Cointegration Pairs Trading Strategy On Derivatives 1

Transcription

1 Cointegration Pairs Trading Strategy On Derivatives Cointegration Pairs Trading Strategy On Derivatives 1 By Ngai Hang CHAN Co-Authors: Dr. P.K. LEE and Ms. Lai Fun PUN Department of Statistics The Chinese University of Hong Kong November 27, Research supported in part by grants from GRF-RGC-HKSAR. Ngai Hang CHAN CUHK / 31

2 Cointegration Pairs Trading Strategy On Derivatives Table of Contents 1 Introduction 2 Cointegration Pairs Trading Strategy on Stocks 3 Cointegration Pairs Trading Strategy on Derivatives 4 Empirical Study with Foreign Exchange Options 5 Further Trading Strategies 6 Conclusion and Further Discussion Ngai Hang CHAN CUHK / 31

3 Cointegration Pairs Trading Strategy On Derivatives Introduction Introduction Ngai Hang CHAN CUHK / 31

4 Cointegration Pairs Trading Strategy On Derivatives Introduction Arbitrage Arbitrage: free lunch; earning extra profit without taking additional risk. The no arbitrage assumption serves as the building block of modern finance. It is the cornerstone of the celebrated Black-Scholes models for option pricing. Statistical Arbitrage: An attempt to profit from the pricing efficiencies that are identified through the use of mathematical models. Statistical arbitrage attempts to profit from the likelihood that prices will trend together toward a historical norm. Unlike arbitrage, statistical arbitrage is not risk-free. Ngai Hang CHAN CUHK / 31

5 Cointegration Pairs Trading Strategy On Derivatives Introduction Style Investing Barberis and Shleifer (2003) discussed style investing. Style: Assets with similar characteristics. Switchers Allocate funds at the level of a style. The amount allocated in each style depends on the past performance of that style relative to other styles. Switchers cause co-movements for the stocks in the same style. Ngai Hang CHAN CUHK / 31

6 Cointegration Pairs Trading Strategy On Derivatives Introduction Arbitrage Opportunity in Short-run Wilson and Marashdeh (2007) argued that co-movements between stock prices imply market efficiency in long-run equilibrium. Co-movements cause arbitrage opportunity in short-run. Inefficiency in the short-run is eliminated by arbitrage activities. Resulting efficiency in the long-run equilibrium. It is reasonable to think that stock has co-integration property in the long-run. We may take advantage of such a property to find statistical arbitrage opportunities in the short-run. Ngai Hang CHAN CUHK / 31

7 Cointegration Pairs Trading Strategy On Derivatives Introduction Introduction Some financial instruments move more or less in sync with each other, because they are driven by similar fundamental (e.g. economic) factors. For example, Stock prices of Coca Cola and Pepsi. Currency Pairs AUD/USD vs NZD/USD. However, they do not move EXACTLY the same because of their individual technical factors, which can be assumed to be noises on top of the common movement. Statistical Approach: Cointegration Assume that the underlying are stochastic processes sharing the same stochastic drift. Filter out the co-movement of pairs of market instruments by identifying possible stationary series which is a linear combination of two non-stationary series (e.g. prices of two stocks). Ngai Hang CHAN CUHK / 31

8 Cointegration Pairs Trading Strategy On Derivatives Introduction Cointegration Definition Two non-stationary time series {X t } and {Y t } are cointegrated if some linear combination ax t + by t, with a and b being constants, is a stationary series. There are two popular approaches to establish the cointegration relationship: Engle-Granger methodology (1987) Johansen methodology (1988) Ngai Hang CHAN CUHK / 31

9 Cointegration Pairs Trading Strategy On Derivatives Cointegration Pairs Trading Strategy on Stocks Cointegration Pairs Trading Strategy on Stocks Ngai Hang CHAN CUHK / 31

10 Cointegration Pairs Trading Strategy On Derivatives Cointegration Pairs Trading Strategy on Stocks Cointegration Pairs Trading Strategy on Stocks The notion of cointegration has been widely used in finance and econometrics, in particular in constructing statistical arbitrage strategy in the stock market. Profitability has been reported using the cointegration strategy on stocks trading, see Chan (2010). Ngai Hang CHAN CUHK / 31

11 Cointegration Pairs Trading Strategy On Derivatives Cointegration Pairs Trading Strategy on Stocks Cointegration Pairs Trading Strategy on Stocks Example: (Chan, 2010) Let X t and Y t be two stock prices at time t. Assume that a log X t + b log Y t is stationary, i.e. log X t and log Y t are cointegrated. By Taylor expansion, a log The X t notion + b logof Ycointegration t a(log X t0 + has X t X been t0 widely ) + b(log usedy in t0 + finance Y t Y and t0 ) X econometrics, in particular in constructing t0 Y statistical arbitrage t0 strategy in the stock = a market. X t + b Y t + a(log X t0 1) + b(log Y t0 1). X t0 Y t0 Because Profitability a(log X has been reported using the cointegration strategy on t0 1) + b(log Y t0 1) is a constant, the stationarity of a log stocks X trading, see Chan (2010). a t + b log Y t implies that X t0 X t + b Y t0 Y t is approximately stationary, a i.e. X t0 X t + b Y t0 Y t should exhibit mean-reverting property. We can initiate a position with c a X t0 shares of Y for any given value c, where c can be considered as the starting initial capital. shares of X and c b Y t0 Ngai Hang CHAN CUHK / 31

12 Cointegration Pairs Trading Strategy On Derivatives Cointegration Pairs Trading Strategy on Stocks Implementation of Trading Strategy on Stocks Ngai Hang CHAN CUHK / 31

13 Cointegration Pairs Trading Strategy On Derivatives Cointegration Pairs Trading Strategy on Stocks Investigate possible cointegrated series from pairs of log(stock prices) [or log(exchange rates)] based on the Johansen test (typically 5% of pairs). For each cointegration pairs, check if the current level of the stationary series (i.e. a log X t + b log Y t )) is too low/high against its historical mean (e.g. Z-score = 2, +2). Enter the trade (i.e. buy/sell a X t of X and b Y t shares of Y at time t 0 and expect the stationary series a log X t + b log Y t ) to mean-revert back to it historical average level. For a portfolio with 46 stocks, there are C 46 2 = 575 possible combination of stock pairs and E(Number of cointegrated series at t 0 ) 575x5% 29 pairs. Works well for stock prices in the same sector, harder to interpret for stocks from different sectors. Not much juice on simple financial instruments, as other statistical techniques based on mean-reversion assumption (e.g. PCA) is able to detect similar trades. Ngai Hang CHAN CUHK / 31

14 Cointegration Pairs Trading Strategy On Derivatives Cointegration Pairs Trading Strategy on Derivatives Would like to extend the cointegration strategy on derivatives. For example, extend from Currency pairs of AUD/USD vs NZD/USD to Option on AUD/USD vs option on NZD/USD. Ngai Hang CHAN CUHK / 31

15 Cointegration Pairs Trading Strategy On Derivatives Cointegration Pairs Trading Strategy on Derivatives Cointegration Pairs Trading Strategy on Derivatives Ngai Hang CHAN CUHK / 31

16 Cointegration Pairs Trading Strategy On Derivatives Cointegration Pairs Trading Strategy on Derivatives Implied volatility is the assessment of the future variation in an underlying asset. Ngai Hang CHAN CUHK / 31

17 Cointegration Pairs Trading Strategy On Derivatives Cointegration Pairs Trading Strategy on Derivatives Implied volatility is the assessment of the future variation in an underlying asset. Even though S X σ 2 X not cointegrated cointegrated S Y σ 2 Y Ngai Hang CHAN CUHK / 31

18 Cointegration Pairs Trading Strategy On Derivatives Cointegration Pairs Trading Strategy on Derivatives Implied volatility is the assessment of the future variation in an underlying asset. Even though S X σ 2 X not cointegrated cointegrated S Y σ 2 Y because they may be affected by the same exogenous event so that people have the same perspective to the variations of the underlying assets. Similar with the cointegration pairs trading strategy on stocks, a divergence from the mean level of the cointegration pairs can be captured to make a profit. Ngai Hang CHAN CUHK / 31

19 Cointegration Pairs Trading Strategy On Derivatives Cointegration Pairs Trading Strategy on Derivatives How to trade volatility? Ngai Hang CHAN CUHK / 31

20 Cointegration Pairs Trading Strategy On Derivatives Cointegration Pairs Trading Strategy on Derivatives How to trade volatility? Straddle!! Ngai Hang CHAN CUHK / 31

21 Cointegration Pairs Trading Strategy On Derivatives Cointegration Pairs Trading Strategy on Derivatives Why Straddle? Ngai Hang CHAN CUHK / 31

22 Cointegration Pairs Trading Strategy On Derivatives Cointegration Pairs Trading Strategy on Derivatives What is Straddle? Definition: A long (short) straddle is long (short) a call option and a put option at the same strike price and expiration date. Ngai Hang CHAN CUHK / 31

23 Cointegration Pairs Trading Strategy On Derivatives Cointegration Pairs Trading Strategy on Derivatives What is Straddle? Definition: A long (short) straddle is long (short) a call option and a put option at the same strike price and expiration date. At-the-money Straddle (ST t ) ST t = C t + P t = 2 π S tσ T t + O(S t σ 2 (T t)) Ngai Hang CHAN CUHK / 31

24 Cointegration Pairs Trading Strategy On Derivatives Cointegration Pairs Trading Strategy on Derivatives How the cointegration strategy works? Suppose the trading signal is TS t = aσ X t bσ Y t I (0), where a, b > 0, the z-score of the trading signal is Z t = TSt µ σ, where µ are σ are the mean and the standard deviation of the trading signal. Ngai Hang CHAN CUHK / 31

25 Cointegration Pairs Trading Strategy On Derivatives Cointegration Pairs Trading Strategy on Derivatives How the cointegration strategy works? Suppose the trading signal is TS t = aσ X t bσ Y t I (0), where a, b > 0, the z-score of the trading signal is Z t = TSt µ σ, where µ are σ are the mean and the standard deviation of the trading signal. TS t << µ σ X t too low comparing to σ Y t, long ST X t and short ST Y t. TS t >> µ σ X t too high comparing to σ Y t, long ST Y t and short ST X t. Ngai Hang CHAN CUHK / 31

26 Cointegration Pairs Trading Strategy On Derivatives Cointegration Pairs Trading Strategy on Derivatives How does the cointegration strategy work? Setting up the Trading Portfolio For t (t 0, T ),the value of the straddles portfolio Π t becomes Π t = A (C X S t0 + P X S t0 ) B (C Y S t0 + P Y S t0 ), where C X S t0, P X S t0, C Y S t0, P Y S t0 at time t 0 are at-the-money options. Based on the Taylor expansion, Π t can be approximated by Π t = Π t t t + Π t S X t S X t + Π t S Y t St Y Π t 2 (St X ) 2 ( S t X ) Π t 2 (St Y ) 2 ( S t Y ) 2 + Π t σt X σ X t + Π t σ Y t σ Y t, Ngai Hang CHAN CUHK / 31

27 Cointegration Pairs Trading Strategy On Derivatives Cointegration Pairs Trading Strategy on Derivatives How does the cointegration strategy work? Setting up the Trading Portfolio For t (t 0, T ),the value of the straddles portfolio Π t becomes Vega Part : I = Π t σ Π t = A (CS X t0 + PS X t X + Π t σ t0 ) B (CS Y t Y t0 + PS Y t0 ), Delta where CS X Part t0, PS X t0, CS Y : II t0, PS Y = Π t t0 at Stime X S t t X + Π t 0 are at-the-money options. t S Y St Y t Gamma Part : III = 1 2 Π t 2 (St X ) 2 ( S t X ) Π t 2 (St Y ) 2 ( S t Y ) 2 Based on the Taylor expansion, Π t can be approximated by Theta Part : IV = Π t t Π t = Π t t + Π t t St X + Π t St Y Π t t 2 (St X ) 2 ( S t X ) 2 S X t σ X t S Y t σ Y t i.e. Π t = I + II + III + IV Π t 2 (St Y ) 2 ( S t Y ) 2 + Π t σt X σt X + Π t σ Y t σ Y t, Ngai Hang CHAN CUHK / 31

28 Cointegration Pairs Trading Strategy On Derivatives Cointegration Pairs Trading Strategy on Derivatives Requirements of the trade The trade will be dominated by vega if the trade fulfills the following requirements: 1 Short-term period of trading; 2 Using long-dated option; 3 The mean of the trading signal is negative (positive), the position is initiated when the trading signal is too low (high). Otherwise, the position should not be initiated. In the derivation below, { it is shown that at time t, TS t Θ = Πt t 2, long the portfolio, T t, short the portfolio. TS t 2 T t where TS t = Z t σ + µ, µ and σ are the mean and standard deviation of the trading signal TS t. Ngai Hang CHAN CUHK / 31

29 Cointegration Pairs Trading Strategy On Derivatives Cointegration Pairs Trading Strategy on Derivatives How does the cointegration strategy work? Because the trading is in a short-term period since time t 0, the Greek Letters can be approximated. Assume that µ(ts) < 0 and the portfolio of longing STt X and shorting STT Y is entered. Let A = π 2 cointegration. a, B = π St X 2 0 By the approximation of Φ(d 1 ) and Φ(d 2 ): b S Y t 0, where a, b are the coefficients of the Φ(d 1 ) = σ T t 2π + O(d 2 1 ), Φ(d 2 ) = 1 2 σ T t 2π + O(d 2 2 ), and the assumption that S t is driven by the Geometric Brownian motion ds t S t = r dt + σ r dw t, where σ r is the realized volatility, I, II, III, IV can be approximated as follows. Ngai Hang CHAN CUHK / 31

30 Cointegration Pairs Trading Strategy On Derivatives Cointegration Pairs Trading Strategy on Derivatives How does the cointegration strategy work? Because the trading is in a short-term period since time t 0, the Greek Letters canapproximation be approximated. Assume that µ(ts) < 0 and the portfolio of longing STt X and shorting STT Y is entered. Let A = E(I ) π a 2, B = T t 0 (E(TS t ) TS t0 ), π b St X 2, where a, b are the coefficients of the 0 St Y 0 cointegration. E(II ) t T t 0 r t0 TS t0, 2 t By the approximation E(III ) of Φ(d 1 ) and Φ(d 2 ): 2 ( ae(σr X )2 T t 0 σt X be(σr Y )2 0 σt Y ), 0 Φ(d 1 ) = σ T t 2π + O(d E(IV ) t TS t 1 2), 0 Φ(d 2 ) 2, = T 1 2 t σ 0 T t 2π + O(d2 2), and the assumption where t = that t S t 0, t is r t0 driven is the by risk-free the Geometric rate, (σx r Brownian )2 and motion (σy r )2 are the average ds t S t = squared r dt + annual σ r dw realized t, volatilities. where σ r is the realized volatility, I, II, III, IV can be approximated as follows. Ngai Hang CHAN CUHK / 31

31 Cointegration Pairs Trading Strategy On Derivatives Cointegration Pairs Trading Strategy on Derivatives How does the cointegration strategy work? Vega Part: I = T t 0 (E(TS t ) TS t0 ) This is the main profit to be captured by the trading strategy. Should the trading signal reverts back to its mean value, then E(I ) > 0. Ngai Hang CHAN CUHK / 31

32 Cointegration Pairs Trading Strategy On Derivatives Cointegration Pairs Trading Strategy on Derivatives How does the cointegration strategy work? Proposition 1 If the trade is in a short-time period, E(II ) + E(IV ) > 0. Vega Part: Proposition I = T 2 t 0 (E(TS t ) TS t0 ) This is the Assume main that profit to be captured by the trading strategy. Should the trading signal 1 Annualized reverts back volatilities to its mean (include value, implied then E(I volatility ) > 0. and realized volatility) of the underlying assets are smaller than 80% ; 1 2 c σi t 0 < σi r < cσt i 0, for i = X and Y, c > 1; 3 σ(ts t ) > 1. Then E(I ) + E(III ) > 0. Ngai Hang CHAN CUHK / 31

33 Cointegration Pairs Trading Strategy On Derivatives Cointegration Pairs Trading Strategy on Derivatives In conclusion If the implied volatilities of the underlying assets are not too high and do not deviate too far from the corresponding realized volatilities, and if E(TS t ) TS t0 is large enough, then E( Π t ) = E(I ) + E(II ) + E(III ) + E(IV ) > 0. Ngai Hang CHAN CUHK / 31

34 Cointegration Pairs Trading Strategy On Derivatives Empirical Study with Foreign Exchange Options Empirical Study with Foreign Exchange Options Ngai Hang CHAN CUHK / 31

35 Cointegration Pairs Trading Strategy On Derivatives Empirical Study with Foreign Exchange Options Empirical Study with Foreign Exchange Options Ngai Hang CHAN CUHK / 31

36 Cointegration Pairs Trading Strategy On Derivatives Empirical Study with Foreign Exchange Options Find out the cointegration pairs In our analysis, FX options with 3-month expiry of 30 currency pairs (e.g. EUR/USD) for all the strikes from 2009Q4 to 2011Q4 were considered. The Johansen test is based on the data from 2009Q4-2011Q3. The cointegration pairs identified will be used as the trading signals for 2011Q4. Ngai Hang CHAN CUHK / 31

37 Cointegration Pairs Trading Strategy On Derivatives Empirical Study with Foreign Exchange Options Find out the cointegration pairs In our analysis, FX options with 3-month expiry of 30 currency pairs (e.g. EUR/USD) for all the strikes from 2009Q4 to 2011Q4 were considered. The Johansen test is based on the data from 2009Q4-2011Q3. The cointegration pairs identified will be used as the trading signals for 2011Q4. According to the results from Johansen test, the implied volatilities of GBPNZD and GBPUSD are significantly cointegrated with the parameters a = and b = , i.e. the trading signal is TS = σ GBPNZD σ GBPUSD. Ngai Hang CHAN CUHK / 31

38 Cointegration Pairs Trading Strategy On Derivatives Empirical Study with Foreign Exchange Options Details of cointegration pairs 4 z score of the Trading Signal /1/2009 4/1/ / 1/2010 4/1/ /1/2011 The z-score of the Trading Signal on Oct 3 rd, 2011 is 3.39 and on Oct 7 th, 2011, it is Ngai Hang CHAN CUHK / 31

39 Cointegration Pairs Trading Strategy On Derivatives Empirical Study with Foreign Exchange Options Details of the trade Cumulative P/L (100) 4 Oct 5 Oct 6 Oct 7 Oct Actual P/L vega delta gamma theta High Order Figure: Change of P/L due to Greek Letters The value of the portfolio is -USD3,465 on 03/10/2011, and then the value increases to -USD2,783 on 07/10/2011. Finally, we gain +USD682 by this strategy. Ngai Hang CHAN CUHK / 31

40 Cointegration Pairs Trading Strategy On Derivatives Empirical Study with Foreign Exchange Options More examples We identify possible cointegration in any two currency pairs, based on the data from 2009Q1-2010Q4. The cointegration pairs identified will be used as trading signals for 2011Q1. We repeated the same procedure to identify possible cointegration pairs during 2009Q2-2011Q1, 2009Q3-2011Q2 and 2009Q4-2011Q3, which would be served as the trading signals for 2011Q2, 2011Q3 and 2011Q4. Ngai Hang CHAN CUHK / 31

41 Cointegration Pairs Trading Strategy On Derivatives Empirical Study with Foreign Exchange Options More examples We identify possible cointegration in any two currency pairs, based on the data from 2009Q1-2010Q4. The cointegration pairs identified will be used as trading signals for 2011Q1. We repeated the same procedure to identify possible cointegration pairs during 2009Q2-2011Q1, 2009Q3-2011Q2 and 2009Q4-2011Q3, which would be served as the trading signals for 2011Q2, 2011Q3 and 2011Q4. Ngai Hang CHAN CUHK / 31

42 Cointegration Pairs Trading Strategy On Derivatives Empirical Study with Foreign Exchange Options More Examples The profit by this strategy in each trade is as follow. 3,000 Q1 Q2 Q3 Q4 2,000 1,000 0 (1,000) (2,000) 1,976 1,290 8,691 3,998 (3,000) Total Profit: -1,427USD Ngai Hang CHAN CUHK / 31

43 Cointegration Pairs Trading Strategy On Derivatives Further Trading Strategies Further Trading Strategies Ngai Hang CHAN CUHK / 31

44 Cointegration Pairs Trading Strategy On Derivatives Further Trading Strategies Strategy Performance: Straddles on HK Stocks Core Part of the trade Mean Reversion of Trading Signal => positive Vega Positive Theta (Carry) Trades with positive theta (P/L over time) were chosen from our trade selection criteria Negative Gamma Opposite to Theta in general, a function of realized volatilities Distribution of Delta roughly symmetric 0 on average 0 for each trade Ngai Hang CHAN CUHK / 31

45 Cointegration Pairs Trading Strategy On Derivatives Further Trading Strategies Strategy Performance: Straddles on FX Rates Why doesn t work on FX Rates? More affected by fundamental factors (vs. technical factors for stocks) What happened in July August 2011? Risk Aversion => Realized vol => Implied vol => Cointegration Opportunities Need a criterion on Implied vol based on forecasts of Realized vol Ngai Hang CHAN CUHK / 31

46 Cointegration Pairs Trading Strategy On Derivatives Modeling and Forecasting Realized Volatilities Modeling and Forecasting Realized Volatilities Ngai Hang CHAN CUHK / 31

47 Cointegration Pairs Trading Strategy On Derivatives Modeling and Forecasting Realized Volatilities Estimation of Realized Volatility 1. Historical Estimate: Log-daily-return r t = log(x t ) log(x t 1 ), H H ˆR t 2 = (r t i+1 r t ) 2 /(H 1), with r t = r t i+1 /H i=1 2. J.P. Morgan Risk Metrics: Exponentially Weighted Moving Average of squared log-daily-return: ˆR t 2 = 1 λ H 1 λ H λ i (r t i+1 r t ) 2, where λ = i=1 3. Garman-Klass Estimates: Assume the underlying X t follows the Geometric Brownian Motion: dx t = µx t dt + σx t dw t, ˆR 2 t = 0.17(O t C t 1 ) 2 f i= (u t d t ) 2 (1 f )4 log 2, where O t, C t, u t and d t are the Open, Close, High and Low of the underlying of the t-th day, and f is fraction of non-trading hours in a trading day. Ngai Hang CHAN CUHK / 31

48 Cointegration Pairs Trading Strategy On Derivatives Modeling and Forecasting Realized Volatilities 4. High Frequency Estimate: Let {r n,t }, n = 1,..., N, t = 1,..., T be the log-return of an underlying asset at the nth minute of the t-th day. Assume that r n,t are i.i.d with mean zero and constant variance σ 2 /N. The High Frequency Estimate is ˆR 2 t = N rn,t 2 for t = 1,..., T. n=1 Advantages of High Frequency Estimate: a) The estimate depends on todays information ONLY No lagging issue as in Historical Estimate and JP Morgan Risk Metrics. b) Unbiased estimator for σ 2. In contrast, Garman-Klass underestimates σ 2, as the high and low observed in discrete time under- and over-estimate the high and low in GBM Ngai Hang CHAN CUHK / 31

49 Cointegration Pairs Trading Strategy On Derivatives Modeling and Forecasting Realized Volatilities Forecasting Realized Volatilities R t Andersen, Bollerslev, Diebold and Labys (2003) proposed the use of ARFIMA(1,1,0) model on the log of realized volatilities of high-frequency estimates y t = log(ˆr t ) Φ(L)(1 L) d y t = ɛ t, ɛ t i.i.d. N(0, σ 2 ɛ ). The model is able to a) capture the asymmetric property of the unconditional distribution of the unconditional distribution of ˆR t ; b) capture the long-memory property of ˆR t ; c) provide the best 1- and 10-day ahead realized volatilities forecasts among 10+ models. Ngai Hang CHAN CUHK / 31

50 Cointegration Pairs Trading Strategy On Derivatives Modeling and Forecasting Realized Volatilities Estimation of Realized Volatility Gamma Part: E(III ) = t 2 T t 0 ( ae(σr X )2 be(σr σt X Y )2 0 σt Y 0 ), where σ r X and σr Y are the average squared future annual realized volatilities for longing the portfolio. Ngai Hang CHAN CUHK / 31

51 Cointegration Pairs Trading Strategy On Derivatives Modeling and Forecasting Realized Volatilities Estimation of Realized Volatility Gamma Part: E(III ) = t 2 T t 0 ( ae(σr X )2 be(σr σt X Y )2 0 σt Y 0 ), where σ r X and σr Y are the average squared future annual realized volatilities for longing the portfolio. Andersen, Bollerslev, Diebold and Labys (2003) introduced the following methods to model and forecast the realized volatility. Modelling: High-Frequency Realized Volatility Estimation, Forecasting: ARFIMA model: s 2 t = N n=1 r 2 n,t, for t = 1, 2,, T, Φ(L)(1 L) d (y t µ) = ɛ t, where y t = log s t, d is the order of integration and ɛ t is a vector white noise process. Ngai Hang CHAN CUHK / 31

52 Cointegration Pairs Trading Strategy On Derivatives Modeling and Forecasting Realized Volatilities Additional Criterion Implied-Realized Criterion K = ae(σr X )2 σ X t 0 / be(σr Y )2 σ Y t 0 { d, if TS < 0, u, if TS > 0, Here, d set as 0.7 and u set as 1.3. Ngai Hang CHAN CUHK / 31

53 Cointegration Pairs Trading Strategy On Derivatives Modeling and Forecasting Realized Volatilities Additional Criterion Implied-Realized Criterion K = ae(σr X )2 σ X t 0 / be(σr Y )2 σ Y t 0 { d, if TS < 0, u, if TS > 0, Here, d set as 0.7 and u set as 1.3. Gamma-Vega Criterion Vega K = Gamma = T t0 (E(TS t ) TS t0 ) t be(σr σt X Y )2 ) 0 σt Y 0 2 T t 0 ( ae(σr X )2 l, Here, l set as 11. Ngai Hang CHAN CUHK / 31

54 Cointegration Pairs Trading Strategy On Derivatives Modeling and Forecasting Realized Volatilities Performance of the Original Strategy Period with High Realized Volatilities (July August 2011) Ngai Hang CHAN CUHK / 31

55 Cointegration Pairs Trading Strategy On Derivatives Modeling and Forecasting Realized Volatilities Performance of Trading Strategy 2 Most of the trades in July August were filtered out Ngai Hang CHAN CUHK / 31

56 Cointegration Pairs Trading Strategy On Derivatives Modeling and Forecasting Realized Volatilities Performance of Trading Strategy 3 Ngai Hang CHAN CUHK / 31

57 Cointegration Pairs Trading Strategy On Derivatives Modeling and Forecasting Realized Volatilities Summary of Trading Strategies Ngai Hang CHAN CUHK / 31

58 Cointegration Pairs Trading Strategy On Derivatives Conclusion and Further Discussion Conclusion and Further Discussion Ngai Hang CHAN CUHK / 31

59 Cointegration Pairs Trading Strategy On Derivatives Conclusion and Further Discussion Conclusion Cointegration strategy has been applied into the derivatives. If the trading signal can show mean-reverting in a short-term and the realized volatility is not too high and deviates too far from the implied volatility, the portfolio makes profit. The additional criteria, Implied-Realized Criterion and Gamma-Vega Criterion, are effective. Ngai Hang CHAN CUHK / 31

60 Cointegration Pairs Trading Strategy On Derivatives Conclusion and Further Discussion Further Discussion The number of traded underlying assets can be more than two. The dynamic hedging can be imposed in the strategy to minimize the impact of delta. This method may weaken the short-term period trading restriction, but costly. The method for modeling and forecasting realized volatility can be revised and the notion of fractional cointegration may be pursued for the further study. Ngai Hang CHAN CUHK / 31

61 Cointegration Pairs Trading Strategy On Derivatives Thank You Ngai Hang CHAN CUHK / 31

62 Cointegration Pairs Trading Strategy On Derivatives Appendix Engle-Granger s Methodology Consider a p-dimensional non-stationary I (1) time series {x t }. x t can be partitioned into (x 1t, x 2t ), where x 1t is a scaler and x 2t is an (p 1) 1 vector. Through the ordinary least squares (OLS) method, we have x 1t = ˆβx 2t + û t. If û t = x 1t ˆβx 2t I (0), it means that there exist ˆθ = (1, ˆβ ) such that ˆθx t I (0), i.e. x t is cointegration. Ngai Hang CHAN CUHK / 31

63 Cointegration Pairs Trading Strategy On Derivatives Appendix Johansen s Methodology Consider a p-dimensional non-stationary I (1) time series {X t }, which follows a VAR(k) process: X t = Φ 1 X t 1 + Φ 2 X t Φ k X t k + ε t, t =..., 1, 0, 1,..., where Φ 1, Φ 2,..., Φ k are p p matrices, and ε t is Gaussian random vector with mean 0 and covariance matrix Ω. Note that the above equation can be rewritten as a Vector Error Correction Model (VECM): X t = ΓX t 1 + Γ 1 X t Γ k 1 X t k+1 + ε t, where Γ = k i=1 Φ i I, Γ l = k j=l+1 Φ j, l = 1,..., k 1. Hence, Γ l, l = 1,..., k 1 are unrestricted. Ngai Hang CHAN CUHK / 31

64 Cointegration Pairs Trading Strategy On Derivatives Appendix Johansen s Methodology If Γ = α p r β p r, where r < p, then β X t is stationary, where α is the adjustment coefficients and β is the cointegration vector. Trace test: L trace = N p i=r+1 log(1 ˆλ i ), H 0 : K c = r vs H 1 : K c = p. Maximum eigenvalue test: L eig = N log(1 ˆλ r+1 ), H 0 : K c = r vs H 1 : K c = r + 1. Here N is the sample size, ˆλ i is the i-th largest canonical correlation and K c is the number of cointegrating vector. Ngai Hang CHAN CUHK / 31

65 Cointegration Pairs Trading Strategy On Derivatives Appendix Appendix A First of all, C t = S t e q(t t) Φ(d 1 ) Ke r(t t) Φ(d 2 ), P t = Ke r(t t) Φ( d 2 ) S t e q(t t) Φ( d 1 ), where d 1 = log( S t K σ2 )+(r q+ 2 σ T t By the Taylor expansion, Then, Hence, )(T t), d 2 = d 1 σ T t. Φ(d 1 ) = Φ(0) + φ(0)d 1 + φ (0)d1 2 + O(d 1 2) = d 1 2π + O(d1 2), Φ(d 2 ) = Φ(0) + φ(0)d 2 + φ (0)d2 2 + O(d 2 2) = d 2 2π + O(d2 2), C t = S q(t t) te σ T t + O(S t σ 2 (T t)), 2π P t = S q(t t) te σ T t + O(S t σ 2 (T t)). 2π 2 ST t = C t + P t = π S tσ T t + O(S t σ 2 (T t)) Ngai Hang CHAN CUHK / 31

66 Cointegration Pairs Trading Strategy On Derivatives Appendix Appendix B Proposition If the trade is in a short-time period, E(II ) + E(IV ) > 0. Proof. In section 3.3.1, we know that theta (IV) is a positive net carry in the trade. Compared the expected value of theta (IV) with that of delta (II), 1 = E(IV ) E(II ) (T t 0 )r t0. Because T t 0 is smaller than one (year) and r t0 is very small (0%-5% in most countries), E(II ) << E(IV ). Hence, E(II ) + E(IV ) E(IV ) E(II ) > 0, i.e. the loss due to delta is likely to be compensated by the positive carry from IV. Ngai Hang CHAN CUHK / 31

67 Cointegration Pairs Trading Strategy On Derivatives Appendix Appendix C Proposition Assume that 1 Annualized volatilities (include implied volatility and realized volatility) of the underlying assets are smaller than 80% ; 1 2 c σi t 0 < σi r < cσt i 0, for i = x, y, c > 1; 3 σ(ts t ) > 1. Then E(I ) + E(III ) > 0. Proof Under assumptions (1) and (2), one can show that E(III ) = t 2 ( ae(σr X )2 be(σr T t 0 σt X Y )2 0 σt Y 0 ) 0.4c2 t T t0. Ngai Hang CHAN CUHK / 31

68 Cointegration Pairs Trading Strategy On Derivatives Appendix cont. Recall that the trade requires using long dated options and to trade in a short-term period. To strike a balance between the trading requirements and options liquidity, the maturity of the options should be at least three months and the trade should not last over one month. i.e. T t 0 t 3. Vega is the main profit in the trade. Compared the expected value of gamma with that of vega, E(I ) E(III ) = T t0 (E(TS t) TS t0 ) t ( ae(σr ) 2 X 2 T t 0 σ t X be(σr )2 Y σ Y ) 0 t c 2 (E(TS t ) TS t0 ). Ngai Hang CHAN CUHK / 31

69 Cointegration Pairs Trading Strategy On Derivatives Appendix cont. The trade is initiated when TS t0 is too negative comparing to the mean level (E(TS t )), and c is close to 1. Hence, and therefore 7.5(E(TS c 2 t ) TS t0 ) > 15 σ(ts c 2 t ) >> 1, E(I ) + E(III ) E(I ) E(III ) > 0. Ngai Hang CHAN CUHK / 31