Optimal Discretization of Hedging Strategies with Directional Views

Abstract

We consider the hedging error of a derivative due to discrete trading in the presence of a drift in the dynamics of the underlying asset. We suppose that the trader wishes to find rebalancing times for the hedging portfolio which enable him to keep the discretization error small while taking advantage of market trends. Assuming that the portfolio is readjusted at high frequency, we introduce an asymptotic framework in order to derive optimal discretization strategies. More precisely, we formulate the optimization problem in terms of an asymptotic expectation-error criterion. In this setting, the optimal rebalancing times are given by the hitting times of two barriers whose values can be obtained by solving a linear-quadratic optimal control problem. In specific contexts such as in the Black-Scholes model, explicit expressions for the optimal rebalancing times can be derived.

Optimal Hedging of a Perpetual American Put with a Single Trade

“…Besides the seminal work by Föllmer and Schweizer [13], which laid the foundations of quadratic hedging for claims in incomplete markets, we also mention here work by Schweizer [30], Schäl [29] and Mercurio and Vorst [22], who focus on approximation of random variables (representing European claims at maturity) via stochastic integrals for discrete-time processes. More recently in the mathematical literature we find numerous papers concerning the asymptotic optimality of discrete-time hedging strategies as the number of hedging opportunities tends to infinity (see, e.g., Fukasawa [14], Gobet and Landon [15], Rosenbaum and Tankov [27], Cai et al [5]). Those papers also approach the problem by approximating random variables with stochastic integrals for discrete-time processes.…”

A Scaling Limit for Utility Indifference Prices in the Discretized Bachelier Model

“…Although the topic of discrete time hedging in the Brownian setting was largely studied, see for instance, [3,17,13,18,14,15,10,16,11,4], these papers studied the optimal discretization of given hedging strategies such as the delta hedging strategy. In the present study, instead of tracking a given hedging strategy we follow the well known approach of utility indifference pricing which is commonly used in the setup of incomplete markets (see [5] and the references therein).…”

The accounting of illiquidity and transaction costs during the delta-hedging

“…Использовались следующие цены исполнения опционов: 𝐾 0 = 0.40, 𝐾 1 = 0.32, 𝐾 2 = 0.48. Интересной переспективой развития полученных в данной работе результатов представляется качественное и численное изучение уравнения (13). Это новое нелинейное уравнение в частных производных типа Блэка -Шоулса, которое учитывает нехватку ликвидности в LOB.…”