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

i

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

ii

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

iii

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

iv

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.

vi

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.

1

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

2

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].

3

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.

4

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.

5

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.

6

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.

7

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.

8

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.

9

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.

10

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.

11

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.

12

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)

13

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.

14

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

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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

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5033

002.

61.

90.

0060

8−

0.00

241

430.

878

0.8

0.10

140.

0010

00.

0008

122

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5033

002.

72.

00.

0061

7−

0.00

259

429.

677

9.9

0.09

860.

0010

50.

0008

822

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836

5036

502.

71.

00.

0061

90.

0021

642

8.8

430.

6

0.09

590.

0010

70.

0009

025

.117

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5032

502.

82.

20.

0065

5−

0.00

272

420.

582

1.0

0.08

770.

0009

30.

0006

823

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0033

002.

92.

50.

0073

7−

0.00

315

394.

979

5.6

0.08

490.

0011

30.

0008

821

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0037

002.

51.

70.

0065

40.

0044

338

8.1

389.

0

0.08

220.

0010

50.

0008

719

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0033

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21.

70.

0052

6−

0.00

217

411.

581

2.0

0.07

950.

0007

20.

0006

318

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5033

002.

11.

50.

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5−

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366.

181

5.7

0.07

670.

0009

10.

0006

121

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002.

72.

30.

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0.00

298

338.

378

8.7

0.06

850.

0006

90.

0004

818

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0033

002.

41.

80.

0077

1−

0.00

228

304.

880

4.2

0.06

580.

0012

00.

0010

517

.217

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0033

502.

32.

00.

0074

6−

0.00

269

307.

275

7.5

0.06

300.

0008

40.

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618

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0033

002.

52.

40.

0080

3−

0.00

295

304.

280

4.9

0.06

030.

0011

10.

0007

217

.122

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0033

002.

52.

70.

0085

8−

0.00

338

293.

479

4.2

0.05

750.

0009

60.

0007

719

.817

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5032

502.

41.

80.

0078

8−

0.00

226

306.

980

6.5

0.04

930.

0014

20.

0008

922

.459

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5036

002.

75.

80.

0079

90.

0149

033

9.9

385.

6

0.04

660.

0021

20.

0019

512

.415

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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

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0032

003.

64.

20.

0043

60.

0051

881

6.3

815.

7

0.35

620.

0012

40.

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736

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5032

504.

65.

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382

4.7

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340.

0012

30.

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185

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8

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504.

35.

00.

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90.

0058

885

1.9

851.

2

0.34

790.

0013

50.

0013

134

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5037

004.

22.

30.

0048

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0.00

517

850.

043

8.1

0.34

520.

0012

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334

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71.

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0044

10.

0021

484

2.5

844.

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700.

0010

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0009

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0033

003.

46.

40.

0041

90.

0077

981

7.9

814.

9

0.33

420.

0010

10.

0009

624

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0033

003.

72.

40.

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00.

0029

880

9.9

811.

2

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150.

0009

70.

0008

236

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004.

11.

60.

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0.00

338

784.

446

4.7

0.32

880.

0007

80.

0006

425

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003.

82.

40.

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3−

0.00

513

836.

546

8.8

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600.

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0033

004.

02.

50.

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50.

0030

880

9.3

810.

8

0.31

780.

0009

80.

0007

632

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004.

26.

80.

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20.

0081

877

5.8

821.

3

0.31

510.

0011

50.

0009

933

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0037

504.

12.

00.

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0.00

487

777.

941

2.1

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230.

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0008

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00.

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0−

0.00

479

777.

241

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0.30

960.

0009

90.

0008

132

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0040

504.

43.

50.

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50.

0204

176

6.5

165.

6

0.30

680.

0013

50.

0012

236

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5036

504.

72.

90.

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0015

70.

0014

137

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0039

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10.

0113

481

2.6

174.

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0015

80.

0014

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0039

504.

52.

10.

0055

90.

0128

080

3.3

163.

6

0.29

320.

0015

30.

0015

036

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5032

504.

12.

90.

0051

10.

0036

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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

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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.

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62