A Robust Approach to Hedging and Pricing in Imperfect Markets †
Abstract
:1. Introduction
2. Preliminaries and Analytical Setup
- S1.
- Normality if ;
- S2.
- Positive homogeneity if , for all ;
- S3.
- Translation-invariance if ;
- S4.
- Sub-additivity if ;
- S5.
- Convexity if .
2.1. Risk Measures
- R1.
- ;
- R2.
- , for all and ;
- R3.
- , for all and ;
- R4.
- , for all and ;
- R5.
- R6.
2.2. Pricing Rules
- P1.
- ;
- P2.
- , for all and ;
- P3.
- , for all and (cash-invariance);
- P4.
- , for all and ;
- P5.
- , for all ;
- P6.
- .
2.3. Projection
3. Main Theoretical Results
3.1. Market Principles
- and are positive homogeneous if ϱ and π are.
- and are translation-invariant if ϱ and π are.
- and are sub-additive if ϱ and π are.
- and are convex if ϱ and π are.Furthermore,
- is monotone if ϱ and π are.
3.2. Positive-Homogeneous and Monotone Risk and Pricing Rules
3.3. Positive-Homogeneous and Sub-Additive Risk and Pricing Rules
- Furthermore, if condition 3 holds for π and ϱ, these statements are equivalent to
- There is no Good Deal in the market.
4. An Application to Hedging Economic Risk
4.1. Estimation Problem
4.2. Data Description
4.3. Results
5. Conclusions
Author Contributions
Conflicts of Interest
Appendix A. Proofs of Propositions and Theorems
Appendix A.1. Proof of Theorem 1
Appendix A.2. Proof of Proposition 1
Appendix A.3. Proof of Theorem 2
Appendix A.4. Proof of Proposition 2
Appendix A.5. Proof of Theorem 3
References
- Artzner, Philippe, Freddy Delbaen, Jean-Marc Eber, and David Heath. 1999. Coherent measures of risk. Mathematical Finance 9: 203–28. [Google Scholar] [CrossRef]
- Assa, Hirbod, and Alejandro Balbás. 2011. Good deals and compatible modification of risk and pricing rule: A regulatory treatment. Mathematics and Financial Economics 4: 253–68. [Google Scholar] [CrossRef] [Green Version]
- Balbás, Alejandro, Beatriz Balbás, and Antonio Heras. 2009. Optimal reinsurance with general risk measures. Insurance: Mathematics and Economics 44: 374–84. [Google Scholar] [CrossRef] [Green Version]
- Balbás, Alejandro, Raquel Balbás, and José Garrido. 2010. Extending pricing rules with general risk functions. European Journal of Operational Research 201: 23–33. [Google Scholar] [CrossRef] [Green Version]
- Balduzzi, Pierluigi, and Hédi Kallal. 1997. Risk premia and variance bounds. Journal of Finance 52: 1913–49. [Google Scholar] [CrossRef]
- Balduzzi, Pierluigi, and Cesare Robotti. 2001. Minimum-Variance Kernels, Economic Risk Premia, and Tests of Multi-Beta Models. Working Paper 2001-24. Federal Reserve Bank of Atlanta. [Google Scholar]
- Bassett, Gilbert W., Roger Koenker, and Gregory Kordas. 2004. Pessimistic portfolio allocation and choquet expected utility. Journal of Financial Econometrics 2: 477–92. [Google Scholar] [CrossRef]
- Bernardi, Mauro, Ghislaine Gayraud, and Lea Petrella. 2015. Bayesian tail risk interdependence using quantile regression. Bayesian Analysis 10: 553–603. [Google Scholar] [CrossRef]
- Breeden, Douglas T., Michael R. Gibbons, and Robert H. Litzenberger. 1989. Empirical test of the consumption-oriented capm. The Journal of Finance 44: 231–62. [Google Scholar] [CrossRef]
- Campbell, John Y. 1996. Understanding risk and return. Journal of Political Economy 104: 298–345. [Google Scholar] [CrossRef]
- Carhart, Mark M. 1997. On persistence in mutual fund performance. The Journal of Finance 52: 57–82. [Google Scholar] [CrossRef]
- Cheridito, Patrick, Freddy Delbaen, and Michael Kupper. 2006. Dynamic monetary risk measures for bounded discrete-time processes. Electronic Journal of Probability 11: 57–106. [Google Scholar] [CrossRef]
- Cochrane, John H., and Jesus Saa-Requejo. 2000. Beyond arbitrage: Good deal asset price bounds in incomplete markets. Journal of Political Economy 108: 79–119. [Google Scholar] [CrossRef]
- Cont, Rama, Romain Deguest, and Giacomo Scandolo. 2010. Robustness and sensitivity analysis of risk measurement procedures. Quantitative Finance 10: 593–606. [Google Scholar] [CrossRef]
- Delbaen, Freddy. 2002. Coherent risk measures on general probability spaces. In Advances in Finance and Stochastics. Berlin: Springer, pp. 1–37. [Google Scholar]
- Duffie, Darrell, and Henry R. Richardson. 1991. Mean-variance hedging in continuous time. The Annals of Applied Probability 1: 1–15. [Google Scholar] [CrossRef]
- El Karoui, Nicole, and Marie-Claire Quenez. 1995. Dynamic programming and pricing of contingent claims in an incomplete market. SIAM Journal on Control and Optimization 33: 29–66. [Google Scholar] [CrossRef]
- Fama, Eugene F., and Keneth R. French. 1992. The cross-section of expected stock returns. The Journal of Finance 47: 427–65. [Google Scholar] [CrossRef]
- Föllmer, Hans, and Peter Leukert. 2000. Efficient hedging: Cost versus shortfall risk. Finance and Stochastics 4: 117–46. [Google Scholar] [CrossRef]
- Föllmer, Hans, and Alexander Schied. 2002. Convex measures of risk and trading constraints. Finance and stochastics 6: 429–47. [Google Scholar] [CrossRef]
- Föllmer, Hans, and Martin Schweizer. 1991. Hedging of contingent claims under incomplete information. In Applied Stochastic Analysis (London, 1989). Stochastics Monographs, New York: Gordon and Breach, vol. 5, pp. 389–414. [Google Scholar]
- Goorbergh, Rob W. J. van den, Frans A. de Roon, and Bas J. M. Werker. 2003. Economic Hedging Portfolios. Discussion Paper 2003-102. Tilburg: Tilburg University, Center for Economic Research. [Google Scholar]
- Gourieroux, Christian, Jean Paul Laurent, and Huyên Pham. 1998. Mean-variance hedging and numéraire. Mathematical Finance 8: 179–200. [Google Scholar] [CrossRef]
- Jouini, Elyès, and Hédi Kallal. 1995a. Arbitrage in securities markets with short-sales constraints. Mathematical Finance 5: 197–232. [Google Scholar] [CrossRef]
- Jouini, Elyès, and Hédi Kallal. 1995b. Martingales and arbitrage in securities markets with transaction costs. Journal of Economic Theory 66: 178–97. [Google Scholar] [CrossRef]
- Jouini, Elyès, and Hédi Kallal. 1999. Viability and equilibrium in securities markets with frictions. Mathematical Finance 9: 275–92. [Google Scholar] [CrossRef]
- Karatzas, Ioannis, John P. Lehoczky, Steven E. Shreve, and Gan-Lin Xu. 1991. Martingale and duality methods for utility maximization in an incomplete market. SIAM Journal on Control and Optimization 29: 702–30. [Google Scholar] [CrossRef]
- Karatzas, Ioannis, and Steven E. Shreve. 1998. Methods of Mathematical Finance. Applications of Mathematics (New York). New York: Springer, vol. 39. [Google Scholar]
- Koenker, Roger. 2005. Quantile Regression. Econometric Society Monographs. Cambridge: Cambridge University Press. [Google Scholar]
- Kusuoka, Shigeo. 2001. On law invariant coherent risk measures. In Advances in Mathematical Economics. Tokyo: Springer, vol. 3, pp. 83–95. [Google Scholar]
- Lamont, Owen A. 2001. Economic tracking portfolios. Journal of Econometrics 105: 161–84. [Google Scholar] [CrossRef]
- Laurent, Jean Paul, and Huyên Pham. 1999. Dynamic programming and mean-variance hedging. Finance and Stochastics 3: 83–110. [Google Scholar] [CrossRef]
- Nakano, Yumiharu. 2004. Efficient hedging with coherent risk measure. Journal of Mathematical Analysis and Applications 293: 345–54. [Google Scholar] [CrossRef]
- Rudloff, Birgit. 2007. Convex hedging in incomplete markets. Applied Mathematical Finance 14: 437–52. [Google Scholar] [CrossRef]
- Rudloff, Birgit. 2009. Coherent hedging in incomplete markets. Quantitative Finance 9: 197–206. [Google Scholar] [CrossRef]
- Schäl, Manfr. 1994. On quadratic cost criteria for option hedging. Mathematics of operations research 19: 121–31. [Google Scholar] [CrossRef]
- Schweizer, Martin. 1992. Mean-variance hedging for general claims. The Annals of Applied Probability 2: 171–79. [Google Scholar] [CrossRef]
- Schweizer, Martin. 1995. Variance-optimal hedging in discrete time. Mathematics of Operations Research 20: 1–32. [Google Scholar] [CrossRef]
- Smith, James E., and Robert F. Nau. 1995. Valuing risky projects: Option pricing theory and decision analysis. Management Science 41: 795–816. [Google Scholar] [CrossRef]
- Vassalou, Maria. 2003. News related to future gdp growth as a risk factor in equity returns. Journal of Financial Economics 68: 47–73. [Google Scholar] [CrossRef]
- Wang, Shaun S., Virginia R. Young, and Harry H. Panjer. 1997. Axiomatic characterization of insurance prices. Insurance: Mathematics and Economics 21: 173–83. [Google Scholar] [CrossRef]
- Yu, Keming, and Rana A. Moye. 2001. Bayesian quantile regression. Statistics & Probability Letters 54: 437–47. [Google Scholar]
1 | All of the results can be easily extended to a probability space with no atoms in an appropriate space—for instance, . |
2 | For technical reasons, we use instead of |
Securities\risk | MV | ||||
---|---|---|---|---|---|
Intercept | 0.0026 | 0.0022 | |||
(0.0001) | (0.0002) | ||||
RF | 0.0072 | −0.6737 | −0.7844 | 0.0058 | 0.0027 |
(0.0558) | (0.0705) | (0.1106) | (0.0041) | (0.0019) | |
MARKET | −0.0048 | −0.0072 | −0.0123 | −0.0038 | −0.0066 |
(0.0029) | (0.0030) | (0.0043) | (0.0013) | (0.0023) | |
SMB | −0.0008 | 0.0131 | 0.0383 | −0.0008 | −0.0003 |
(0.0030) | (0.0048) | (0.0065) | (0.0005) | (0.0002) | |
HML | 0.0022 | 0.0013 | −0.0009 | 0.0038 | 0.0031 |
(0.0042) | (0.0013) | (0.0006) | (0.0016) | (0.0023) | |
UMD | 0.0015 | 0.0006 | 0.0002 | 0.0019 | 0.0023 |
(0.0027) | (0.0006) | (0.0001) | (0.0012) | (0.0015) | |
TERM | 0.0084 | −0.1427 | −0.1440 | 0.0025 | 0.0104 |
(0.0109) | (0.0162) | (0.0263) | (0.0018) | (0.0070) | |
DEF | −0.0265 | 0.1063 | 0.1370 | −0.0246 | −0.0227 |
(0.0242) | (0.0209) | (0.0369) | (0.0080) | (0.0117) | |
Price | 0.0023 | 0.0077 | 0.0093 | 0.0025 | 0.0037 |
(0.0000) | (0.0000) | (0.0001) | (0.0000) | (0.0001) |
Securities\risk | MV | ||||
---|---|---|---|---|---|
Intercept | 0.0021 | 0.0035 | |||
(0.0001) | (0.0002) | ||||
RF | −0.0304 | −0.1473 | −0.6805 | −0.0147 | −0.0296 |
(0.0563) | (0.0621) | (0.1000) | (0.0063) | (0.0380) | |
MARKET | 0.0049 | −0.0020 | 0.0094 | 0.0048 | 0.0038 |
(0.0028) | (0.0017) | (0.0045) | (0.0015) | (0.0032) | |
SMB | 0.0013 | −0.0009 | 0.0042 | 0.0009 | 0.0013 |
(0.0029) | (0.0007) | (0.0033) | (0.0003) | (0.0020) | |
HML | −0.0029 | −0.0142 | 0.0188 | −0.0031 | −0.0031 |
(0.0042) | (0.0042) | (0.0072) | (0.0012) | (0.0037) | |
UMD | −0.0008 | −0.0346 | −0.0011 | −0.0007 | −0.0007 |
(0.0027) | (0.0031) | (0.0008) | (0.0003) | (0.0013) | |
TERM | −0.0167 | −0.0664 | −0.1187 | −0.0284 | −0.0166 |
(0.0109) | (0.0123) | (0.0239) | (0.0065) | (0.0133) | |
DEF | 0.0205 | 0.1095 | 0.2810 | 0.0222 | 0.0226 |
(0.0244) | (0.0194) | (0.0305) | (0.0073) | (0.0186) | |
Price | 0.0023 | 0.0075 | 0.0110 | 0.0027 | 0.0035 |
(0.0000) | (0.0000) | (0.0001) | (0.0000) | (0.0001) |
Securities\risk | MV | ||||
---|---|---|---|---|---|
Intercept | −0.0000 | −0.0000 | |||
(0.0012) | (0.0018) | ||||
RF | 0.3958 | 0.3782 | 0.3696 | 0.3886 | 0.3883 |
(0.0922) | (0.0877) | (0.1094) | (0.0526) | (0.0898) | |
MARKET | 0.0011 | −0.0023 | −0.0016 | 0.0018 | 0.0006 |
(0.0043) | (0.0061) | (0.0095) | (0.0005) | (0.0006) | |
SMB | 0.0034 | 0.0047 | 0.0033 | 0.0026 | 0.0028 |
(0.0048) | (0.0092) | (0.0138) | (0.0008) | (0.0016) | |
HML | 0.0071 | 0.0010 | 0.0013 | 0.0091 | 0.0078 |
(0.0053) | (0.0227) | (0.0230) | (0.0021) | (0.0042) | |
UMD | −0.0038 | −0.0017 | −0.0015 | −0.0065 | −0.0051 |
(0.0042) | (0.0069) | (0.0099) | (0.0018) | (0.0026) | |
TERM | 0.1346 | 0.1055 | 0.1024 | 0.1277 | 0.1532 |
(0.0163) | (0.0194) | (0.0242) | (0.0126) | (0.0251) | |
DEF | −0.0691 | −0.0789 | −0.0756 | −0.0926 | −0.0571 |
(0.0296) | (0.0307) | (0.0341) | (0.0138) | (0.0268) | |
Price | 0.0033 | 0.0057 | 0.0082 | 0.0029 | 0.0044 |
(0.0000) | (0.0001) | (0.0002) | (0.0001) | (0.0002) |
Securities\risk | MV | ||||
---|---|---|---|---|---|
Intercept | 0.0010 | 0.0011 | |||
(0.0000) | (0.0001) | ||||
RF | −0.0616 | 0.0358 | −0.0017 | −0.0572 | −0.0359 |
(0.0335) | (0.0214) | (0.0010) | (0.0119) | (0.0133) | |
MARKET | −0.0008 | 0.0002 | 0.0115 | −0.0011 | −0.0005 |
(0.0018) | (0.0001) | (0.0026) | (0.0003) | (0.0002) | |
SMB | −0.0015 | 0.0030 | −0.0090 | −0.0016 | −0.0027 |
(0.0014) | (0.0015) | (0.0036) | (0.0005) | (0.0011) | |
HML | −0.0005 | 0.0091 | 0.0177 | −0.0004 | −0.0001 |
(0.0023) | (0.0015) | (0.0040) | (0.0001) | (0.0001) | |
UMD | −0.0003 | 0.0026 | 0.0097 | −0.0002 | −0.0000 |
(0.0012) | (0.0010) | (0.0026) | (0.0001) | (0.0000) | |
TERM | −0.0147 | −0.0153 | −0.0169 | −0.0135 | −0.0005 |
(0.0049) | (0.0051) | (0.0082) | (0.0025) | (0.0003) | |
DEF | 0.0503 | 0.1078 | 0.0965 | 0.0498 | 0.0882 |
(0.0125) | (0.0063) | (0.0158) | (0.0052) | (0.0079) | |
Price | 0.0011 | 0.0036 | 0.0048 | 0.0009 | 0.0016 |
(0.0000) | (0.0000) | (0.0001) | (0.0000) | (0.0001) |
Securities\risk | MV | ||||
---|---|---|---|---|---|
Intercept | 0.0007 | 0.0011 | |||
(0.0001) | (0.0001) | ||||
RF | −0.0735 | −0.0323 | −0.0794 | −0.0774 | −0.0736 |
(0.0148) | (0.0291) | (0.0352) | (0.0074) | (0.0104) | |
MARKET | −0.0309 | −0.0313 | −0.0343 | −0.0293 | −0.0313 |
(0.0010) | (0.0015) | (0.0017) | (0.0012) | (0.0016) | |
SMB | −0.0000 | 0.0004 | 0.0000 | −0.0000 | −0.0000 |
(0.0012) | (0.0007) | (0.0000) | (0.0000) | (0.0000) | |
HML | −0.0028 | −0.0026 | −0.0020 | −0.0029 | −0.0032 |
(0.0014) | (0.0021) | (0.0019) | (0.0004) | (0.0005) | |
UMD | −0.0012 | 0.0016 | −0.0028 | −0.0013 | −0.0015 |
(0.0008) | (0.0012) | (0.0015) | (0.0002) | (0.0002) | |
TERM | −0.0036 | 0.0146 | −0.0163 | −0.0035 | −0.0040 |
(0.0029) | (0.0069) | (0.0067) | (0.0006) | (0.0008) | |
DEF | −0.0038 | −0.0152 | 0.0276 | −0.0037 | 0.0010 |
(0.0044) | (0.0083) | (0.0105) | (0.0008) | (0.0002) | |
Price | 0.0007 | 0.0018 | 0.0028 | 0.0007 | 0.0010 |
(0.0000) | (0.0000) | (0.0000) | (0.0000) | (0.0000) |
Securities\risk | MV | ||||
---|---|---|---|---|---|
Intercept | 0.0059 | 0.0080 | |||
(0.0003) | (0.0005) | ||||
RF | −0.2533 | −0.0510 | 0.0696 | −0.2882 | −0.1159 |
(0.1112) | (0.0718) | (0.1184) | (0.0672) | (0.0788) | |
MARKET | 0.0082 | 0.0079 | −0.0067 | 0.0074 | 0.0048 |
(0.0056) | (0.0062) | (0.0102) | (0.0023) | (0.0037) | |
SMB | 0.0256 | 0.0315 | 0.0467 | 0.0288 | 0.0237 |
(0.0080) | (0.0097) | (0.0169) | (0.0051) | (0.0094) | |
HML | 0.0132 | 0.0079 | −0.0096 | 0.0034 | 0.0108 |
(0.0080) | (0.0089) | (0.0141) | (0.0020) | (0.0075) | |
UMD | −0.0034 | 0.0334 | 0.0546 | −0.0032 | −0.0032 |
(0.0052) | (0.0073) | (0.0120) | (0.0015) | (0.0028) | |
TERM | −0.0509 | −0.0581 | −0.0070 | −0.0835 | −0.0819 |
(0.0241) | (0.0223) | (0.0160) | (0.0165) | (0.0263) | |
DEF | −0.0568 | 0.0005 | 0.1904 | −0.0562 | −0.0802 |
(0.0312) | (0.0007) | (0.0648) | (0.0178) | (0.0332) | |
Price | 0.0054 | 0.0159 | 0.0223 | 0.0064 | 0.0083 |
(0.0000) | (0.0001) | (0.0002) | (0.0001) | (0.0002) |
© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Assa, H.; Gospodinov, N. A Robust Approach to Hedging and Pricing in Imperfect Markets. Risks 2017, 5, 36. https://doi.org/10.3390/risks5030036
Assa H, Gospodinov N. A Robust Approach to Hedging and Pricing in Imperfect Markets. Risks. 2017; 5(3):36. https://doi.org/10.3390/risks5030036
Chicago/Turabian StyleAssa, Hirbod, and Nikolay Gospodinov. 2017. "A Robust Approach to Hedging and Pricing in Imperfect Markets" Risks 5, no. 3: 36. https://doi.org/10.3390/risks5030036
APA StyleAssa, H., & Gospodinov, N. (2017). A Robust Approach to Hedging and Pricing in Imperfect Markets. Risks, 5(3), 36. https://doi.org/10.3390/risks5030036