Bertram's Pairs Trading Strategy with Bounded Risk
BBertram’s Pairs Trading Strategy with Bounded Risk
Vladimír Holý
Prague University of Economics and BusinessWinston Churchill Square 1938/4, 130 67 Prague 3, Czech [email protected]
Michal Černý
Prague University of Economics and BusinessWinston Churchill Square 1938/4, 130 67 Prague 3, Czech [email protected]
February 9, 2021
Abstract:
Finding Bertram’s optimal trading strategy for a pair of cointegrated assets following theOrnstein–Uhlenbeck price difference process can be formulated as an unconstrained convex optimiza-tion problem for maximization of expected profit per unit of time. We generalize this model to theform where the riskiness of profit, measured by its per-time-unit volatility, is controlled (e.g. in caseof existence of limits on riskiness of trading strategies imposed by regulatory bodies). The resultingoptimization problem need not be convex. In spite of this undesirable fact, we demonstrate that itis still efficiently solvable. We also investigate the problem critical for practice that parameters ofthe price difference process are never known exactly and are imprecisely estimated from an observedfinite sample. We show how the imprecision affects the optimal trading strategy and quantify theloss caused by the imprecise estimate compared to a theoretical trader knowing the parameters exactly.
Keywords:
Pairs Trading, High-Frequency Trading, Bounded Risk Trading, Ornstein–UhlenbeckProcess
Pairs trading is a market-neutral trading strategy which exploits a long-term balance between twoassets and makes profit when they are temporarily out of balance. The strategy assumes that thespread between the two assets is a stationary process implying mean reversion in finite time. It simplysuffices to wait until the spread process is “far” enough from its mean value and then to bet it willrevert to the mean by opening a long position in the underperforming asset and a short position inthe overperforming asset. When the spread process indeed returns close to its mean, the positionsare closed and the profit is made. In the traditional sense, this is a kind of “free-lunch”, or arbitragetrading, where the profit is guaranteed. The tricky point is that mean reversion can be slow, meaningthat waiting times for the collection of the almost sure profit can be long. Intuitively, one couldhardly say that there is a “free-lunch” profit once it is necessary to wait for it for an extremely longtime period.There is a significant body of financial literature dealing with the pairs trading strategy. Themain methods include the distance approach of Gatev et al. (2006), the cointegration approach ofVidyamurthy (2004), the stochastic spread approach of Elliott et al. (2005), the stochastic controlapproach of Jurek and Yang (2007), the machine learning approach of Huck (2009), the copulaapproach of Liew and Wu (2013), and the principal components analysis approach of Avellaneda andLee (2010). For a literature review on pairs trading, see Krauss (2017).We take the stochastic spread approach and focus on determining the optimal values of thresholdscontrolling the course of the strategy. Elliott et al. (2005) proposed to model the spread between thetwo assets in a pair as the Ornstein–Uhlenbeck process which is a stationary Gauss–Markov process incontinuous time. Bertram (2009, 2010) then suggested to find the optimal entry and exit thresholdsof the spread by maximizing the expected profit per unit of time. This can be formulated as an1 a r X i v : . [ q -f i n . M F ] F e b nconstrained optimization problem. The Bertram’s approach is further utilized by Cummins andBucca (2012), Zeng and Lee (2014), Göncü and Akyildirim (2016), and Holý and Tomanová (2019).We generalize the work of Bertram (2009, 2010) by taking into account the variance of profitand consider a version augmented by a constraint formalizing an assumption that there exists anexogenous limit on maximal admissible riskiness (e.g., imposed by a regulatory authority). Thisconstraint can decrease the expected per-time-unit profit, but it also decreases the probability thatthe profit-collection time would be extremely long. The idea of pairs trading with bounded risk wasoriginally hinted by Holý and Tomanová (2019) while the current paper offers an in-depth treatmentof the model including the impact of misspecification of the spread process. Assumption 1 (price processes) (a) There are two assets A and B , with zero risk-free yields,the prices of which are driven by continuous-time price processes A s , B s ; s ≥ .(b) There exists a constant η , called cointegration coefficient , such that the price differenceprocess X s := A s − ηB s is stationary. Assumption 2 (trading environment) (a) An investor is allowed to open positions in the assets A , B both long and short.(b) Trading can be performed in continuous time at arbitrary volumes (in particular, the assets aredivisible).(c) The numéraire is cash, which can be held long or short freely with zero yield/cost.(d) There is an amount c A > ( c B > , respectively) called transaction cost per unit of asset A (asset B , respectively), which is paid for an adjustment of a position in asset A (asset B ,respectively) by one unit. For example, a purchase of three units of A is associated with transaction costs c A . Similarly,a sale of three units of A also costs c A dollars. In the sequel, it will be useful to introduce theshorthand ˜ c := 2 c A + 2 ηc B and refer to ˜ c as transaction cost for short. Remark.
The assumptions 1 and 2 can be reformulated to accomodate for transaction costsper dollar (or unit of any currency). In that case, the price difference process would have the form X s := ln A s − η ln B s and the sequel would be the same.Depending on particular properties of X s , there may exist many “free-lunch” trading strategies.For example, the process might admit this strategy:(i) wait until time s with X s + ˜ c/ < µ := E X s ;(ii) buy the portfolio (1 , − η ) := ( A long , η units of B short ) and pay the cost ˜ c/ ;(iii) wait until time s (cid:48) > s such that X s (cid:48) ≥ µ + ˜ c/ ;(iv) buy the portfolio ( − , η ) to close the positions and pay the cost ˜ c/ .The overall profit, collected in time s (cid:48) , is X s (cid:48) − X s − ˜ c > , and it is achieved in finite time.To be able to derive more detailed results, we need a particular form of the price difference process.2 ssumption 3 (Ornstein–Uhlenbeck process) The price difference process is of the Ornstein–Uhlenbeck form following the equation d X s = τ ( µ − X s ) + σ d W s , (1) where µ stands for the mean E X s , σ > stands for the volatility, τ > measures the speed of meanreversion and W s is the standard Wiener process. Recall that equation (1) has a solution X s = X + µ (1 − e − τs ) + σ (cid:90) s e − τ ( s − t ) d W t . In addition, when X ∼ N ( µ, σ / (2 τ )) and X is independent of W s , the process is stationary. Recallthat this process is utilized in finance frequently, e.g. in Vasicek’s interest rate model. Assumption 4 (no need to handle errors from econometric estimates)
The constants η , µ , τ , σ are known (and need not be estimated from observable finite-sample data ( s i , A s i , B s i ) i =1 ,...,N ). This is a usual assumption in portfolio management theory; recall, for instance, that Markowitzalso assumes the knowledge of exact mean returns and exact covariances of assets to be included ina portfolio (and not their econometric estimates, such as sample means and sample covariances).The knowledge of parameters allows us, without loss of generality, to perform the transformation Y t = (cid:114) τσ ( X s − µ ) , t = τ s, leading to the standardized price difference process Y t with zero mean and unit volatility. The trans-action cost in the reparametrized model is then c = (cid:114) τσ ˜ c. (2) Remark.
In Section 5 we will relax this assumption and study how an estimate of parameters,suffering from statistical imprecision, affects the “optimal” pairs trading strategy and what is the costof the error induced by the fact that from finite samples it is never possible to retrieve the values ofparameters η, µ, τ, σ with full precision.
Steps (ii) and (iv) in the above sample strategy trigger trades (opening/closing of positions). Thequestion is what are the “best” levels of Y t to open, close or adjust positions. Let us formalize thisquestion in terms of so-called Bertram’s trading strategy . Let a > stand for the entry level and b < a for the exit level in the following strategy.(i) Start in time and wait until time t ≥ when Y t = a (“entry”), buy portfolio ( − , η ) and paycost c/ .(ii) Wait until time t (cid:48) > t when Y t (cid:48) = b (“exit”) and buy portfolio (2 , − η ) (i.e., close the positionsfrom (i) and open the opposite positions), collect trading profit a − b and pay transaction cost c .(iii) Wait until time t > t (cid:48) when Y t = a (“entry”) and buy portfolio ( − , η ) (i.e., close the positionsfrom (ii) and open the opposite positions), collect profit a − b , pay cost c and iterate forever.3 ime t price µ = 0 ab = − a (exit level)(entry level) t = 0 Y t t (cid:48) t t (cid:48) T ( − , η ) (1 , − η ) ( − , η ) ( η, − T Figure 1: Bertram’s trading strategy: a sample trajectory of the price difference process Y t , entry-exitlevels a, b , trade cycles T , T , . . . and portfolios ( − , η ) and (1 , − η ) held in the corresponding cycles.For the sake of simplicity we will assume that t = 0 , meaning that there is no “idle-time” in (i).This simplification does not affect the behavior of profit in the limit t → ∞ , which will be studied inthe sequel. For the same reason we can also neglect the cost c/ from (i).The time T i := t i +1 − t i is referred to as trade cycle . The profit π := 2( a − b − c ) (3)per trade cycle is deterministic. What is random here is the time T i to collection of the profit at theend of a cycle. In Bertram’s strategy, profit is measured per time unit . If N t is the counting processfor the number of trade cycles in the time window [0 , t ] , then Π( a, b ) ≡ Π := lim t →∞ π E N t t is the expected profit per time unit . Theorem 1 Π( a, b ) = π E T i , where E T i = ∞ (cid:88) k =1 Γ( k − ) (2 / a ) k − − (2 / b ) k − (2 k − . Now the task is: given c (still assuming standardized Y t ), solve max a,b Π( a, b ) subject to a ≥ , a − b − c ≥ . (4)This is traditional Bertram’s profit-maximization problem . Observe that the constraint a − b − c ≥ formalizes the natural assumption that we require strategies with profit at least zero.The optimal solution ( a ∗ , b ∗ ) is known to satisfy the property b ∗ = − a ∗ , referred here to as symmetry of the strategy (or symmetry of entry-exit thresholds). The situation isdepicted in Figure 1.The optimal solution ( a ∗ , b ∗ ) is not known explicitly (and possibly is not an elementary functionof c ), and thus it is obtained numerically. The geometry of the problem is depicted in Figure 2. Remark.
It is interesting to study the dependence of the optimal trading strategy (entry-exitthresholds) ( a ∗ , b ∗ = − a ∗ ) and the resulting expected profit Π( a ∗ , b ∗ ) as a function of transactioncosts c . This is depicted in Figure 3. Recall that wider gaps a ∗ − b ∗ correspond to longer trade cycles T i and thus to less frequent trades (i.e., less frequently paid transaction costs c ).4 b b = − a ( a ∗ , b ∗ ) a − b − c ≥ a − b − c < Figure 2: Contour lines of expected profit-per-time-unit Π as a function of the entry-exit thresholds a, b . The optimal thresholds, maximizing Π , are denoted by ( a ∗ , b ∗ ) . Here, it is assumed that theprocess Y t is standardized and that c = 0 . . c ca ∗ b ∗ = − a ∗ Π( a ∗ , b ∗ ) Figure 3: The optimal trading strategy ( a ∗ , b ∗ ) as a function of transaction costs c and the resultingprofit Π( a ∗ , b ∗ ) . 5 b a ∗ , b ∗ ) low-variancestrategieshigh-varstrategies V ( a , b ) = v b = − a Figure 4: Contour lines of variance (riskiness) V ( a, b ) of a trading strategy (the pair of entry-exitthresholds) a, b . In this example the risk constraint V ( a, b ) ≤ v is inactive: the optimal solution ( a ∗ , b ∗ ) of Bertram’s unconstrained profit maximization problem is also an optimal solution of therisk-constrained problem. Recall that the profit-per-time-unit Π is subject to randomness through the random duration of atrade cycle. Thus, the variance of profit also reflects the variance of the length of a trade cycle T i .Recall also that N t stands for the counting process for the number of trades in the time interval [0 , t ] .The overall profit over that period is πN t (where π = 2( a − b − c ) is the deterministic amount ofprofit per cycle). The variance of profit, standardized to a time unit, is then V ( a, b ) ≡ V := lim t →∞ var ( πN t ) t . (5)This is the risk measure to be controlled. (Separate section 4.2 will be devoted to the question whether t − var ( πN t ) is a ‘good’ measure of risk in the long run t → ∞ .)Assume that there is an exogenously given bound v on the variance and that the task is to finda profit-maximizing strategy ( a, b ) under the risk constraint ( (cid:63) ) : max a,b Π( a, b ) subject to a ≥ , a − b − c ≥ , V ( a, b ) ≤ v (cid:124) (cid:123)(cid:122) (cid:125) ( (cid:63) ) . (6)Let us illustrate the geometry behind (6). If ( a ∗ , b ∗ ) is an optimal solution of (4) and V ( a ∗ , b ∗ ) ≤ v , then the constraint ( (cid:63) ) is redundant (inactive) and ( a ∗ , b ∗ ) is also an optimal solution of (6).This situation is depicted in Figure 4. However, if v is smaller, the constraint ( (cid:63) ) might “cut-off”the point ( a ∗ , b ∗ ) from the feasible region. Observe that the feasible region need not be convex ingeneral, and thus (6) is not guaranteed to be a convex optimization problem. We return to this issuein Section 4.3. It is a natural question how to measure the risk of a strategy ( a, b ) . Profit per time unit is given by Π t := πN t /t . The objective function in (6) maximizes the expectation thereof in the long run t → ∞ .Thus it would be a natural choice to consider var Π t as a risk measure and push it to the limit t → ∞ .However, var Π t = π t var N t ≈ π t · t · var T i ( E T i ) t →∞ −→ , since it is a known property of the Ornstein–Uhlenbeck process that var N t ≈ t ( E T i ) − var T i for large t . Thus, lim t →∞ var Π t is a trivial risk measure. 6 .3 0.4 0.5 0.6 0.7 0.8 0.9-0.9-0.8-0.7-0.6-0.5-0.4-0.3 ab b = − a Π S ∗ S ∗ S ∗ S (cid:48) S ∗ S ∗ S ∗ v (cid:48) v v v v v v V Figure 5: Contour lines of the expected profit function Π and variance V on a neighborhood of themaximizer of Π when c = 0 . . If the constraint ( (cid:63) ) is in the form V ( a, b ) ≤ v (cid:48) , then the strategy S ∗ = ( a ∗ , b ∗ ) from (4) is optimal and the constraint ( (cid:63) ) is redundant. The strategy S (cid:48) is never optimal.If the constraint ( (cid:63) ) is V ( a, b ) ≤ v i for an i ∈ { , . . . , } , then the constraint ( (cid:63) ) is active and theoptimal strategy is S i . The optimal solutions (4) and (6) coincide only in case i = 0 .More generally, it is easy to see that lim t →∞ var ( πN t ) /t α is a nontrivial function of a, b only for α = 1 (the limit is for α > and it is ∞ for α < ). This justifies why (5) is the ‘right’ choiceof the risk measure if it should reflect volatility and should be nontrivial in the long run (i.e., in thelimit t → ∞ ). Recall that T i is the duration of a trade cycle. From Bertram (2010) we have: Theorem 2 V ( a, b ) = π ( E T i ) var T i , where var T i = w ( a ) − w ( b ) − w ( a ) + w ( b ) ,w ( ξ ) = (cid:32) ∞ (cid:88) k =1 Γ( k ) (2 / ξ ) k k ! (cid:33) − (cid:32) ∞ (cid:88) k =1 Γ( k ) ( − / ξ ) k k ! (cid:33) ,w ( ξ ) = ∞ (cid:88) k =1 Γ( k − ) ϕ ( k − ) (2 / ξ ) k − (2 k − , and ϕ ( · ) is the digamma function. The shape of contour lines of V ( a, b ) in Figure 4 are symmetric around the line b = − a . Theprofit function Π( a, b ) together with the contour lines of V ( a, b ) — the boundary of the feasible regionconstrained by ( (cid:63) ) — are plotted in Figure 5.The shape of contour lines from Figure 5 suggests that even in case (6), the symmetry conditionis satisfied. We will therefore restrict ourselves only to the symmetric trading strategies ( a, b = − a ) .It is interesting to visualize such strategies in Figure 6, where we plot Π( a, − a ) and V ( a, − a ) as afunction of a . Observe that for a = c/ we have Π = 0 , and thus V = 0 . Moreover, the figure7 a Π( a, − a ) V ( a, − a ) v = 0 . V ≤ v feasible strategies for V ≤ v Figure 6: Symmetric strategies with c = 1 , the graph of profit Π( a, − a ) and variance V ( a, − a ) as afunction of a > . In a = c/ / , both profit and its variance is zero. An example of the riskconstraint V ≤ v = 0 . and the optimal strategy. Π Vab ( a ∗ , b = − a ∗ ) b = − a Π EF Figure 7: Left subplot: the ( a, b ) -space of profitable strategies (satisfying a − b − c ≥ with c = 1 )and the profit function Π . Right subplot: the mean-var ( V, Π) -space and an illustration how thespace of strategies maps to the mean-var space. The efficient frontier (in red) is the image of ( a, − a ) with c/ ≤ a ≤ a ∗ .illustrates the interesting property lim a (cid:38) Π( a, − a ) = −∞ , lim a →∞ Π( a, − a ) = 0 , lim a (cid:38) V ( a, − a ) = + ∞ , lim a →∞ V ( a, − a ) = 0 . Finally, Figure 7 shows how the space ( a, b ) of strategies (not necessarily symmetric) maps to themean-var space ( V, Π) . More specifically, restricting only to the profitable strategies a − b − c ≥ ,the mean-var space visualizes the set of strategies { [ V ( a, b ) , Π( a, b )] | a − b − c ≥ } and the efficientfrontier. Not surprisingly, the efficient frontier has the form EF := (cid:110) [ V ( a, − a ) , Π( a, − a )] (cid:12)(cid:12)(cid:12) c ≤ a ≤ a ∗ (cid:111) , (7)where ( a ∗ , − a ∗ ) is the optimal solution of (4). 8 .4 Solving the Risk-Constrained Problem By symmetry, the optimization problem (6) reduces to a problem in a single variable a , which can besolved easily by binary search (Algorithm 1). It is sufficient that the algorithm restricts the searchspace only to the strategies on the efficient frontier (7), see also Figure 7. Algorithm 1.
Solving risk-constrained trading problem (6){1}
Input: cost c > , risk bound v > , numerical tolerance ε > {2} Solve the convex optimization problem (4) to find ( a ∗ , − a ∗ ) {3} If V ( a ∗ , − a ∗ ) ≤ v then stop, ( a ∗ , − a ∗ ) is optimal for (6){4} a := a ∗ , a := c/ {5} While | V ( ( a + a ) , − ( a + a )) − v | > ε do {6} If V ( ( a + a ) , − ( a + a )) > v then a := ( a + a ) else a := ( a + a ) {7} End while {8}
Output (cid:0) ( a + a ) , − ( a + a ) (cid:1) as an ε -optimal strategy for (6) So far, we have considered the standardized price process Y t with zero mean and unit volatility. Now,we return to the general parametrization of the Ornstein–Uhlenbeck process X s with parameters µ , τ and σ . The entry and exit levels as well as transaction cost in general parametrization are obtainedas ˜ a = (cid:114) σ τ a + µ, ˜ b = (cid:114) σ τ b + µ, ˜ c = (cid:114) σ τ c. (8)All three parameters therefore affect the entry and exit levels. Furthermore, the expected profit andthe variance in general parametrization are obtained as ˜Π(˜ a, ˜ b ) = (cid:114) τ σ a, b ) , ˜ V (˜ a, ˜ b ) = σ V ( a, b ) . (9) µ , τ and σ We investigate the effects of misspecified parameters. We consider a benchmark case with µ = 1 , τ = 10 , and σ = 0 . and set the transaction cost ˜ c = 0 . . In figures 8–10, we visualize howmisspecification of these parameters affect the efficient frontier. When the long-term mean parameter µ is misspecified, the efficient frontier is believed to be exactly the same as for the correctly specifiedparameter. Misspecification, however, leads to suboptimal ˜ a with lower expected profit and incorrectvariance constraint.When the speed of reversion τ is overestimated, the expected profit is believed to be higher thanfor the correctly specified parameter for any variance constraint. If the variance constraint in themisspecified model is binding, the expected profit is optimal but for larger variance constraint thandesired. If the variance constraint is not binding, ˜ a is lower than the unrestricted optimum and theexpected profit is stuck at a suboptimal value. When the speed of reversion τ is understimated, theexpected profit is believed to be lower than for the correctly specified parameter. If the varianceconstraint in the related correctly specified model is binding, the expected profit is optimal but forsmaller variance constraint. If the variance constraint is not binding, ˜ a is higher than the unrestrictedoptimum and the expected profit is suboptimal.Finally, when the variance σ is overestimated, the expected profit is believed to be higher than forthe correctly specified parameter for any variance constraint. If the variance constraint in the relatedcorrectly specified model is binding, the expected profit is optimal but for smaller variance constraintthan desired. If the variance constraint is not binding, ˜ a is higher than the unrestricted optimum9 .0000.0050.0100.015 0e+00 2e−05 4e−05 6e−05 Variance Constraint M a x i m u m M ean Slightly Misestimated Long−Term Mean m = 1.001 Variance Constraint M a x i m u m M ean Considerably Misestimated Long−Term Mean m = 1.003 Parameters / Performance
Correct / Assumed & Real Incorrect / Assumed Incorrect / Real
Figure 8: The efficient frontier under misspecification of µ with true value µ = 1 .and the expected profit is suboptimal. When the variance σ is underestimated, the expected profitis believed to be lower than for the correctly specified parameter. If the variance constraint in themisspecified model is binding, the expected profit is optimal but for larger variance constraint. If thevariance constraint is not binding, ˜ a is lower than the unrestricted optimum and the expected profitis stuck at a suboptimal value. η Recall that the existence of η , the cointegration coefficient, assures that X s is a stationary process. Awrong choice of η might have devastating consequences, depending on the properties of the individualprice processes A s , B s . It can happen that a wrong choice of η can lead to non-stationarity of X s . Asa result, it could happen — for example — that the mean-reversion feature disappears and the tradingstrategy would lead to unbounded losses, either with nonzero probability, or even with probabilitytending to one. An example of a less serious (but still harmful) consequence is that that the tradecycles can be extremely long, say E T i → ∞ with i → ∞ . Then the speed of growth of E T i wouldbe critical for practical considerations. And many more undesirable situations can occur. A detailedanalysis of such phenomena deserves further investigation, based on an inspection of possible formsof the price processes A s , B s , namely in case when both of them are non-stationary. We have introduced the risk-constrained version of Bertram’s trading strategy with a pair of cointe-grated assets, studied its geometry and proposed a solution method for finding the profit-maximizingstrategy respecting the risk constraint. We have also studied the effect of misspecification of param-eters of the underlying price difference process of the Ornstein–Uhlenbeck type. This is essential for10 .0000.0050.0100.015 0e+00 2e−05 4e−05 6e−05
Variance Constraint M a x i m u m M ean Overestimated Speed of Reversion t = 50 Variance Constraint M a x i m u m M ean Underestimated Speed of Reversion t = 2 Parameters / Performance
Correct / Assumed & Real Incorrect / Assumed Incorrect / Real
Figure 9: The efficient frontier under misspecification of τ with true value τ = 10 .practice since the parameters are never known exactly and are always estimated from a finite sampleof observations.In this model, risk has been measured by variance of profit normalized to a time unit. Recallthat the variance of profit is driven by the variance of the length of a trade cycle. Further researchshould focus on other risk measures and the geometry of the optimization problem with multiple riskconstraints (e.g., when both the average length of a trade cycle and its volatility are bounded). Inaddition, the trading strategy can be expected to be generalized to a wider class of stationary priceprocesses. Finally, the profit π per cycle has been considered in the form (3). However, it wouldbe suitable to treat π ( a, b, c ) as a more general function modeling other constructions of transactioncosts. Funding
The work was supported by Czech Science Foundation under grant 19-02773S and the Internal GrantAgency of the Prague University of Economics and Business under grant F4/27/2020.
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