Tug-of-war, market manipulation, and option pricing. (English) Zbl 1541.91247
Summary: We develop an option pricing model based on a tug-of-war game. This two-player zero-sum stochastic differential game is formulated in the context of a multidimensional financial market. The issuer and the holder try to manipulate asset price processes in order to minimize and maximize the expected discounted reward. We prove that the game has a value and that the value function is the unique viscosity solution to a terminal value problem for a parabolic partial differential equation involving the nonlinear and completely degenerate infinity Laplace operator.
{© 2014 Wiley Periodicals, Inc.}
{© 2014 Wiley Periodicals, Inc.}
MSC:
| 91G20 | Derivative securities (option pricing, hedging, etc.) |
| 35Q91 | PDEs in connection with game theory, economics, social and behavioral sciences |
| 91A15 | Stochastic games, stochastic differential games |
| 91A05 | 2-person games |
| 91A80 | Applications of game theory |
Keywords:
infinity Laplace; nonlinear parabolic partial differential equation; option pricing; stochastic differential game; tug-of-warReferences:
| [1] | Alfonsi, A., A.Fruth, and A.Schied (2010): Optimal Execution Strategies in Limit Order Books with General Shape Functions, Quant. Financ. 10(2), 143-157. · Zbl 1185.91199 |
| [2] | Atar, R., and A.Budhiraja (2010): A Stochastic Differential Game for the Inhomogeneous ∞‐Laplace Equation, Ann. Probab. 38(2), 498-531. · Zbl 1192.91025 |
| [3] | Atar, R., and A.Budhiraja (2011): On Near Optimal Trajectories for a Game Associated with the ∞‐Laplacian, Probab. Theory Rel. Fields151(3-4), 509-528. · Zbl 1233.91026 |
| [4] | Avellaneda, M., ALevy, and A.Paras (1995): Pricing and Hedging Derivative Securities in Markets with Uncertain Volatilities, Appl. Math. Financ. 2, 73-88. · Zbl 1466.91323 |
| [5] | Avellaneda, M., and A.Paras (1996): Managing the Volatility Risk of Portfolios of Derivative Securities: The Lagrangian Uncertain Volatility Model, Appl. Math. Financ. 3, 31-52. · Zbl 1097.91514 |
| [6] | Back, K., H.Cao, and G.Willard (2000): Imperfect Competition Among Informed Traders, J. Financ. 55, 2117-2155. |
| [7] | Banerjee, A., and N.Garofalo (2012): Gradient Bounds and Monotonicity of the Energy for Some Nonlinear Singular Diffusion Equations, preprint arXiv:1202.5068. |
| [8] | Bernhard, P., J. CEngwerda, B.Roorda, J. M.Schumacher, V.Kolokoltsov, P.Saint‐Pierre, and J.‐P.Aubin (2012): The Interval Market Model in Mathematical Finance: Game‐Theoretic Methods, Basel: Birkhäuser. |
| [9] | Black, F., and M.Scholes (1973): The Pricing of Options and Corporate Liabilities, J. Pol. Econ. 81, 637-659. · Zbl 1092.91524 |
| [10] | Brunnermeier, M., and L.Pedersen (2005): Predatory Trading, J. Financ. 60(4), 1825-1863. |
| [11] | Buckdahn, R., and J.Li (2008): Stochastic Differential Games and Viscosity Solutions of Hamilton‐Jacobi‐Bellman‐Isaacs Equations, SIAM J. Control Optim. 47(1), 444-475. · Zbl 1157.93040 |
| [12] | Buckdahn, R., P.Cardaliaguet, and C.Rainer (2004): Nash Equilibrium Payoffs for Nonzerosum Stochastic Differential Games, SIAM J. Control Optim. 43(2), 624-642. · Zbl 1101.91010 |
| [13] | Carlin, B. I., M. S.Lobo, and S.Viswanathan (2007): Episodic Liquidity Crises: Cooperative and Predatory Trading, J. Financ. 65(5), 2235-2274. |
| [14] | Cetin, U., R.Jarrow, and P.Protter (2004): Liquidity Risk and Arbitrage Pricing Theory, Financ. Stoch. 8(3), 311-341. · Zbl 1064.60083 |
| [15] | Chau, M., and D.Vayanos (2008): Strong‐Form Efficiency with Monopolistic Insiders, Rev. Financ. Stud. 21, 2275-2306. |
| [16] | Chen, Y. G., Y.Giga, and S.Goto (1991): Uniqueness and Existence of Viscosity Solutions of Generalized Mean Curvature Flow Equations, J. Differ. Geom. 33(3), 749-786. · Zbl 0696.35087 |
| [17] | Crandall, M. G., H.Ishii, and P.‐L.Lions (1992): User’s Guide to Viscosity Solutions of Second Order Partial Differential Equations, Bull. Am. Math. Soc. 27(1), 1-67. · Zbl 0755.35015 |
| [18] | Does, K. (2011): An Evolution Equation Involving the Normalized p‐Laplacian, Commun. Pure Appl. Anal. 10(1), 361-396. · Zbl 1229.35132 |
| [19] | Evans, L. C., and R. F.Gariepy (1992): Measure Theory and Fine Properties of Functions, Boca Raton, FL: CRC Press. · Zbl 0804.28001 |
| [20] | Evans, L. C., and J.Spruck (1991): Motion of Level Sets by Mean Curvature I, J. Differ. Geom. 33(3), 635-681. · Zbl 0726.53029 |
| [21] | Fleming, W. H., and P. E.Souganidis (1989): On the Existence of Value Functions of Two‐Player, Zero‐Sum Stochastic Differential Games, Indiana Univ. Math. J. 38(2), 293-314. · Zbl 0686.90049 |
| [22] | Foster, F. D., and S.Viswanathan (1996): Strategic Trading When Agents Forecast the Forecasts of Others, J. Finance51, 1437-1478. |
| [23] | Friedman, A. (1972): Stochastic Differential Games, J. Differ. Equat. 11, 79-108. · Zbl 0208.39402 |
| [24] | Gallmeyer, M., and D.Seppi (2000): Derivative Security Induced Price Manipulation, Working Paper, Carnegie Mellon University. |
| [25] | Giga, Y. (2006): Surface Evolution Equations. A Level Set Approach, Basel: Birkhäuser. · Zbl 1096.53039 |
| [26] | Giga, Y., S.Goto, H.Ishii, and M.‐H.Sato (1991): Comparison Principle and Convexity Preserving Properties for Singular Degenerate Parabolic Equations on Unbounded Domains, Indiana Univ. Math. J. 40(2), 443-470. · Zbl 0836.35009 |
| [27] | Hamadene, S., and J. P.Lepeltier (1995): Zero‐Sum Stochastic Differential Games and Backward Equations, Syst. Control Lett. 24, 259-263. · Zbl 0877.93125 |
| [28] | Ishii, H. (1995): On the Equivalence of Two Notions of Weak Solutions, Viscosity Solutions and Distribution Solutions, Funkcial. Ekvac. 38(1), 101-120. · Zbl 0833.35053 |
| [29] | Jarrow, R. (1994): Derivative Security Markets, Market Manipulation, and Option Pricing Theory, J. Financ. Quant. Anal. 29(2), 241-261. |
| [30] | Julin, V., and P.Juutinen (2012): A New Proof for the Equivalence of Weak and Viscosity Solutions for the p‐Laplace Equation, Comm. Partial Differ. Equat. 37(5), 934-946. · Zbl 1260.35069 |
| [31] | Juutinen, P., and B.Kawohl (2006): On the Evolution Governed by the Infinity Laplacian, Math. Ann. 335(4), 819-851. · Zbl 1110.35037 |
| [32] | Kawohl, B., J. J.Manfredi, and M.Parviainen (2012): Solutions of Nonlinear PDEs in the Sense of Averages, J. Math. Pures Appl. 97(2), 173-188. · Zbl 1236.35083 |
| [33] | Kolokoltsov, V. N. (2013): Game Theoretic Analysis of Incomplete Markets: Emergence of Probabilities, Nonlinear and Fractional Black‐Scholes Equations, Risk Decision Anal. 4, 131-161. · Zbl 1409.91239 |
| [34] | Krylov, N. V. (1976): Sequences of Convex Functions and Estimates of the Maximum of the Solution of a Parabolic Equation, Siberian Math. J. 17(2), 226-236. · Zbl 0362.35038 |
| [35] | Krylov, N. V. (2009): Controlled Diffusion Processes, Berlin: Springer‐Verlag. · Zbl 1171.93004 |
| [36] | Kumar, P., and D.Seppi (1992): Futures Manipulation with Cash Settlement, J. Financ. 47(4), 1485-1502. |
| [37] | Kyle, A. S. (1985): Continuous Auctions and Insider Trading, Econometrica53(6), 1315-1335. · Zbl 0571.90010 |
| [38] | Lindqvist, P. (2012): Regularity of Supersolutions, Lecture Notes Math. 2045, 73-131. · Zbl 1252.35002 |
| [39] | Lyons, T. (1995): Uncertain Volatility and the Risk‐Free Synthesis of Derivatives, Appl. Math. Financ. 2, 117-133. · Zbl 1466.91347 |
| [40] | Manfredi, J. J., M.Parviainen, and J. D.Rossi (2010): An Asymptotic Mean Value Characterization for a Class of Nonlinear Parabolic Equations Related to Tug‐of‐War Games, SIAM J. Math. Anal. 42(5), 2058-2081. · Zbl 1231.35107 |
| [41] | Manfredi, J. J., M.Parviainen, and J. D.Rossi (2012): On the Definition and Properties of p‐Harmonious Functions, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 11(2), 215-241. · Zbl 1252.91014 |
| [42] | McEneaney, W. M. (1997): A Robust Control Framework for Option Pricing, Math. Oper. Res. 22, 201-221. · Zbl 0871.90010 |
| [43] | Merton, R. C. (1973): Theory of Rational Option Pricing, Bell J. Econ. Manag. Sci. 4, 141-183. · Zbl 1257.91043 |
| [44] | Nisio, M. (1988): Stochastic Differential Games and Viscosity Solutions of Isaacs Equations, Nagoya Math. J. 110, 163-184. · Zbl 0850.93741 |
| [45] | Peres, Y., O.Schramm, S.Sheffield, and D.Wilson (2009): Tug‐of‐War and the Infinity Laplacian, J. Am. Math. Soc. 22, 167-210. · Zbl 1206.91002 |
| [46] | Peres, Y., and S.Sheffield (2008): Tug‐of‐War with Noise: A Game Theoretic View of the p‐Laplacian, Duke Math. J. 145(1), 91-120. · Zbl 1206.35112 |
| [47] | Pirrong, C. (2001): Manipulation of Cash‐Settled Futures Contracts, J. Bus. 74(2): 221-244. |
| [48] | Rogers, L. C. G., and S.Singh (2010): The Cost of Illiquidity and Its Effects on Hedging, Math. Financ. 20(4), 597-615. · Zbl 1232.91635 |
| [49] | Schied, A., and T.Schöneborn (2007): Liquidation in the Face of Adversity: Stealth vs. Sunshine Trading, Predatory Trading vs. Liquidity Provision. Working Paper. |
| [50] | Swiech, A. (1996): Another Approach to the Existence of Value Functions of Stochastic Differential Games, J. Math. Anal. Appl. 204(3), 884-897. · Zbl 0870.90107 |
| [51] | Ziegler, A. (2004): A Game Theory Analysis of Options. Corporate Finance and Financial Intermediation in Continuous Time, Berlin: Springer‐Verlag. · Zbl 1103.91042 |
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