.
Λ(t; K) =3∑
i=1
xi(t; Ki) · Λ(t; Ki)
Ξ(t; K) =3∑
i=1
xi(t; Ki) · Ξ(t; Ki) (4.1)
Ψ(t; K) =3∑
i=1
xi(t; Ki) ·Ψ(t; Ki)
or in matrix notation
v = A · x . (4.1′)
Proposition 1The system (4.1′) admits a unique solution x = A−1 · v, with xi given by
x1(t; K) =Λ(t; K)
Λ(t; K1)
ln K2
Kln K3
K
ln K2
K1ln K3
K1
x2(t; K) =Λ(t; K)
Λ(t; K2)
ln KK1
ln K3
K
ln K2
K1ln K3
K2
(4.2)
x3(t; K) =Λ(t; K)
Λ(t; K3)
ln KK1
ln KK2
ln K3
K1ln K3
K2
Proof
|A| = −Λ(t; K1)Λ(t; K2)Λ(t; K3)
Stσ2√
τ·[d−(K3)d+(K2)d−(K2) + d−(K1)d+(K3)d−(K3)
− d+(K1)d−(K1)d−(K3)− d+(K3)d−(K3)d−(K2)− d−(K1)d+(K2)d−(K2)
+ d+(K1)d−(K1)d−(K2)]
= −Λ(t; K1)Λ(t; K2)Λ(t; K3)
Stσ5τ 2ln
K2
K1
lnK3
K1
lnK3
K2
. (4.3)
For positive K1 < K2 < K3, |A| < 0 and the unique solution for (4.1′) follows fromCramer’s rule.
23
The option price with an illiquid strike K is then given by
C(t; K) = CBS(t; K) +3∑
i=1
xi(t; K) · [CMK(t; Ki)− CBS(t; Ki)] . (4.4)
4.2 Implied Volatility
The implied volatility σt;K, corresponding to the pricing formula (4.4) is approximatedby the sum of the reference volatility σ and a term including the basic volatilities σ1, σ2
and σ3
σt;K ≈ σ +−σ +
√σ2 + d+(K)d−(K)(2σD+(K) + D−(K))
d+(K)d−(K)(4.5)
where d+(K) and d−(K) are as in (3.5) and
D+(K) : =ln K2
Kln K3
K
ln K2
K1ln K3
K1
σ1 +ln K
K1ln K3
K
ln K2
K1ln K3
K2
σ2 +ln K
K1ln K
K2
ln K3
K1ln K3
K2
σ3 − σ ,
D−(K) : =ln K2
Kln K3
K
ln K2
K1ln K3
K1
d+(K1)d−(K1)(σ1 − σ)2
+ln K
K1ln K3
K
ln K2
K1ln K3
K2
d+(K2)d−(K2)(σ2 − σ)2 +ln K
K1ln K
K2
ln K3
K1ln K3
K2
d+(K3)d−(K3)(σ3 − σ)2.
As pointed out by Castagna and Mercurio in [3], the above approximation for EUR/USDexchange rate options is extremely accurate for 0.9 < m < 1.1, although been asymp-totically constant at extreme strikes. Another drawback is that it cannot be definedwithout the square root. However the radicand is positive in most applications.
4.3 Vanna-Volga Result for EURO STOXX 50
We calculated the resulting option price, together with the implied volatility approxima-tion for all
(|Kt|3
)combinations of strikes. The best-fitting curve for the implied volatility
and the corresponding option premiums are depicted in Figure 4.1 and Figure 4.2. Aswe see, the approximation delivers very good results for around at-the-money options;Asymptotically constant volatility for deep in-the-money options is obvious. An evalua-tion of Vanna-Volga and its comparison to the extended method are given in Chapter 6.
24
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.250
0.1
0.2
0.3
0.4
Moneyness
Vola
tility
Vanna−VolgaMarket
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.250
200
400
600
800
Moneyness
Opt
ion
Prem
ium
Vanna−VolgaMarket
Figure 4.1: The Vanna-Volga method for OESX-1206, τ = 0.1205.The upper graph shows the volatility approximation (the light blue line) compared to themarket implied volatility (red line), where the markers give the positions of the ankerpoints.The lower, the corresponding option premiums.
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.20
0.1
0.2
0.3
0.4
Moneyness
Vola
tility
Vanna−VolgaMarket
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.20
200
400
600
800
1000
Moneyness
Opt
ion
Prem
ium
Vanna−VolgaMarket
Figure 4.2: The Vanna-Volga method for OESX-0307, τ = 0.3999.The upper graph shows the volatility approximation (the light blue line) compared to themarket implied volatility (red line), where the markers give the positions of the ankerpoints.The lower, the corresponding option premiums.
25
5 Investigating ∆∆∆-neutrality
As we have observed, the Vanna-Volga method also delivers good results for index op-tions. Asymptotically constant volatility for extreme strikes could come from the lack in∆-neutrality. A hedge against movements in the underlying is the most “natural” prac-tice among traders. Thus we will investigate consequences of extending the Vanna-Volgamethod by introducing ∆-neutrality.
5.1 Portfolio Construction
Let us now construct a portfolio of four liquid options with strikes K1, K2, K3, K4.We can easily derive the corresponding implied volatilities σi, i = 1 . . . 4. WithCMK(t; Ki), Ki ∈ K we denote option premiums of liquid options. As in Vanna-Volgamethod we are to find time-dependent weights x1(t; K), x2(t; K), x3(t; K), x4(t; K), suchthat the constructed portfolio remains delta, vega, vanna and volga neutral with respectto an illiquid option with strike K
∆(t; K) =4∑
i=1
xi(t; K) ·∆(t; Ki)
Λ(t; K) =4∑
i=1
xi(t; K) · Λ(t; Ki) (5.1)
Ξ(t; K) =4∑
i=1
xi(t; K) · Ξ(t; Ki)
Ψ(t; K) =4∑
i=1
xi(t; K) ·Ψ(t; Ki)
or using matrix notation, with column vectors v and x
v = A · x . (5.1′)
26
Being delta, vega, volga and vanna-neutral, the portfolio is furthermore also gamma-neutral. This arises from the Vega-Gamma relationship for European plain-vanilla op-tions (see [12])
1
2σΛ =
τ
2σ2S2Γ . (5.2)
From the Black-Scholes Differential Equation (3.20) we derive, that it is also Θ neutral.Thus this portfolio is hedged against all Greeks up to the second order.
Now we are to find the appropriate interest rate. This is done by minimizing theput-call parity in r under the assumption of arbitrage-free forward valuation (see [1], p.91).
The arbitrage-free forward price Ft at time t with maturity T is given by
Ft = exp(rτ)St. (5.3)
Then the put-call parity written as a function of r is:
p(r) = Ct − Pt + exp(−rτ)(K − Ft). (5.4)
Minimizing p(r)2 in r by OLS-method we retrieve the interest rate consistent with themarket prices
minr
[∑K∈Kt
(Ct − Pt + exp(−rτ)(K − Ft))2
]. (5.5)
The goodness of this method can be proved by comparing the corresponding futuresprices∗ derived from the recovered interest rate according to (5.3). For that see Table A.1.
The Vanna-Volga method used three strikes to calculate the option price: a 0∆ strad-dle, a 25∆risk-reversal and a vega-weighted butterfly. These particular options werechosen, because the FX market has very few active quotes.
This is not the case for index options – being frequently traded, the market becomeseven more liquid through market makers.
One important question to address is the appropriate choice of the four strikes. Letkt := |Kt| be the number daily strikes that match the moneyness condition (in our casek was about 30). Thus we have to test all
(kt
4
)combinations of strikes. Do quotes within
the strike price windows of PML and PMM-intervals deliver better results?
∗Under deterministic interest rates futures and forward prices coincide, see [1], p. 92.
27
5.2 The Pricing FormulaProposition 2The system (5.1) admits always a unique solution.
ProofAt first consider that
Λ
∆=
∂ ln ∆
∂sS2τσ .
Since ∆ is strictly increasing in S, ln ∆ is also strictly increasing in S and ∂ ln ∆∂s
strictlydecreasing. From the duality of S and K follows(
S ↑⇔ K ↓)
=⇒(
K ↓⇔ ∂ ln ∆
∂s↓)
. (5.6)
Applying Proposition 1 we write the determinant of A as
|A| = −(
∆1Λ2Λ3Λ4
Sσ5τ 2ln
K3
K2
lnK4
K2
lnK4
K3
−∆2Λ1Λ3Λ4
Sσ5τ 2ln
K3
K1
lnK4
K1
lnK4
K3
+∆3Λ1Λ2Λ4
Sσ5τ 2ln
K2
K1
lnK4
K1
lnK4
K2
−∆4Λ1Λ2Λ3
Sσ5τ 2ln
K2
K1
lnK3
K1
lnK3
K2
)= −Λ1Λ2Λ3Λ4
Sσ5τ 2
(∆1
Λ1
lnK3
K2
lnK4
K2
lnK4
K3
− ∆2
Λ2
lnK3
K1
lnK4
K1
lnK4
K3
+∆3
Λ3
lnK2
K1
lnK4
K1
lnK4
K2
− ∆4
Λ4
lnK2
K1
lnK3
K1
lnK3
K2
)(5.7)
= −Λ1Λ2Λ3Λ4
S3σ6τ 3
(∂s
∂ ln ∆1
lnK3
K2
lnK4
K2
lnK4
K3
− ∂s
∂ ln ∆2
lnK3
K1
lnK4
K1
lnK4
K3
+∂s
∂ ln ∆3
lnK2
K1
lnK4
K1
lnK4
K2
− ∂s
∂ ln ∆4
lnK2
K1
lnK3
K1
lnK3
K2
), (5.7′)
where ∆i := ∆(St, t; Ki) and Λi := Λ(St, t; Ki).Simple algebra shows
lnK3
K2
lnK4
K2
lnK4
K3
− lnK3
K1
lnK4
K1
lnK4
K3
+ lnK2
K1
lnK4
K1
lnK4
K2
− lnK2
K1
lnK3
K1
lnK3
K2
= 0 .
Summing up, the term before parenthesis in (5.7′) is negative and for the coefficients
28
before logarithm terms in parenthesis we observe with (5.6)
∂s
∂ ln ∆i
>∂s
∂ ln ∆j
, for i < j
since Ki < Kj , for i < j.Thus, |A| < 0 .
The unique solution for (5.1) follows from Cramer’s Rule with (5.7)
x1(t;K) =ΛK
Λ1
ln K4K3
(∆KΛK
ln K3K2
ln K4K2
− ∆2Λ2
ln K3K ln K4
K
)+ ln K2
K
(∆3Λ3
ln K4K ln K4
K2− ∆4
Λ4ln K3
K ln K3K2
)ln K4
K3
(∆1Λ1
ln K3K2
ln K4K2
− ∆2Λ2
ln K3K1
ln K4K1
)+ ln K2
K1
(∆3Λ3
ln K4K1
ln K4K2
− ∆4Λ4
ln K3K1
ln K3K2
)x2(t;K) =
ΛK
Λ2
ln K4K3
(∆1Λ1
ln K3K ln K4
K − ∆KΛK
ln K3K1
ln K4K1
)+ ln K
K1
(∆3Λ3
ln K4K1
ln K4K − ∆4
Λ4ln K3
K1ln K3
K
)ln K4
K3
(∆1Λ1
ln K3K2
ln K4K2
− ∆2Λ2
ln K3K1
ln K4K1
)+ ln K2
K1
(∆3Λ3
ln K4K1
ln K4K2
− ∆4Λ4
ln K3K1
ln K3K2
)x3(t;K) =
ΛK
Λ3
ln K4K
(∆1Λ1
ln KK2
ln K4K2
− ∆2Λ2
ln KK1
ln K4K1
)+ ln K2
K1
(∆KΛK
ln K4K1
ln K4K2
− ∆4Λ4
ln KK1
ln KK2
)ln K4
K3
(∆1Λ1
ln K3K2
ln K4K2
− ∆2Λ2
ln K3K1
ln K4K1
)+ ln K2
K1
(∆3Λ3
ln K4K1
ln K4K2
− ∆4Λ4
ln K3K1
ln K3K2
)x4(t;K) =
ΛK
Λ4
ln KK3
(∆1Λ1
ln K3K2
ln KK2
− ∆2Λ2
ln K3K1
ln KK1
)+ ln K2
K1
(∆3Λ3
ln KK1
ln KK2
− ∆KΛK
ln K3K1
ln K3K2
)ln K4
K3
(∆1Λ1
ln K3K2
ln K4K2
− ∆2Λ2
ln K3K1
ln K4K1
)+ ln K2
K1
(∆3Λ3
ln K4K1
ln K4K2
− ∆4Λ4
ln K3K1
ln K3K2
)(5.8)
where ∆i := ∆(St, t;Ki), ∆K := ∆(St, t;K) and Λi := Λ(St, t;Ki), ΛK := Λ(St, t;K).
Then the option premium C(t; K) of the illiquid option with strike K is:
C(t; K) = CBS(t; K) +4∑
i=1
xi(t; K) · [CMK(t; Ki)− CBS(t; Ki)] (5.9)
or substituting from (5.1′) and y a column vector, with yi := CMK(t; Ki)− CBS(t; Ki)
C(t; K) = CBS(t; K) + (A−1v)′ y = CBS(t; K) + v′w , (5.9′)
where w := (A′)−1 y .
Properties of (5.9):
1. The option premium approximation formula is a inter or extrapolation formula ofC(t; K). Thus on the one hand we are able to price far out-of-the-money, as well
29
as deep in-the-money options. On the other hand we retrieve premiums even foroptions that are not offered by the market place.
2. The four anker points CMK(t; Ki), i = 1 . . . 4 are matched exactly, since forK = Kj we have (compare Table A.2)
xi(t; K) =
1 for i = j,
0 otherwise
3. However, the pricing formula delivers not always arbitrage-free prices, that is
C(t; Ki) < C(t; Kj), for some i < j, Ki ∈ K .
4. Following no-arbitrage conditions still hold
a) C(t; K) ∈ C2((0, +∞))
b) limK→+∞ C(t; K) = 0
This is an economic interpretation of the pricing formula (5.9′):The vector w is interpreted as a vector of premiums of the market prices that must beattached to the Greeks in order to adjust the Black-Scholes price of liquid options. Thisadjustment is called an over-hedge. With this interpretation, ∆, Λ, Ξ and Ψ can be seenas proxies for certain risks – volga correction for the kurtosis and vanna correction for theskew. Traders, willing to offload these risks to another party, should compensate them;those bringing the risks into the market, should pay for them. The market cost of sucha protection form the weighted excess one has to add to the theoretical Black-Scholesprice.
5.3 Derivation of the Implied Volatility
To emphasize the dependance on the volatility we rewrite (5.9) as
C(t; K) = CBS(t; K; σ) +4∑
i=1
xi(t; K) · [CMK(t; Ki; σi)− CBS(t; Ki; σ)] . (5.10)
30
We approximate the option premium C(t; K) by the second order Taylor expansion of(5.10) in σ:
C(t; K;σ1,2,3,4) ≈ CBS(t; K; σ) +4∑
i=1
xi(t; K) ·[
CMK(t; Ki; σ)− CBS(t; Ki; σ)︸ ︷︷ ︸(∗)
]
+∂CBS(t; K; σ)
∂σ(σ − σ)
+4∑
i=1
xi(t; K) ·[∂CMK(t; Ki; σ)
∂σ(σi − σ)− ∂CBS(t; Ki; σ)
∂σ(σ − σ)
]+
1
2
∂2CBS(t; K; σ)
∂σ2(σ − σ)2
+1
2
4∑i=1
xi(t; K) ·[∂2CMK(t; Ki; σ)
∂σ2(σi − σ)2 − ∂2CBS(t; Ki; σ)
∂σ2(σ − σ)2
]
=CBS(t; K; σ) +4∑
i=1
xi(t; K) · ∂CMK(t; Ki; σ)
∂σ(σi − σ)
+1
2·
4∑i=1
xi(t; K) · ∂2CMK(t; Ki; σ)
∂σ2(σi − σ)2
=CBS(t; K; σ) +4∑
i=1
xi(t; K) ·[Λ(t; Ki; σ)(σi − σ) +
1
2· Ξ(t; Ki; σ)(σi − σ)2
],
(5.11)
with (∗) vanishing, since the market price of options under constant volatility σ equalsthe Black-Scholes price (compare (7.13)).
On the over hand the market price of the illiquid option is:
C(t; K;σt;K) ≈ CMK(t; K; σ) +∂CMK(t; K; σ)
∂σ(σt;K − σ) +
1
2· ∂2CMK(t; K; σ)
∂σ2(σt;K − σ)2
=CBS(t; K; σ) + Λ(t; K; σ)(σt;K − σ) +1
2· Ξ(t; K; σ)(σt;K − σ)2 ,
(5.12)
where CMK(t; K; σ) turns into CBS(t; K; σ) for the same reason as in (∗). The same istrue for its derivatives.
The implied volatility σt;K follows by equating (5.11) and (5.12) and solving the second-
31
order algebraic equation
σ±t;K ≈ σ +−Λ(t; K; σ)±
√Λ(t; K; σ)2 + 2 · Ξ(t; K; σ) · κΞ(t; K; σ)
, (5.13)
where
κ :=4∑
i=1
xi(t; K) ·[Λ(t; Ki; σ)(σi − σ) +
1
2· Ξ(t; Ki; σ)(σi − σ)2
].
Matching the anker volatilities exactly, the formula above gives an easy to implementapproximation of the implied volatility. Empirical tests have shown, that the secondsolution σ−t;K delivers a flat structure.
The formula above gives us an comfortable way to derive implied volatilities. Anoticeable drawback is its dependence on rigorously derived option premiums. As alreadymentioned, option premiums are not always arbitrage-free. This has severe consequenceson the derivation of the implied volatility. However, we are able to provide a controltool for the slope of the implied volatility curve.
We use the definition and properties of implied volatility, which itself is a function ofmoneyness. For a fixed t and thus constant St equation (3.21) can be written in termsof moneyness as
CMK = CBS(I(m); m) .
Taking the derivative with respect to m and noting that CMK is decreasing in K weobtain
∂CMK
∂m=
∂CBS(I(m); m)
∂σ· ∂I
∂m+
∂CBS(I(m); m)
∂m≤ 0
S√
τφ(d+) · ∂I
∂m− S exp(−rτ)Φ(d−) ≤ 0
giving us the upper bound for the slope of the implied volatility curve. Similar derivationfor P MK, which is increasing in K, provides us with a lower bound. Altogether we get:
−√
2π
τexp
(− rτ +
d+
2
)Φ(d−) ≤ ∂I
∂m≤√
2π
τexp
(− rτ +
d+
2
)Φ(d−) .
5.4 Justification
In this section we give a justification for (5.9) using Itô’s-formula.
32
Let us assume, a function Ct depends on St, τ = T − 0, K and σt. We allow σt to benot only time-dependand but also possibly stochastic. Applying Itô’s Lemma we get forthe value of a vanilla option C(St, t; K; σt)
C(ST ; K; σt) = C(S0; K; σ0) +
∫ T
0
∂C
∂sdS +
∫ T
0
∂C
∂tdt +
∫ T
0
1
2
∂2C
∂s2d〈S, S〉
+
∫ T
0
∂C
∂σdσ +
∫ T
0
∂2C
∂s∂σd〈S, σ〉+
∫ T
0
1
2
∂2C
∂σd〈σ, σ〉 .
The first three integral terms form the Itô-expansion of the Black-Scholes price, thetheoretical price. The latter three come from the stochastic volatility and give an ad-justment to the theoretical price. Hence for an arbitrary option C(St; Ki; σt, τ)
CMK(St; Ki; σt, τ)− CBS(St; Ki; σt, τ)
gives its stochastic part, which can be approximated by a delta, vega, vanna and volganeutral portfolio
4∑i=1
xi(t; K)·[CMK(t; Ki)− CBS(t; Ki)] .
33
6 Comparison
6.1 Results
Time series from 1.-30. November 2006 were chosen as a basis for the comparison – atime span containing 22 trading days. In order to compare the results of both pricingprocedures December 2006 (OESX-0612) and March 2007 (OESX-0307) were selected asoption expiries – 44-15 and 135-106 days before expiry respectively. This corresponds tovery short-term and midterm maturities. Furthermore, the number of data point of thetwo sets was approximately equal. Comparing the estimates for the implied volatility andthe option premiums of both methods, as well as the performance of the approximationwith increasing maturity, we were able to show the superiority of the ∆-neutral methodto the Vanna-Volga. The pronounced supremacy is although decreasing with increasingtime to maturity.
We are interested in the goodness of both methods. The following results are validfor a fixed t. To focus on high vega strikes, the daily data set containing the receivedvolatility estimates is vega-weighted. Vega takes its maximum for at-the-money options.A good approximation for the volatility in the region around m = 1 is more desirable,than those in the wings. Thus we introduce weighting coefficients for each strike Kj ∈ Kt
Λ(Kj)∑Ki∈Kt
Λ(Ki).
Table 6.1 and Table 6.2 compare the methods for the first (0.1205 ≤ τ ≤ 0.0411) andthe second (0.2904 ≤ τ ≤ 0.3699) data set respectively. Boxes with two columns givethe corresponding data for both methods.The total volatility deviation is computed by vega-weighting
∑Kj∈Kt
|σ(Kj)− I(Kj)|Λ(Kj)∑
Ki∈KtΛ(Ki)
. (6.1)
34
Total premium deviation is the sum over deviations of premium estimates∑Kj∈Kt
|C(Kj)− CMK(Kj)|
followed by the strike at which the maximal premium deviation was attained.Maximal premium deviation
maxKj
|C(Kj)− CMK(Kj)| .
Premium ratio is calculated for the maximal deviation
C(Kj)− CMK(Kj)
CMK(Kj)
as well as the estimated premium.Figure 6.1 and Figure 6.3, show the best estimates, that is those with minimal total
volatility deviation, the premium estimates for the same set of strikes follow. Figure 6.2as well as Figure 6.4 depicts the corresponding residuals:
the top graph: (I − σ)
the middle graph: (CMK − C)
the bottom graph: (CMK − C)/CMK .
35
τTo
talV
olat
ility
Tota
lPre
miu
mM
ax.
Pre
miu
mM
ax.
Pre
miu
mP
rem
ium
Est
imat
edD
evia
tion
Dev
iation
atta
ined
atD
evia
tion
Rat
ioP
rem
ium
0.12
050.
0013
10.
0012
326
.615
.635
0038
003.
01.
70.
0057
4−
0.00
735
520.
224
0.2
0.11
780.
0013
10.
0011
926
.413
.735
0032
002.
81.
60.
0057
0−
0.00
210
482.
778
2.4
0.11
510.
0010
30.
0008
628
.020
.835
5032
003.
43.
00.
0075
4−
0.00
375
443.
879
4.6
0.10
680.
0009
30.
0006
828
.411
.336
5032
503.
21.
30.
0078
6−
0.00
159
404.
280
3.0
0.10
410.
0010
00.
0008
422
.313
.536
5033
002.
61.
90.
0060
8−
0.00
241
430.
878
0.8
0.10
140.
0010
00.
0008
122
.213
.636
5033
002.
72.
00.
0061
7−
0.00
259
429.
677
9.9
0.09
860.
0010
50.
0008
822
.69.
836
5036
502.
71.
00.
0061
90.
0021
642
8.8
430.
6
0.09
590.
0010
70.
0009
025
.117
.036
5032
502.
82.
20.
0065
5−
0.00
272
420.
582
1.0
0.08
770.
0009
30.
0006
823
.117
.037
0033
002.
92.
50.
0073
7−
0.00
315
394.
979
5.6
0.08
490.
0011
30.
0008
821
.212
.237
0037
002.
51.
70.
0065
40.
0044
338
8.1
389.
0
0.08
220.
0010
50.
0008
719
.114
.137
0033
002.
21.
70.
0052
6−
0.00
217
411.
581
2.0
0.07
950.
0007
20.
0006
318
.412
.337
5033
002.
11.
50.
0056
5−
0.00
186
366.
181
5.7
0.07
670.
0009
10.
0006
121
.218
.737
5033
002.
72.
30.
0078
2−
0.00
298
338.
378
8.7
0.06
850.
0006
90.
0004
818
.114
.538
0033
002.
41.
80.
0077
1−
0.00
228
304.
880
4.2
0.06
580.
0012
00.
0010
517
.217
.138
0033
502.
32.
00.
0074
6−
0.00
269
307.
275
7.5
0.06
300.
0008
40.
0004
618
.721
.038
0033
002.
52.
40.
0080
3−
0.00
295
304.
280
4.9
0.06
030.
0011
10.
0007
217
.122
.438
0033
002.
52.
70.
0085
8−
0.00
338
293.
479
4.2
0.05
750.
0009
60.
0007
719
.817
.037
5032
502.
41.
80.
0078
8−
0.00
226
306.
980
6.5
0.04
930.
0014
20.
0008
922
.459
.936
5036
002.
75.
80.
0079
90.
0149
033
9.9
385.
6
0.04
660.
0021
20.
0019
512
.415
.936
0036
001.
41.
80.
0038
20.
0047
638
0.0
379.
7
0.04
380.
0017
00.
0017
09.
97.
037
0037
001.
20.
60.
0037
70.
0019
432
7.2
327.
8
0.04
110.
0024
60.
0023
910
.212
.637
0032
001.
21.
10.
0040
−0.
0014
429
5.0
794.
0
Tab
le6.
1:A
com
pari
son
ofm
etho
dsfo
r0.
1205≤
τ≤
0.04
11.
Blo
cks
oftw
ogi
veth
esa
me
char
acte
rist
ics
for
both
met
hods
.T
hefir
stco
lum
nco
ntai
nsda
tafo
rVan
na-V
olga
met
hod;
the
seco
ndfo
r∆
-neu
tral
.D
ata
isgi
ven
onda
ilyba
sis.
Vol
atili
tyde
viat
ions
com
pute
dby
vega
-wei
ghti
ng.
36
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.250.1
0.15
0.2
0.25
0.3
0.35
0.4
Moneyness
Impl
ied
Vola
tility
Vanna−Volga∆−neutralMarket
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.250
200
400
600
800
Moneyness
Opt
ion
Prem
ium
Vanna−Volga∆−neutralMarket
Figure 6.1: Best volatility and premium estimates OESX-1206, τ = 0.1205The upper graph shows the volatility approximations for Vanna-Volga (the light blue line)and its extension (the blue line) compared to the market implied volatility (red line), wherethe colored markers give the positions of the anker points.The lower, the corresponding option premiums.
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25−0.4
−0.2
0
0.2
0.4
in P
erce
nt P
oint
s
Implied Volatility Residuals
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25−2
0
2
4
Inde
x Po
ints
Option Premium Residuals
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25−50
0
50
100
Perc
ent
Moneyness
Relative Option Premium Residuals
Vanna−Volga σimp
−σest
∆−neutral σimp
−σest
Vanna−Volga CMK−Cest
∆−neutral CMK−Cest
Vanna−Volga (CMK−Cest
)/(CMK)
∆−neutral (CMK−Cest
)/(CMK)
Figure 6.2: Volatility and premium residuals OESX-1206, τ = 0.1205
37
τTo
talV
olat
ility
Tota
lPre
miu
mM
ax.
Pre
miu
mM
ax.
Pre
miu
mP
rem
ium
Est
imat
edD
evia
tion
Dev
iation
atta
ined
atD
evia
tion
Rat
ioP
rem
ium
0.36
990.
0014
00.
0012
536
.022
.432
0042
504.
22.
20.
0049
7−
0.04
465
842.
450
.5
0.36
710.
0015
10.
0014
636
.531
.032
0032
003.
22.
60.
0040
10.
0031
680
6.5
807.
1
0.36
440.
0014
80.
0014
535
.236
.332
0032
003.
64.
20.
0043
60.
0051
881
6.3
815.
7
0.35
620.
0012
40.
0011
736
.354
.132
5032
504.
65.
20.
0056
00.
0062
382
4.7
824.
1
0.35
340.
0012
30.
0011
641
.341
.732
5032
504.
03.
90.
0046
60.
0045
185
1.7
851.
8
0.35
070.
0013
60.
0013
435
.942
.032
5032
504.
35.
00.
0049
90.
0058
885
1.9
851.
2
0.34
790.
0013
50.
0013
134
.527
.832
5037
004.
22.
30.
0048
7−
0.00
517
850.
043
8.1
0.34
520.
0012
70.
0012
334
.224
.832
5032
503.
71.
80.
0044
10.
0021
484
2.5
844.
4
0.33
700.
0010
20.
0009
735
.942
.833
0033
003.
46.
40.
0041
90.
0077
981
7.9
814.
9
0.33
420.
0010
10.
0009
624
.621
.933
0033
003.
72.
40.
0045
00.
0029
880
9.9
811.
2
0.33
150.
0009
70.
0008
236
.818
.333
0037
004.
11.
60.
0052
1−
0.00
338
784.
446
4.7
0.32
880.
0007
80.
0006
425
.018
.533
0037
003.
82.
40.
0045
3−
0.00
513
836.
546
8.8
0.32
600.
0011
00.
0010
029
.022
.533
0033
004.
02.
50.
0049
50.
0030
880
9.3
810.
8
0.31
780.
0009
80.
0007
632
.445
.033
0033
004.
26.
80.
0054
20.
0081
877
5.8
821.
3
0.31
510.
0011
50.
0009
933
.419
.833
0037
504.
12.
00.
0052
9−
0.00
487
777.
941
2.1
0.31
230.
0010
50.
0008
932
.119
.233
0037
504.
12.
00.
0053
0−
0.00
479
777.
241
2.1
0.30
960.
0009
90.
0008
132
.526
.433
0040
504.
43.
50.
0056
50.
0204
176
6.5
165.
6
0.30
680.
0013
50.
0012
236
.328
.532
5036
504.
72.
90.
0056
8−
0.00
631
827.
746
2.0
0.29
860.
0015
70.
0014
137
.828
.032
0039
504.
62.
00.
0056
10.
0113
481
2.6
174.
3
0.29
590.
0015
80.
0014
337
.024
.732
0039
504.
52.
10.
0055
90.
0128
080
3.3
163.
6
0.29
320.
0015
30.
0015
036
.132
.232
5032
504.
12.
90.
0051
10.
0036
479
9.7
800.
9
0.29
040.
0014
50.
0012
836
.325
.332
0036
004.
32.
10.
0052
6−
0.00
470
815.
944
9.4
Tab
le6.
2:A
com
pari
son
ofm
etho
dsfo
r0.
2904≤
τ≤
0.36
99.
Blo
cks
oftw
ogi
veth
esa
me
char
acte
rist
ics
for
both
met
hods
.T
hefir
stco
lum
nco
ntai
nsda
tafo
rVan
na-V
olga
met
hod;
the
seco
ndfo
r∆
-neu
tral
.D
ata
isgi
ven
onda
ilyba
sis.
Vol
atili
tyde
viat
ions
com
pute
dby
vega
-wei
ghti
ng.
38
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.20.1
0.15
0.2
0.25
0.3
0.35
0.4
Moneyness
Impl
ied
Vola
tility
Vanna−Volga∆−neutralMarket
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.20
200
400
600
800
1000
Moneyness
Opt
ion
Prem
ium
Vanna−Volga∆−neutralMarket
Figure 6.3: Best volatility and premium estimates OESX-0307, τ = 0.3699.The upper graph shows the volatility approximations for Vanna-Volga (the light blue line)and its extension (the blue line) compared to the market implied volatility (red line), wherethe colored markers give the positions of the anker points. The lower, the correspondingoption premiums.
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2−0.4
−0.2
0
0.2
0.4
in P
erce
nt P
oint
s
Implied Volatility Residuals
Vanna−Volga σimp
−σest
∆−neutral σimp
−σest
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2−5
0
5
Inde
x Po
ints
Option Premium Residuals
Vanna−Volga CMK−Cest
∆−neutral CMK−Cest
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2−20
0
20
40
Perc
ent
Moneyness
Relative Option Premium Residuals
Vanna−Volga (CMK−Cest
)/(CMK)
∆−neutral (CMK−Cest
)/(CMK)
Figure 6.4: Volatility and premium residuals OESX-0307, τ = 0.3699.
39
6.2 Discussion
As the first result we can point out that the Vanna-Volga method as well as its extensionare applicable to equity index options.Main results for both data sets are similar:
Over-all results become less precise, that is the accuracy region becoming more tight(compare Figure B.4), with declining maturity.
Out-of-the-money region
• Volatility approximation by both methods flattening.
• Premium approximation by Vanna-Volga is more precise – due to very smallpremiums, even tiny deviations have extreme consequences for the price andcorrespondingly to the ratio.
At-the-money region
• Volatility and premium approximation by both methods are very good.
In-the-money region
• Volatility approximation by Vanna-Volga extension is more precise – due tothe fourth anker point, the moneyness range was extended by 0.1, whichcorresponds to approximately 8 strikes.
• Vanna-Volga extension produces much better premiums in absolute values;Due to the high premium prices for deep-in-the-money options, the ratios areapproximately equal.
6.3 Choice of Anker Points
The choice of anker points in the Vanna-Volga method was driven by the fact, that therisk reversal and the vega-weighted butterfly belong to the few liquid options in theFX market. Surely, this choice does not deliver good results in all cases. The empiricalresults have shown, that the range of strikes becomes more tight with declining maturity– thus we are not able to give the set of strikes, delivering the best result for all maturities.Nevertheless there are regions delivering good results for almost all maturities (compareFigure B.3). Evaluating the corresponding histogram (Table A.3) we are able to give a
40
recommendation for the choice of strikes in terms of moneyness:
K1 ≈ 0.875 K2 ≈ 0.975 K3 ≈ 1.025 K4 ≈ 1.125 . (6.2)
Thus a pretty good choice would be equidistant distributed diametral strikes (compareTable A.4).
41
7 Pricing Under StochasticVolatility
Though, the Vanna-Volga method is widely spread among traders, there is no mathemat-ical explanation for the option pricing formula (5.9) – only heuristic justification withItô exists. While searching for a theory which could explain the pricing formula, oneparticular model seemed to deliver interesting results: under mean-reverting volatilitythe appropriate option price could be represented by a Black-Scholes price adjusted bya sum of Gamma and Speed. Further investigation has shown, this model delivers thedesired explanation.
7.1 Dynamics
We assume an underlying depending on volatility, which itself is a function of a mean-reverting Ornstein-Uhlenbeck (OU) process. Under a high rate of mean-reversion volatil-ity is pulled back to its “natural” mean level on a shorter time scale than the remainingtime to expiration of a particular option. The results of the first two sections werederived by Fouque et al. [7].
At first consider the dynamics:
St − S0 =
∫ t
0
µSudu +
∫ t
0
f(Yu)SudWu (7.1)
Yt − Y0 =
∫ t
0
α(m− Yu)du +
∫ t
0
βdZu (7.2)
Zt := ρWt +√
1− ρ2Zt
where Wt and Zt are independent Brownian motions, α is the rate of mean reversion, m
long run mean of Yt, β is the volatility of volatility – VolVol, |ρ| < 1 is the correlationcoefficient between price and volatility shocks,∗ f(·) some positive function. At discrete∗The case ρ = 0 implies smile effect, as shown by Renault and Touzi in [15].
42
times the price of the underlying is observable, volatility σt := f(Yt) is not observeddirectly and is subject to a hidden Markov process. The solution to (7.2) is
Yt = m + exp(−αt)(Y0 −m) + β
∫ t
0
exp(−α(t− u))dZu
and given Y0, Yt is Gaussian
Yt − exp(−αt)Y0 ∼ N(
m(1− exp(−αt)),β2
2α(1− exp(−2αt))
).
The unique invariant distribution for Y is then N (m, β2
2α) (see [10]), providing a simple
building-block for stochastic volatility models with arbitrary f(·). The existence of aunique invariant distribution means, that Y is pulled towards its mean value m and thevolatility of (7.1) approximately towards f(m) as t →∞. In distribution it is the sameas if α, the rate of mean reversion, tends to infinity.
7.2 Pricing
The following risk-neutral pricing is also valid for non-Markovian models.By Girsanov’s Theorem we introduce independent Brownian motions under an equivalentmartingale measure Qλ
W ∗t :=Wt +
∫ t
0
µ− r
f(Yu)du,
Z∗t :=Zt +
∫ t
0
λudu
(7.3)
assuming ( µ−rf(Yt)
, λt) satisfy the Novikov’s condition. Since the market is incomplete(volatility is assumed to be a non-fungible asset; compare the Complete Market Theoremand the “Meta-theorem” in [1] ) we denote this inability to derive a unique equivalentmartingale measure by the dependance of Q on the market price of volatility risk λ;µ−rf(Yt)
is called excess return-to-risk ratio. The approach here is that the derivative shouldbe priced in order not to introduce any arbitrage into the market, – thus according to(3.6) and that the market, that is supply and demand, selects the unique equivalentmartingale measure represented by λ to price derivatives.
43
Under new measure with the Radon-Nikodym derivative given by
dQλ
dP= exp
(−1
2
∫ T
0
((µ− r)2
f(Yu)2+ (λu)
2
)du−
∫ T
0
µ− r
f(Yu)dWu −
∫ T
0
λudZu
)the equations (7.1) and (7.2) are written:
St − S0 =
∫ t
0
rSudu +
∫ t
0
f(Yu)SudW ∗u (7.4)
Yt − Y0 =
∫ t
0
[α(m− Yu)− β
(ρ(µ− r)
f(Yu)+ λu
√1− ρ2
)]du +
∫ t
0
βdZ∗u (7.5)
Z∗t := ρW ∗
t +√
1− ρ2Z∗t .
Any allowable choice of λ leads to an equivalent martingale measure and by the Feynman-Kač Stochastic Representation Formula the option premium V writes as:
Vt = EQλ
(exp(−rτ)h(ST )|Ft
), (7.6)
where h(x) denotes the derivative payoff function.If λ = λ(t, St, Yt) the setting is Markovian. In following we derive the partial differentialequation corresponding this case.
Since the market lacks enough underlyings to price options in terms of those, theprice of a particular derivative is not completely determined by the dynamics of itsunderlying and the requirement that the market is arbitrage-free. Thus a valuation with(7.6) requires a benchmark option G. Let G be an option with same parameters as V ,but with a different strike:
Vt = v(t, St, Yt), with VT = (ST −K)+ ,
Gt = g(t, St, Yt), with VT = (ST −K ′)+ , K 6= K ′ .
The price for V should satisfy market internal consistency relations, in order not tointroduce any arbitrage opportunities. Taking the price of the benchmark option as apriory given, the prices of other derivatives are then uniquely determined – in consistencywith the Meta-theorem.
A riskless portfolio consists now of two options and the underlying:
Π = V −∆1S −∆2G ,
44
which change over a time interval t is
∫ t
0
dΠu =
∫ t
0
dVu −∫ t
0
∆1
udSu −∫ t
0
∆2
udGu . (7.7)
Applying the Itô’s Lemma on v and g, substituting from (7.1) and recombining the terms(7.7) becomes
∫ t
0
dΠu =
∫ t
0
(∂tv +
1
2f(Yu)
2S2u ∂ssv +
1
2β2 ∂yyv + ρf(Yu)Suβ ∂syv
)du
−∫ t
0
∆2
u
(∂tg +
1
2f(Yu)
2S2u ∂ssg +
1
2β2 ∂yyg + ρf(Yu)Suβ ∂syg
)du
+
∫ t
0
(∂sv −∆2
u ∂sg −∆1
u) dSu +
∫ t
0
(∂yv −∆2
u ∂yg) dYu .
(7.8)
A choice
∆1 = ∂sv −∂sg ∂yv
∂yg,
∆2 =∂yv
∂yg
(7.9)
makes the portfolio risk-free, eliminating the integrands of dSt and dYt.At the same time the portfolio earns at a risk-free rate r in absence of arbitrage
opportunities: ∫ t
0
dΠu =
∫ t
0
rΠudu.
A substitution from (7.9) and the above consideration lead to:(∂tv +
1
2f(y)2s2 ∂ssv +
1
2β2 ∂yyv + ρf(y)sβ ∂syv − rv + rs ∂sv
)/(∂yv)
=
(∂tg +
1
2f(y)2s2 ∂ssg +
1
2β2 ∂yyg + ρf(y)sβ ∂syg − rg + rs ∂sg
)/(∂yg) ,
(7.10)
where v, g and their derivatives are evaluated at (t, St, Yt).Each side of (7.10) depends only on v or g respectively. Thus, both sides should be equalto some option-independent function (for we also could have taken an option g′, similar
45
to v, but with a different maturity, instead of a different strike)
−γ(y) := α(m− y)− β
(ρµ− r
f(y)+ λ(t, s, y)
√1− ρ2
), (7.11)
where λ(t, s, y) is an arbitrary function.The model parameters α, m, β, ρ, µ and λ are not constant in general. But identifying
intervals of underlying stationarity we are able to take the parameters as constant. Theprice of volatility risk is determined solemnly by the benchmark option G, that is by themarket itself.
The PDE corresponding to (7.6) is then written as
∂tv +1
2f(y)2s2 ∂ssv +
1
2β2 ∂yyv + ρf(y)sβ ∂syv − rv + rs ∂sv + γ(t, s, y) ∂yv = 0 (7.12)
subject to the terminal condition v(T, s, y) = h(s).†
The rate of mean-reversion α is crucial for validating the applicability of the asymp-totic analysis. Thus we are to prove the volatility of the DJ EURO STOXX 50 to befast mean-reverting.‡
A research by Dotsis et al. (see [5]§) explores several models describing the dynamicsof implied volatility of the main American and European volatility indexes – among themVSTOXX. The estimation period covers a time span from 4/01/1999 to 24/03/2004. Thelikelihood function is estimated by the maximum-likelihood method from the densityfunction of the process following (7.2).
An estimation of the volatility parameters states that volatility of EURO STOXX 50 iswell described by a mean-reverting Gaussian process as in (7.2). Although other models,especially those, based on a jump diffusion model, produced better explanations for thetime series, the estimates of the parameters α, m and β for a simple mean-revertingGaussian process are still significant. This allows us to apply the asymptotic analysisdescribed above.
Derived from the Black-Scholes price V BS described in (3.5), with the underlyingfollowing the dynamics as in (3.4) and the volatility σ given by constant volatility, thefollowing formula corrects the Black-Scholes price by a term containing the option Γ and
†For further details on the PDE see [19] and [17].‡Fast mean-reverting in terms of the lifetime of the option, slow mean-reverting compared to the
intraday data.§A compact version of the paper is to appear in the Journal of Banking and Finance, lacking three
models and some indexes, presented in the preprint.
46
Υ (compare Table 3.1)
V c(t, St, σ) = V BS(t, St, σ)− τ ·H(t, St, σ) , (7.13)
where
H(t, St, σ) = c2S2t
∂2V BS
∂s2(t, St, σ) + c3S
3∂3V BS
∂s3(t, St, σ) (7.14)
with c2 and c3 being constants related to the model parameters α, m, β, ρ and thefunctions f and λ. Containing information about the market, the coefficients c2 and c3
are not specific to any contract.The coefficients are given by
c2 :=σ
((σ − b
)− a(r +
3
2σ2))
,
c3 :=− aσ3 .
The estimates for a and b are derived from least-squares fitting to a linear function
I(t, St; K, T ) = a
(ln m
τ
)+ b .
where I(t, St; K, T ) are the implied volatilities of liquid near-the-money European calloptions of various strikes and maturities.The variable
(ln mτ
)is referred to as log-moneyness-to-maturity-ratio, which states that
volatility for longer maturities is linear function. This is often referred to as a skew andis observable in a market; Right before the expiration volatility is commonly U-shaped(compare Figure B.1).
7.3 Application on the Pricing Formula
Now we can derive the pricing formula from Section 5.2 under the assumption of themean-reverting stochastic volatility.
Proposition 3The choice of xi(t; K) as in (5.1) implies speed neutrality.
47
ProofBy the construction of the portfolio observe
Λ(t; K) =4∑
i=1
xi(t; K) · Λ(t; Ki)
∂Λ(t; K)
∂s=
4∑i=1
∂xi(t; K)
∂s· Λ(t; Ki) +
4∑i=1
xi(t; K) · ∂Λ(t; Ki)
∂s
and by vanna-neutrality
0 =4∑
i=1
∂xi(t; K)
∂s· Λ(t; Ki)
which applying (5.2) yields
0 = τσS2 ·
(4∑
i=1
∂xi(t; K)
∂s· Γ(t; Ki)
)
0 =4∑
i=1
∂xi(t; K)
∂s· Γ(t; Ki) . (7.15)
By vega-gamma neutrality for portfolios of European plain vanilla options stated in (5.2)it holds, that
Γ(t; K) =4∑
i=1
xi(t; K) · Γ(t; Ki)
which differentiated with respect to S becomes
Υ(t; K) =4∑
i=1
∂xi(t; K)
∂s· Γ(t; Ki) +
4∑i=1
xi(t; K) ·Υ(t; Ki) .
The assertion follows then with (7.15).
We form a hypothesis, that with λ being an independent parameter, the model de-scribed in Section 7.1 can be adjusted to match CMK(t; Kj) for all Kj ∈ Kt. In otherwords, the market selects a unique equivalent martingale measure to price the derivativesand provides us with appropriate prices observable in the market. The value of market’s
48
price of volatility risk can thus be seen only in derivatives prices. This viewpoint iscalled selecting an approximating complete market. The consistency of (5.9) with thestochastic mean-reverting volatility model follows then by:
C(t; K; σ) = CBS(t; K; σ) +4∑
i=1
xi(t; K) · [CMK(t; Ki; σi)− CBS(t; Ki; σ)]
= CBS(t; K; σ) +4∑
i=1
xi(t; K) · [CBS(t; Ki; σ)− τH(t; Ki; σ)− CBS(t; Ki; σ)]
= CBS(t; K; σ)− τ
4∑i=1
xi(t; K)H(t; Ki; σ)
= CBS(t; K; σ)− τ ·H(t; K; σ)
(7.16)
with the last step following from the choice of xi and Proposition 3.That is, we replace the adjustment option to the Black-Scholes price in (7.13) withweighted sensitivities of liquid options.
49
8 Evaluation
The Vanna-Volga method as well as its extension are applicable to index options toadjust for a skew as well as for a smile. Both adjustments represent real extra andinterpolation formulas, reproducing exactly the inputs. Although, neither the originalmethod, nor the extension guarantee for convex premiums.
The valuating procedure is for instance applicable on illiquid deep in-the-money op-tions, whose premiums need to be known for the evaluation of volatility indexes. For thesame reason, one could use it for longer dated maturities, especially in the wings, sincethose are not covered by the obligations for market makers and thus are poorly traded.This could enable an extension of volatility-subindexes out to 5 years.
The formula for option prices (5.9), as well as the approximation of the implied volatil-ity (5.13) both are easily implementable requiring no sophisticated algorithm and thusno special software for their derivation – a simple excel sheet would suffice.
50
A Tables
Expiration 200612 Expiration 200703Date Market OLS-Forward Market OLS-Forward
11/01/2006 4022 4022 4051 4051.2
11/02/2006 3983 3983 4012 4011.9
11/03/2006 3994 3994 4023 4023.0
11/06/2006 4054 4054 4083 4083.0
11/07/2006 4081 4081 4111 4110.2
11/08/2006 4080 4080 4110 4110.5
11/09/2006 4079 4079 4109 4108.6
11/10/2006 4071 4071 4100 4100.5
11/13/2006 4095 4095 4125 4124.5
11/14/2006 4088 4088 4116 4116.8
11/15/2006 4112 4112 4141 4140.7
11/16/2006 4116 4116 4145 4144.7
11/17/2006 4088 4088 4116 4116.3
11/20/2006 4104 4104 4132 4132.0
11/21/2006 4107 4107 4135 4134.5
11/22/2006 4104 4104 4133 4133.6
11/23/2006 4093 4093 4122 4122.6
11/24/2006 4056 4056 4085 4084.6
11/27/2006 3989 3989 4018 4017.4
11/28/2006 3980 3980 4008 4008.5
11/29/2006 4027 4027 4056 4055.6
11/30/2006 3994 3994 4023 4022.5
Table A.1: Comparison of market futures prices with obtained forward prices.
51
Strike x1 x2 x3 x4
3250 1.03244127 −0.06798949 0.06747687 −0.12158149
3300 1.03213298 −0.06728704 0.06669937 −0.12006348
3350 1.03106641 −0.06490269 0.06412412 −0.11513322
3400 1.02785631 −0.05788665 0.05676569 −0.10135903
3450 1.01941976 −0.039943 0.03861244 −0.06825608
3500 1 2 .43E − 17 −4 .33E − 18 1 .46E − 17
3550 0.96075955 0.07737073 −0.07045937 0.11991063
3600 0.89106768 0.20742045 −0.17948611 0.296661
3650 0.78228683 0.39575872 −0.31885102 0.5078738
3700 0.63336433 0.62710768 −0.45650514 0.69437209
3750 0.45546439 0.85970073 −0.53792578 0.77247959
3800 0.27197992 1.03291294 −0.50370776 0.67259018
3850 0.1120851 1.08906021 −0.319133 0.38777758
3900 1 .03E − 16 1 0 −9 .17E − 17
3950 −0.05450027 0.78325375 0.38330196 −0.34199154
4000 −0.05899325 0.49692847 0.73004275 −0.48872944
4050 −0.03333633 0.21548568 0.95022704 −0.36362382
4100 6 .92E − 17 −6 .92E − 17 1 2 .31E − 17
4150 0.02465001 −0.12143942 0.89321655 0.47874992
4200 0.03417112 −0.1555234 0.68654465 0.92008895
4250 0.03057972 −0.13109841 0.45022459 1.20690971
4300 0.02002475 −0.08193401 0.24123217 1.29346233
4350 0.0085912 −0.03386264 0.08974422 1.2030917
4400 −8 .66E − 18 1 .73E − 17 0 1
4450 −0.00466712 0.01739399 −0.04025493 0.75558854
4500 −0.00607199 0.02214908 −0.04893456 0.52487838
4600 −0.00416339 0.01467601 −0.03026727 0.20291469
4700 −0.00167121 0.00573973 −0.01127501 0.06047165
4800 −0.00047885 0.00161136 −0.00305251 0.01432504
Table A.2: Typical set of coefficients. The anker points are set in italic.
52
0.29
04≤
τ≤
0.36
990.
2904≤
τ≤
0.36
99C
umul
ativ
em
K1
K2
K3
K4
mK
1K
2K
3K
4m
K1
K2
K3
K4
0.80
510
310
00
0.80
529
190
00
0.80
539
500
00
0.81
416
450
00
0.81
544
6760
00
0.81
561
1260
00
0.82
310
200
00
0.82
538
9325
70
00.
825
5573
257
00
0.83
285
60
00
0.83
519
6328
13
00.
835
3055
281
30
0.84
115
690
00
0.84
517
5541
00
00.
845
3181
410
00
0.85
019
180
00
0.85
515
2674
80
00.
855
2691
748
00
0.85
982
10
00
0.86
530
4917
320
00.
865
5201
1732
00
0.86
813
310
00
0.87
524
4921
7791
00.
875
4850
2177
910
0.87
724
010
00
0.88
515
7810
1651
00.
885
3272
1016
510
0.88
726
630
00
0.89
586
669
528
00.
895
2630
700
280
0.89
679
55
00
0.90
564
517
61
00.
905
2634
176
10
0.90
519
890
00
0.91
575
721
1719
41
0.91
532
7021
7819
61
0.91
425
1361
20
0.92
510
811
5016
04
0.92
522
0518
2516
04
0.92
320
9767
50
00.
935
6223
2248
45
0.93
520
5547
8948
45
0.93
217
338
40
00.
945
4811
8234
24
0.94
521
322
6336
24
0.94
118
2020
830
00.
955
163
2710
263
20.
955
1006
5356
279
20.
950
1008
3727
360
0.96
550
3643
896
240.
965
527
7663
1111
240.
959
477
4020
215
00.
975
7117
4722
12
0.97
526
675
0011
362
0.96
90
00
00.
985
2619
7010
5827
0.98
597
7110
2820
370.
978
195
5753
915
00.
995
211
837
46
0.99
55
2068
2734
110.
987
7151
4017
6210
1.00
50
00
01.
005
00
00
0.99
63
1950
2360
51.
015
144
631
2641
1.01
51
1800
8354
145
1.00
50
00
01.
025
024
231
3130
1.02
54
1170
8446
549
Con
tinu
edne
xtpa
ge
53
0.29
04≤
τ≤
0.36
990.
2904≤
τ≤
0.36
99C
umul
ativ
em
K1
K2
K3
K4
mK
1K
2K
3K
4m
K1
K2
K3
K4
1.02
34
928
5315
519
1.04
50
355
3033
231.
045
039
356
6431
521.
032
036
1642
1146
1.05
50
093
920
1.05
50
010
4493
1.04
20
209
1924
304
1.06
50
194
2636
123
1.06
50
209
4802
4010
1.05
10
3827
3632
021.
075
073
1092
405
1.07
50
9522
4646
371.
060
015
2166
3887
1.08
50
1114
9914
401.
085
011
2152
6058
1.06
90
043
1861
1.09
50
1892
932
341.
095
018
1103
6825
1.07
80
2211
1123
711.
105
00
239
3264
1.10
50
027
439
741.
087
00
653
4618
1.11
50
040
045
161.
115
00
444
6188
1.09
60
017
435
911.
125
00
520
5674
1.12
50
052
066
931.
105
00
3571
01.
135
00
114
1174
1.13
50
015
318
561.
114
00
4416
721.
145
00
00
1.14
50
00
501
1.12
40
00
1019
1.15
50
00
881.
155
00
088
1.13
30
00
537
1.16
50
00
01.
165
00
019
81.
142
00
3914
51.
175
00
018
181.
175
00
018
181.
151
00
050
11.
185
00
035
821.
185
00
035
821.
160
00
019
81.
195
00
083
51.
195
00
083
5
Tab
leA
.3:
Dis
trib
utio
nhi
stog
ram
for
stri
kes
inte
rms
ofm
oney
ness
for
both
sets
ofda
ta(fi
rst
two
bloc
ks)
and
the
cum
ulat
ion
(las
tbl
ock)
.E
ach
bloc
kgi
ves
the
mon
eyne
ssan
dth
enu
mbe
rof
occu
rren
ces
ofth
eco
rres
pond
ing
stri
ke.
Mon
eyne
sssp
ecifi
esth
ece
nter
ofan
inte
rval
.
54
OE
SX-1
206
OE
SX-0
307
Dat
eB
est
estim
ates
Rec
omm
enda
tion
Bes
tes
tim
ates
Rec
omm
enda
tion
Vol
atili
tySe
ttle
men
tV
olat
ility
Sett
lem
ent
Vol
atili
tySe
ttle
men
tVol
atili
tySe
ttle
men
t11
/01/
2006
0.00
1233
2015
.60.
0012
4504
15.2
0.00
1251
1122
.40.
0018
8050
40.4
11/0
2/20
060.
0011
8573
13.7
0.00
1220
1113
.90.
0014
6464
31.0
0.00
1975
3838
.3
11/0
3/20
060.
0008
5674
20.8
0.00
0936
8415
.60.
0014
4936
36.3
0.00
1627
4436
.6
11/0
6/20
060.
0006
8015
11.3
0.00
0995
2612
.70.
0011
6553
54.1
0.00
1432
7140
.7
11/0
7/20
060.
0008
3615
13.5
0.00
1360
6410
.80.
0011
6432
41.7
0.00
1683
8938
.7
11/0
8/20
060.
0008
0996
13.6
0.00
1473
2011
.00.
0013
3955
42.0
0.00
1838
1246
.8
11/0
9/20
060.
0008
8162
9.8
0.00
1489
0310
.90.
0013
1471
27.8
0.00
1855
2740
.1
11/1
0/20
060.
0009
0477
17.0
0.00
1198
3812
.10.
0012
2863
24.8
0.00
1736
9436
.4
11/1
3/20
060.
0006
7526
17.0
0.00
0959
4312
.80.
0009
7429
42.8
0.00
1022
0927
.5
11/1
4/20
060.
0008
7818
12.2
0.00
1290
8511
.90.
0009
6220
21.9
0.00
1156
9923
.4
11/1
5/20
060.
0008
7285
14.2
0.00
1907
1314
.00.
0008
2121
18.3
0.00
1281
3732
.6
11/1
6/20
060.
0006
3299
12.3
0.00
1594
5015
.50.
0006
3809
18.5
0.00
1204
7327
.2
11/1
7/20
060.
0006
0913
18.7
0.00
5997
6569
.90.
0009
9930
22.5
0.00
1245
8124
.2
11/2
0/20
060.
0004
8440
14.5
0.00
3691
5745
.90.
0007
5904
45.0
0.00
1070
5425
.5
11/2
1/20
060.
0010
5295
17.1
0.00
6669
8474
.40.
0009
8561
19.8
0.00
1472
0731
.0
11/2
2/20
060.
0004
6173
21.0
0.00
6375
0210
2.5
0.00
0888
7619
.20.
0013
9242
30.8
11/2
3/20
060.
0007
2028
22.4
0.00
9457
0423
4.1
0.00
0809
1326
.40.
0010
7509
25.7
11/2
4/20
060.
0007
7051
17.0
0.00
5514
9640
.30.
0012
1868
28.5
0.00
1474
9536
.0
11/2
7/20
060.
0008
8939
59.8
0.00
5120
4427
.80.
0014
1102
28.0
0.00
1488
7928
.4
11/2
8/20
060.
0019
4807
15.9
0.00
3956
9911
.60.
0014
3435
24.7
0.00
1732
8326
.6
11/2
9/20
060.
0017
0057
7.0
0.00
7871
7533
.10.
0014
9990
32.2
0.00
1695
9430
.4
11/3
0/20
060.
0023
9487
12.6
0.01
3669
1594
.10.
0012
8437
25.3
0.00
1688
9440
.1
Tab
leA
.4:
Dev
iati
ons
ofth
ebe
stes
tim
ates
and
the
reco
mm
ende
dse
tof
stri
kes
(as
in6.
2)fr
omth
em
arke
tda
taon
daily
basi
sfo
rbo
thex
piri
es.
Vol
atili
tyde
viat
ions
calc
ulat
edby
vega
-wei
ghti
ng.
55
B Figures
32003400
36003800
40004200
44004600
48005000
1038
73129
220318
409591
773955
11371501
18652236
26002964
3328
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
11/07/2006
Strikesτ in days
Vo
lati
lity
Figure B.1: Estimated volatility term structure.
56
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15−50
−9
4
50
100
150
200
Figure B.2: Bounds for implied volatility slope OESX-0307, τ = 0.3699.
57
0.8
0.8
20
.84
0.8
60
.88
0.9
0.9
20
.94
0.9
60
.98
11
.02
1.0
41
.06
1.0
81
.11
.12
1.1
41
.16
1.1
81
.20
20
00
40
00
60
00
K1
K2
K3
K4
0.8
0.8
20
.84
0.8
60
.88
0.9
0.9
20
.94
0.9
60
.98
11
.02
1.0
41
.06
1.0
81
.11
.12
1.1
41
.16
1.1
81
.20
20
00
40
00
60
00
0.8
0.8
20
.84
0.8
60
.88
0.9
0.9
20
.94
0.9
60
.98
11
.02
1.0
41
.06
1.0
81
.11
.12
1.1
41
.16
1.1
81
.20
50
00
10
00
0
Fig
ure
B.3
:C
umul
ativ
est
rike
sdi
stri
buti
onof
the
first
1200
sets
per
mat
urity.
Top
–0.
1205≤
τ≤
0.04
11,M
iddl
e–
0.29
04≤
τ≤
0.36
99,B
otto
m–
cum
ulat
ive
for
both
expi
ries
.
58
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.250.1
0.15
0.2
0.25
0.3
0.35
0.4
Moneyness
Impl
ied
Vola
tility
Vanna−Volga∆−neutralMarket
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.250
200
400
600
800
Moneyness
Opt
ion
Prem
ium
Vanna−Volga∆−neutralMarket
Figure B.4: Best volatility and premium estimates OESX-1206, τ = 0.0411.The upper graph shows the volatility approximations for Vanna-Volga (the light blue line)and its extension (the blue line) compared to the market implied volatility (red line), wherethe colored markers give the positions of the anker points.The lower, the corresponding options’ premiums.
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25−0.4
−0.2
0
0.2
0.4
in P
erce
nt P
oint
s
Implied Volatility Residuals
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25−2
−1
0
1
2
Inde
x Po
ints
Option Premium Residuals
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25−50
0
50
100
Perc
ent
Moneyness
Relative Option Premium Residuals
Vanna−Volga σimp
−σest
∆−neutral σimp
−σest
Vanna−Volga CMK−Cest
∆−neutral CMK−Cest
Vanna−Volga (CMK−Cest
)/(CMK)
∆−neutral (CMK−Cest
)/(CMK)
Figure B.5: Volatility and premium residuals OESX-1206, τ = 0.0411.
59
C Matlab
A large part of the work was the implementation of the theoretical results in Matlabcode. Here we give a summary of the Matlab procedure for a fixed t.Matlab allows scalars as well as vectors as input. Scalars DATE, TAU, FUTURE andUNDERLYING (underlying close price) and vectors STRIKES, CALLS were easily re-ceived from the data delivered in matrix form – division by variables in vector form,taking to a power and multiplication have a special implementation:
The interest rate was derived in a variable RATE by the OLS-approximation accordingto (5.5)
RATE =@(X) sum((CALLS-PUTS+exp(-X*TAU).*(STRIKES-FUTURE)).^2) .
Next step was the calculation of Black-Scholes vectors ∆, Λ, Φ, Ξ with a constantreference volatility σ. Financial toolbox relieve the handling with the Greeks – the mostprominent of them are available:
blsprice, blsdelta, blsgamma, blsvega
giving the corresponding sensitivities.To accomplish the calculation of volatility we had to check all
(|Kt|4
)possibilities. A
command
randerr(m,n,errors)
generates an m-by-n binary matrix, where errors determines how many nonzero entriesare in each row. This gave us a matrix which applied to a vector, e.g. STRIKES, CALLS,∆, Λ, Φ, Ξ, chose four variables as anker points. For all
(|Kt|4
)combinations we calculated
the coefficients xi(t; K), i = 1 . . . 4 subject to (5.8).The calculation of the premium estimates according to (5.9) followed, which delivered
with (5.13) the volatility estimates. As already mentioned the radicand in (5.13) is notalways positive. Thus we had to filter the volatility estimates with an imaginary part,what reduced the number of daily combinations by more that a half. Vega-weighting asin (6.1) ordered the results.
60
Bibliography
[1] Tomas Bjørk. Arbitrage Theory in Continuous Time. Oxford University Press,1998.
[2] Fischer Black and Myron Scholes. The Pricing of Options and Corporate Liabilities.Journal of Political Economy, pages 637–654, 1973.
[3] Antonio Castagna and Fabio Mercurio. Consistent Pricing of FX Options. Risk,pages 106–111, January 2007.
[4] Freddy Delbaen and Walter Schachermayer. Non-arbitrage and theFundamental Theorem of Asset Pricing: Summary of Main Results.http://citeseer.ist.psu.edu/282878.html.
[5] George Dotsis, Dimitris Psychoyios, and George Skiadopoulos. ImpliedVolatility Processes: Evidence from the Volatility Derivatives Market.http://www2.warwick.ac.uk/fac/soc/wbs/research/wfri/rsrchcentres/forc/preprintseries/pp_06-151.pdf.
[6] Richard Durrett. Brownian Motion and Martingales in Analysis. Wadsworth Ad-vanced Books & Software, 1984.
[7] Jean-Pierre Fouque, George Papanicolaou, and K. Ronnie Sircar. Derivatives inFinancial Markets with Stochastic Volatility. Cambridge University Press, 2001.
[8] J.M. Harrison and S.R. Pliska. Martingales and stochastic integrals in the theory ofcontinuous trading. Stochastic Processes and their Applications (11), pages 215–260,1981.
[9] John C. Hull. Options, Futures and Other Derivatives. Prentice Hall, 2002.
[10] Samuel Karlin and Howard M. Taylor. A Second Course in Stochastic Processes,page 221. Academic Press, 1981.
61
[11] Lyndon Lyons. Volatility and its Measurements: The Design of a VolatilityIndex and the Evaluation of its Historical Time Series at the Deutsche BörseAG. http://www.eurexchange.com/download/documents/publications/ Volatil-ity_and_its_Measurements.pdf.
[12] Fabio Mercurio. A Vega-Gamma Relationship for European-Style or barrier options in the black-scholes model.http://www.fabiomercurio.it/VegaGammaRelationship.pdf.
[13] Sheldon Natenberg. Option Volatility & Pricing. McGraw-Hill, 1994.
[14] Oliver Reiss and Uwe Wystup. Efficient Computation of Option Price SensitivitiesUsing Homogenity and other Tricks. Journal of Derivatives, (Vol. 9, Num. 2), 2001.
[15] Eric Renault and Nizar Touzi. Option Hedging and Implied Volatilities in a Stochas-tic Volatility Model. Mathematical Finance 6 (3), pages 279–302, 1996.
[16] L.C.G Rogers and D. Williams. Diffusions, Markov Processes and Martingales,volume II. 2000.
[17] Rainer Schöbel and Jianwei Zhu. Stochastic Volatility With an Ornstein-UhlenbeckProcess: An Extension. European Finance Review 3(1), pages 23–46, 1999.
[18] Albert N. Shiryaev. Essentials of Stochastic Finance. World Scientific, 2001.
[19] Elias M. Stein and Jeremy C. Stein. Stock Price Distributions with StochasticVolatility: An Analytic Approach. The Review of Financial Studies 4(4), pages727–752, 1992.
[20] Bernt K. Øksendal. Stochastic Differential Equations: an Introduction with Appli-cations. Springer-Verlag, 2000.
[21] Uwe Wystup. FX Options and Structured Products. Wiley, 2006.
62