3,708 research outputs found
Optimal hedging of Derivatives with transaction costs
We investigate the optimal strategy over a finite time horizon for a
portfolio of stock and bond and a derivative in an multiplicative Markovian
market model with transaction costs (friction). The optimization problem is
solved by a Hamilton-Bellman-Jacobi equation, which by the verification theorem
has well-behaved solutions if certain conditions on a potential are satisfied.
In the case at hand, these conditions simply imply arbitrage-free
("Black-Scholes") pricing of the derivative. While pricing is hence not changed
by friction allow a portfolio to fluctuate around a delta hedge. In the limit
of weak friction, we determine the optimal control to essentially be of two
parts: a strong control, which tries to bring the stock-and-derivative
portfolio towards a Black-Scholes delta hedge; and a weak control, which moves
the portfolio by adding or subtracting a Black-Scholes hedge. For simplicity we
assume growth-optimal investment criteria and quadratic friction.Comment: Revised version, expanded introduction and references 17 pages,
submitted to International Journal of Theoretical and Applied Finance (IJTAF
``String'' formulation of the Dynamics of the Forward Interest Rate Curve
We propose a formulation of the term structure of interest rates in which the
forward curve is seen as the deformation of a string. We derive the general
condition that the partial differential equations governing the motion of such
string must obey in order to account for the condition of absence of arbitrage
opportunities. This condition takes a form similar to a fluctuation-dissipation
theorem, albeit on the same quantity (the forward rate), linking the bias to
the covariance of variation fluctuations. We provide the general structure of
the models that obey this constraint in the framework of stochastic partial
(possibly non-linear) differential equations. We derive the general solution
for the pricing and hedging of interest rate derivatives within this framework,
albeit for the linear case (we also provide in the appendix a simple and
intuitive derivation of the standard European option problem). We also show how
the ``string'' formulation simplifies into a standard N-factor model under a
Galerkin approximation.Comment: 24 pages, European Physical Journal B (in press
Options hedging under liquidity costs
Following the framework of Cetin, Jarrow and Protter (CJP) we study the problem of super-replication in presence of liquidity costs under additional restrictions on the gamma of the hedging strategies in a generalized Black-Scholes economy. We find that the minimal super-replication price is different than the one suggested by the Black-Scholes formula and is the unique viscosity solution of the associated dynamic programming equation. This is in contrast with the results of CJP who find that the arbitrage free price of a contingent claim coincides with the Black-Scholes price. However, in CJP a larger class of admissible portfolio processes is used and the replication is achieved in the L^2 approximating
sense
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