Abstract
Hedging down-and-out puts (and up-and-out calls), where the maximum payoff is reached just before a barrier is hit that would render the claim worthless afterwards, is challenging. All hedging methods potentially lead to large errors when the underlying is already close to the barrier and the hedge portfolio can only be adjusted in discrete time intervals. In this paper, we analyze this hedging situation, especially the case of overnight trading gaps. We show how a position in a short-term vanilla call option can be used for efficient hedging. Using a mean-variance hedging approach, we calculate optimal hedge ratios for both the underlying and call options as hedge instruments. We derive semi-analytical formulas for optimal hedge ratios in a Black–Scholes setting for continuous trading (as a benchmark) and in the case of trading gaps. For more complex models, we show in a numerical study that the semi-analytical formulas can be used as a sufficient approximation, even when stochastic volatility and jumps are present.
Highlights
This paper analyzes time-discrete hedging of European down-and-out put options near the barrier in the case of overnight trading gaps
We show that the existence of trading gaps has a significant impact on mean-variance optimal hedge ratios compared to discrete hedging in continuous time
We introduce the idea of using opposite vanilla options to dynamically hedge barrier options
Summary
This paper analyzes time-discrete hedging of European down-and-out put options near the barrier in the case of overnight trading gaps. Even robust techniques may lead to very high hedging errors when this is not possible, for example, in the case of trading gaps Maruhn et al (2011) Another strand of literature considers delta hedging for more complex models. This finding raises the question of whether the comparably simple semi-analytical formulas can yield reasonable results in more complicated situations Such an approach—assuming geometric Brownian motion the actual process is different—leads to fairly small hedging errors, despite the not correctly specified model. The remainder of this paper is structured as follows: Section 2 describes the hedging problem and our approach for down-and-out puts and provides a total of seven different hedging strategies These include no hedging, time-continuous Black–Scholes delta hedging, one static approach as well as mean-variance delta hedging using the underlying or vanilla calls with different maturities.
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