[OSAKA University, 1/12/2005]

Jump Process Models in Mathematical Finance

Yoshio Miyahara∗

Graduate School of Economics

Nagoya City University

December 1, 2005

Abstract

In this talk we survey the jump process models in mathematical finance, and we put our focus on the geometric

Levy process(GLP) models. The GLP models are incomplete market models in general, and so there are many

equivalent martingale measures. Therefore we have to select a suitable martingale measure for the option pricing.

When we have selected a pair of a Levy process and an equivalent martingale measure, then we have established

an option pricing model. This is the first topic of this talk. The next problem is how to decide the best (or better)

model. This problem is related to the calibration or fitness analysis. We next study this problem. And finally we

discuss what kinds of search may be important in the future from the theoretical and practical points of view.∗E-mail: y-miya@econ.nagoya-cu.ac.jp

1

Key words: Jump process, Geometric Levy process, Martingale measure, Calibration, Fitness analysis, Utility

indifference price, Risk measure.

JEL Classification: G12, G13

Contents

1 Introduction 4

1.1 The reasons why jump processes? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 New models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Geometric Levy process models 6

2.1 Distribution of log return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Candidates for the suitable Levy process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Equivalent martingale measures for geometric Levy process 9

3.1 Candidates for the suitable equivalent martingale measures . . . . . . . . . . . . . . . . . . . . . . . . 9

3.2 Esscher transformed MM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.2.1 Esscher transform and Esscher transformed martingale measure . . . . . . . . . . . . . . . . . 10

3.2.2 Corresponding Risk Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2.3 Examples of Esscher transformed martingale measure . . . . . . . . . . . . . . . . . . . . . . . 13

3.3 Minimax MM, Minimal distance MM, Utility-MM, . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.3.1 Utility functions and Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.3.2 Existence of Minimax Martingale measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2

3.3.3 Minimax (or Minimal Distance) MM for Geometric Levy Processes . . . . . . . . . . . . . . . 16

3.4 Summary of the Explicit Forms of MM for Geometric Levy Processes . . . . . . . . . . . . . . . . . . 18

3.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.5.1 Geometric Variance Gamma Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.5.2 Merton Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.5.3 Geometric Stable Process Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4 Properties of the geometric Levy process pricing models 24

4.1 Risk neutral pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.2 Volatility smile/smirk property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5 Calibration and fitness analysis 28

5.1 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5.2 Fitness analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5.3 Empirical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

6 Concluding Remarks 29

3

1 Introduction

1.1 The reasons why jump processes?

1) The compound Poisson process is very popular in the risk theory (see [2] and [18]).

2) Some of the underlying asset processes seem to be jump process. (Ex. real estate.)

3) Black-Scholes model

St = S0e(µ−1

2σ2)t+σWt (1.1)

dSt = St (µdt + σdWt) . (1.2)

does not explain all properties of the market. (fat tail property and asymmetry of the

distribution of log return, volatility smile/smirk, etc.)

4

1.2 New models

Candidates for the new model

1) geometric Levy process model

2) random volatility model

3) SDE model, jump-diffusion model

4) econometric model (GARCH)

——————————————————————

•jump process model (→ Levy process model→ 1))

•There are many equivalent martingale measures.

•option pricing problems in the incomplete market

•Selection of a suitable equivalent martingale measure.

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2 Geometric Levy process models

•A probability space (Ω,F , P ) and a filtration Ft, 0 ≤ t ≤ T are given.

• A geometric Levy precess (GLP) is

St = S0eZt (2.1)

where Zt is a Levy process with the generating triplet (σ2, ν(dx), b).

2.1 Distribution of log return

•The characteristic function of Zt:

φt(u) = E[eiuZt] = exp

t

−σ2

2u2 + ibu +

∫

|x|≤1(eiux − 1− iux)ν(dx) +

∫

|x|>1(eiux − 1)ν(dx)

(2.2)

Infinitely divisible distribution

•The price process St has the following another expression

St = S0E(Z)t (2.3)

6

where E(Z)t is the Doleans-Dade exponential of Zt, and the generating triplet of Zt,

say (σ2, ν(dx), b), is

σ2 = σ2 (2.4)

ν(dx) = (ν J−1)(dx), J(x) = ex − 1, (2.5)

( i.e. ν(A) =∫

1A(ex − 1)ν(dx) )

b = b +1

2σ2 +

∫ ((ex − 1)1|ex−1|≤1 − x1|x≤1

)ν(dx). (2.6)

Remark 1 (i) St satisfies

dSt = St−dZt. (2.7)

(ii) supp ν ⊂ (−1,∞).

(iii) If ν(dx) has the density n(x), then ν(dx) has the density n(x) and n(x) is

given by

n(x) =1

1 + xn(log(1 + x)). (2.8)

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2.2 Candidates for the suitable Levy process

(1) Stable process (Mandelbrot, Fama(1963))

(2) Jump diffusion process (Merton(1973))

(3) Variance Gamma process (Madan(1990))

(4) Generalized Hyperbolic process (Eberlein(1995)

(5) CGMY process (Carr-Geman-Madam-Yor(2000))

(6) Normal inverse Gaussian process (Barndorff-Nielsen)

(7) finite moment log stable process(Carr-Wu(2003))

For the selection of the suitable process we need the empirical analysis, and to do

this we have to develop the estimation methods for the Levy processes.

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3 Equivalent martingale measures for geometric Levy process

3.1 Candidates for the suitable equivalent martingale measures

(1) Minimal Martingale Measure (MMM) (Follmer-Schweizer(1991))

(2) Variance Optimal Martingale Measure (VOMM)(Schweizer(1995))

(3) Mean Correcting Martingale Measure (MCMM)

(4) Esscher Martingale Measure (ESMM) (Gerber-Shiu(1994), B-D-E-S(1996))

(5) Minimal Entropy Martingale Measure (MEMM) (Miyahara(1996), Frittelli(2000))

(6) Utility Based Martingale Measure (U-MM)

For the selection of the suitable martingale measure, we have to study the goodness

of the models. This should be done from two kinds of view points. One is the theoretical

point of view, and the other is the empirical point of view.

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3.2 Esscher transformed MM

Esscher transformed MM,　

• Esscher transformed MM:

(3) Mean Correcting Martingale Measure (MCMM),

(4) Esscher Martingale Measure,

(5) Minimal Entropy Martingale Measure (MEMM)

Esscher(’32), Gerber-Shiu(’94), B-D-E-S(’96), Kallsen-Shiryaev(’02), etc.

3.2.1 Esscher transform and Esscher transformed martingale measure

Definition 1 Let R be a risk variable and h be a constant. Then the probability

measure P(ESS)R,h defined by

dP(ESS)R,h

dP|F =

ehR

E[ehR](3.1)

is called the Esscher transformed measure of P by the random variable R and h,

and this measure transformation is called the Esscher transform by the random

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variable R and h.

Definition 2 Let Rt, 0 ≤ t ≤ T, be a risk process. Then the Esscher transformed

measure of P by the process Rt and a constant h is the probability measure P(ESS)R[0,T ],h

,

which is defined by

dP(ESS)R[0,T ],h

dP|F =

ehRT

E[ehRT ](3.2)

( Remark that P(ESS)R[0,T ],h

= P(ESS)RT ,h . )

and this measure transformation is called the Esscher transform by the process Rt and

a constant h.

Definition 3 In the above definition, if the constant h is chosen so that the

P(ESS)R[0,T ],h

is a martingale measure of St, then P(ESS)R[0,T ],h

is called the Esscher trans-

formed martingale measure of St by the process Rt, and it is denoted by P(ESS)R[0,T ]

or

P(ESS)RT

.

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3.2.2 Corresponding Risk Processes

When we give a certain risk process Rt, we obtain a corresponding Esscher transformed

martingale measure if it exists. As we have seen in the previous section, the GLP has

two kinds of representation such that

St = S0eZt = S0E(Z)t.

The processes Zt and Zt are candidates for the risk process.

Definition 4 The process Zt is called the simple return process of St, and the

process Zt is called the compound return process of St.

Remark 2 The terms ‘simple return’ and ‘compound return’ were introduced in

B-D-E-S(’96) p.294.

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3.2.3 Examples of Esscher transformed martingale measure

(1) ESMM

Definition 5 If the Esscher transformed martingale measure P(ESS)ZT

is well-defined,

then this measure is called the Compound Return Esscher transformed martingale

measure or the Esscher Martingale Measure (ESMM), and is denoted by P (ESMM).

(2) MEMM

Definition 6 If the Esscher transformed martingale measure P(ESS)

ZTis well-defined,

then this measure is called the Simple Return Esscher transformed martingale

measure, or based on the following Proposition, the Minimal Entropy Martingale

Measure (MEMM), and is denoted by P (MEMM) or P ∗.

Definition 7 (MEMM) If an equivalent martingale measure P ∗ satisfies

H(P ∗|P ) ≤ H(Q|P ) ∀Q : EMM, (3.3)

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then P ∗ is called the minimal entropy martingale measure (MEMM) of St. Where

H(Q|P ) is the relative entropy of Q with respect to P

H(Q|P ) =

∫Ω log[dQ

dP ]dQ, if Q ¿ P,

∞, otherwise,

. (3.4)

From the proof of Fujiwara-Miyahara(’03) Theorem 3.1, it follows that

Proposition 1 The simple return Esscher transformed martingale measure P(ESS)

Z[0,T ]

of St is the minimal entropy martingale measure (MEMM) of St.

(3) MCMM

Definition 8 Suppose σ > 0, and let σWt be the diffusion part of Zt. If the

Esscher transformed martingale measure P(ESS)WT

is well-defined, then this measure

is called the Mean Correcting Martingale Measure (MCMM), and is denoted by

P (MCMM).

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3.3 Minimax MM, Minimal distance MM, Utility-MM,

(2) Variance Optimal Martingale Measure (VOMM),

(5) Minimal Entropy Martingale Measure (MEMM),

(6) Utility Based Martingale Measure (U-MM)

He-Pearson(’91), Goll-Ruschendorf(’01), Bellini-Frittelli(’02), Kallsen(’02)

3.3.1 Utility functions and Duality

The problem to obtain the minimax martingale measure for a given utility function

u(x) is equivalent to the problem to obtain the minimal distance martingale measure

for the dual distance function u∗(y) defined by

u∗(y) = supx

(u(x)− xy) . (3.5)

• minimization

E[F

dP ∗

dP

] = min

Q:EMM

E[F

dQ

dP

]

(3.6)

Examples of distance functions and minimal distance MM(See [1] or [20]):

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1) F (x) = x log x, minimal relative entropy martingale measure (MEMM)

2) F (x) = − log x, minimal reverse relative entropy MM

3) F (x) = |x− 1|, minimal total variation MM

4) F (x) = −√x, minimal Hellinger distance MM

5) F (x) = |x− 1|p, minimal p-moment MM

6) F (x) = |x− 1|2, minimal variance MM (variance minimal MM (VMMM))

3.3.2 Existence of Minimax Martingale measures

The existence conditions in the general form have been studied in Bellini-Frittelli(’02)

and Gundel(’05).

3.3.3 Minimax (or Minimal Distance) MM for Geometric Levy Processes

We will see the explicit form of VMMM for Geometric Levy processes. (Jeanblanc-

Miyahara [22])

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Theorem 1 (i) For the existence of the VMMM, it is sufficient that the following

equation for (f, g(x), µ) has a solution.

f = µσ (3.7)

(eg(x) − 1) = µ(ex − 1) (3.8)

µσ2 +∫(µ(ex − 1)2 + (ex − 1)− x1|x|≤1)ν(dx) = r − b− 1

2σ2β. (3.9)

(ii) When the above equation has a solution (f ∗, g∗(x), µ∗), then the martingale

measure QLT (f∗,g∗) is the VMMM, where

dQLT (f∗,g∗)

dP= LT (f ∗, g∗). (3.10)

• Explicit form of L∗t = LT (f ∗, g∗) is

L∗t = exp

f ∗Wt −

1

2(f ∗)2 +

∫(eg∗(x) − 1− g∗(x)1|x|≤1)ν(dx)

t

+∫ t ∫

|x|>1 g∗(x)N(ds, dx) +∫ t ∫

|x≤1 g∗(x) N(ds, dx))

(3.11)

• The Levy measure of Zt under P (V MMM) is

ν(V MMM)(dx) = (1 + µ∗(ex − 1)) ν(dx) (3.12)

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• The σ(V MMM) is σ(V MMM) = σ, and the b(V MMM) is determined from the martingale

condition.

• The proof is based on Kunita’s representation theorem for positive martingale pro-

cesses.

• The idea of the proof can be applied to the power function distance.

3.4 Summary of the Explicit Forms of MM for Geometric Levy Processes

The Levy measures of Zt (if exist).

• MEMM:

ν(MEMM)(dx) = eθ∗(ex−1)ν(dx) (3.13)

where θ∗ is the solution of

b + (1

2+ θ)σ2 +

∫ ∞−∞

((ex − 1)eθ(ex−1) − x1|x|≤1(x)

)ν(dx) = r. (3.14)

Usually θ∗ < 0

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• ESMM:

ν(ESMM)(dx) = eh∗xν(dx) (3.15)

where h∗ is the solution of

b + (1

2+ h)σ2 +

∫ ∞−∞

((ex − 1)ehx − x1|x|≤1

)ν(dx) = r. (3.16)

• MCMM:

ν(MCMM)(dx) = ν(dx) (3.17)

• VMMM:

ν(V MMM)(dx) = (1 + µ∗(ex − 1)) ν(dx) (3.18)

where

µ∗ =r − b− 1

2σ2 − ∫

(ex − 1− x1|x|≤1)ν(dx)

σ2 +∫(ex − 1)2ν(dx)

. (3.19)

• Power-function MM:

Distance function: F (x) = axp, where (a > 0, p > 1) or (a < 0, p < 1).

ν(PfMM)(dx) = eg∗(x)ν(dx) (3.20)

19

where g∗ is given by

p(e(p−1)g∗(x) − 1) = 2µ∗(ex − 1) (3.21)

and µ∗ is the solution of

2µσ2

p(p− 1)+

∫(1 +

2µ(ex − 1)

p)

1p−1(ex − 1)− x1|x|≤1

ν(dx) = r − b− 1

2σ2. (3.22)

3.5 Examples

3.5.1 Geometric Variance Gamma Model

ν(dx) = C(Ix<0 exp(−c1|x|) + Ix>0 exp(−c2|x|)

)|x|−1dx, (3.23)

where C, c1, c2 are positive constants.

• MEMM:

ν(MEMM)(dx) = eθ∗(ex−1)C(Ix<0 exp(−c1|x|) + Ix>0 exp(−c2|x|)

)|x|−1dx (3.24)

(c2 < 1), or (c2 > 1 + some condition)

• ESMM:

ν(ESMM)(dx) = eh∗xC(Ix<0 exp(−c1|x|) + Ix>0 exp(−c2|x|)

)|x|−1dx (3.25)

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h∗ must be in (−c1, c2)

• MCMM: not exists.

• VMMM:

ν(V MMM)(dx) = (1 + µ∗(ex − 1)) C(Ix<0 exp(−c1|x|) + Ix>0 exp(−c2|x|)

)|x|−1dx

(3.26)

where

µ∗ =r − b− 1

2σ2 − ∫

(ex − 1− x1|x|≤1)ν(dx)

σ2 +∫(ex − 1)2ν(dx)

. (3.27)

and µ∗ must be in (0,1).

• Power-function MM

ν(PfMM)(dx) = eg∗(x)ν(dx) (3.28)

where

p(e(p−1)g∗(x) − 1) = 2µ∗(ex − 1).

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3.5.2 Merton Model

ν(dx) = cg(x; m, v)dx = c1√2πv

exp

−(x−m)2

2v

dx (3.29)

c > 0, m ∈ (−∞,∞), v > 0.

g(u) =∫ ∞−∞ eiuxg(x; m, v)dx = exp

imu− 1

2vu2

. (3.30)

• MEMM:

ν(MEMM)(dx) = eθ∗(ex−1)cg(x; m, v)dx (3.31)

• ESMM:

ν(ESMM)(dx) = eh∗xcg(x; m, v)dx (3.32)

• MCMM: Exists.

ν(MCMM)(dx) = ν(dx) (3.33)

• VMMM:

ν(V MMM)(dx) = (1 + µ∗(ex − 1)) cg(x; m, v)dx (3.34)

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• Power-function MM

ν(PfMM)(dx) = eg∗(x)cg(x; m, v)dx (3.35)

where

p(e(p−1)g∗(x) − 1) = 2µ∗(ex − 1).

3.5.3 Geometric Stable Process Model

ν(dx) =(c1Ix<0(x) + c2Ix>0(x)

)|x|−(α+1)dx, (3.36)

where 0 < α < 2 and we assume that c1 > 0, c2 > 0.

• MEMM:

ν(MEMM)(dx) = eθ∗(ex−1)(c1Ix<0(x) + c2Ix>0(x)

)|x|−(α+1)dx. (3.37)

• Other martingale measures do not exist in general.

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4 Properties of the geometric Levy process pricing models

In this section we discuss the properties of the model from the theoretical point of view.

4.1 Risk neutral pricing

Risk neutral pricing, i.e. martingale measure pricing.

0) Law of Zt under MMs given above

They are all Levy processes.

1) Corresponding risk process relating to Esscher transformed MM

• MEMM: Simple return process Zt

• ESMM: Compound return process Zt

• MCMM: Wiener process Wt

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2) Conditions for the Existence

• MEMM P (MEMM):

∫

|x|>1 |(ex − 1)|eθ∗(ex−1) ν(dx) < ∞. (4.1)

This condition is satisfied for wide class of Levy measures, if θ∗ < 0.

• ESMM P (ESMM):∫

|x|>1 |ex − 1| eh∗xν(dx) < ∞ (4.2)

• MCMM P (MCMM):

σ > 0, and∫

|x|>1 | (ex − 1) |ν(dx) < ∞ (4.3)

• VMMM P (V MMM):∫

|x|>1(ex − 1)2ν(dx) < ∞. (4.4)

• PfMM P (PfMM):

∫

|x|>1(1 +2µ∗(ex − 1)

p)

1p−1(ex − 1)ν(dx) < ∞. (4.5)

25

3) Corresponding utility functions

• The MEMM is corresponding to the exponential utility function. (Frittelli(’00) or

Goll-Ruschendorf(’01)).

• The ESMM is corresponding to power utility function or logarithm utility function.

(Gerber-Shiu(’94) or Goll-Ruschendorf(’01)). But the corresponding power parameter

depends on the process.

Remark 3 In the case of MEMM, the relation of the MEMM to the utility in-

difference price is known. (See Fujiwara-Miyahara(’03, §4), etc.). This result is

generalized by C. Stricker(′04).

4) Minimal distance to the original probability

• The relative entropy is called Kullback-Leibler Information Number(see Ihara(’93,

p.23) or Kullback-Leibler distance (see Cover-Thomas(’91, p.18) in the field of informa-

tion theory. We can state that the MEMM is the nearest equivalent martingale measure

to the original probability P in the sense of Kullback-Leibler distance.

26

• The VMMM is the nearest equivalent martingale measure to the original probability

P in the sense of variance.

• The ESMM is the nearest equivalent martingale measure to the original probability

P in the sense of power function metric. But the minimal distance MM for the power

function is not always ESMM.

5) Large deviation property of MEMM: The large deviation theory is closely

related to the minimum relative entropy analysis, and the Sanov’s theorem or Sanov

property is well-known (see, e.g. Cover-Thomas(’91 pp.291-304) or Ihara(’93, pp.110-

111)). This theorem results to saying that the MEMM is the most possible empirical

probability measure of paths of price process in the class of the equivalent martingale

measures.

4.2 Volatility smile/smirk property

Computer simulation results shows us that [GLP & EMM] model has the volatility

smile/smirk property. [35], [43]

27

5 Calibration and fitness analysis

In this section we see the method to study the goodness of the models from the empirical

point of view.

5.1 Calibration

Estimate the parameter of the model. (Cf. [34, §8], [43].)

• The class of model is fixed. (Ex. Variance Gamma model)

• The parameters of model are estimated from the data of options in the market. (The

sequential data of the prices of underlying asset are not used.)

5.2 Fitness analysis

The sequential data of the prices of underlying asset and the data of the market prices

of the options are given.

Which model is most fitting to the given market data? (Cf. [34, §7]

28

• The classes of model are not fixed. ( B-S model, Variance Gamma model, CGMY

model, Stable model, Merton model, etc. are all under the consideration.)

• Both data of the prices of the underlying asset and the market prices of the options

are used.

5.3 Empirical analysis

• Index option

• equity option

• foreign exchange option

6 Concluding Remarks

[mathematical finance] vs [financial engineering]

(1) Methods for Option Pricing: Risk-neutral pricing and utility indifference pricing.

29

• If the market is efficient enough, the risk neutral pricing (i.e. martingale measure

pricing) is reasonable.

• a) The Esscher transformed MM, b) Minimal distance MM, c) Minimax MM

Definition 9 Set

UQ(x) = supE[u(Y )]; Y ∈ L1(Q), EQ[Y ] ≤ x,E[u(y)]− < ∞ (6.1)

Q∗ = Q∗(x) is called minimax MM for x if it satisfies

UQ∗(x) = inf UQ(x). (6.2)

The economical implication of Minimax Pricing is: “It produces prices which are least

favourable for an investor with a given utility profile, i.e., the maximal expected utility

with respect to prices based on a martingale measure is minimal.” ([20, p.564])

Remark 4 A minimax MM is corresponding to a minimal distance martingale

measure. (See [20])

• For the pricing of derivatives based on the non-tradable asset, the utility indifference

pricing is effective.

30

Definition 10 The buyer’s utility indifference price of X, πb[X ] = πb[X|W ], is

defined by

E[u((W − πb[X ]) + X)] = E[u(W )] (6.3)

The seller’s utility indifference price of X, πs[X ] = πs[X|W ], is defined by

E[u((W + πs[X ])−X)] = E[u(W )] (6.4)

• This pricing has strong relations with risk measures.

(2) Risk measures

Definition 11 A convex risk measure ρ(Φ) is

a) ρ(λΦ + (1− λ)Ψ) ≤ λρ(Φ) + (1− λ)ρ(Ψ)

b) Φ ≤ Ψ ⇒ ρ(Φ) ≥ ρ(Ψ)

c) ρ(Ψ + m) = ρ(Ψ)−m.

If ρ(Φ) satisfy a), b), c) and d)

d) ∀λ ∈ R+ρ(λΦ) = λρ(Φ),

then it is called a coherent.

31

(3) Real option

(4) Hedging

(5) Multi-dimensional model

References

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