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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

$$Y = \sum\limits_{j=0}^{n}{k}_{ j}{C}_{j} - K{C}_{0},$$

and the problem is then to find \({k}_{0},\ldots,{k}_{n}\) such that attain the minimum

$$ \begin{array}{rcl} \min\limits_{{k}_{0},\ldots,{k}_{n}}\left ({\bf E}[Y ] + \lambda {\bf V}[Y ]\right ),& & \\ \end{array}$$

for some λ > 0. Assuming the derivative C is smooth in terms of its underlying S, it follows from the Taylor series expansion that

$$\begin{array}{rcl}{ C}_{j} = f({S}_{0}) + f^{\prime}({S}_{0})(\tilde{{S}}_{j} - {S}_{0}) + \frac{1} {2}f^{\prime\prime}({S}_{0}){(\tilde{{S}}_{j} - {S}_{0})}^{2} + {R}_{ 3},& & \\ \end{array}$$

where \(\tilde{S}\) is the effective price and R 3 is the remainder which is \(o({(\tilde{{S}}_{j} - {S}_{0})}^{3})\). Hence,

$$\begin{array}{rlrlrl} \sum\limits_{j=0}^{n}{k}_{ j}{C}_{j}& = \sum\limits_{j=0}^{n}{k}_{ j}f({S}_{0}) + \sum \limits_{j=0}^{n}f^{\prime}({S}_{ 0}){k}_{j}(\tilde{{S}}_{j} - {S}_{0}) + \frac{1} {2}f^{\prime\prime}({S}_{0})\!\sum \limits_{j=0}^{n}{k}_{ j}{(\tilde{{S}}_{j} - {S}_{0})}^{2} + \sum\limits_{j=0}^{n}{k}_{ j}{R}_{3}&& \\ & = K{C}_{0} + f^{\prime}({S}_{0})\left (\sum\limits_{j=0}^{n}{k}_{ j}\tilde{{S}}_{j} - K{S}_{0}\right )s + \frac{1} {2}f^{\prime\prime}({S}_{0})\sum\limits_{j=0}^{n}{k}_{ j}{(\tilde{{S}}_{j} - {S}_{0})}^{2} + \sum\limits_{j=0}^{n}{k}_{ j}{R}_{3}. && \end{array}$$

That is,

$$\begin{array}{rlrlrl} Y & = \sum\limits_{j=0}^{n}{k}_{ j}{C}_{j} - K{C}_{0} & & \\ & = f^{\prime}({S}_{0})\left (\sum \limits_{j=0}^{n}{k}_{ j}\tilde{{S}}_{j} - K{S}_{0}\right ) + \frac{1} {2}f^{\prime\prime}({S}_{0})\sum\limits_{j=0}^{n}{k}_{ j}{(\tilde{{S}}_{j} - {S}_{0})}^{2} + \sum\limits_{j=0}^{n}{k}_{ j}{R}_{3}. &\end{array}$$
(9.1)

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

$$\begin{array}{rcl}{ S}_{j}& =& {S}_{j-1} - \tau g\left (\frac{{k}_{j}} {\tau } \right ) + \sigma {\tau }^{1/2}{\xi }_{ j}, \\ \tilde{{S}}_{j}& =& {S}_{j} - h\left (\frac{{k}_{j}} {\tau } \right ), \\ \end{array}$$

and the permanent and temporary market impact will be modeled, for simplicity, as

$$\begin{array}{rcl} g\left (\frac{{k}_{j}} {\tau } \right ) = \alpha \frac{{k}_{j}} {\tau },\qquad h\left (\frac{{k}_{j}} {\tau } \right ) = \beta \frac{{k}_{j}} {\tau },& & \\ \end{array}$$

for some constant α and β. Hence, letting

$$\begin{array}{rcl} {x}_{j}& :=& K -\sum\limits_{m=0}^{j}{k}_{ m}\quad \mbox{ and} \\ {W}_{j}& :=& \sum\limits_{m=1}^{j}{\xi }_{ m},\quad \mbox{ i.e. }{W}_{j} \sim {\bf N}(0,j), \mbox{ Cov}({W}_{j},{W}_{i}) =\min (i,j), \\ \end{array}$$

it follows that

$$\begin{array}{rcl} \tilde{{S}}_{j} - {S}_{0} = \sigma {\tau }^{1/2}{W}_{ j} - \alpha (K - {x}_{j}) -\frac{\beta } {\tau }{k}_{j}.& &\end{array}$$
(9.2)

9.3.2 Second-Order Approximation

From (9.1) and (9.2), the second-order approximation of the execution shortfall Y is given by

$$\begin{array}{rcl} Y & \approx & f^{\prime}({S}_{0})\sum\limits_{j=0}^{n}{k}_{ j}\left (\sigma {\tau }^{1/2}{W}_{ j} - \alpha (K - {x}_{j}) -\frac{\beta } {\tau }{k}_{j}\right ) \\ & & +\frac{1} {2}f^{\prime\prime}({S}_{0})\sum\limits_{j=0}^{n}{k}_{ j}{\left (\sigma {\tau }^{1/2}{W}_{ j} - \alpha (K - {x}_{j}) -\frac{\beta } {\tau }{k}_{j}\right )}^{2}.\end{array}$$
(9.3)

Next, expanding the squared term, we get

$$\begin{array}{rlrlrl} {\left (\sigma {\tau }^{1/2}{W}_{ j} - \alpha (K - {x}_{j}) -\frac{\beta } {\tau }{k}_{j}\right )}^{2}& = {\sigma }^{2}\tau {W}_{ j}^{2} + {\alpha }^{2}{(K - {x}_{ j})}^{2} + \frac{{\beta }^{2}} {{\tau }^{2}} {k}_{j}^{2} - 2 \frac{\beta \sigma } {{\tau }^{1/2}}{k}_{j}{W}_{j}&& \\ &\quad - 2\alpha \sigma {\tau }^{1/2}(K - {x}_{ j}){W}_{j} + 2\frac{\alpha \beta } {\tau } {k}_{j}(K - {x}_{j}), && \end{array}$$

Thus the expected value of Y is approximately

$$\begin{array}{rcl} {\bf E}[Y ]& =& f^{\prime}({S}_{0})\sum\limits_{j=0}^{n}{k}_{ j}\left (-\alpha (K - {x}_{j}) -\frac{\beta } {\tau }{k}_{j}\right ) \\ & & +\frac{1} {2}f^{\prime\prime}({S}_{0})\sum\limits_{j=0}^{n}{k}_{ j}\left[{\sigma }^{2}\tau j + {\alpha }^{2}{(K - {x}_{ j})}^{2} + \frac{{\beta }^{2}} {{\tau }^{2}} {k}_{j}^{2} + 2\frac{\alpha \beta } {\tau } {k}_{j}(K - {x}_{j})\right], \\ & & \end{array}$$
(9.4)

to compute the variance V of Y we rearrange (9.3) as

$$\begin{array}{rcl} Y \approx \sum\limits_{j=0}^{n}{\nu }_{ j}{k}_{j}{W}_{j} + \sum\limits_{j=0}^{n}{\eta }_{ j}{k}_{j}{W}_{j}^{2} + D,& & \\ \end{array}$$

where D are all the deterministic terms and

$$\begin{array}{rcl}{ \nu }_{j}& :=& f^{\prime}({S}_{0})\sigma {\tau }^{1/2} - f^{\prime\prime}({S}_{ 0})\left [\alpha \sigma {\tau }^{1/2}(K - {x}_{ j}) + \frac{\beta \sigma } {{\tau }^{1/2}}{k}_{j}\right ] \\ {\eta }_{j}& :=& \frac{1} {2}f^{\prime\prime}({S}_{0}){\sigma }^{2}\tau.\end{array}$$

It follows that the variance of Y is

$$\begin{array}{rcl} {\bf V}[Y ]& =& {\bf V}\left[\sum\limits_{j=0}^{n}{\nu }_{ j}{k}_{j}{W}_{j}\right ] +{\bf V}\left[\sum\limits_{j=0}^{n}{\eta }_{ j}{k}_{j}{W}_{j}^{2}\right ]+ 2\,\mbox{ Cov}\left (\sum\limits_{j=0}^{n}{\nu }_{j}{k}_{j}{W}_{j},\sum\limits_{j=0}^{n}{\eta }_{j}{k}_{j}{W}_{j}^{2}\right ) \\ & =& \sum\limits_{j=0}^{n}{v}_{j}^{2}{k}_{ j}^{2}j + 2\sum\limits_{0\leq i<j\leq n}{v}_{i}{k}_{i}{v}_{j}{k}_{j}i + \sum\limits_{j=0}^{n}{\eta }_{j}^{2}{k}_{ j}^{2} \cdot 2{j}^{2} + 2\sum\limits_{0\leq i<j\leq n}{\eta }_{i}{k}_{i}{\eta }_{j}{k}_{j} \cdot 2{i}^{2} \\ & &\end{array}$$
(9.5)

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

$$\begin{array}{rcl} {\bf E}[Y ] + \lambda {\bf V}[Y ]& & \\ \end{array}$$

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,

$$\begin{array}{rlrlrl} {\bf E}[Y ] + \lambda {\bf V}[Y ]& = f^{\prime}({S}_{0})\sum\limits_{j=0}^{n}{k}_{ j}\left [\alpha ({x}_{j} - K) -\frac{\beta } {\tau }{k}_{j}\right ] && \\ &\quad + \frac{1} {2}f^{\prime\prime}({S}_{0})\sum\limits_{j=0}^{n}{k}_{ j}\left [{\sigma }^{2}\tau j + {\alpha }^{2}{(K - {x}_{ j})}^{2} + \frac{{\beta }^{2}} {{\tau }^{2}} {k}_{j}^{2} + \frac{2\alpha \beta } {\tau } (K - {x}_{j}){k}_{j}\right ] && \\ &\quad + \lambda \sum\limits_{j=0}^{n}j{k}_{ j}\left[{v}_{j}^{2}{k}_{ j} + 2j{k}_{j}{\eta }_{j}^{2} + 2{v}_{ j} \sum\limits_{m=j+1}^{n}{v}_{ m}{k}_{m} + 4j{\eta }_{j} \sum\limits_{m=j+1}^{n}{\eta }_{ m}{k}_{m}\right].&& \end{array}$$

9.4 Numerical Solution

For Y as in (9.3), the optimization problem we aim to solve is

$$ \begin{array}{rcl} \min\limits_{{k}_{0},{k}_{1},\ldots,{k}_{n}}\left ({\bf E}[Y ] + \lambda {\bf V}[Y ]\right )& & \\ \end{array}$$

subject to

$$\begin{array}{rcl} \sum\limits_{j=0}^{n}{k}_{ j} = K.& & \\ \end{array}$$

We solve the problem using fmincon in the Matlab.

Example 9.4.1.

For this example let

$$\begin{array}{rcl} & & n = 2;\quad K = 1000;\quad \alpha = 0.1;\quad \beta = 0.5;\quad \lambda = 0.4;\quad \tau = 1; \\ & & \delta = f^{\prime}({S}_{0}) = 0.5;\quad \gamma = f^{\prime}({S}_{0}) = 0.2;\quad \sigma = 0.5, \\ \end{array}$$

the optimal trading strategy is

$$\begin{array}{rcl}{ k}_{0} = 333.3348,\quad {k}_{1} = 333.3336,\quad {k}_{2} = 333.3316,& & \\ \end{array}$$

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.