Volatility Smile/Smirk Properties of [GLP & MEMM] Models
Yoshio Miyahara∗ and Naruhiko Moriwaki
Graduate School of Economics
Nagoya City University

Abstract
The [GLP(Geometric Lévy Process) & MEMM(Minimal Entropy Martingale Measure)]pricing model was first introduced in [13] as one of the pricing models for the incomplete
market. We first explain the structure of this model, and next we investigate the volatility
smile/smirk properties of this model by the use of computer simulation method.

1Introduction

It is well-known that the implied volatility surface has smile or smirk properties in the realmarkets, and this property is called volatility smile/smirk property (or smile/skew property).
This fact suggests us that the construction of a new option pricing model, which possesses thissmile/smirk property, is inevitable.
Several kinds of models have been proposed and investigated. The [GLP & MEMM] pricing
model is one of them, and it is known that this model have many good properties as an option
pricing model for the incomplete market. (See [13] and [15]). In this paper we investigate thevolatility smile/smirk properties of the [GLP & MEMM] pricing models by the use of computer
simulation method.
In §2 we survey the volatility smile/smirk problems. In §3 we explain the [GLP & MEMM]
model briefly and in §4 we give several examples of it. In §5 we see that the [GLP & MEMM]
model possesses the volatility smile/smirk properties in various forms. Finally in §6 we discuss
the calibration results of of Nikkei 225 options by the [GLP & MEMM] models.
The results obtained in this paper show us that the [GLP & MEMM] model is a very strong
candidate for the new model which should have the volatility smile/smirk property and that
this model can be applied to the calibration problems.

∗Corresponding author. E-mail: y-miya@econ.nagoya-cu.ac.jp12

Volatility Smile/Smirk Problems

2.1

Historical Volatility and Implied Volatility for Black-Scholes Model

The Black-Scholes model is the special case of GLP model with no jump part, namely the process
is given by
1

2

St = S0 e(µ− 2 σ )t+σWt ,(2.1)

or in the form of stochastic differential equationdSt = St (µdt + σdWt ),(2.2)

where µ is called the drift parameter and σ is called the volatility parameter.
The theoretical B-S price of the European call option CK with the strike price K and thefixed maturity T is given by the following formula
CK = S0 Φ(d1 ) − e−rT KΦ(d2 ),(2.3)

where Φ(d) is the normal distribution function and22√
log SK0 + (r − σ2 )Tlog SK0 + (r + σ2 )T√√, d2 == d1 − σ T(2.4)
σ Tσ TUnder the above setting, the historical volatility of the process is the estimated value of σd1 =based on the sequential data of the price process St . We denote it by σb. On the other hand theimplied volatility is defined as what follows. Suppose that the market price of the European call(m)option with the strike K, say CK , were given. Then the value of σ which satisfies the following
equation(m)

S0 Φ(d1 ) − e−rT KΦ(d2 ) = CK ,(2.5)(im)

is the implied volatility, and this value is denoted by σK . We remark here that the implied(im)volatility σKdepends on the strike value K, and on the contrary the historical volatility σbdoes not depend on K.

2.2Volatility Smile/Smirk Properties of Market Option Prices

We first consider the case where the market value of options obey to the Black-Scholes model, and(m)so the market price CK is equal to the theoretical B-S price CK of (2.3). In this case the solution(im)of the equation (2.5) is equal to the original σ and it holds true that σK= σ = constant. Thismeans that if the market obeys exactly to the Black-Scholes model, then the implied volatility(im)σK

should be equal to the historical volatility σb (= σ).

But in the real world this is not true. It is well-known that the implied volatility is not equal(im)to the historical volatility, and the implied volatility curve σK

is sometimes a convex functionof K, and sometimes the combination of convex part and concave part. These properties are
so-called volatility smile or smirk properties.
The Figure 1, 2, 3, and Table 1, 2 illustrate this situation, where a new variable moneyness(= SK0 ) is introduced.
2

Figure 1: Implied volatility surface for Nikkei 225 index options. (July-2002, time to maturity
is 20-day or more)

Figure 2: 07/02/2001, time to maturity = 30-

Figure 3: 18/01/2000, time to maturity = 52-

day, volatility smileday, volatility smirk3

Table 1: Historical Volatility vs Implied Volatility
To consider the volatility smile/smirk properties of market option prices, we use the data ofNikkei 225 call options traded at OSE (Osaka Stock Exchange), whose time to maturity is lessthan or equal to 40-days and whose trading volume is more than 50. In this case, we haveobtained such a result that the Nikkei 225 call option has the volatility smile property for eachyear.
err ratio: mean during 1-year of (implied volatility - historical volatility) / historical volatility.
The historical volatility is estimated from the log returns of recent 245 days.
1998

1999

2000

2001

2002

2003

2004

moneyness ≤ 0.9

1.27951.6432

1.77482.05392.25752.84763.0841

0.9 < moneyness ≤ 0.95

0.46600.43580.6954

0.5861

0.36230.33230.39220.95 < moneyness ≤ 1.05

0.0587-0.0576

0.14670.16880.0711

0.0668-0.0356

1.05 < moneyness ≤ 1.10

0.0494-0.0746

0.13440.16700.11250.13480.0162

1.10 < moneyness

0.1148-0.02500.2756

0.34770.26390.29490.1201

Table 2: Historical Volatility vs Implied Volatility
To consider the volatility smile/smirk properties of market option prices, we use the data ofNikkei 225 call options traded at OSE (Osaka Stock Exchange), whose time to maturity is morethan 40-days and whose trading volume is more than 50. In this case, we have obtained such aresult that the Nikkei 225 call option has the volatility smirk property for each year.
err ratio: mean during 1-year of (implied volatility - historical volatility) / historical volatility.
The historical volatility is estimated from the log returns of recent 245 days.
1998

1999

2000

2001

2002

2003

2004

moneyness ≤ 0.9

0.09390.20040.33140.34680.49500.41690.4272

0.9 < moneyness ≤ 0.95

-0.0607-0.03250.12020.0503-0.0390-0.0324-0.08210.95 < moneyness ≤ 1.05

-0.0450-0.1071

0.05420.0105-0.0459-0.0562-0.1037

1.05 < moneyness ≤ 1.10

-0.0684-0.1347

0.0253-0.0205-0.0649-0.0545-0.11031.10 < moneyness-0.0216-0.14630.0865-0.0208-0.0461-0.0209-0.10424

2.3

Volatility Smile/Smirk Properties of Models

Based on the above facts, new models, which have the volatility smile/smirk property, arerequired. A pricing model is said to have the volatility smile property or volatility smirk property
if the implied volatility function for the theoretical option prices of that model has the volatility(im)∗∗ be the theoretical option price of the new model and let σsmile or smirk property. Let CKK

be the solution of∗S0 Φ(d1 ) − e−rT KΦ(d2 ) = CK.
(im)∗Then σK(2.6)(im)∗is the implied volatility related to the new model. If σK

has the volatility smile(im)∗property, then the model is said to have the volatility smile property, and if σK

has thevolatility smirk property, then the model is said to have the volatility smirk property.

3[GLP & MEMM] Pricing Model

The [GLP & MEMM] Pricing Model is such a model:(A) The price process St is a geometric Lévy process (GLP).
(B) The price of an option X is defined by e−rT EP ∗ [X], where P ∗ is the minimal entropy
martingale measure (MEMM).
This model was first introduced in [13], and the properties of this model are summarised in[15].

3.1

Geometric Lévy Process (GLP)The price process St of a stock is assumed to be defined as what follows. We suppose that aprobability space (Ω, F, P ) and a filtration {Ft , 0 ≥ t ≥ T } are given, and that the price process
St = S0 eZt of a stock is defined on this probability space and given in the form
St = S0 eZt ,0 ≤ t ≤ T,(3.1)

where Zt is a Lévy process. We call such a process St the geometric Lévy precess (GLP), andwe denote the generating triplet of Zt by (σZ2 , νZ (dx), bZ ) or simply by (σ 2 , ν(dx), b).

3.2

Minimal Entropy Martingale Measure (MEMM)

We first give the definition of the MEMM.
Definition 1 (minimal entropy martingale measure (MEMM)) If an equivalent martingale measure P ∗ satisfies
H(P ∗ |P ) ≤ H(Q|P )∀Q :equivalent

5martingalemeasure,(3.2)

then P ∗ is called the minimal entropy martingale measure (MEMM) of St . Where H(Q|P ) isthe relative entropy of Q with respect to P( R)dQ
log[]dQ,
ifQ
¿
P,dP
Ω
H(Q|P ) =.
∞,otherwise,

3.3

(3.3)

Sufficient Conditions for the Existence of the MEMM

The existence problem of the MEMM of geometric Lévy processes has been studies in [12], [3]and [13], and finally those results are generalized in [7] as the following form.
Theorem 1 (Fujiwara-Miyahara [7, Theorem 3.1]) Suppose that the following condition(C) holds
Condition (C)There exists θ∗ ∈ R which satisfies the following conditions :(C)1Rx θ∗ (ex −1) ν(dx) < ∞,{x>1} e eR

∗

(3.4)

x(C)2 b + ( 21 + θ∗ )σ 2 + {|x|>1} (ex − 1)eθ (e −1) ν(dx)¡¢R∗ x+ {|x|≤1} (ex − 1)eθ (e −1) − x ν(dx) = r.

(3.5)

Then the probability measure P ∗ is well defined and it holds that(i)(MEMM): P ∗ is the MEMM of St .
(ii)(Lévy process): Zt is also a Lévy process w.r.t. P ∗ , and the generating triplet (A∗ , ν ∗ , b∗ ) ofZt under P ∗ isA∗ = σ 2 ,∗(3.6)
xν ∗ (dx) = eθ (e −1) ν(dx),Z∗∗ 2b = b+θ σ +R\{0}

3.4

(3.7)
xI{|x|≤1} d(ν ∗ − ν).

(3.8)

Prices of European Call Options

We investigate the European call option. The price of European call option isC(S0 , K, T ) = e−rT EP ∗ [(ST − K)+ ].

(3.9)

It is known that this value is computed as follows.
The characteristic function φ∗t (u) of Zt under the MEMM P ∗ isφ∗t (u) = φ∗Zt (u) = EP ∗ [eiuZt ] = exp(ψt∗ (u)) = exp(tψ ∗ (u)),

i=√−1,(3.10)

where ψ ∗ (u) = ψ1∗ (u).
Setζ(v; S0 , T ) = S0e−rT φ∗T (v − i) − eivrTiv(1 + iv)
6(3.11)and1c̃(k; S0 , T ) =2π

Z ∞−∞e−ikv ζ(v; S0 , T )dv(3.12)

Then the price of the European call option C(S0 , K, T ) is computed asC(S0 , K, T ) = c̃(log(K/S0 ); S0 , T ) + (S0 − e−rT K)+ .

(3.13)(See [15].) This method is used in §5 for the computation of the theoretical option prices.

4Examples of [GLP & MEMM] Pricing ModelIn this section we see several examples of [GLP & MEMM] Pricing Models. To do this, we haveto check the existence of the MEMM, i.e. we have to examine that the given geometric Lévy
process St = S0 exp Zt satisfies the Condition (C). Set
Z1x2f (θ) = b + ( + θ)σ +(ex − 1)eθ(e −1) ν(dx)2{|x|>1}
Z³´x+(ex − 1)eθ(e −1) − x ν(dx).

(4.1){|x|≤1}

Then the condition (C)2 is equivalent to that θ∗ is the solution off (θ) = r.

4.1

(4.2)

[Geometric Stable Process & MEMM] Model

We consider the stable model. Suppose that Zt is a stable process and let (0, ν(dx), b) be its
generating triplet. The Lévy measure isν(dx) = c1 I{x<0} |x|−(α+1) dx + c2 I{x>0} |x|−(α+1) dx,(4.3)

where 0 < α < 2 and we assume thatc1 > 0,c2 > 0.

(4.4)

It is shown that the equation (4.2) has a unique solution θ∗ and that θ∗ is negative. (See [7]or [16].) So the geometric stable process model is an example of the [GLP & MEMM] Pricing
Model.

4.2[Geometric CGMY Process & MEMM] Model

The Lévy measure of the CGMY process is¡¢
ν(dx) = C I{x<0} exp(−G|x|) + I{x>0} exp(−M |x|) |x|−(1+Y ) dx,(4.5)

where C > 0, G ≥ 0, M ≥ 0, Y < 2 (see [1]). If Y ≤ 0, then G > 0 and M > 0 are assumed.
We mention here that the case Y = 0 is the VG process case, and the case G = M = 0 and
0 < Y < 2 is the symmetric stable process case. In the sequel we assume that G, M > 0.
For this model the following results are obtained (see [16]).
7

Proposition 1 (1) If M ≤ 1, then the equation f (θ) = r has a unique solution θ∗ , and thesolution is negative.
(2) If M > 1 and f (0) ≥ r, then the equation f (θ) = r has a unique solution θ∗ , and the solution
is non-positive.
(3) If M > 1 and f (0) < r, then the equation f (θ) = r has no solution.

4.3[Geometric Variance Gamma Process & MEMM] ModelThe Lévy measure of Variance Gamma process is of the following form (see [8]).
¡¢ν(dx) = C I{x<0} exp(−c1 |x|) + I{x>0} exp(−c2 |x|) |x|−1 dx,(4.6)

where C, c1 , c2 are positive constants.
The following results are obtained (see[7] or [16]).
Proposition 2 (1) If c2 ≤ 1, then the equation f (θ) = r has a unique solution θ∗ , and thesolution is negative.
(2) If c2 > 1 and f (0) ≥ r, then the equation f (θ) = r has a unique solution θ∗ , and the solution
is non-positive.
(3) If c2 > 1 and f (0) < r, then the equation f (θ) = r has no solution.

5Volatility Smile/Smirk Properties of [GLP & MEMM] Models: Simulation Results.

For the models described in the previous sections we have carried on the simulation analysis,and we have obtained many results which support that our model has the volatility smile/smirkproperty. The following figures of implied surface are typical examples where the volatility
smile/smirk property can be seen.

85.1[Geometric Stable Process & MEMM] Model(a) b = −0.4(b) b = 0(c) b = 0.6

Figure 4: [G-Stable Process & MEMM] Model (α = 1.2, c1 = 0.2, c2 = 0.2)(a) α = 0.8(b) α = 1.4(c) α = 1.8

Figure 5: [G-Stable Process & MEMM] Model (b = 0, c1 = 0.01, c2 = 0.01)(a) c1 = 0.005, c2 = 0.095(b) c1 = 0.095, c2 = 0.005(c) c1 = 0.1, c2 = 0.1

Figure 6: [G-Stable Process & MEMM] Model (b = 0, α = 1.2)

95.2

[Geometric CGMY Process & MEMM] Model(a) b = −0.1(b) b = 0(c) b = 0.2

Figure 7: [G-CGMY Process & MEMM] Model (C = 0.05, G = 0.5, M = 0.5, Y = 1.2)(a) C = 0.05(b) C = 0.1(c) C = 0.3

Figure 8: [G-CGMY Process & MEMM] Model (b = 0, G = 0.5, M = 0.5, Y = 1.2)(a) G = 5, M = 0.05(b) G = 0.05, M = 0.05(c) G = 0.05, M = 3

Figure 9: [G-CGMY Process & MEMM] Model (b = 0, C = 0.05, Y = 1.2)

10

(a) Y = 0.5(b) Y = 1(c) Y = 1.5

Figure 10: [G-CGMY Process & MEMM] Model (b = 0, C = 0.05, G = 0.5, M = 0.5)

5.3

[Geometric Variance Gamma Process & MEMM] Model(a) b0 = −0.4(b) b0 = 0(c) b0 = 0.6

Figure 11: [G-VG Process & MEMM] Model (C = 5, c1 = 15, c2 = 0.5)(a) C = 5(b) C = 10(c) C = 20

Figure 12: [G-VG Process & MEMM] Model (b0 = 0, c1 = 15, c2 = 15)11

(a) c1 = 15, c2 = 15(b) c1 = 28, c2 = 2(c) c1 = 10, c2 = 10

Figure 13: [G-VG Process & MEMM] Model (b0 = 0, C = 10)

6Implied Volatility Level Calibration of Nikkei225 Index Option by [GLP & MEMM] Models(im)Let {σj(im)∗} be the set of implied volatilities obtained in the market, and let {σj(Θ)} bethe set of the corresponding implied volatilities of the model with the parameter Θ. Then theimplied volatility level calibration is to solve the following minimization problem.
minΘX(im)∗|σj(im) 2(Θ) − σj| .

j(a) 07/02/2001, time to maturity = 30-day(b) 18/01/2000, time to maturity = 52-day

Figure 14: the implied volatility curve of [Geometric VG Process & MEMM] Model and Nikkei
225 call option.

12

(a) 07/02/2001, time to maturity = 30-day(b) 18/01/2000, time to maturity = 52-day

Figure 15: the implied volatility curve of [Geometric CGMY Process & MEMM] Model andNikkei 225 call option.

(a) 07/02/2001, time to maturity = 30-day(b) 18/01/2000, time to maturity = 52-day

Figure 16: the implied volatility curve of [Geometric Stable Process & MEMM] Model andNikkei 225 call option.

137Concluding Remarks

We have only done the implied volatility level calibration. For the selection of suitable model,
we have to do the option price level calibration and the fitness analysis. These analyses are thesubjects for us to do next.

References[1] Carr, P., Geman, H., Madan, D.B., and Yor, M. (2000), The Fine Structure of Asset
Returns: An Empirical Investigation. (Preprint).
[2] Carr, P. and Madan, D. (1999), Option valuation using the fast Fourier transform. Journalof Computational Finance, 2, 61-73.
[3] Chan, T. (1999), Pricing Contingent Claims on Stocks Derived by Lévy Processes. The
Annals of Applied Probability, v. 9, No. 2, 504- 528 .
[4] Cont, R. and Tankov, P. (2002), Calibration of Jump-Diffusion Option-Pricing Models: A
Robust Non-parametric Approach. (preprint).
[5] Fama, E. F. (1963), Mandelbrot and the Stable Paretian Hypothesis. J. of Business, 36,420-429.
[6] Frittelli, M. (2000), The Minimal Entropy Martingale Measures and the Valuation Problem
in Incomplete Markets, Mathematical Finance 10, 39-52.
[7] Fujiwara, T. and Miyahara, Y. (2003), The Minimal Entropy Martingale Measures forGeometric Lévy Processes, Finance and Stochastics 7(2003), pp.509-531.
[8] Madan, D. and Seneta, E. (1990), The variance gamma (vg) model for share market returns.
Journal of Business, v.63(4), 511-524.
[9] Madan, D., Carr, P. and Chang, E. (1998), The variance gamma process and option pricing.
European Finance Review, v.2, 79-105.
[10] Mandelbrot, B. (1963), The variation of certain speculative prices. J. of Business, 36,394-419.
[11] Miyahara, Y.(1996), Canonical Martingale Measures of Incomplete Assets Markets, in Probability Theory and Mathematical Statistics: Proceedings of the Seventh Japan-Russia Symposium, Tokyo 1995 (eds. S. Watanabe et al.), pp.343-352.
[12] Miyahara, Y. (1999), Minimal Entropy Martingale Measures of Jump Type Price Processes
in Incomplete Assets Markets. Asian-Pacific Financial Markets, Vol. 6, No. 2, pp. 97-113.

14

[13] Miyahara, Y.(2001), [Geometric Lévy Process & MEMM] Pricing Model and Related Estimation Problems, Asia-Pacific Financial Markets 8, No. 1, pp. 45-60.
[14] Miyahara, Y.(2002), Estimation of Lévy Processes, Discussion Papers in Economics,Nagoya City University No. 318, pp. 1-36.
[15] Miyahara, Y.(2004), [GLP & MEMM] Pricing Model and its Calibration, (preprint).
[16] Miyahara, Y. and Novikov, A. (2002), Geometric Lévy Process Pricing Model, Proceedingsof Steklov Mathematical Institute, Vol.237(2002), pp.176-191.
[17] Rachev, S. and Mittnik, S. (2000), ”Stable Paretian Models in Finance”, Wiley.
[18] Sato, K. (1999), ”Lévy Processes and Infinitely Divisible Distributions”. Cambridge University Press.

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