Victoria Steblovskaya, Ph.D., Lucy Kimball, Ph.D., and N. Josephy, Ph.D. School of Mathematical Sciences, Bentley University, 175 Forest Street, Waltham, MA 02452
Significant interest and effort has been applied to the pricing and hedging of financial derivatives in incomplete markets, where no-arbitrage economic conditions produce a multiplicity of 'fair' derivative prices in both discrete and continuous financial models. Typical approaches to hedging in such markets impose a self-financing condition on the set of potential hedge portfolios, and then attempt to optimize a constraint on the economic characteristics of the portfolios in this restricted set. An alternative approach is to widen the set of potential hedging portfolios to include non-self-financing portfolios, which results in intermediate time cash flows. The statistical characteristics of the cash flows can be optimized in the process of selecting a satisfactory hedging portfolio.
Based on theoretical results of Nagaev and Nagaev [NN], we construct a discrete time stock price model where stock price jumps are assumed to be distributed over a bounded interval. This assumption results in an incomplete market. In this setting, a non-self-financing hedging strategy produces a residual cash flow at each time step. We present a two-stage algorithm that first identifies the contour of market-calibrated model parameters, and then constructs a hedge portfolio by numerically solving an economic optimization problem within the contour of model parameters. Our approach to hedge portfolio selection accommodates a wide variety of optimization criteria and can be applied to both path independent and path dependent options with a convex payoff function. Along with theoretical description of our model and algorithm, encouraging numerical results will be presented.
[JKNPS] Josephy, N., Kimball, L., Nagaev, A.V., Pasniewski, M., and Steblovskaya, V.: An Algorithmic Approach to Non-self-financing Hedging in a Discrete Time Incomplete Market, Discrete Mathematics and Applications, 17, 2, 189-207 (2007).
[JKSa] Josephy, N., Kimball, and Steblovskaya, V.: A Time Series Approach to Non-self-financing Hedging in a Discrete Time Incomplete Market, Journal of Applied Mathematics and Stochastic Analysis, vol. 2008, Article ID 275217, 20 pages, 2008. doi:10.1155/2008/275217,
http://www.hindawi.com/getarticle.aspx?doi=10.1155/2008/275217.
[JKSb] Josephy, N., Kimball, L., and Steblovskaya, V.: Optimal Hedging of Path-Dependent Options in Discrete Time Incomplete Market, Communications on Stochastic Analysis, 2, 3, 385-404 (2008).
[NN] Nagaev, A.V. and Nagaev, S.A.: Asymptotics of riskless profit under selling of discrete time call options. Applicationes Mathematicae, 30, 2, 173-191 (2003).
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www.bentley.edu/academics_research/faculty_research/faculty_database/faculty_detail.cfm?id=758323