Hedging Strategies
Apr 4, 2025
The Quanted Round-up is a curated summary that covers relevant research on key topics in quantitative financial decision-making.
Highlights
This edition looks at how quants are refining hedging strategies to deal with real-world frictions, from market impact to instrument selection, in pursuit of more effective risk control. Across the board, the focus is on making hedging more robust under realistic trading conditions.
Gamma and vega hedging using deep distributional reinforcement learning
Jay Cao, Jacky Chen, Soroush Farghadani, John Hull, Zissis Poulos, Zeyu Wang & Jun Yuan
We show how reinforcement learning can be used in conjunction with quantile regression to develop a hedging strategy for a trader responsible for derivatives that arrive stochastically and depend on a single underlying asset. We assume that the trader makes the portfolio delta-neutral at the end of each day by taking a position in the underlying asset. We focus on how trades in options can be used to manage gamma and vega. The option trades are subject to transaction costs. We consider three different objective functions. We reach conclusions on how the optimal hedging strategy depends on the trader's objective function, the level of transaction costs, and the maturity of the options used for hedging. We also investigate the robustness of the hedging strategy to the process assumed for the underlying asset.
Delta hedging bitcoin options with a smile
Carol Alexander & Arben Imeraj
We analyse robust dynamic delta hedging of bitcoin options using a set of smile-implied and other smile-adjusted deltas that are either model-free, in the sense that they are the same for every scale-invariant stochastic and/or local volatility model, or they are based on simple regime- dependent parameterisations of local volatility. These deltas are popular with option market makers in traditional assets because they are very easy to implement. Previous empirical research on dynamic delta hedging is based solely on equity index options, but analysis of our unique data on hourly historical bitcoin option prices reveals that bitcoin implied volatility curves behave very differently from those of equity index options. For call and put options with a wide range of moneyness and with synthetic constant maturities of 10, 20 and 30 days, we compare the dynamic hedging performance of different smile-adjusted deltas over two one-year periods. We also examine the use of the perpetual contract rather than the standard futures as hedging instrument because the basis risk for the perpetual is very much smaller than it is for calendar futures. Results are presented as testable statistics of hedging error variance ratios. In certain periods the use of smile-implied hedge ratios can significantly out-perform the simple Black– Scholes delta hedge, especially when using the perpetual swap as hedging instrument, where efficiency gains can exceed 30% for out-of-the-money puts, and reach an average of 15% when hedging short-term out-of-the money calls during periods when the implied volatility curve slopes upwards. The advantage of using the perpetual contract is especially evident during 2021, for the longer-term contracts for which the basis is still rather large.
The Best Strategies for FX Hedging
Pedro Castro, Carl Hamill, John Harber, Campbell R. Harvey & Otto Van Hemert
The question of whether, when, and how to hedge foreign exchange risk has been a vexing one for investors since the end of the Bretton Woods system in 1973. Our study provides a comprehensive empirical analysis of dynamic FX hedging strategies over several decades, examining various domestic and foreign currency pairs. While traditional approaches often focus on risk mitigation, we explore the broader implications for expected returns, highlighting the interplay between hedging and strategies such as the carry trade. Our findings reveal that incorporating additional factors-such as trend (12-month FX return), value (deviation from purchasing power parity), and carry (interest rate differential) - into hedging decisions delivers significant portfolio benefits. By adopting a dynamic, active approach to FX hedging, investors can enhance returns and manage risk more effectively than with static hedged or unhedged strategies.
Deep Bellman Hedging
Hans Buehler, Phillip Murray & Ben Wood
We present an actor-critic-type reinforcement learning algorithm for solving the problem of hedging a portfolio of financial instruments such as securities and over-the- counter derivatives using purely historic data. The key characteristics of our approach are: he ability to hedge with derivatives such as forwards, swaps, futures, options; incorporation of trading frictions such as trading cost and liquidity constraints; applicability for any reasonable portfolio of financial instruments; realistic, continuous state and action spaces; and formal risk-adjusted return objectives. Most importantly, the trained model provides an optimal hedge for arbitrary initial portfolios and market states without the need for re-training. We also prove existence of finite solutions to our Bellman equation, and show the relation to our vanilla Deep Hedging approach.
To Hedge or Not to Hedge: Optimal Strategies for Stochastic Trade Flow Management
Philippe Bergault, Olivier Guéant & Hamza Bodor
This paper addresses the trade-off between internalisation and externalisation in the management of stochastic trade flows. We consider agents who must absorb flows and manage risk by deciding whether to warehouse it or hedge in the market, thereby incurring transaction costs and market impact. Unlike market makers, these agents cannot skew their quotes to attract offsetting flows and deter risk-increasing ones, leading to a fundamentally different problem. Within the Almgren- Chriss framework, we derive almost-closed-form solutions in the case of quadratic execution costs, while more general cases require numerical methods. In particular, we discuss the challenges posed by artificial boundary conditions when using classical grid-based numerical PDE techniques and propose reinforcement learning methods as an alternative.
Option Expected Hedging Demand
Xiaoxiao Tang, Guofu Zhou & Zhaoque (Chosen) Zhou
Options market makers' delta hedging has an increasing impact on underlying stock prices as both the option volume and the ratio of option volume to stock volume grow drastically in recent years. We introduce a novel approach utilizing real-time option information to calculate the spot elasticity of delta (ED) and expected hedging demand (EHD), and find that the EHD significantly predicts future stock returns in the cross section. The positive impact of EHD on stock prices lasts up to five trading days, and then a reversal follows. The empirical evidence of heterogeneous EHD-return relationship, influenced by ED, leads to varied option market maker behaviors, and is consistent with conventional economic theory. Moreover, we find that EHD has a little correlation with other popular firm characteristics, representing a new risk that is not captured by conventional factor models.
References
Deep Bellman Hedging. June 2022. Buehler, H.; Murray, P. and Wood, B. Available at SSRN: http://dx.doi.org/10.2139/ssrn.4151026
Delta hedging bitcoin options with a smile. March 2023. Alexander, C. and Imeraj, A. Quantitative Finance, 23(5), 799–817. Available at Taylor & Francis: https://doi.org/10.1080/14697688.2023.2181205
Gamma and vega hedging using deep distributional reinforcement learning. February 2023. Cao, J.; Chen, J.; Farghadani, S.; Hull, J.; Poulos, Z.; Wang, Z. and Yuan, J. Front. Artif. Intell. (6) 1129370. Available at Frontiers: https://doi.org/10.3389/frai.2023.1129370
Option Expected Hedging Demand. March 2024. Tang, X.; Zhou, G. and Zhou, Z.
Available at SSRN: http://dx.doi.org/10.2139/ssrn.4729672
The Best Strategies for FX Hedging. February 2025. Harvey, C. R. and van Hemert, O. Castro, P.; Hamill, C.; Harber, J.; Available at SSRN: http://dx.doi.org/10.2139/ssrn.5047797
To Hedge or Not to Hedge: Optimal Strategies for Stochastic Trade Flow Management. March 2025. Bergault, P.; Guéant, O. and Bodor, H. Available at arXiv: https://doi.org/10.48550/arXiv.2503.02496
The Quanted Round-up is a curated summary that covers relevant research on key topics in quantitative financial decision-making.
Highlights
This edition looks at how quants are refining hedging strategies to deal with real-world frictions, from market impact to instrument selection, in pursuit of more effective risk control. Across the board, the focus is on making hedging more robust under realistic trading conditions.
Gamma and vega hedging using deep distributional reinforcement learning
Jay Cao, Jacky Chen, Soroush Farghadani, John Hull, Zissis Poulos, Zeyu Wang & Jun Yuan
We show how reinforcement learning can be used in conjunction with quantile regression to develop a hedging strategy for a trader responsible for derivatives that arrive stochastically and depend on a single underlying asset. We assume that the trader makes the portfolio delta-neutral at the end of each day by taking a position in the underlying asset. We focus on how trades in options can be used to manage gamma and vega. The option trades are subject to transaction costs. We consider three different objective functions. We reach conclusions on how the optimal hedging strategy depends on the trader's objective function, the level of transaction costs, and the maturity of the options used for hedging. We also investigate the robustness of the hedging strategy to the process assumed for the underlying asset.
Delta hedging bitcoin options with a smile
Carol Alexander & Arben Imeraj
We analyse robust dynamic delta hedging of bitcoin options using a set of smile-implied and other smile-adjusted deltas that are either model-free, in the sense that they are the same for every scale-invariant stochastic and/or local volatility model, or they are based on simple regime- dependent parameterisations of local volatility. These deltas are popular with option market makers in traditional assets because they are very easy to implement. Previous empirical research on dynamic delta hedging is based solely on equity index options, but analysis of our unique data on hourly historical bitcoin option prices reveals that bitcoin implied volatility curves behave very differently from those of equity index options. For call and put options with a wide range of moneyness and with synthetic constant maturities of 10, 20 and 30 days, we compare the dynamic hedging performance of different smile-adjusted deltas over two one-year periods. We also examine the use of the perpetual contract rather than the standard futures as hedging instrument because the basis risk for the perpetual is very much smaller than it is for calendar futures. Results are presented as testable statistics of hedging error variance ratios. In certain periods the use of smile-implied hedge ratios can significantly out-perform the simple Black– Scholes delta hedge, especially when using the perpetual swap as hedging instrument, where efficiency gains can exceed 30% for out-of-the-money puts, and reach an average of 15% when hedging short-term out-of-the money calls during periods when the implied volatility curve slopes upwards. The advantage of using the perpetual contract is especially evident during 2021, for the longer-term contracts for which the basis is still rather large.
The Best Strategies for FX Hedging
Pedro Castro, Carl Hamill, John Harber, Campbell R. Harvey & Otto Van Hemert
The question of whether, when, and how to hedge foreign exchange risk has been a vexing one for investors since the end of the Bretton Woods system in 1973. Our study provides a comprehensive empirical analysis of dynamic FX hedging strategies over several decades, examining various domestic and foreign currency pairs. While traditional approaches often focus on risk mitigation, we explore the broader implications for expected returns, highlighting the interplay between hedging and strategies such as the carry trade. Our findings reveal that incorporating additional factors-such as trend (12-month FX return), value (deviation from purchasing power parity), and carry (interest rate differential) - into hedging decisions delivers significant portfolio benefits. By adopting a dynamic, active approach to FX hedging, investors can enhance returns and manage risk more effectively than with static hedged or unhedged strategies.
Deep Bellman Hedging
Hans Buehler, Phillip Murray & Ben Wood
We present an actor-critic-type reinforcement learning algorithm for solving the problem of hedging a portfolio of financial instruments such as securities and over-the- counter derivatives using purely historic data. The key characteristics of our approach are: he ability to hedge with derivatives such as forwards, swaps, futures, options; incorporation of trading frictions such as trading cost and liquidity constraints; applicability for any reasonable portfolio of financial instruments; realistic, continuous state and action spaces; and formal risk-adjusted return objectives. Most importantly, the trained model provides an optimal hedge for arbitrary initial portfolios and market states without the need for re-training. We also prove existence of finite solutions to our Bellman equation, and show the relation to our vanilla Deep Hedging approach.
To Hedge or Not to Hedge: Optimal Strategies for Stochastic Trade Flow Management
Philippe Bergault, Olivier Guéant & Hamza Bodor
This paper addresses the trade-off between internalisation and externalisation in the management of stochastic trade flows. We consider agents who must absorb flows and manage risk by deciding whether to warehouse it or hedge in the market, thereby incurring transaction costs and market impact. Unlike market makers, these agents cannot skew their quotes to attract offsetting flows and deter risk-increasing ones, leading to a fundamentally different problem. Within the Almgren- Chriss framework, we derive almost-closed-form solutions in the case of quadratic execution costs, while more general cases require numerical methods. In particular, we discuss the challenges posed by artificial boundary conditions when using classical grid-based numerical PDE techniques and propose reinforcement learning methods as an alternative.
Option Expected Hedging Demand
Xiaoxiao Tang, Guofu Zhou & Zhaoque (Chosen) Zhou
Options market makers' delta hedging has an increasing impact on underlying stock prices as both the option volume and the ratio of option volume to stock volume grow drastically in recent years. We introduce a novel approach utilizing real-time option information to calculate the spot elasticity of delta (ED) and expected hedging demand (EHD), and find that the EHD significantly predicts future stock returns in the cross section. The positive impact of EHD on stock prices lasts up to five trading days, and then a reversal follows. The empirical evidence of heterogeneous EHD-return relationship, influenced by ED, leads to varied option market maker behaviors, and is consistent with conventional economic theory. Moreover, we find that EHD has a little correlation with other popular firm characteristics, representing a new risk that is not captured by conventional factor models.
References
Deep Bellman Hedging. June 2022. Buehler, H.; Murray, P. and Wood, B. Available at SSRN: http://dx.doi.org/10.2139/ssrn.4151026
Delta hedging bitcoin options with a smile. March 2023. Alexander, C. and Imeraj, A. Quantitative Finance, 23(5), 799–817. Available at Taylor & Francis: https://doi.org/10.1080/14697688.2023.2181205
Gamma and vega hedging using deep distributional reinforcement learning. February 2023. Cao, J.; Chen, J.; Farghadani, S.; Hull, J.; Poulos, Z.; Wang, Z. and Yuan, J. Front. Artif. Intell. (6) 1129370. Available at Frontiers: https://doi.org/10.3389/frai.2023.1129370
Option Expected Hedging Demand. March 2024. Tang, X.; Zhou, G. and Zhou, Z.
Available at SSRN: http://dx.doi.org/10.2139/ssrn.4729672
The Best Strategies for FX Hedging. February 2025. Harvey, C. R. and van Hemert, O. Castro, P.; Hamill, C.; Harber, J.; Available at SSRN: http://dx.doi.org/10.2139/ssrn.5047797
To Hedge or Not to Hedge: Optimal Strategies for Stochastic Trade Flow Management. March 2025. Bergault, P.; Guéant, O. and Bodor, H. Available at arXiv: https://doi.org/10.48550/arXiv.2503.02496
The Quanted Round-up is a curated summary that covers relevant research on key topics in quantitative financial decision-making.
Highlights
This edition looks at how quants are refining hedging strategies to deal with real-world frictions, from market impact to instrument selection, in pursuit of more effective risk control. Across the board, the focus is on making hedging more robust under realistic trading conditions.
Gamma and vega hedging using deep distributional reinforcement learning
Jay Cao, Jacky Chen, Soroush Farghadani, John Hull, Zissis Poulos, Zeyu Wang & Jun Yuan
We show how reinforcement learning can be used in conjunction with quantile regression to develop a hedging strategy for a trader responsible for derivatives that arrive stochastically and depend on a single underlying asset. We assume that the trader makes the portfolio delta-neutral at the end of each day by taking a position in the underlying asset. We focus on how trades in options can be used to manage gamma and vega. The option trades are subject to transaction costs. We consider three different objective functions. We reach conclusions on how the optimal hedging strategy depends on the trader's objective function, the level of transaction costs, and the maturity of the options used for hedging. We also investigate the robustness of the hedging strategy to the process assumed for the underlying asset.
Delta hedging bitcoin options with a smile
Carol Alexander & Arben Imeraj
We analyse robust dynamic delta hedging of bitcoin options using a set of smile-implied and other smile-adjusted deltas that are either model-free, in the sense that they are the same for every scale-invariant stochastic and/or local volatility model, or they are based on simple regime- dependent parameterisations of local volatility. These deltas are popular with option market makers in traditional assets because they are very easy to implement. Previous empirical research on dynamic delta hedging is based solely on equity index options, but analysis of our unique data on hourly historical bitcoin option prices reveals that bitcoin implied volatility curves behave very differently from those of equity index options. For call and put options with a wide range of moneyness and with synthetic constant maturities of 10, 20 and 30 days, we compare the dynamic hedging performance of different smile-adjusted deltas over two one-year periods. We also examine the use of the perpetual contract rather than the standard futures as hedging instrument because the basis risk for the perpetual is very much smaller than it is for calendar futures. Results are presented as testable statistics of hedging error variance ratios. In certain periods the use of smile-implied hedge ratios can significantly out-perform the simple Black– Scholes delta hedge, especially when using the perpetual swap as hedging instrument, where efficiency gains can exceed 30% for out-of-the-money puts, and reach an average of 15% when hedging short-term out-of-the money calls during periods when the implied volatility curve slopes upwards. The advantage of using the perpetual contract is especially evident during 2021, for the longer-term contracts for which the basis is still rather large.
The Best Strategies for FX Hedging
Pedro Castro, Carl Hamill, John Harber, Campbell R. Harvey & Otto Van Hemert
The question of whether, when, and how to hedge foreign exchange risk has been a vexing one for investors since the end of the Bretton Woods system in 1973. Our study provides a comprehensive empirical analysis of dynamic FX hedging strategies over several decades, examining various domestic and foreign currency pairs. While traditional approaches often focus on risk mitigation, we explore the broader implications for expected returns, highlighting the interplay between hedging and strategies such as the carry trade. Our findings reveal that incorporating additional factors-such as trend (12-month FX return), value (deviation from purchasing power parity), and carry (interest rate differential) - into hedging decisions delivers significant portfolio benefits. By adopting a dynamic, active approach to FX hedging, investors can enhance returns and manage risk more effectively than with static hedged or unhedged strategies.
Deep Bellman Hedging
Hans Buehler, Phillip Murray & Ben Wood
We present an actor-critic-type reinforcement learning algorithm for solving the problem of hedging a portfolio of financial instruments such as securities and over-the- counter derivatives using purely historic data. The key characteristics of our approach are: he ability to hedge with derivatives such as forwards, swaps, futures, options; incorporation of trading frictions such as trading cost and liquidity constraints; applicability for any reasonable portfolio of financial instruments; realistic, continuous state and action spaces; and formal risk-adjusted return objectives. Most importantly, the trained model provides an optimal hedge for arbitrary initial portfolios and market states without the need for re-training. We also prove existence of finite solutions to our Bellman equation, and show the relation to our vanilla Deep Hedging approach.
To Hedge or Not to Hedge: Optimal Strategies for Stochastic Trade Flow Management
Philippe Bergault, Olivier Guéant & Hamza Bodor
This paper addresses the trade-off between internalisation and externalisation in the management of stochastic trade flows. We consider agents who must absorb flows and manage risk by deciding whether to warehouse it or hedge in the market, thereby incurring transaction costs and market impact. Unlike market makers, these agents cannot skew their quotes to attract offsetting flows and deter risk-increasing ones, leading to a fundamentally different problem. Within the Almgren- Chriss framework, we derive almost-closed-form solutions in the case of quadratic execution costs, while more general cases require numerical methods. In particular, we discuss the challenges posed by artificial boundary conditions when using classical grid-based numerical PDE techniques and propose reinforcement learning methods as an alternative.
Option Expected Hedging Demand
Xiaoxiao Tang, Guofu Zhou & Zhaoque (Chosen) Zhou
Options market makers' delta hedging has an increasing impact on underlying stock prices as both the option volume and the ratio of option volume to stock volume grow drastically in recent years. We introduce a novel approach utilizing real-time option information to calculate the spot elasticity of delta (ED) and expected hedging demand (EHD), and find that the EHD significantly predicts future stock returns in the cross section. The positive impact of EHD on stock prices lasts up to five trading days, and then a reversal follows. The empirical evidence of heterogeneous EHD-return relationship, influenced by ED, leads to varied option market maker behaviors, and is consistent with conventional economic theory. Moreover, we find that EHD has a little correlation with other popular firm characteristics, representing a new risk that is not captured by conventional factor models.
References
Deep Bellman Hedging. June 2022. Buehler, H.; Murray, P. and Wood, B. Available at SSRN: http://dx.doi.org/10.2139/ssrn.4151026
Delta hedging bitcoin options with a smile. March 2023. Alexander, C. and Imeraj, A. Quantitative Finance, 23(5), 799–817. Available at Taylor & Francis: https://doi.org/10.1080/14697688.2023.2181205
Gamma and vega hedging using deep distributional reinforcement learning. February 2023. Cao, J.; Chen, J.; Farghadani, S.; Hull, J.; Poulos, Z.; Wang, Z. and Yuan, J. Front. Artif. Intell. (6) 1129370. Available at Frontiers: https://doi.org/10.3389/frai.2023.1129370
Option Expected Hedging Demand. March 2024. Tang, X.; Zhou, G. and Zhou, Z.
Available at SSRN: http://dx.doi.org/10.2139/ssrn.4729672
The Best Strategies for FX Hedging. February 2025. Harvey, C. R. and van Hemert, O. Castro, P.; Hamill, C.; Harber, J.; Available at SSRN: http://dx.doi.org/10.2139/ssrn.5047797
To Hedge or Not to Hedge: Optimal Strategies for Stochastic Trade Flow Management. March 2025. Bergault, P.; Guéant, O. and Bodor, H. Available at arXiv: https://doi.org/10.48550/arXiv.2503.02496
