Abstract
In this chapter, we derive the Markowitz-optimal trading trajectory for a trader who wishes to sell a large position of Kunits on some contingent claim. To do so, we first use a Taylor expansion of the derivative with respect to the price of the underlying asset at time zero. We then use up to the second-order approximation to solve the mean-variance optimization problem.
Access provided by Autonomous University of Puebla. Download chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
9.1 Introduction
The problem of optimal execution is a very general problem in which a trader who wishes to buy or sell a large position K of a given asset S—for instance, wheat, shares, derivatives, etc.—is confronted with the dilemma of executing slowly or as quick as possible. In the first case, he/she would be exposed to volatility, and in the second, to the laws of offer and demand. Thus, the trader must hedge between the market impact (due to his trade) and the volatility (due to the market).
The main aim of this chapter is to study and characterize the so-called Markowitz-optimal open-loop execution trajectory of contingent claims.
The problem of minimizing expected overall liquidity costs has been analyzed using different market models by [1, 6, 8], and [2], just to mention a few. However, some of these approaches miss the volatility risk associated with time delay. Instead, [3, 4] suggested studying and solving a mean-variance optimization for sales revenues in the class of deterministic strategies. Further on, [5] allowed for intertemporal updating and proved that this can strictly improve the mean-variance performance. Nevertheless, in [9], the authors study the original problem of expected utility maximization with CARA utility functions. Their main result states that for CARA investors there is surprisingly no added utility from allowing for intertemporal updating of strategies. Finally, we mention that the Hamilton-Jacobi-Bellman approach has also recently been studied in [7].
The chapter is organized as follows: in Sect. 9.2, we state the optimal execution contingent claim problem. Next, in Sect. 9.3, we provide its closed form solution. In Sect. 9.4, a numerical example is studied, and finally we conclude in Sect. 9.5 with some final remarks and comments.
9.2 The Problem
The Model. A trader wishes to execute \(K = {k}_{0} + \cdots + {k}_{n}\) units of a contingent claimC with underlying S by time T. The quantity to optimize is given by the so-called execution shortfall, defined as
and the problem is then to find \({k}_{0},\ldots,{k}_{n}\) such that attain the minimum
for some λ > 0. Assuming the derivative C is smooth in terms of its underlying S, it follows from the Taylor series expansion that
where \(\tilde{S}\) is the effective price and R 3 is the remainder which is \(o({(\tilde{{S}}_{j} - {S}_{0})}^{3})\). Hence,
That is,
Note that if we use only the first-order approximation, then our optimization problem has already been solved and corresponds to [4] trading trajectory.
9.3 Second-Order Taylor Approximation
In this section, we extend [4] market impact model for the case of a contingent claim. We provide our main result which is the closed form objective function by adapting a second order-Taylor approximation.
9.3.1 Effective Price Process
Let \({\xi }_{1},{\xi }_{2},\ldots \) be a sequence of i.i.d. Gaussian random variables with mean zero and variance 1, and let the execution times be equally spaced, that is, \(\tau := T/n\). Then, the price and “effective” processes are respectively defined as
and the permanent and temporary market impact will be modeled, for simplicity, as
for some constant α and β. Hence, letting
it follows that
9.3.2 Second-Order Approximation
From (9.1) and (9.2), the second-order approximation of the execution shortfall Y is given by
Next, expanding the squared term, we get
Thus the expected value of Y is approximately
to compute the variance V of Y we rearrange (9.3) as
where D are all the deterministic terms and
It follows that the variance of Y is
and the last term equals zero.
9.3.3 Optimal Trading Schedule for the Second-Order Approximation
To find the optimal trading schedule for the second-order approximation of Y, we need find the sequence of \({k}_{0},\ldots,{k}_{n}\) such that
is minimized for a given λ and where \({\bf E}[Y ]\) and \({\bf V}[Y ]\) are as in (9.4) and (9.5), respectively. After some simplification,
9.4 Numerical Solution
For Y as in (9.3), the optimization problem we aim to solve is
subject to
We solve the problem using fmincon in the Matlab.
Example 9.4.1.
For this example let
the optimal trading strategy is
and the optimal objective function is 5. 5736 ×108.
Remark 9.4.1.
The trading trajectory has a downward trend. Intuitively, and on contrast to executing a large size at a single transaction, our result suggests to split the overall position in almost even trades. The linear assumption that we made on the temporary and the permanent impacts seems to explain the almost equal execution quantities.
9.5 Concluding Remarks
In this work, we study the Markowitz-optimal execution trajectory of contingent claims. In order to do so, we use a second-order Taylor approximation with respect to the contingent claim C evaluated at the initial value of the underlying S. We obtain the closed form objective function given a risk averse criterion. Our approach allows us to obtain the explicit numerical solution and we provide an example.
References
A. Alfonsi, A. Fruth and A. Schied (2010). Optimal execution strategies in limit order books with general shape functions, Quantitative Finance, 10, no. 2, 143–157.
A. Alfonsi and A. Schied (2010). Optimal trade execution and absence of price manipulations limit order books models, SIAM J. on Financial Mathematics, 1, pp. 490–522.
R. Almgren and N. Chriss (1999). Value under liquidation, Risk, 12, pp. 61–63.
R. Almgren and N. Chriss (2000). Optimal execution of portfolio transactions, J. Risk, 3 (2), pp. 5–39.
R. Almgren and J. Lorenz (2007). Adaptive arrival price, Trading, no. 1, pp. 59–66.
D. Bertsimas and D. Lo (1998). Optimal control of execution costs, Journal of Financial Markets, 1(1), pp. 1–50.
P.A. Forsyth (2011). Hamilton Jacobi Bellman approach to optimal trade schedule, Journal of Applied Numerical Mathematics, 61(2), pp. 241–265.
A. Obizhaeva, and J. Wang (2006). Optimal trading strategy and supply/demand dynamics. Journal of Financial Markets, forthcoming.
A. Schied, T. Schöneborn and M. Tehranchi (2010). Optimal basket liquidation for CARA investors is deterministic, Applied Mathematical Finance, 17, pp. 471–489.
Acknowledgements
The authors were partially supported by Algorithmic Trading Management LLC.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Hernandez-del-Valle, G., Sun, Y. (2012). Optimal Execution of Derivatives: A Taylor Expansion Approach. In: Hernández-Hernández, D., Minjárez-Sosa, J. (eds) Optimization, Control, and Applications of Stochastic Systems. Systems & Control: Foundations & Applications. Birkhäuser, Boston. https://doi.org/10.1007/978-0-8176-8337-5_9
Download citation
DOI: https://doi.org/10.1007/978-0-8176-8337-5_9
Published:
Publisher Name: Birkhäuser, Boston
Print ISBN: 978-0-8176-8336-8
Online ISBN: 978-0-8176-8337-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)