Risk Minimizing Option Pricing for a Class of Exotic Options in a Markov-Modulated Market

We address risk minimizing option pricing in a regime switching market where the floating interest rate depends on a finite state Markov process. The growth rate and the volatility of the stock also depend on the Markov process. Using the minimal martingale measure, we show that the locally risk minimizing prices for certain exotic options satisfy a system of Black-Scholes partial differential equations with appropriate boundary conditions. We find the corresponding hedging strategies and the residual risk. We develop suitable numerical methods to compute option prices.

[1]  E. Süli,et al.  Numerical Solution of Partial Differential Equations , 2014 .

[2]  Paul Wilmott,et al.  Paul Wilmott on Quantitative Finance , 2010 .

[3]  S. Shreve,et al.  Methods of Mathematical Finance , 2010 .

[4]  Barrier option pricing for assets with Markov-modulated dividends , 2006 .

[5]  L. C. G. Rogers,et al.  Option Pricing With Markov-Modulated Dynamics , 2006, SIAM J. Control. Optim..

[6]  PricingDavid D. Yao,et al.  A Regime-Switching Model for European Option , 2006 .

[7]  Rogemar S. Mamon,et al.  Explicit solutions to European options in a regime-switching economy , 2005, Oper. Res. Lett..

[8]  Gang George Yin,et al.  Markowitz's mean-variance portfolio selection with regime switching: from discrete-time models to their continuous-time limits , 2004, IEEE Transactions on Automatic Control.

[9]  Xin Guo,et al.  Closed-Form Solutions for Perpetual American Put Options with Regime Switching , 2004, SIAM J. Appl. Math..

[10]  Constrained Stochastic Estimation Algorithms for a Class of Hybrid Stock Market Models , 2003 .

[11]  Robert J. Elliott,et al.  Robust parameter estimation for asset price models with Markov modulated volatilities , 2003 .

[12]  Gang George Yin,et al.  Markowitz's Mean-Variance Portfolio Selection with Regime Switching: A Continuous-Time Model , 2003, SIAM J. Control. Optim..

[13]  Robert J. Elliott,et al.  American options with regime switching , 2002 .

[14]  M. Musiela,et al.  Martingale Methods in Financial Modelling , 2002 .

[15]  D. Heath,et al.  A Comparison of Two Quadratic Approaches to Hedging in Incomplete Markets , 2001 .

[16]  Xin Guo,et al.  Information and option pricings , 2001 .

[17]  Hailiang Yang,et al.  European option pricing when the riskfree interest rate follows a jump process , 2000 .

[18]  M. Schweizer.  A guided tour through quadratic hedging approaches , 1999 .

[19]  G. Kallianpur,et al.  Introduction to option pricing theory , 1999 .

[20]  Huyên Pham,et al.  Mean-variance hedging for continuous processes: New proofs and examples , 1998, Finance Stochastics.

[21]  Wolfgang J. Runggaldier,et al.  Mean-variance hedging of options on stocks with Markov volatilities , 1995 .

[22]  H. Föllmer,et al.  Hedging of contingent claims under incomplete in-formation , 1991 .

[23]  M. Mariton,et al.  Jump Linear Systems in Automatic Control , 1992 .

[24]  Darrell Duffie,et al.  Implementing Arrow-Debreu equilbria by continuous trading of a few long-lived securities , 1985 .

[25]  D. Sondermann.  Hedging of non-redundant contingent claims , 1985 .

[26]  L. Rogers.  Stochastic differential equations and diffusion processes: Nobuyuki Ikeda and Shinzo Watanabe North-Holland, Amsterdam, 1981, xiv + 464 pages, Dfl.175.00 , 1982 .

[27]  A. Bensoussan.  Stochastic control by functional analysis methods , 1982 .

[28]  John Tydeman,et al.  A note on the Kounias and Marin Method of Best Linear Bonferroni Bounds , 1980 .

[29]  O. Ladyženskaja.  Linear and Quasilinear Equations of Parabolic Type , 1968 .

[30]  G. Hedstrom,et al.  Numerical Solution of Partial Differential Equations , 1965 .