9 Arbitrage trading strategies 9 1 Introduction Law

9. Arbitrage trading strategies 9. 1 Introduction Law of One Price: equivalent assets (with the same payoff) must have the same price. Violation => (deterministic) arbitrage: risk-free profiteering by buying low and selling high on different markets. Usually doesn’t last long. . . Examples: Drugs in US and Canada (persistent due to regulation) Triangle FX arbitrage: (EUR/USD)*(USD/JPY)/(EUR/JPY) = 1 (? ) Statistical arbitrage: statistical deviation of asset prices from expected values. Many arbitrage strategies are based on hedging: combining long & short positions in the same portfolio. Market-neutral portfolio => pair trading. Company A is ‘better’ than B; which shares to long/short? Mean reversion: divergence in returns is a temporary effect 1

9. Arbitrage trading strategies 9. 2 Hedging strategies (Stefanini (2006)) Equity hedge. Also index arbitrage, ADR arbitrage, 130/30 funds Equity market-neutral strategy & statistical arbitrage Khandani & Lo (2007): statistical arbitrage is “highly technical short-term mean-reversion strategies involving large numbers of securities (hundreds to thousands, depending on the amount of risk capital), very short holding periods (measured in days to seconds), and substantial computational, trading, and IT infrastructure”. Equity market-neutral strategy is a less demanding approach that may involve lower frequency of trading and fewer securities. Economic parameters also may be incorporated into predicting models derived under umbrella of market-neutral strategies. 2

9. Arbitrage trading strategies 9. 2 Hedging strategies (continued) Convertible arbitrage. Convertible bonds are bonds that can be converted into shares of the same company. Convertible bonds often decline less in a falling market than shares of the same company do. Fixed-income arbitrage. This strategy implies taking long and short positions in different fixed-income securities. Issuance-driven arbitrage, e. g. “on-the-run” versus “off-the-run” US Treasury bonds. Newly issued (on-the-run) Treasuries usually have yields lower than older off-the-run Treasuries but both yields are expected converge with time. A more generic yield curve arbitrage is based on anomalies in dependence of bond yield on maturity. Other opportunities may appear in comparison of yields for Treasuries, corporate bonds, and municipal bonds. Mortgage-backed securities (MBS): a form of fixed income with prepayment option. Namely, mortgage borrowers can prepay their loans fully or partially prior to the mortgage term, which increases uncertainty of the MBS value. 3

9. Arbitrage trading strategies 9. 2 Hedging strategies (continued 2) Relative value arbitrage versus event-driven arbitrage Merger arbitrage (also called risk arbitrage). This form of arbitrage involves buying shares of a company that is expected to be bought and short selling the shares of the acquirer. Distressed asset arbitrage Multi-strategy hedge funds Global macro hedge funds 4

9. Arbitrage trading strategies 9. 3 Cointegration and causality How to choose trading pairs? Correlation analysis can be used only for stationary variables (returns) – measures only short-range relationships that affected by volatility. Cointegration can be used for long-range trends. Vidyamurthy (2004) Two non-stationary time series are cointegrated if their linear combination is stationary. While the difference between two arbitrary prices series (the spread) may vary unpredictably, it is stationary for cointegrated series. If the spread deviates from its stationary value, it is expected that mean reversion will bring it back, or, in other words, mispricing will be eliminated. Formally, two time series x, y ~ I(1) are cointegrated if there is such a constant α that z = x – αy ~ I(0) The standard technique (Engle & Granger) for finding if two time series are cointegrated is to construct the linear regression xt = αyt+ c + εt If residuals are stationary (check with augumented Dickey-Fuller test), x and y are cointegrated. 5

9. Arbitrage trading strategies 9. 3 Cointegration and causality (continued) Granger causality. Consider error correction model (ECM) Δxt = δ 1 + Δyt = δ 2 + Granger representation theorem: cointegration and ECM are equivalent. If xt and yt are cointegrated, then γ 1 < 0 and αγ 2 > 0. When x and y are cointegrated and lagged terms of one time series (e. g. x) are significant in the dynamics of the other time series (y), it is said that x “Granger-causes” y. Granger causality is not true causality. It may be simply that both x and y depend on a common factor but have different lagging models. 6

9. Arbitrage trading strategies 9. 4 Pairs trading Simple rule: Buy portfolio (long A with log price x, short B with log price y) when xt - αyt = c – Δ; Sell portfolio when xt - αyt = c + Δ; How to select A, B, and Δ? Arbitrage pricing theory (APT): similar companies have the same risk factors/premiums: E[Ri(t)] = λ 0 + βi 1λ 1 +. . . + βi. K λK Grinold & Kahn (2000): “APT is an art, not a science”. Hence, check the same industry first, then sectors (Materials, Finance, etc. ), or even beyond (Gatev et al (2006)). 7

9. Arbitrage trading strategies 9. 4 Pairs trading (continued) Practitioners: Δ = 2σ (? ) Vidyamurthy (2004): Δ must maximize the trading profits W. If the cumulative distribution function of the spread is Pr(Δ), then W ~ Δ(1 - Pr(Δ)). For the normal distribution, Δ = 0. 75σ. Another problem: xt - αyt or yt - αxt ? Vidyamurthy (2004): choosing a variable with lower volatility as the independent one. Gatev (2006): choose such trading pairs that minimize the sum of squared deviations between the two normalized price series (“the distance measure”). Trading must be done within acceptable time horizon. Hence analysis of mean-reversion time. Elliott (2005): the Ornstein-Ulenbeck equation for modeling mean reversion of the spread X(t): d. X(t) = ρ(µ – X(t))dt + σd. W 8

9. Arbitrage trading strategies 9. 5 Arbitrage risks Liu & Longstaff (2004) quote a bond trader: “So there’s an arbitrage. So what? This desk has lost a lot of money on arbitrages”. Event-driven arbitrage may go wrong simply because event does not happen or happens in an unexpected way. Relative value arbitrage: past mean reversion may be broken. The spread may widen before converging => margin calls: - Long-Term Capital Management (Lowenstein (2000)) - Unprecedented losses that long/short hedge funds had in August of 2007 (Khandani and Lo (2007)). Jurek & Yang (2007): arbitrageurs with longer reporting horizon are more aggressive. Liu & Timmermann (2009): risk aversion may yield unbalanced long and short positions. 9