Time Is Money: The Equilibrium Trading Horizon and Optimal Arrival Price

Kevin Darby

doi:doi:10.11648/j.ajam.20221003.12

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Time Is Money: The Equilibrium Trading Horizon and Optimal Arrival Price

Kevin Darby

Published in American Journal of Applied Mathematics (Volume 10, Issue 3)

Received: 6 April 2022 Accepted: 10 May 2022 Published: 26 May 2022

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Abstract

Executing even moderately large derivatives orders can be expensive and risky; it’s hard to balance the uncertainty of working an order over time versus paying a liquidity premium for immediate execution. Here, we introduce the Time Is Money model, which calculates the Equilibrium Trading Horizon over which to execute an order within the adversarial forces of variance risk and liquidity premium. We construct a hypothetical at-the-money option within Arithmetic Brownian Motion and invert the Bachelier model to compute an inflection point between implied variance and liquidity cost as governed by a central limit order book, each in real time as they evolve. As a result, we demonstrate a novel, continuous-time Arrival Price framework. Further, we argue that traders should be indifferent to choosing between variance risk and liquidity cost, unless they have a predetermined bias or an exogenous position with a convex payoff. We, therefore, introduce half-life factor asymptotics to the model based on a convexity factor and compare results to existing models. We also describe a specialization of the model for trading a basket of correlated instruments, as exemplified by a futures calendar spread. Finally, we establish groundwork for microstructure optimizations as well as explore short term drift and conditional expected slippage within the Equilibrium Horizon framework.

Published in	American Journal of Applied Mathematics (Volume 10, Issue 3)
DOI	10.11648/j.ajam.20221003.12
Page(s)	93-99
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Derivatives, Optimal Execution, Volatility, Arrival Price, Trading

References

[1]	Almgren, Robert, and Neil Chriss. 2000. “Optimal execution of portfolio transactions.” Risk 3: 5-39.
[2]	Almgren, Robert, and Julian Lorenz. 2007. “Adaptive Arrival Price.” Institutional Investor Journals Algorithmic Trading III.
[3]	Bachelier, Louis. 1900. “The Theory Of Speculation (translated).” Annales scientifiques de l’Ecole Normale Superieur 3 (17): 21-86.
[4]	Garman, Mark, and Michael Klass. 1980 “On the Estimation of Security Price Volatilities from Historical Data” The Journal of Business 53 no. 1: 67-78.
[5]	Gatheral, Jim, and Alexander Schied. 2011. “Optimal Trade Execution under Geometric Brownian Motion in the Almgren and Chriss Framework.” International Journal of Theoretical and Applied Finance 14 (3): 353-368.
[6]	Grunspan, Cyril. 2011. “A Note on the Equivalence between the Normal and the Lognormal Implied Volatility: A Model Free Approach.” ESILV, Department of Financial Engineering, (December).
[7]	Jäckel, Peter. 2017. “Implied Normal Volatility.” Wilmott, no. July 2017.
[8]	LeFloc'h, Fabien. 2016. “Fast and Accurate Analytic Basis Point Volatility.” SSRN, (June).
[9]	Parkinson, Michael. 1980 “The Extreme Value Method for Estimating the Variance of the Rate of Return” The Journal of Business 53, no. 1: 61-65.
[10]	Perold, Andre. 1988. “The implementation shortfall: Paper versus reality.” Journal of Portfolio Management 14, no. 3 (Spring): 4-9.
[11]	Rogers, L. C. G. and S. E. Satchell. 1991. “Estimating Variance From High Low and Closing Prices” The Annals of Appllied Probability 1, no. 4: 504-512.
[12]	Schaefer, Matthew P. 2002. “Pricing and Hedging European Options On Futures Spreads Using the Bachelier Spread Option Model.” Conference on Applied Commodity Price Analysis, Forecasting, and Market Risk Management, (April).
[13]	Shimko, David. 1994. “Options on Futures Spreads: Hedging, Speculation and Valuation.” Journal of Futures Markets 14 (2): 183-213.
[14]	Wood, Greg. 2011. “Transaction Cost Analysis for Futures.” https://www.cmegroup.com/education/files/TCA-4.pdf.
[15]	Yang, Dennis, and Qiang Zhang. 2000. “Drift-Independent Volatility Estimation Based on High, Low, Open, and Close Prices" The Journal of Business 73, no. 3: 477 – 491.

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APA Style

Kevin Darby. (2022). Time Is Money: The Equilibrium Trading Horizon and Optimal Arrival Price. American Journal of Applied Mathematics, 10(3), 93-99. https://doi.org/10.11648/j.ajam.20221003.12

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ACS Style

Kevin Darby. Time Is Money: The Equilibrium Trading Horizon and Optimal Arrival Price. Am. J. Appl. Math. 2022, 10(3), 93-99. doi: 10.11648/j.ajam.20221003.12

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AMA Style

Kevin Darby. Time Is Money: The Equilibrium Trading Horizon and Optimal Arrival Price. Am J Appl Math. 2022;10(3):93-99. doi: 10.11648/j.ajam.20221003.12

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@article{10.11648/j.ajam.20221003.12,
  author = {Kevin Darby},
  title = {Time Is Money: The Equilibrium Trading Horizon and Optimal Arrival Price},
  journal = {American Journal of Applied Mathematics},
  volume = {10},
  number = {3},
  pages = {93-99},
  doi = {10.11648/j.ajam.20221003.12},
  url = {https://doi.org/10.11648/j.ajam.20221003.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20221003.12},
  abstract = {Executing even moderately large derivatives orders can be expensive and risky; it’s hard to balance the uncertainty of working an order over time versus paying a liquidity premium for immediate execution. Here, we introduce the Time Is Money model, which calculates the Equilibrium Trading Horizon over which to execute an order within the adversarial forces of variance risk and liquidity premium. We construct a hypothetical at-the-money option within Arithmetic Brownian Motion and invert the Bachelier model to compute an inflection point between implied variance and liquidity cost as governed by a central limit order book, each in real time as they evolve. As a result, we demonstrate a novel, continuous-time Arrival Price framework. Further, we argue that traders should be indifferent to choosing between variance risk and liquidity cost, unless they have a predetermined bias or an exogenous position with a convex payoff. We, therefore, introduce half-life factor asymptotics to the model based on a convexity factor and compare results to existing models. We also describe a specialization of the model for trading a basket of correlated instruments, as exemplified by a futures calendar spread. Finally, we establish groundwork for microstructure optimizations as well as explore short term drift and conditional expected slippage within the Equilibrium Horizon framework.},
 year = {2022}
}

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TY  - JOUR
T1  - Time Is Money: The Equilibrium Trading Horizon and Optimal Arrival Price
AU  - Kevin Darby
Y1  - 2022/05/26
PY  - 2022
N1  - https://doi.org/10.11648/j.ajam.20221003.12
DO  - 10.11648/j.ajam.20221003.12
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 93
EP  - 99
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20221003.12
AB  - Executing even moderately large derivatives orders can be expensive and risky; it’s hard to balance the uncertainty of working an order over time versus paying a liquidity premium for immediate execution. Here, we introduce the Time Is Money model, which calculates the Equilibrium Trading Horizon over which to execute an order within the adversarial forces of variance risk and liquidity premium. We construct a hypothetical at-the-money option within Arithmetic Brownian Motion and invert the Bachelier model to compute an inflection point between implied variance and liquidity cost as governed by a central limit order book, each in real time as they evolve. As a result, we demonstrate a novel, continuous-time Arrival Price framework. Further, we argue that traders should be indifferent to choosing between variance risk and liquidity cost, unless they have a predetermined bias or an exogenous position with a convex payoff. We, therefore, introduce half-life factor asymptotics to the model based on a convexity factor and compare results to existing models. We also describe a specialization of the model for trading a basket of correlated instruments, as exemplified by a futures calendar spread. Finally, we establish groundwork for microstructure optimizations as well as explore short term drift and conditional expected slippage within the Equilibrium Horizon framework.
VL  - 10
IS  - 3
ER  -

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Author Information

Kevin Darby

CQG Inc. Denver, USA

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Plain Text BibTeX RIS

APA Style

Kevin Darby. (2022). Time Is Money: The Equilibrium Trading Horizon and Optimal Arrival Price. American Journal of Applied Mathematics, 10(3), 93-99. https://doi.org/10.11648/j.ajam.20221003.12

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ACS Style

Kevin Darby. Time Is Money: The Equilibrium Trading Horizon and Optimal Arrival Price. Am. J. Appl. Math. 2022, 10(3), 93-99. doi: 10.11648/j.ajam.20221003.12

Copy | Download

AMA Style

Kevin Darby. Time Is Money: The Equilibrium Trading Horizon and Optimal Arrival Price. Am J Appl Math. 2022;10(3):93-99. doi: 10.11648/j.ajam.20221003.12

Copy | Download

@article{10.11648/j.ajam.20221003.12,
  author = {Kevin Darby},
  title = {Time Is Money: The Equilibrium Trading Horizon and Optimal Arrival Price},
  journal = {American Journal of Applied Mathematics},
  volume = {10},
  number = {3},
  pages = {93-99},
  doi = {10.11648/j.ajam.20221003.12},
  url = {https://doi.org/10.11648/j.ajam.20221003.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20221003.12},
  abstract = {Executing even moderately large derivatives orders can be expensive and risky; it’s hard to balance the uncertainty of working an order over time versus paying a liquidity premium for immediate execution. Here, we introduce the Time Is Money model, which calculates the Equilibrium Trading Horizon over which to execute an order within the adversarial forces of variance risk and liquidity premium. We construct a hypothetical at-the-money option within Arithmetic Brownian Motion and invert the Bachelier model to compute an inflection point between implied variance and liquidity cost as governed by a central limit order book, each in real time as they evolve. As a result, we demonstrate a novel, continuous-time Arrival Price framework. Further, we argue that traders should be indifferent to choosing between variance risk and liquidity cost, unless they have a predetermined bias or an exogenous position with a convex payoff. We, therefore, introduce half-life factor asymptotics to the model based on a convexity factor and compare results to existing models. We also describe a specialization of the model for trading a basket of correlated instruments, as exemplified by a futures calendar spread. Finally, we establish groundwork for microstructure optimizations as well as explore short term drift and conditional expected slippage within the Equilibrium Horizon framework.},
 year = {2022}
}

Copy | Download

TY  - JOUR
T1  - Time Is Money: The Equilibrium Trading Horizon and Optimal Arrival Price
AU  - Kevin Darby
Y1  - 2022/05/26
PY  - 2022
N1  - https://doi.org/10.11648/j.ajam.20221003.12
DO  - 10.11648/j.ajam.20221003.12
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 93
EP  - 99
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20221003.12
AB  - Executing even moderately large derivatives orders can be expensive and risky; it’s hard to balance the uncertainty of working an order over time versus paying a liquidity premium for immediate execution. Here, we introduce the Time Is Money model, which calculates the Equilibrium Trading Horizon over which to execute an order within the adversarial forces of variance risk and liquidity premium. We construct a hypothetical at-the-money option within Arithmetic Brownian Motion and invert the Bachelier model to compute an inflection point between implied variance and liquidity cost as governed by a central limit order book, each in real time as they evolve. As a result, we demonstrate a novel, continuous-time Arrival Price framework. Further, we argue that traders should be indifferent to choosing between variance risk and liquidity cost, unless they have a predetermined bias or an exogenous position with a convex payoff. We, therefore, introduce half-life factor asymptotics to the model based on a convexity factor and compare results to existing models. We also describe a specialization of the model for trading a basket of correlated instruments, as exemplified by a futures calendar spread. Finally, we establish groundwork for microstructure optimizations as well as explore short term drift and conditional expected slippage within the Equilibrium Horizon framework.
VL  - 10
IS  - 3
ER  -

Copy | Download