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74 CHAPTER 2. THE CALIBRATION PROBLEM where α is a risk-aversion coefficient. Given some set of admissible trading strategies Θ, the utility indifference price p α (c, H) of a claim H for this investor is defined as the solution of the following equation: ∫ sup E[U α (c + p α (c, H) + θ∈Θ (0,T ] ∫ θ u dS u − H)] = sup E[U α (c + θ u dS u )]. θ∈Θ (0,T ] Due to the special form (2.16) of the utility function, the initial endowment c cancels out of the above equation and we see that p α (c, H) = p α (0, H) := p α (H). Using the results in [33], Miyahara and Fujiwara [73] established the following properties of utility indifference price in exponential Lévy models. Similar results have been obtained by El Karoui and Rouge [39] in the setting of continuous processes. Proposition 2.8. Let (X, P ) be a Lévy process such that P ∈ L + B ∩ L NA, and let Q ∗ be the MEMM defined by (2.13). Let S t := e Xt and let Θ include all predictable S-integrable processes θ such that ∫ (0,t] θ udS u is a martingale for each local martingale measure Q, with I(Q|P ) < ∞. Then the corresponding utility indifference price p α (H) of a bounded claim H has the following properties: 1. p α (H) ≥ E Q∗ [H] for any α > 0. 2. If 0 < α < β then p α (H) ≤ p β (H). 3. lim α↓0 p α (H) = E Q∗ [H]. The price of a claim H computed under the MEMM thus turns out to be the highest price at which all investors with exponential utility function will be willing to buy this claim. Kallsen [58] defines neutral derivative prices for which the optimal trading strategy consists in having no contingent claim in one’s portfolio. This approach to valuation in incomplete markets corresponds to the notion of fair price introduced by Davis [31]. Kallsen further shows that such prices are unique and correspond to a linear arbitrage-free pricing system, defined by an equivalent martingale measure (neutral pricing measure). If the utility function of an investor has the form (2.16), the neutral pricing measure coincides with the minimal entropy
2.3. RELATIVE ENTROPY IN THE LITERATURE 75 martingale measure. The neutral pricing measure Q ∗ also corresponds to the least favorable market completion from the point of view of the investor in the following sense. Let V (c, Q) := sup{E[U(c + X − E Q [X])] : X is F T -measurable} be the maximum possible expected utility that an investor with initial endowment c and utility function U can get by trading in a market where the price of every contingent claim X is equal to E Q [X]. Then V (c, Q ∗ ) = inf V (c, Q), Q∈EMM(P ) that is, the neutral pricing measure minimizes the maximum possible expected utility that an investor can get in a completed market, over all possible arbitrage-free completions. The notion of neutral pricing measure thus coincides with the minimax measures studied in [16], [49] and other papers. 2.3.2 Calibration via relative entropy minimization Despite its analytic tractability and interesting economic interpretation, the MEMM has an important drawback which makes it impossible to use for pricing in real markets: it does not take into account the information obtained from prices of traded options. To tackle this problem, Goll and Rüschendorf [48] introduced the notion of minimal distance martingale measure consistent with observed market prices. In particular, given prices of call options {C M (T i , K i )} N i=1 , a probability measure Q ∗ ∈ M is called consistent minimal entropy martingale measure (CMEMM) if I(Q ∗ |P ) = min I(Q|P ), Q where the minimum is taken over all martingale measures Q such that C M (T i , K i ) = C Q (T i , K i ) for i = 1, . . . , N. Kallsen [58] shows that consistent minimal distance martingale measures correspond to constant demand derivative pricing (when the optimal portfolio must contain not zero but a fixed amount of each traded derivative), thus providing an economic rationale for distance minimization under constraints, and, in particular, for entropy minimization. A drawback of CMEMM is that it may be very difficult to compute numerically. In particular, if X is a Lévy process under the historical measure P , it will in general no longer be
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Thèse présentée pour obtenir le
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Abstract This thesis deals with the
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Contents Remarks on notation 11 Int
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CONTENTS 9 II Multidimensional mode
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12 For {P n } n≥1 , P ∈ P(Ω) t
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14 INTRODUCTION EN FRANCAIS corresp
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16 INTRODUCTION EN FRANCAIS Lévy p
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18 INTRODUCTION EN FRANCAIS Calibra
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20 INTRODUCTION EN FRANCAIS • Si
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22 INTRODUCTION EN FRANCAIS et Scai
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Page 28 and 29: 28 INTRODUCTION 7500 Taux de change
Page 30 and 31: 30 INTRODUCTION Lévy models by Fou
Page 33 and 34: Chapter 1 Lévy processes and exp-L
Page 35 and 36: 1.1. LEVY PROCESSES 35 Proposition
Page 37 and 38: 1.1. LEVY PROCESSES 37 The characte
Page 39 and 40: 1.1. LEVY PROCESSES 39 On the other
Page 41 and 42: 1.2. EXPONENTIAL LEVY MODELS 41 (b)
Page 43 and 44: 1.3. EXP-LEVY MODELS IN FINANCE 43
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Page 47 and 48: 1.4. PRICING EUROPEAN OPTIONS 47 wi
Page 49 and 50: 1.4. PRICING EUROPEAN OPTIONS 49 Pr
Page 51 and 52: 1.4. PRICING EUROPEAN OPTIONS 51 Nu
Page 53 and 54: 1.4. PRICING EUROPEAN OPTIONS 53 By
Page 55 and 56: 1.4. PRICING EUROPEAN OPTIONS 55 Si
Page 57 and 58: Chapter 2 The calibration problem a
Page 59 and 60: 2.1. LEAST SQUARES CALIBRATION 59 m
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Page 85 and 86: 2.5. REGULARIZING THE CALIBRATION P
Page 95 and 96: Chapter 3 Numerical implementation
Page 97 and 98: 3.1. DISCRETIZING THE CALIBRATION P
Page 99 and 100: 3.2. CHOICE OF THE PRIOR 99 Since
Page 101 and 102: 3.2. CHOICE OF THE PRIOR 101 twice
Page 103 and 104: 3.3. CHOICE OF THE REGULARIZATION P
Page 111 and 112: 3.4. CHOICE OF THE WEIGHTS 111 Then
Page 113 and 114: 3.5. NUMERICAL SOLUTION 113 have th
Page 115 and 116: 3.5. NUMERICAL SOLUTION 115 of X t
Page 117 and 118: 3.5. NUMERICAL SOLUTION 117 since
Page 119 and 120: 3.6. NUMERICAL AND EMPIRICAL TESTS
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3.6. NUMERICAL AND EMPIRICAL TESTS
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Part II Multidimensional modelling
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134 CHAPTER 4. DEPENDENCE OF LEVY P
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166 CHAPTER 5. APPLICATIONS OF LEVY
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188 CONCLUSIONS AND PERSPECTIVES ge
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190 BIBLIOGRAPHY [9] O. E. Barndorf
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192 BIBLIOGRAPHY [33] F. Delbaen, P
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194 BIBLIOGRAPHY [57] P. Jorion, On
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196 BIBLIOGRAPHY [81] S. Raible, L
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198 BIBLIOGRAPHY
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200 INDEX tightness, 40 levycalibra
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