파생상품의 최적 헤지에 관한 연구
Essays on Optimal Hedging of Contingent Claims
- 주제(키워드) Hedging , Knightian Uncertainty , G-Brownian Motion
- 발행기관 The Graduate School, Ajou University
- 지도교수 Koo, Hyeng-Keun
- 발행년도 2011
- 학위수여년월 2011. 2
- 학위명 박사
- 학과 및 전공 일반대학원 경영학과
- 실제URI http://www.dcollection.net/handler/ajou/000000011664
- 본문언어 영어
- 저작권 아주대학교 논문은 저작권에 의해 보호받습니다.
초록/요약
The first chapter studies indifference pricing with Knightian uncertainty in an incomplete market. A one period trinomial model is analysed assuming an exponential utility function. Indifference pricing and hedging methods are extended to the case of Knightian uncertainty represented by the maxmin expected utility criterion by I. Gilboa & D. Schmeidler (1989, [35]). Since the issuance of a contingent claim can change the reference belief, one must take into account how the worst case scenarios change with or without the contingent claim. In this paper, the optimal selection rule is characterized firstly, and some examples are presented to show how to calculate indifference prices and optimal hedge ratios. The second chapter investigates the problem of optimal hedging and pricing with Knightian uncertainty a la I. Gilboa & D. Schmeidler. We start with a simple one period model and extend it to multi-period models as Cox et al. did for the Black-Scholes model (1979, [15]). We assume that the set of beliefs is convex and closed to utilize the min-max theorem. We also assume that for multi-period models, the set of beliefs satisfies the rectangularity defined as in Epstein & Schneider (2003, [30]) to ensure the time consistency of value functions. In discrete time models, given the second moment of the risky asset the agent would choose his belief from risk neutral ones. So the hedging demand for the risky asset would be zero. If the set of beliefs does not contain any risk neutral belief, he would choose the one which is closest to the set of risk neutral beliefs, which gives rise to hedging demand for the risky asset. If we admit the second moment of the risky asset vary, the problem would be more complex. For the simplest case in which the set of beliefs consist of only risk neutral ones, the agent would choose one of the extreme beliefs according to comparision between the certainty equivalent values at future nodes. It turns out that the comparison is related to the second order derivative of the certainty equivalent value along spatial dimension. We could use the multi-period results and Taylor series expansion to derive the limitting equations, which are formulated as a conjecture by the author. In doing this, we have to scale the set of beliefs according to the unit size of time steps. In the limiting case, it is demonstrated by an example that the process of indifference price satisfies the Black-Scholes-Barenblatt equation, which is a kind of G-equation introduced by Shige Peng (2006, [63]). The third chapter is about the Pareto optimal allocation of Knightian uncertainty in a single period discrete model. Maxmin expected utility by Gilboa and Schmeidler is also adopted to deal with ambiguity of two agents. If two agents have the same belief on ambiguity in a complete market, ambiguity does not seem to be involved in their decisions. But if their beliefs are differerent, ambiguity might play some roles in their decision. If the market is not complete, the two agents could decide according to their beliefs even though they share the same belief on the common state. All the proofs can be found in the appendix except the shorter ones located in the main text.
more목차
1 Pricing and Hedging under Knightian Uncertainty: A Discrete-Time Model 2
1.1 Introduction 2
1.2 A trinomial model without uncertainty 3
1.3 A simple maxmin utility model 7
1.3.1 Indi erence pricing under Knightian uncertainty 8
1.3.2 A simple example - selection rule change with a contigent claim 9
1.4 The minmax theorem and indifference prices 12
1.4.1 How many times the two expected utility meet as α changes 13
1.4.2 The cusp point condition 14
1.4.3 Case: (p1;u - p2;u) (p1;d - p2;d) < 0 15
1.4.4 Case: (p1;u - p2;u) (p1;d - p2;d) = 0 17
1.4.5 Case: (p1;u - p2;u) (p1;d - p2;d) > 0 18
1.5 A convex set of beliefs 19
1.6 Conclusion 23
2 Optimal hedging under Knightian uncertainty 25
2.1 Introduction 25
2.2 Single-period models 27
2.2.1 Settings 27
2.2.2 An investment component and a hedging component in a complete market 29
2.2.3 An incomplete market with ambiguity 32
2.2.4 Examples 35
2.3 Multi-period models 38
2.3.1 A two-period model 38
2.3.2 Certainty equivalent formulation 42
2.4 Infinitesimal analysis 43
2.4.1 Infinitesimal beliefs 44
2.4.2 Approximating equation with CARA utility 45
2.4.3 The limiting equations 48
2.4.4 Examples 51
2.5 Conclusion 59
3 Pareto optimal allocation under Knightian uncertainty 60
3.1 Introduction 60
3.2 Two-state model (complete) 61
3.2.1 No ambiguities 61
3.2.2 Equal attitude toward ambiguity 62
3.2.3 Different attitudes toward ambiguity 64
3.3 Three-state model (incomplete) 65
3.4 Conclusion 67
A Proofs in Chapter 1 69
A.1 Derivatives of some functions 69
A.2 Proofs 70
B Proofs in Chapter 2 81
B.1 Proofs 81
C Proofs in Chapter 3 91
C.1 Proofs 91
References 99
