Super-replication in stochastic volatility models under portfolio constraints

We study a financial market with incompleteness arising from two sources: stochastic volatility and portfolio constraints. The latter are given in terms of bounds imposed on the borrowing and short-selling of a `hedger' in this market, and can be described by a closed convex set K. We find explicit characterizations of the minimal price needed to super-replicate European-type contingent claims in this framework. The results depend on whether the volatility is bounded away from zero and/or infinity, and also, on if we have linear dynamics for the stock price process, and whether volatility process depends on the stock price. We use a previously known representation of the minimal price as a supremum of the prices in the corresponding shadow markets, and we derive a PDE characterization of that representation.

[1]  丸山 徹.  Convex Analysisの二,三の進展について , 1977 .

[2]  N. Krylov.  Controlled Diffusion Processes , 1980 .

[3]  P. Lions.  Optimal control of diffusion processes and hamilton–jacobi–bellman equations part 2 : viscosity solutions and uniqueness , 1983 .

[4]  P. L. Linos.  Optimal control of diffustion processes and hamilton-jacobi-bellman equations part I: the dynamic programming principle and application , 1983 .

[5]  James B. Wiggins.  Option values under stochastic volatility: Theory and empirical estimates , 1987 .

[6]  Ioannis Karatzas,et al.  Brownian Motion and Stochastic Calculus , 1987 .

[7]  Alan G. White,et al.  The Pricing of Options on Assets with Stochastic Volatilities , 1987 .

[8]  W. Fleming,et al.  Controlled Markov processes and viscosity solutions , 1992 .

[9]  P. Lions,et al.  User’s guide to viscosity solutions of second order partial differential equations , 1992, math/9207212.

[10]  Jakša Cvitanić,et al.  Hedging Contingent Claims with Constrained Portfolios , 1993 .

[11]  M. Avellaneda,et al.  Pricing and hedging derivative securities in markets with uncertain volatilities , 1995 .

[12]  N. Karoui,et al.  Dynamic Programming and Pricing of Contingent Claims in an Incomplete Market , 1995 .

[13]  E. Jouini,et al.  ARBITRAGE IN SECURITIES MARKETS WITH SHORT-SALES CONSTRAINTS , 1995 .

[14]  H. Soner,et al.  Optimal Replication of Contingent Claims Under Portfolio Constraints , 1996 .

[15]  Bruce D. Grundy,et al.  General Properties of Option Prices , 1996 .

[16]  I. Karatzas,et al.  On the pricing of contingent claims under constraints , 1996 .

[17]  Ying Hu,et al.  Pricing of American Contingent Claims with Jump Stock Price and Constrained Portfolios , 1998, Math. Oper. Res..

[18]  S. Shreve,et al.  Robustness of the Black and Scholes Formula , 1998 .

[19]  R. Frey,et al.  Bounds on European Option Prices under Stochastic Volatility , 1999 .