Friday, October 9, 2009: 4:55 PM
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).
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).