Malliavin Calculus for Stochastic Strings with Applications To Barrier Options and Optimal Portfolios

Interest rate option hedging portfolios without bank account

“…To apply expression (3), we will need to know the Malliavin derivative of discounted bond prices. The following result corresponds to Proposition 6 of Bueno-Guerrero et al (2017).…”

“…The problem that arises with the application of Theorem 1 is that usually we do not know the process j(t, u) in the martingale representation of V t .To solve this problem, Bueno-Guerrero et al (2015b) develops a Malliavin calculus valid for stochastic string models that allows to obtain the martingale representation (2) in terms of the Malliavin derivative of the payoff [2]. Applying this result, Theorem 1 can be rewritten as follows (Theorem 5 of Bueno-Guerrero et al, 2017).…”

“…In Bueno-Guerrero et al (2017), the hedging portfolio for a bond barrier option is obtained under the same framework presented here, but the authors do not determine whether the hedging portfolio has a bank account part. Their result, which we reproduce here without proof, is the following.…”

“…Theorem 4. (Bueno-Guerrero et al, 2017) Consider an up-and-out call option with maturity T 0 , strike K, barrier H ! K, and written on a zero-coupon bond maturing at T 1 > T 0 .…”

“…Nevertheless, in our framework the source of uncertainty is not Brownian motion but the socalled stochastic string shock [Santa-Clara and Sornette (2001), Bueno-Guerrero et al (2015a)]. 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.…”