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# 数学代写|金融数学代写Financial Mathematics代考|ACTS201 Practical Considerations

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## 数学代写|金融数学代写Financial Mathematics代考|Practical Considerations

After identifying a potential pairs strategy via co-integration testing for instance, traders need to pay particular attention to the expected profitability of the strategy once real-life trading frictions have been taken into consideration. In practice, a significant portion of potential pairs trading strategies experience negative $\mathrm{P} \& \mathrm{~L}$ in production due to large execution slippage, whether originating from aggressive hedging at trade initiation, or position closing out.

When back testing a pairs strategy, one of the immediate costs that might render the strategy unprofitable is the spread cost. In other words, how much will it cost to extract liquidity when the signal triggers, in particular if the signal is too short-lived to allow for sourcing liquidity passively, via limit order book posting. For intraday pairs trading, it is also valuable to analyze if there is any asymmetry of cost between the initiation trade and the unwind trade. Oftentimes, it is not uncommon to witness significantly higher exit costs once the market has normalized following a short-lived entry trade opportunity.

The individual instrument’s microstructure also plays an important role in the strategy profitability. For instance long queue names (instruments for which the top of book liquidity is large in comparison to daily volume) are harder to trade, because the time it takes for an order to reach the top of the queue and trade passively is generally larger than the time that is acceptable for the strategy to mitigate its legging risk. $^{1}$ As a result, most of the executions on the leg that displays a long queue will end up having to pay the full spread to be executed.

## 数学代写|金融数学代写Financial Mathematics代考|Cross-Sectional Momentum Strategies

The trading strategies discussed thus far depend exclusively on the existence of time-series patterns in returns or on the relationships between returns and the risk factors. Time series strategies generally exploit the fact that stock prices may not at times follow a random walk, at least in the short run. Even if price follows a random walk, it has been shown that strategies based on past performance contain a crosssectional component that can be favorably exploited. For instance, momentum strategies demonstrate that the repeated purchase of past winners, from the proceeds of the sale of losers can result in profits. This is equivalent to buying securities that have high average returns at the expense of securities that yield low average returns. Therefore if there is some cross-sectional variance in the mean returns in the universe of the securities, a momentum strategy is profitable. On the other hand, a contrarian strategy will not be profitable even when the price follows a random walk. For the demonstration of the momentum strategy in the low frequency setting, refer to Jegadeesh and Titman (1993) [215], 2002 [217]) and Conrad and Kaul (1998) [82].

To better understand the sources of profit from a momentum strategy Lo and MacKinlay (1990) [249] decompose the profit function into three components. We will follow the set-up as outlined in Lewellen (2002) [242]. Let $r_{i, t-1}^{k}$ be the asset i’s, ‘ $k$ ‘ month return ending in ‘ $t-1$ ‘ and let $r_{m, t-1}^{k}$ be the return from the equal-weighted (market) index’s ‘ $k$ ‘ month return ending in ‘ $t-1$ ‘. The allocation scheme over ‘ $N$ ‘ assets under the momentum strategy is,
$$w_{i, t}^{k}=\frac{1}{N}\left(r_{i, t-1}^{k}-r_{m, t-1}^{k}\right), \quad i=1, \ldots, N .$$

## 数学代写|金融数学代写Financial Mathematics代考|Cross-Sectional Momentum Strategies

$$w_{i, t}^{k}=\frac{1}{N}\left(r_{i, t-1}^{k}-r_{m, t-1}^{k}\right), \quad i=1, \ldots, N .$$

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