Risk Measures beyond Frictionless Markets
Abstract.
We develop a general theory of risk measures to determine the optimal amount of capital to raise and invest in a portfolio of reference traded securities in order to meet a prespecified regulatory requirement. The distinguishing feature of our approach is that we embed portfolio constraints and transaction costs into the securities market. As a consequence, the property of translation invariance, which plays a key role in the classical theory, ceases to hold. We provide a comprehensive analysis of relevant properties, such as star shapedness, positive homogeneity, convexity, quasiconvexity, subadditivity, and lower semicontinuity. In addition, we establish dual representations for convex and quasiconvex risk measures. In the convex case, the absence of a special kind of arbitrage opportunity allows one to obtain dual representations in terms of pricing rules that respect market bid-ask spreads and assign a strictly positive price to each nonzero position in the regulator’s acceptance set.
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References
C. Aliprantis and K. Border (2006), Infinite Dimensional Analysis: A Hitchhiker’s Guide, Springer.
M. Arduca and C. Munari (2023), Fundamental theorem of asset pricing with acceptable risk in markets with frictions, Finance Stoch., 27, pp. 831–862.
P. Artzner, F. Delbaen, J. Eber, and D. Heath (1999), Coherent measures of risk, Math. Finance, 9, pp. 203–228.
P. Artzner, F. Delbaen, and P. Koch-Medina (2009), Risk measures and efficient use of capital, Astin Bull., 39, pp. 101–116.
F. Astic and N. Touzi (2007), No arbitrage conditions and liquidity, J. Math. Econ., 43, pp. 692–708.
M. Baes, P. Koch-Medina, and C. Munari (2020), Existence, uniqueness, and stability of optimal payoffs of eligible assets, Math. Finance, 30, pp. 128–166.
V. Barbu and T. Precupanu (2012), Convexity and Optimization in Banach Spaces, 4th ed., Springer.
F. Bellini, B. Klar, A. Müller, and E. Rosazza Gianin (2014), Generalized quantiles as risk measures, Insurance, 54, pp. 41–48.
F. Bellini, P. Koch-Medina, C. Munari, and G. Svindland (2021), Law-invariant functionals that collapse to the mean, Math. Econ., 98, pp. 83–91.
A. Bernardo and O. Ledoit (2000), Gain, loss, and asset pricing, J. Polit. Econ., 108, pp. 144–172.
V. Bignozzi, M. Burzoni, and C. Munari (2020), Risk measures based on benchmark loss distributions, J. Risk Insurance, 87, pp. 437–475.
J. Borwein and Q. Zhu (2004), Techniques of Variational Analysis, Springer.
J. M. Borwein (1987), Automatic continuity and openness of convex relations, Proc. Amer. Math. Soc., 99, pp. 49–55.
B. Bouchard, Y. M. Kabanov, and N. Touzi (2001), Option pricing by large risk aversion utility under transaction costs, Decis. Econ. Finance, 24, pp. 127–136.
M. Broadie, J. Cvitanić, and H. M. Soner (1998), Optimal replication of contingent claims under portfolio constraints, Rev. Financial Stud., 11, pp. 59–79.
M. Burzoni, C. Munari, and R. Wang (2022), Adjusted expected shortfall, J. Bank. Finance, 134, 106297.
P. Carr, H. Geman, and D. B. Madan (2001), Pricing and hedging in incomplete markets, J. Financ. Econ., 62, pp. 131–167.
E. Castagnoli, G. Cattelan, F. Maccheroni, C. Tebaldi, and R. Wang, (2022), Star-shaped risk measures, Oper. Res., 70, pp. 2637–2654.
U. Çetin and L. C. G. Rogers (2007), Modeling liquidity effects in discrete time, Math. Finance, 17, pp. 15–29.
A. Černý and S. Hodges (2002), The theory of good-deal pricing in financial markets, in Mathematical Finance—Bachelier Congress 2000: Selected Papers from the First World Congress of the Bachelier Finance Society, Springer Finance, H. Geman et al., eds., Springer, pp. 175–202.
S. Cerreia-Vioglio, F. Maccheroni, M. Marinacci, and L. Montrucchio (2011), Risk measures: Rationality and diversification, Math. Finance, 21, pp. 743–774.
P. Cheridito, M. Kupper, and L. Tangpi (2017), Duality formulas for robust pricing and hedging in discrete time, SIAM J. Financial Math., 8, pp. 738–765.
A. S. Cherny (2008), Pricing with coherent risk, Theory Probab. Appl., 52, pp. 389–415.
S. A. Clark (1993), The valuation problem in arbitrage price theory, J. Math. Econ., 22, pp. 463–478.
J. Cochrane and J. Saa-Requejo (2000), Beyond arbitrage: Good-deal asset price bounds in incomplete markets, J. Polit. Econ., 108, pp. 79–119.
R. Cont, R. Deguest, and G. Scandolo (2010), Robustness and sensitivity analysis of risk measurement procedures, Quant. Finance, 10, pp. 593–606.
F. Delbaen (2002), Coherent risk measures on general probability spaces, in Advances in Finance and Stochastics, K. Sandmann and P. J. Schönbucher, eds., Springer, pp. 1–37.
A. Doldi and M. Frittelli (2023), Entropy martingale optimal transport and nonlinear pricing–hedging duality, Finance Stoch., 27, pp. 255–304.
S. Drapeau and M. Kupper (2013), Risk preferences and their robust representation, Math. Oper. Res., 38, pp. 28–62.
W. Farkas, P. Koch-Medina, and C. Munari (2014a), Beyond cash-additive risk measures: When changing the numéraire fails, Finance Stoch., 18, pp. 145–173.
W. Farkas, P. Koch-Medina, and C. Munari (2014b), Capital requirements with defaultable securities, Insurance, 55, pp. 58–67.
W. Farkas, P. Koch-Medina, and C. Munari (2015), Measuring risk with multiple eligible assets, Math. Financ. Econ., 9, pp. 3–27.
H. Föllmer and A. Schied (2002), Convex measures of risk and trading constraints, Finance Stoch., 6, pp. 429–447.
H. Föllmer and A. Schied (2016), Stochastic Finance: An Introduction in Discrete Time, 4th ed., De Gruyter.
M. Frittelli and E. Rosazza Gianin (2002), Putting order in risk measures, J. Bank. Finance, 26, pp. 1473–1486.
M. Frittelli and G. Scandolo (2006), Risk measures and capital requirements for processes, Math. Finance, 16, pp. 589–612.
N. Gao, D. Leung, C. Munari, and F. Xanthos (2018), Fatou property, representations, and extensions of law-invariant risk measures on general Orlicz spaces, Finance Stoch., 22, pp. 395–415.
S. Jaschke and U. Küchler (2001), Coherent risk measures and good-deal bounds, Finance Stoch., 5, pp. 181–200.
E. Jouini and H. Kallal (1999), Viability and equilibrium in securities markets with frictions, Math. Finance, 9, pp. 275–292.
E. Jouini, W. Schachermayer, and N. Touzi (2006), Law invariant risk measures have the Fatou property, in Advances in Mathematical Economics, S. Kusuoka and A. Yamazaki, eds., Springer, pp. 49–71.
Y. Kabanov (1999), Hedging and liquidation under transaction costs in currency markets, Finance Stoch., 3, pp. 237–248.
K. Kaval and I. Molchanov (2006), Link-save trading, J. Math. Econ., 42, pp. 710–728.
D. Kreps (1981), Arbitrage and equilibrium in economies with infinitely many commodities, J. Math. Econ., 8, pp. 15–35.
F.-B. Liebrich and G. Svindland (2019), Risk sharing for capital requirements with multidimensional security markets, Finance Stoch., 23, pp. 925–973.
I. Molchanov (2005), Theory of Random Sets, Springer.
T. Pennanen (2011a), Arbitrage and deflators in illiquid markets, Finance Stoch., 15, pp. 57–83.
T. Pennanen (2011b), Dual representation of superhedging costs in illiquid markets, Math. Financ. Econ., 5, pp. 233–248.
T. Pennanen and A.-P. Perkkiö (2018), Convex duality in optimal investment and contingent claim valuation in illiquid markets, Finance Stoch., 22, pp. 733–771.
J.-P. Penot and M. Volle (1990), On quasi-convex duality, Math. Oper. Res., 15, pp. 597–625.
A. L. Peressini (1967), Ordered Topological Vector Spaces, Harper & Row.
H. Pham and N. Touzi (1999), The fundamental theorem of asset pricing with cone constraints, J. Math. Econ., 31, pp. 265–279.
R. T. Rockafellar and R. J.-B. Wets (2009), Variational Analysis, Grundlehren Math. Wiss. 317, corrected 3rd printing, Springer-Verlag.
W. Schachermayer (2004), The fundamental theorem of asset pricing under proportional transaction costs in finite discrete time, Math. Finance, 14, pp. 19–48.
J. Staum (2004), Fundamental theorems of asset pricing for good deal bounds, Math. Finance, 14, pp. 141–161.
J.-A. Yan (1980), Caractérisation d’une classe d’ensembles convexes de \(L^1\) ou \(H^1\), in Séminaire de Probabilités (Strasbourg) XIV: 1978/79, Lecture Notes in Math. 784, J. Azéma and M. Yor, eds., Springer-Verlag, pp. 220–222.
C. Zălinescu (2002), Convex Analysis in General Vector Spaces, World Scientific.
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Published In
SIAM Journal on Financial Mathematics
Pages: 537 - 570
DOI: 10.1137/22M1540090
ISSN (online): 1945-497X
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© 2024 Society for Industrial and Applied Mathematics.
History
Submitted: 8 December 2022
Accepted: 16 February 2024
Published online: 7 June 2024
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