FinMarketsTrading

is defined as D ¼ P – BO. Let’s introduce a loss function for an order of size V placed at distance D from the best price:

The first term within the brackets of (13.38) is an estimate of potential loss due to volatility, s; T(V, D) is the expected order execution time and l is the risk-aversion coefficient. The second term is the order P/L in respect to the current market best price. The scaling parameter a in (13.38) depends on the units of V and D.

Schmidt (2010b) found that in the FX market, the loss function might have minima and therefore point at optimal placement of the limit order. This approach can be expanded for the optimal slicing of large orders. Consider a large amount N partitioned into n child orders of amount V (N ¼ nV). Each child order is placed immediately after the former one is filled. Within this strategy, the nth order is on hold during the time it takes to execute (n – 1) previous orders as well as the nth order itself. Therefore, the potential loss, Ln, for the nth order is:

Then, the loss function for the total amount, N, is the sum of the individual loss functions L 1 through Ln:

Searching the global minimum of (13.40) on the V-D plane for a given risk aversion and total order size N can define the optimal size of child orders and their price. For example, if N is 100, the minimum of (13.40) may answer the question whether to trade 10,000 units using, for example, 1,000 child orders of size 10 at the best price, or using 5 child units with size 2,000 at a price one tick behind the best price.

The critical element in calculations of optimal child order size and price using the relation (13.38) is an accurate estimation of the expected execution time T(V, D). This is a complicated task. First, the values of T(V, D) depend on the market volatility. Another problem is the treatment of those orders that are cancelled prior to their execution or after they are partially filled. Order cancellation can occur for various reasons. In particular, when price moves in an adverse direction, traders (or automated trading software) may decide that price will not revert within an acceptable time horizon and

therefore resubmit an order at a new price (see discussion of a relevant strategy below). Cancelled orders constitute a significant percentage of submitted orders and ignoring them can notably skew the results towards shorter execution times (Lo et al. 2002; Eisler et al. 2009).

T h e R a n d o m W a l k M o d e l

An interesting contribution to the problem of limit-order execution time estimation was given by Lo et al. (2002). In this work, price is modeled using the geometric Brownian motion with drift

In (13.41), a and s are constant and dW is the standard Brownian motion. Let’s denote the current time and price with t0 and P0, respectively. Consider the time interval [t0; t0 þ t], where Pmin denotes the lowest price observed in this interval. A bid order with price Pl will be executed within

the given interval if and only if Pmin is less than or equal to Pl. Thus, the probability of filling the bid within the interval [t0; t0 þ t] is the probability

that Pmin is less than or equal to Pl. This probability can be formulated in terms of the first-passage time (FPT). Namely,

where m ¼ a – s2/2, F() is the standard normal cumulative distribution function. If T is the bid execution time, the cumulative distribution function F(t) for T equals

The theoretical distribution F(t) can be compared with empirical data for the limit-order execution time. The values m and s that define F(t) can be calculated from a given sample of returns using the maximum likelihood estimator. Then, the histogram of the empirical limit-order execution times is computed.

Lo et al. (2002) employed the methods of survival analysis to estimate the probability that limit orders will not be cancelled by the time t. They chose a parametric survival distribution in the form of the generalized

gamma distribution and approximated it using empirical data on order cancellations for the 100 largest stocks in the S&P 500 for 1994 to 1995. Then, this survival distribution was used for censoring the empirical distribution of the limit-order execution times. Namely, if the order cancellation and execution are independent stochastic processes, the probability that the time to fill (TTF) for the limit order is t equals (Eisler et al. 2009):

PFPTðtÞPLT ð> tÞ

PTTFðtÞ ¼ 1 ð13:44Þ

R

PFPTðtÞPLT ð> tÞdt

0

In (13.44), PLT(>t) is the probability that the limit order will not be cancelled by the time t.

Unfortunately, for practical applications, Lo et al. (2002) found that the first-passage time based on the random walk model does not describe accurately the empirical limit-order execution times. This conclusion was confirmed by Eisler et al. (2009) with further analysis of empirical firstpassage times, order cancellation times, and limit-order execution times. In particular, Eisler et al. have shown that the statistical distributions for all three variables follow the power law with varying scaling exponents, and the first-passage time distribution decays notably slower than the execution time distribution.

S i m u l a t i o n s o f t h e E x e c u t i o n C o s t s

In general, the problem of estimating the limit-order execution time cannot be reduced to estimation of first-passage time since limit orders are placed in the order book in price/time priority and hence must reside at the top of the order book prior to their execution. Therefore, the model of limit-order execution should describe the order book depletion that depends on both the filling and cancellation of orders. One such model was offered by Schmidt (2009a) for describing a maker strategy in the institutional FX market. This strategy is much more aggressive than the one described with the utility function (13.39). Namely, rather than submit a bid order at best bid price and wait until the market comes and takes it, it is suggested to cancel and resubmit the order each time the best bid moves in an adverse direction prior to the order execution.6

In simulations, historical data for best bid prices and aggregated order volumes at best bid price were used. The distribution of the order book depletion rate was estimated using empirical data on transactions and order cancellations. For EUR/USD, this distribution fits well with the gamma distribution. In the beginning of each simulation, a virtual order was placed at the end of the order queue present at best bid. If the new best bid was higher

than the current one, the virtual order was cancelled and resubmitted at the new bid price. If the new bid was the same, the order book was depleted using the simulated depletion rate. Depending on whether the virtual order was or was not on top of the order book, the depletion rate was determined only by the order filling rate or by both the order filling and cancellation rates. Finally, if the new best bid was lower than the current one, two scenarios were considered. Namely, if the virtual order was not on top of the order book, it was brought on top. If the virtual order was already on top, it was depleted with a simulated depletion rate. The simulation process for a given virtual order continued until it was completely filled. Such simulations were repeated for several thousand times using the random entry protocol (see Chapter 12). Finally, the averaged execution time and loss were estimated. The main conclusion from these simulations is that the maker strategy described above has an average loss per unit of order size (i.e., the cost in respect to initial best bid) lower than the bid/ask spread, and hence, it has statistical advantage over the taker strategy. However, order execution time remains an important risk factor in this strategy.

SUMMARY

&Execution strategies focus on minimizing losses associated with the trading process.

&Implementation shortfall is the generic measure of transactional costs.

&Market impact is the main cause of execution costs.

&Execution strategies can be partitioned into the benchmark-based schedules and cost-driven schedules.

&Benchmark-based schedules (most notably VWAP) are based on simple measures of market dynamics.

&Cost-driven schedules can be formulated as risk-neutral or riskaverse protocols.

&Risk-neutral protocols minimize market impact.

&Risk-averse protocols minimize utility function that includes market impact and market risk. The latter risk is determined by volatility.

&Estimation of market impact remains a considerable challenge in deriving and implementing optimal execution strategies.

Two characteristics are used to describe the most probable values of random variables: (1) mean (or expectation), and (2) median. Mean of X is the average of all possible values of X that are weighed with the probability density f(x):

Z

m ¼ E½X& ¼ xf ðxÞdx ðA:5Þ

Median of X is the value M for which

Expectation of a random variable calculated using some available information It (that may change with time t) is named conditional expectation. The conditional probability density is denoted by f(xjIt). Conditional expectation equals

Z

E½XtjIt& ¼ xf ðxjItÞdx ðA:7Þ

Variance, Var, and the standard deviation, s, are the conventional estimates of the deviations from the mean values of X:

Z

Var½X& s2 ¼ ðx mÞ2f ðxÞdx ðA:8Þ

In financial literature, the standard deviation of price is used to characterize volatility (see Chapter 8). Higher-order moments of the probability distributions are defined as

Z

mn ¼ E½Xn& ¼ xnf ðxÞdx ðA:9Þ

According to this definition, mean is the first moment (m m1), and variance can be expressed via the first two moments, s2 ¼ m2 m2. Two other important parameters, skewness S and kurtosis K, are related to the third and fourth moments, respectively:

Both parameters S and K are dimensionless. Zero skewness implies that the distribution is symmetrical around its mean value. The positive and negative values of skewness indicate long positive tails and long

negative tails, respectively. Kurtosis characterizes the distribution peakedness. Kurtosis of the normal distribution (defined in the next section) equals three. The excess kurtosis, Ke ¼ K 3, is often used as a measure of deviation from the normal distribution. In particular, positive excess kurtosis (or leptokurtosis) indicates more frequent large deviations from the mean value than is typical for the normal distribution. Leptokurtosis leads to the flatter central part as well as to the so-called fat tails in the distribution. Negative excess kurtosis indicates frequent small deviations from the mean value. In this case, the distribution sharpens around its mean value while the distribution tails decay faster than the tails of the normal distribution.

The joint distribution of two random variables X and Y is the generalization of the cumulative distribution (A.3):

In (A.11), h(x, y) is the joint density that satisfies the normalization condition

11

ZZ

1 1

Two random variables are independent if their joint density function is simply the product of the univariate density functions: h(x, y) ¼ f(x) g(y).

Covariance between two variates provides a measure of their simultaneous change. Consider two variates X and Y that have the means mX and mY, respectively. Their covariance equals

Covðx; yÞ ¼ sXY ¼ E½ðx mXÞðy mY Þ& ¼ E½xy& mXmY ðA:13Þ

Clearly, covariance reduces to variance if X ¼ Y : sXX ¼ s2X. Positive covariance between two variates implies that these variates tend to change simultaneously in the same direction rather than in opposite directions. Conversely, negative covariance between two variates implies that when one variate grows, the other one tends to fall and vice versa. Another popular measure of simultaneous change is the correlation coefficient:

The values of the correlation coefficient are within the range [ 1, 1]. In the general case with N variates X1, . . . , XN (where N > 2), correlations

The factor CNn is called the binomial coefficient. Mean and higherorder moments for the binomial distribution are equal, respectively:

In the case of large N and large (N n), the binomial distribution approaches the normal (or Gaussian) distribution:

N ! 1; ðN nÞ ! 1

T h e P o i s s o n D i s t r i b u t i o n

The Poisson distribution can be considered as the limiting case of the binomial distribution in the case with p 1. The former describes the probability of n successes in N trials assuming that the fraction of successes n is proportional to the number of trials (n ¼ pN)

When the number of trials N becomes very large (N ! 1), the Poisson distribution approaches the limit

Mean, variance, skewness, and excess kurtosis of the Poisson distribution are equal, respectively:

T h e N o r m a l D i s t r i b u t i o n

The normal (Gausian) distribution has the form

f NðxÞ ¼ p 1 exp ðx mÞ2=2s2 ðA:25Þ

2ps

It is often denoted N(m, s). Skewness and excess kurtosis of the normal distribution equal zero. The transform z ¼ (x m)/s converts the normal distribution into the standard normal distribution

The integral over the standard normal distribution within the interval [0, z] can be used as the definition of the error function erf(x):

Then, the cumulative distribution function for the standard normal distribution equals

According to the central limit theorem, the probability density distribution for a sum of N independent and identically distributed random variables with finite variances and finite means approaches the normal distribution as N grows to infinity. The Box-Miller method is often used for modeling the normal distribution (see, e.g., Press et al. 1992). It is based on drawings from the uniform distribution that is available for simulations in many computer languages. Namely, if two numbers x1 and x2 are drawn from the standard uniform distribution, then y1 and y2 are the standard normal variates:

y1 ¼ ½ 2 ln x1Þ&1=2 cosð2px2Þ; y2 ¼ ½ 2 ln x1Þ&1=2 sinð2px2Þ ðA:29Þ

Mean and variance of the multivariate normal distribution with N variates can be easily calculated via the univariate means mi and covariances sij

NN

XX

T h e L o g n o r m a l D i s t r i b u t i o n

In the lognormal distribution, the logarithm of a variate has the normal form