# Hedge or Rebalance: Optimal Risk Management with Transaction Costs

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Model

## 4. The Frictionless Case

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

## 5. Approximately Optimal Strategies with Small Transaction Costs

**Proposition**

**1.**

**Proposition**

**2.**

- If $\alpha >r+\gamma {\sigma}^{2}$, then ${C}^{*}$ is monotone increasing in α and monotone decreasing in σ;
- if $\alpha >r+\gamma {\sigma}^{2}$ and ${\gamma}^{-1}\mu /{\sigma}_{F}>\beta {\sigma}_{F},$ then ${C}^{*}$ is monotone decreasing in $\gamma ;$
- If ${\gamma}^{-1}\mu /{\sigma}_{F}>\beta {\sigma}_{F},$ then ${C}^{*}$ is monotone decreasing in both β and ${\sigma}_{F}$;
- ${C}^{*}$ is monotone decreasing in μ for $\mu \in (-\infty ,\beta {\sigma}_{F}^{2}\gamma )$, and is monotone increasing in μ otherwise.

#### The Response of the No-Trading Zone to Shocks

**Proposition**

**3.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Corollary**

**2.**

**Proof.**

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Proofs

**Proof**

**of**

**Lemma**

**1.**

**Proof**

**of**

**Lemma**

**2.**

**Proof**

**of**

**Proposition**

**1.**

**Proof**

**of**

**Proposition**

**3.**

**Proof**

**of**

**Corollary**

**1.**

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1 | For processes with infinite total variation, losses from transaction costs are infinite. |

**Figure 1.**Simulated discrete time approximation of the P&L of the two strategies (the frictionless one, ${s}_{t}$, from Lemma 2; and the optimal one, ${\tilde{s}}_{t},$ of Proposition 1), net after transaction costs. Parameter values are: $\gamma =5,\phantom{\rule{4pt}{0ex}}c=0.2,\phantom{\rule{4pt}{0ex}}\mu =0.05,\phantom{\rule{4pt}{0ex}}r=0,\phantom{\rule{4pt}{0ex}}\sigma =0.02,\phantom{\rule{4pt}{0ex}}{\sigma}_{F}=0.2,\alpha =0.01,\phantom{\rule{4pt}{0ex}}\beta =0.6,\phantom{\rule{4pt}{0ex}}\epsilon =0.01,\phantom{\rule{4pt}{0ex}}{W}_{0}=1.$ Given the simulated paths of the two Brownian motions, the wealth curve reports the cumulative wealth of an agent following the respective strategy, net of costs. Since the frictionless strategy trades too frequently, losses due to costs are larger, and hence, the wealth of an agent following the frictionless strategy is lower than the wealth of an agent following the optimal strategy that account for costs.

**Figure 2.**The dependence of ${C}_{*}$ on the model parameters. In each plot, except for the varying parameter, all other parameters are fixed at their values of $\gamma =5,\phantom{\rule{4pt}{0ex}}c=0.2,\phantom{\rule{4pt}{0ex}}\mu =0.05,\phantom{\rule{4pt}{0ex}}r=0,\phantom{\rule{4pt}{0ex}}\sigma =0.02,\phantom{\rule{4pt}{0ex}}{\sigma}_{F}=0.2,$$\alpha \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.01,\phantom{\rule{4pt}{0ex}}\beta =0.6,\phantom{\rule{4pt}{0ex}}\epsilon =0.01,\phantom{\rule{4pt}{0ex}}{W}_{0}=1.$

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**MDPI and ACS Style**

Gallien, F.; Kassibrakis, S.; Malamud, S.
Hedge or Rebalance: Optimal Risk Management with Transaction Costs. *Risks* **2018**, *6*, 112.
https://doi.org/10.3390/risks6040112

**AMA Style**

Gallien F, Kassibrakis S, Malamud S.
Hedge or Rebalance: Optimal Risk Management with Transaction Costs. *Risks*. 2018; 6(4):112.
https://doi.org/10.3390/risks6040112

**Chicago/Turabian Style**

Gallien, Florent, Serge Kassibrakis, and Semyon Malamud.
2018. "Hedge or Rebalance: Optimal Risk Management with Transaction Costs" *Risks* 6, no. 4: 112.
https://doi.org/10.3390/risks6040112