FinMarketsTrading

Therefore, risk-averse practitioners are advised to use a value of f lower than the Kelly’s criterion suggests.

The total number of trades for a given period is also important. Indeed, frequent trading may lead to more volatile outcomes. Multiple trades with a given strategy generate a probability distribution that can be used for hypothesis testing, in particular for comparing different strategies. Hence, the average return, m, and its variance, s, are very important performance measures. Note that a positive return being accompanied with a high variance does not guarantee the strategy’s quality. Providing the return distribution is normal, one can use the t-statistic for testing the hypothesis that the distribution mean is zero. Namely, for a given number of round-trip trades, N, one calculates the t-value:

Then, the t-value can be used for finding statistical significance (also called p-value) from the Student’s distribution. Note that this distribution is the function of degrees of freedom, which in our case equal (N – 1). Usually, the null hypothesis in an analysis of trading strategies is that the strategy return is zero. For example, if p is less than 0.05, the hypothesis of zero return can be rejected in more than 95 percent of cases.

Obviously, if two strategies have the same average return, the one with lower variance is more attractive. Providing that the return distributions are normal, two trading strategies A and B can be compared using the t-statistic:

In this case, the degrees of freedom equal NA þ NB – 2. The t-statistic (12.5) can also be used for analysis of profitability of a single strategy if indexes A and B refer to buy and sell signals, respectively (Brock et al. 1992).

The Sharpe ratio (8.24) is often used in performance analysis. Sometimes, the Sortino ratio is chosen instead. In the latter ratio, only negative returns are included in calculating the standard deviation s. If a trading strategy (or portfolio) performance is compared with performance of an index (or buy-and-hold strategy), the information ratio that looks similar to the Sharpe ratio can be used:

In (12.6), ri is given return, r0 is return of an index, and si0 is the tracking

error (standard deviation between returns of the strategy and returns of the index4).

The maximum drawdown (MD) is another important risk measure, particularly for leveraged trades. For a process X(t) on the interval [0, T],

t 2 ½0; T&s 2 ½0; t&

In other words, MD is the largest drop of price after its peak. For the general random walk with drift, estimation of MD is quite cumbersome. However, if drift can be neglected, expectation of MD has a simple analytic form (Magdon-Ismail & Atiya 2004):

In some performance criteria, such as the Calmar ratio, the Sharpe ratio is modified by replacing the standard deviation of returns with MD.

Finally, no trading strategy works forever. A possible signal for dropping a strategy that used to be profitable in the past is a growing number of consecutive losing trades. Hence, this number may be handy for tracking the strategy performance.

R E S A M P L I N G T E C H N I Q U E S

In this section, I describe the most widely used resampling techniques: the bootstrap and MCMC. Then I address a problem of resampling a system of two coupled time series with the random entry protocol.

B o o t s t r a p

The t-statistic can be misleading when it is applied to non-normal distributions. Brock et al. (1992) suggest using bootstrap for analyzing the performance of trading strategies. In the simple case, the bootstrap protocol is based on picking up at random an element of a given sample of size N, copying it into the new sample, and putting it back (replacement). This random selection continues until the new sample has the same number of elements as the original one. In recent literature, application of bootstrap to back-testing the trading strategies is discussed by Hsu et al. (2009) and Fusai & Roncoroni (2008).

Sometimes, blocks of several sequential elements of a given sample are picked at once (block bootstrap). Usually, the block bootstrap is implemented with replacement and blocks are not overlapping. Such an approach may preserve short-range autocorrelations present in the original sample. While a simple estimate of an optimal block size L N–1/3 can be used, choice of L in the general case is not trivial (Davidson & Hinkley 1997). Politis & Romano (1994) offered an expansion of the block bootstrap. This method ensures the stationarity of samples bootstrapped from the stationary data—and is called accordingly: stationary bootstrap. The block size L in stationary bootstrap is randomly drawn from the geometric distribution

The average block size for the stationary bootstrap case equals 1/p, which can serve as a bridge between the size of the simple block bootstrap and the stationary bootstrap parameter p.

A more sophisticated approach implies estimating a mathematical model that fits the given sample and bootstrapping the model’s residuals. Typical models used for stock prices are the random walk with drift, the AR models and the GARCH models (Brock et al. 1992). Note that residuals within this approach are assumed to be IID and must be standardized for obtaining unbiased estimates.

Consider an example with the AR(1) model for logarithmic returns r ¼ log(p):

First, the coefficients a and b < 1 in (12.10) are estimated using OLS. Then, the standardized residuals are calculated as

A new sample of residuals, e9t, is generated using the bootstrap and used for constructing a new sample of returns:

Then, a new price sample calculated via sampled returns, p^t ¼ expðr‘ tÞ can be used for another round of testing the trading strategy.

The number of bootstrapped samples needed for good accuracy may reach from several hundred (Brock et al. 1992) to several thousand (Schmidt 2009a). The fraction of bootstrapped samples for which a given

strategy yields return that is higher than for the original sample can be interpreted as the p-value. If this fraction is small (a few percentage points), then either the model chosen for simulations of returns is inadequate or the return obtained with a given strategy on the given sample is the result of data snooping.

M a r k o v C h a i n M o n t e C a r l o

Markov process is a generic stochastic process determined with relationships between its future, present, and past values. Here, discrete Markov processes called Markov chains are discussed. In a nutshell, future value for the Markov chain of the 1st order is determined by its present value; future value for the Markov chain of the 2nd order is determined by its present value and the most recent past value, and so on. The Markov processes of the 1st order cover a very wide class of dynamic short-memory phenomena including diffusional transfer. In fact, the equation for Brownian motion can be derived directly from the definition of the Markov process.5

By definition, the Markov chain of the kth order is such a sequence of random variables X1, X2, . . . , which satisfies the following equation:

PrðXn ¼ xjXn 1 ¼ xn 1; Xn 2 ¼ xn 2; . . . ; X1 ¼ x1Þ

¼ PrðXn ¼ xjXn 1 ¼ xn 1; Xn 2 ¼ xn 2; . . . ; Xn k ¼ xn kÞ ð12:13Þ

In other words, only k past values (sometimes called initial conditions) determine the present value. In particular, for k ¼ 1, only one initial condition is needed:

The Markov chain is stationary (or time-homogeneous) if probability in the left-hand side of (12.13) does not depend on index n.

Generally, Markov variables can assume only a finite number of values (states). Stationary Markov chains of the 1st order with N states are deter-

mined with N2 probabilities Pr(Xn ¼ xk j Xn–1 ¼ xi) ¼ pik, i, k ¼ 1, 2, . . . , N. These probabilities are called the transition kernel, and the complete set of

values pik is called the transition matrix. Note that for each k

N

X

i¼1

Similarly, stationary Markov chains of the 2nd order are determined

with N3 probabilities Pr(Xn ¼ xk j Xn–1 ¼ xi, Xn–2 ¼ xj) ¼ pijk, i, j, k ¼ 1, 2, . . . , N, and

N

X

i;j¼1

In MCMC-based resampling, the transition matrix is assumed stationary and is calculated using the original sample. Then, drawings from the uniform distribution are mapped onto transition probabilities for generating new samples. For example, consider a two-state Markov chain with

p11 ¼ p, p22 ¼ q (which implies that p12 ¼ 1 – p and p21 ¼ 1 – q). Say the current state is 1. If a drawing from the uniform distribution is less than or

equal to p, then the next state is 1; otherwise, it is 2. If the current state is 2 and a drawing from the uniform distribution is less than or equal to q, then the next state is 2; otherwise, it is 1.

Since the transition matrix size grows with the Markov chain’s order as the power law, the use of higher orders for multi-state models can become a computational challenge. Also, the original sample may be not long enough for reliable estimates of all transition probabilities. Luckily, financial returns do not have a long memory. Hence, low-order Markov chains should suffice for their resampling. In particular, Schmidt (2009a) found that MCMC with Markov chains of the first and second order used in resampling of FX returns on a grid with one-second timescale yielded practically the same results.

R a n d o m E n t r y P r o t o c o l

So far, we have discussed resampling of a single time series (i.e., returns, as far as trading strategies are concerned). In the general case, trading strategies can be determined not only by price dynamics but also by some liquidity measure(s), such as the bid/ask spread and the asset amount available at best price, which also varies with time. Depending on the problem addressed with resampling, one may either want to preserve or destroy correlations between two time series. If correlations between two samples are weak, MCMC can be implemented for both coupled samples independently, and bootstrap for coupled samples can be reduced to picking up pairs of variables at the same time. However, if both samples have autocorrelations of varying strength, choosing the bootstrap block size becomes tricky.

On the other hand, it may not be desirable to preserve all autocorrelations in some applications. A case in point is the simulation of execution

sequence xt ¼ {Xt–R, Xt–Rþ1, . . . , XT} into the market positions. The trading signal functions can assume values of 0 (cash), 1 (long position),

and –1 (short position). Buy-and-hold strategy or cash position are usually chosen as the benchmark models. In the latter case, b0 is the risk-free interest rate.

The null hypothesis in BRC is that the performance of the best technical trading rule is no better than the performance of the benchmark. In terms of the average returns for each strategy k, f k ¼ Eðf kÞ, this hypothesis states

Rejection of the null hypothesis implies that performance of the best technical trading is superior to the benchmark. The stationary bootstrap is used in BRC for resampling the observed values of fk,t. This yields the average bootstrapped values f k;i for each strategy k where i ¼ 1, . . . , N indexes the bootstrap samples. Then, the following statistics are calculated:

Finally, the percentile of Vi is compared with V for estimating p-value of BRC.

Sullivan et al. (1999) have investigated a wide range of technical trading strategies using BRC on 100 years (1897 to 1996) of daily data for the Dow Jones Industrial Average. They have shown, in particular, that the strategy that is the best (according to BRC) for a given sample outperforms holding cash. However, the best in-sample strategy is not superior to the benchmark when tested out of sample.

N e w D e v e l o p m e n t s

The ideas used in BRC for avoiding data snooping were expanded further in several directions. First, Hansen (2005) has shown that BRC may have a lower power due to the possible presence of poorly performing strategies in the test. Note that the power of the statistical test relates to the ability of rejecting false null hypotheses. Hansen enhanced BRC with two contributions. The first is studentizing the performance statistics, that is, dividing

f k by an estimator of their standard deviation sk. Secondly, Hansen suggested including in the test only promising strategies with performance that satisfy the condition

Furthermore, Romano and Wolf (2005), and Hsu et al. (2009) suggested using a stepwise protocol for enhancing the data snooping technique. This approach is focused on the family wide error (FWE) rate that is defined as the probability of rejecting at least one correct null hypothesis. Namely, the stepwise protocol identifies the strategies that violate the null hypothesis at a predefined significance level.

In a nutshell, this protocol starts with calculating a critical value, q, for all bootstrapped strategy performances using a chosen significance level. Next, all f k are rearranged in descending order. The top model k is rejected if its performance exceeds q. If no rejection occurs, the process stops. Otherwise, the model is eliminated and the critical value is recalculated. The process continues until no model is rejected.

An interesting result Hsu et al. (2009) obtained with the stepwise bootstrap is that technical trading strategies that used to be profitable for emerging market indices in the past have deteriorated after exchange-traded funds replicating these indexes were introduced.

SUMMARY

&Simple back-testing protocols divide available data sample into two parts, one of which is applied for in-sample model calibration, with the other one used for out-of-sample testing.

&Akaike information criterion can be used for choosing the optimal number of model parameters.

&Important trading performance measures include, among others, total return and its variance, percentage of winning trades, and maximum drawdown.

&Resampling, most notably bootstrap and MCMC, is widely used as a measure against data snooping.

&Bootstrap implies creating multiple data sets by randomly drawing the elements of a given sample (with replacement). Block

bootstrap may preserve autocorrelations in samples but choosing the block size is not trivial.

&A more sophisticated form of bootstrap implies estimating a mathematical model that fits the given sample (e.g., the random walk with drift) and bootstrapping the model’s residuals.

&MCMC is based on the assumption that time series is a Markov chain. In MCMC, the probabilities of given return conditioned on past returns are calculated for the original data sample. Then, new samples are generated using these conditional probabilities.

&The higher is the fraction of generated samples for which given trading strategy yields returns exceeding return for the original sample—the better is the chance that the strategy is profitable.

&White’s BRC can be used for comparing the efficiency of multiple trading strategies.

&The BRC can be enhanced with studentizing performance statistics and filtering out poorly performing strategies.

&The strategy that is the best for a given sample does not guarantee superior performance in the future.