Bond Market Completeness Under Stochastic Strings with Distribution-Valued Strategies

Interest rate option hedging portfolios without bank account

“…The main theorem in Bueno-Guerrero et al (2015b), stated as follows, gives the explicit expression for the bond part of the hedging portfolio.…”

“…Theorem 1. In the stochastic string model of Bueno-Guerrero et al (2015b), the market is complete and the generalized function h(t,•) in the hedging portfolio is given by:…”

“…This fact led to the development of infinitedimensional models under which natural strategies (those that are composed of bonds with maturities less than or equal to the longest maturity of the bonds underlying the claim) could be obtained. Examples of these infinite-dimensional models are Björk et al (2002), Carmona andTehranchi (2004), De Donno andPratelli (2004), Privault and Teng (2012) and Bueno-Guerrero et al (2015b). In the models in which an explicit expression for the hedging portfolio is available (the last two of the previous references), only the bond part is given and the bank account part is obtained by difference with the current value of the derivative.…”

“…In Bueno-Guerrero et al (2015b), an explicit expression for the hedging portfolio of any contingent claim is obtained. In this expression appears the integrand in the martingale representation of the conditional expectation of the discounted payoff under the equivalent martingale measure.…”

“…Fortunately, in Bueno-Guerrero et al (2017), a Malliavin calculus for stochastic string shocks has been developed. In that paper, the corresponding version of the Clark-Ocone formula is obtained and applied to rewrite the hedging portfolio result of Bueno-Guerrero et al (2015b) in terms of Malliavin derivatives of the discounted payoff. We use this last approach to obtain a necessary and sufficient condition for the hedging portfolio to have no bank account part.…”