EExcursion Risk
Anna ANANOVA, Rama CONT and Renyuan XU
Mathematical Institute, University of Oxford
October 2020
Abstract
The risk and return profiles of a broad class of dynamic trading strategies, including pairstrading and other statistical arbitrage strategies, may be characterized in terms of excursions ofthe market price of a portfolio away from a reference level. We propose a mathematical frameworkfor the risk analysis of such strategies, based on a description in terms of price excursions, first ina pathwise setting, without probabilistic assumptions, then in a Markovian setting.We introduce the notion of δ -excursion , defined as a path which deviates by δ from a referencelevel before returning to this level. We show that every continuous path has a unique decompositioninto δ -excursions, which is useful for the scenario analysis of dynamic trading strategies, leadingto simple expressions for the number of trades, realized profit, maximum loss and drawdown. As δ is decreased to zero, properties of this decomposition relate to the local time of the path.When the underlying asset follows a Markov process, we combine these results with Ito’s excur-sion theory to obtain a tractable decomposition of the process as a concatenation of independent δ -excursions, whose distribution is described in terms of Ito’s excursion measure. We provide an-alytical results for linear diffusions and give new examples of stochastic processes for flexible andtractable modeling of excursions. Finally, we describe a non-parametric scenario simulation methodfor generating paths whose excursion properties match those observed in empirical data. Keywords : excursion theory, Markov process, local time, regenerative processes, mean-reversionstrategies, diffusion processes, Ornstein-Uhlenbeck process, drawdown risk, statistical arbitrage.1 a r X i v : . [ q -f i n . M F ] N ov ontents δ − excursions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 Decomposition of a path into δ − excursions . . . . . . . . . . . . . . . . . . . . . . . . . . 73.3 Scenario analysis for mean-reversion strategies . . . . . . . . . . . . . . . . . . . . . . . . 93.4 Irregular price paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 δ -excursions . . . . . . . . . . . . . . . . . . . . 184.3 Distributional properties of mean-reversion strategies . . . . . . . . . . . . . . . . . . . . 19 δ -excursions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.2 Linear diffusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 δ -excursions 23 δ -excursions . . . . . . . . . . . . . . . . . . . . . . . . . 256.2 Non-parametric scenario simulation using excursions . . . . . . . . . . . . . . . . . . . . . 28 A Technical proofs 35
A.1 Proof of Proposition 4.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35A.2 Proof of Proposition 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
A broad class of trading strategies may be described in terms of the relation between the market price P t of an asset –a stock, bond, commodity, a spread between two such assets, or a basket of assets– and a reference level A t , which may refer to an assessment of the portfolio’s fundamental value by an analyst,or a forecast of the portfolio’s value based on ’technical’ indicators, such as moving average estimatorsused in pairs trading [33] or ’technical indicators’ used in statistical arbitrage strategies [1, 2, 3, 20].The deviation of the market price from the reference value then represents a trading signal: S t = P t − A t . If S falls below some negative threshold − δ <
0, this represents a buying opportunity, while if S exceedsa positive threshold δ >
0, this represents an opportunity for entering a short position. A wide range oftrading strategies – pairs trading [33, 17], mean-reversion strategies [3, 27], statistical arbitrage strategiesbased on cointegration [1], index arbitrage [2] and other statistical arbitrage strategies [3, 20]– fall underthis description. The reference level A t is computed differently in each of these examples, but once thesignal S = P − A is constructed all these strategies follow the description given above.Regardless of how the reference value A t is arrived at, e.g. using fundamental valuation principles,or statistical forecasts, this leads to similar features across all such trading strategies: a long position is2ntered when the signal S crosses − δ and held until S crosses 0; similarly, a short position is enteredwhen S crosses δ and held until S crosses zero. The holding periods of positions thus coincide with excursions of the signal S above (or below) certain levels.It turns out that this remark has interesting implications: it implies that the risk and return profileof such trading strategies may be described in terms of the properties of excursions of the process S .For example, the profit of such a strategy is linked to the number of the excursions alluded to above,while the magnitude of drawdown risk may be linked to the height of the excursions.Analytical and probabilistic properties of excursions of stochastic processes have been studied indetail, starting with P. L´evy [29] for Brownian motion, K. Ito [21] for Markov processes. Ito’s seminalcontribution was to note that the excursions of a Markov process from a level may be viewed as acollection of independent random variables in a (infinite-dimensional) space of excursions. This infinite-dimensional viewpoint has proved extremely fruitful for the theoretical study of stochastic processes[6, 14, 31, 36, 35, 43, 45]. In the present work, we show that this approach is also very relevant forapplications, in particular for the analysis of dynamic trading strategies.The construction and empirical performance of pairs trading [33, 17] and ’mean-reversion’ tradingstrategies [3, 27] considered in this paper have been studied by Avellaneda & Lee [3], Gatev et al.[17] and others [33, 20]. Leung and Li [27] study mean-reversion strategies from the perspective ofoptimal control, in the setting of the Ornstein-Uhlenbeck model. The connection between statisticalarbitrage and cointegration has been discussed by many authors, including Alexander [1] and Alexander& Dimitriu [2]. Our approach provides a different perspective on these results through the angle ofexcursion theory and explains the common features observed across the variety of strategies consideredin these studies.Excursion theory has also been applied in mathematical finance, for the pricing of certain path-dependent options involving barrier crossings of a price process, such as Parisian options [10, 13], barrieroptions [30] or ’occupation time derivatives’ [8]. These studies focus on analytical results for specialmodels such as Brownian motion [10] or certain L´evy processes [8, 30]. Lakner et al [25] apply excursiontheory to the study of scaling limits for a limit order book model.A related topic is the modeling of drawdown risk for trading strategies [19]. The literature on thistopic has focused on the analytical study of drawdown risk and optimal investment under drawdownconstraints in specific models. Zhang [46] uses excursion theory for one-dimensional diffusion models toderive formulas for drawdown risk of static portfolios. On the other hand empirical studies of drawdownrisk indicate that commonly used stochastic models do not correctly quantify drawdown risk even forpassive index portfolios [23], suggesting that better, more flexible stochastic models are needed. InSection 6 we propose a new approach to stochastic modeling which addresses this issue. Outline
We propose a mathematical framework for the risk analysis of such strategies, based on adescription in terms of price excursions. We present our approach, first in a pathwise setting, withoutprobabilistic assumptions, then in a probabilistic setting, when the price is modeled as a Markov process.We start in Section 2 by describing how properties of a large class of trading strategies may beexpressed in terms of excursions of a reference process away from zero. We then introduce in Section 3the notion of δ -excursion , defined as a path which deviates by δ from a reference level before returningto this level. We show that every continuous path has a unique decomposition into such δ -excursions,which turns out to be useful for the scenario analysis of dynamic trading strategies, leading to simpleexpressions for the number of trades, realized profit, maximum loss and drawdown (Section 3.3). Inthe case of irregular paths which possess a local time, we describe in Section 3.4 the relation between δ -excursions and local time at zero of the path.When the underlying asset follows a Markov process, we combine these results with Ito’s excursiontheory for Markov processes in Section 4 to obtain a tractable decomposition of the process as a con-catenation of independent, identically distributed δ -excursions, whose distribution is described in terms3f Ito’s excursion measure (Theorem 4.7).In the case of Brownian motion and linear diffusions, the decomposition into δ − excursions may befruitfully combined with analytical properties of the Ito excursion measure to obtain analytical resultson the properties of mean-reversion strategies, such as the distribution of maximum loss or drawdownrisk. Examples of such results are given in Section 5.In Section 6 we propose an approach for building stochastic models based on the properties of their δ − excursions, and provide examples of flexible parametric models for modeling price excursions. Ourapproach extends the Ito construction [21] and leads to new examples of regenerative processes withasymmetric upward and downward excursions. Finally, we describe in Section 6.2 a non-parametricscenario simulation method for generating paths whose excursions match those observed in a data set. Many trading strategies are based on the assumption that the market price P t of a reference asset revertsto a ’target value’ or forecast A t over a certain horizon, although it may deviate from it in the shortterm. The examples below illustrate the generality of this concept. Example 2.1 (Value trading) . An investor who believes that the price of the asset will eventually revertto a ’fundamental’ value
A > P t drops below A and short the assetwhen P t exceeds A . This ’fundamental’ value can be a book value or a valuation by a financial analyst.The deviation S t = P t − A from the fundamental value then plays the role of trading signal. Example 2.2 (Pairs trading) . Pairs trading is a relative-value trading strategy which looks for pairs ofassets whose prices P , P are cointegrated [1], i.e. there exists a stationary combination P t = P t − wP t . w is typically estimated using regression techniques [17]. If A is the stationary mean of P t then thedeviation S t = P t − wP t − A is expected to revert to zero and is used as a trading signal. In practicethis mean A is estimated via a moving average [33] which leads to a time-dependent but slowly varyingreference level A t . Example 2.3 (Mean-reversion strategies) . Many statistical arbitrage strategies [3, 20] are based onidentifying combinations of assets (portfolios) whose market price follows a stationary, mean-revertingprocess [2, 3], using methods such as index tracking or cointegration [2].The market price P t = (cid:80) w i P it of such a stationary combination is then expected to revert to itsmean A , which may be estimated using for instance a moving average estimator A t , leading to thetrading signal S t = (cid:80) w i P it − A t which is expected to revert to zero.These strategies, while distinct in their design, share a common feature: they are based on theassumption that a trading signal S t = P t − A t , defined as the deviation of the market price P t of areference asset from a target value A t , reverts to zero over some time horizon. This assumption impliesthat if S t < S t >
0) one should take a long (resp. short) position in the portfolio P .In presence of transaction costs, such transactions will be entered only if the signal reaches somethreshold ± δ , leading to the following strategy:(i) Enter a long position in the reference portfolio when S t drops below − δ ; unwind the long positionwhen S t crosses zero;(ii) Enter a short position in the portfolio when S t exceeds δ ; unwind the short position when S t crosses zero. 4uch a strategy may be implemented through limit orders placed at the appropriate price levels, resultingin transactions when the market price P t crosses these levels.We now describe the associated trading strategies and their properties in more detail. Regardless of how the signal S is constructed, the trading strategies in the above examples share somecommon features, which may be described in terms of the level crossings of the signal S .We define the following level crossing times of S : we set τ +0 = 0, θ +0 = 0 and ∀ i ≥ , τ + i = inf { t > θ + i − , S t ≥ δ } θ + i = inf { t > τ + i , S t ≤ } . (1)The intervals ( τ + i , θ + i ) , ( θ + i , τ + i +1 ) are the down-crossing and up-crossing intervals of the interval [0 , δ ] . Each interval [ θ + i , θ + i +1 ], corresponds to an excursion of S from 0 to δ and back to zero.It is readily observed that the intervals ( τ + i , θ + i ) , ( θ + i , τ + i +1 ) form a partition of [0 , ∞ ) and, if the pathis continuous, they are all non-empty. One can also define similar quantities for downward excursions: τ − = θ − i = 0 , and ∀ i ≥ , τ − i = inf { t > θ − i − , S t ≤ − δ } θ − i = inf { t > τ − i , S t ≥ } . (2)A mathematical description of the trading strategies described in Section 2.1 can now be given interms of the level crossing times defined above: • buy the reference portfolio when the trading signal drops below − δ , sell when it returns to 0: φ − = (cid:88) k ≥ [ τ − k ,θ − k ) . (3) • short the reference portfolio when the signal exceeds δ , unwind the position when it reaches 0: φ + = − (cid:88) k ≥ [ τ + k ,θ + k ) . (4)We refer to φ + , φ − as one-sided strategies.Combining the two strategies we obtain what is usually called a ’mean-reversion strategy’ or ’con-vergence trade’ based on the trading signal S : φ ( t ) = φ + ( t ) + φ − ( t ) = (cid:88) k ≥ [ τ − k ,θ − k ) − (cid:88) k ≥ [ τ + k ,θ + k ) . (5)One may also considering a position size modulated as a function of S . For example, (5) has unboundedexposure to price movements and in most cases portfolios are subject to position limits or exposurelimits (’stop loss’). A maximum exposure limit of M on short positions in (4) leads to unwinding theposition if S reaches δ + M during the holding period: φ + M ( t ) = − (cid:88) k ≥ [ τ + k ,θ + k ∧ κ k ) κ k = inf { t > τ + k , S ( t ) ≥ δ + M } . (6)In addition to the position in the risky asset S , each portfolio has a cash component, which is adjustedto reflect the gains and losses from trading, so that the strategy is self-financing. Denoting by V t ( φ ) thesum of the cash holdings and the market value of a position φ in the risky asset, we have V t ( φ ) = V t (0) + (cid:90) t φ ( u − ) dS u . (7) In the case of c`adl`ag paths a finite number of these intervals could be empty, i.e. τ + i = θ + i , if the process jumps acrossthe interval (0 , δ ). In this paper we focus on the case of continuous trajectories. S isrequired to define the integral in (7).As the sets ∪ k ≥ [ τ − k , θ − k ] and ∪ k ≥ [ τ + k , θ + k ] are disjoint we may study the properties of φ + , φ − sepa-rately. In the following sections we will focus on φ + , but it is clear that properties of φ − are analogouslyobtained by replacing S by − S .Let us now examine further the properties of the one-sided strategy (4). Each transaction cycle[ θ + k − , θ + k ] is decomposed into a waiting period [ θ + k − , τ + k ] followed by a holding period [ τ + k , θ + k ]. Thestrategy generates a profit of δ over each transaction cycle, leading to a portfolio value V t ( φ + ) = V ( φ + ) + δ D δt ( S ) + S ( t ∧ θ + D δt ( S )+1 ) − S ( t ∧ τ + D δt ( S )+1 ) where D δt ( S ) = (cid:88) i ≥ θ + i ≤ t (8)represents the number of transactions in [0 , t ]. The first term δ D δt ( S ) represents the realized profit whilethe second term corresponds to the market value of the current position. If the path of S wanders highabove δ then the portfolio can incur a large market loss. It is therefore clear that the gains and losses ofthe trading strategy φ + are linked to the frequency, duration and amplitude of positive excursions of S which exceed the level δ . Similarly, one can readily observe that the gains and losses of φ − are linked tothe frequency, duration and height of negative excursions of S which reach − δ . In the following sectionswe build on this insight and study in more detail the structure of such excursions in order to model therisk and return profile of such portfolios. δ − excursions Let E = C ([0 , ∞ ) , R ) be the space of continuous functions equipped with the Borel measurable structureinduced by the uniform norm and E = { f ∈ E , f (0) = 0 } . Denote, for f ∈ E ,T x ( f ) = inf { t > , f ( t ) = x } , T xt ( f ) = inf { u > t, f ( u ) = x } . (9)Let δ ∈ R . We will call an excursion from 0 to δ a path which starts from zero, reaches δ in a finitetime, and stops when it reaches δ : E ,δ = { f : C ([0 , ∞ ) → R , f (0) = 0 , T δ ( f ) < ∞ ; ∀ t ≥ T δ ( f ) , f ( t ) = δ } . (10)Note that by this definition an excursion from 0 to δ is stopped at the first time it reaches δ . Inparticular, E , is the space of excursions from 0 to 0.Define the concatenation at T > u, v ∈ E as the element u ⊕ T v ( t ) := u ( t ) 1 [0 ,T ) + v ( t − T ) 1 [ T, ∞ ) . (11)Note that if u ∈ E ,a , v ∈ E a, then for T ≥ T a ( u ) , u ⊕ T v ∈ E , .We define a δ − excursion as an excursion from 0 to δ , followed by an excursion from δ back to 0: Definition 3.1 ( δ -excursion) . A δ − excursion is a path f ∈ E such that ∃ ( u, v ) ∈ E ,δ × E δ, , f = u ⊕ T δ ( u ) v, i . e . f ( t ) = u ( t ) 1 [0 ,T δ ( u )) + v (cid:0) t − T δ ( u ) (cid:1) [ T δ ( u ) , ∞ ) (12)The decomposition (12) is then unique and we denote Λ( f ) = T δ ( u ) + T ( v ) the duration of f .We denote by U δ the set of δ − excursions. The map f ∈ U δ (cid:55)→ ( u, v, Λ( f )) ∈ E ,δ × E δ, × [0 , ∞ ) ismeasurable. 6xamples of δ − excursions are excursions from 0 to 0 which reach δ :Γ δ = (cid:110) f ∈ E , : max( f ) ≥ δ (cid:111) = U δ ∩ E , . (13)But the inclusion Γ δ ⊂ U δ is strict, as a typical δ − excursion may reach zero (infinitely) many timesbefore reaching δ and we may have Λ( f ) > T ( f ) for f ∈ U δ . In particular U δ is not a subset of E , .However, each path in U δ contains exactly one excursion of type Γ δ : Lemma 3.2 (Last exit decomposition of δ -excursions) . Any δ -excursion f ∈ U δ has a unique decompo-sition into a path from to which does not reach δ followed by an excursion γ ∈ Γ δ from to whichreaches δ : ∀ f ∈ U δ , ∃ !( T, g, γ ) ∈ [0 , ∞ ) × E × Γ δ , f = g ⊕ T γ with g (0) = g ( T ) = 0 , max( g ) < δ. (14) Proof.
Consider a δ -excursion f ∈ U δ . Then f has a decomposition (12) for some ( u, v ) ∈ E ,δ × E δ, wehave f ( t ) = 0 for t ≥ Λ( f ) = T δ ( u ) + T ( v ). Now define T as the last zero of u before T δ ( u ): T = sup { t < T δ ( u ) , u ( t ) = 0 } . Then by continuity of u , T < T δ ( u ) and therefore max { u ( t ) , ≤ t ≤ T } < δ . Setting g = f [0 ,T ] and γ ( t ) = f (( t − T ) + ), it is readily verified that γ ∈ Γ δ and g satisfy the required conditions.0 20 40 60 80 100 120 140 − . .
51 time S t Figure 1: Example of last exit decomposition of a δ -excursion f ∈ U δ with δ = 1: f = g ⊕ T γ where g is in purple and γ ∈ Γ δ (in orange) is the last excursion. δ − excursions The following proposition gives the decomposition of any path starting from zero into a sequence ofexcursions from 0 to δ and back to 0: Proposition 3.3.
Let S ∈ C ([0 , ∞ ) , R ) with S = 0 . Define the level crossing times θ +0 = 0 , τ +0 = 0 , τ + i = T δθ + i − ( S ) , θ + i = T τ + i ( S ) . Then ∀ t ≥ , D δt ( S ) = (cid:80) i ≥ θ + i ≤ t < ∞ and (15) ∀ t ≥ , S t = (cid:80) D δt ( S )+1 i =1 (cid:104) u i (cid:0) t − θ + i − (cid:1) [ θ + i − ,τ + i ) + v i (cid:0) t − τ + i (cid:1) [ τ + i ,θ + i ) (cid:105) , (16) where u i ∈ E ,δ and v i ∈ E δ, . roof. To prove the first assertion, we first note that S is continuous, thus uniformly continuous on[0 , T ] for any T >
0. If D δt = ∞ for some t > , then the set { k ∈ N , θ + k ≤ t } is infinite. Since byconstruction the intervals ( τ + k , θ + k ) are disjoint, we have (cid:88) { k,θ + k ≤ t } | θ + k − τ + k | ≤ t < ∞ , so inf θ + k ≤ t | θ + k − τ + k | = 0 while | S θ + k − S τ + k | = δ which contradicts the uniform continuity of S on [0 , t ]. Therefore D δt < ∞ for all t ≥
0. Starting from: S t = D δt ( S )+1 (cid:88) i =1 (cid:104) ( S t ∧ τ + i − S t ∧ θ + i − ) + ( S t ∧ θ + i − S t ∧ τ + i ) (cid:105) . Note that( S t ∧ τ + i − S t ∧ θ + i − ) + ( S t ∧ θ + i − S t ∧ τ + i ) = ( S t ∧ τ + i − S t ∧ θ + i − )1 [ θ + i − ,τ + i ) + ( δ + S t ∧ θ + i − S t ∧ τ + i )1 [ τ + i ,θ + i ) , it remains to set u i ( t ) := S ( t + θ + i − ) ∧ τ + i − S ( t + θ + i − ) ∧ θ + i − , i ≥ , v i ( t ) := δ + S ( t + τ + i ) ∧ θ + i − S ( t + τ + i ) ∧ τ + i , i ≥ . The above results translate into a (measurable) decomposition of any continuous path into δ -excursions: Proposition 3.4 (Decomposition of a path into δ − excursions) . Let δ > and S ∈ C ([0 , ∞ ) , R ) with S = 0 , and define D δt ( S ) as in (15) .(i) If sup t ≥ D δt ( S ) = ∞ there exists a unique sequence ( e k ) k ≥ of δ − excursions e k ∈ U δ such that ∀ t ≥ , S t = (cid:88) k ≥ e k (cid:0) ( t − θ + k − ) + (cid:1) where θ +0 = 0 , θ + k = k (cid:88) i =1 Λ( e i ) . (17) (ii) If d = sup t ≥ D δt ( S ) < ∞ then there exist ( e , ..., e d ) ∈ ( U δ ) d and e d +1 ∈ E such that S t = d +1 (cid:88) k =1 e k (cid:0) ( t − θ + k − ) + (cid:1) where θ +0 = 0 , θ + k = k (cid:88) i =1 Λ( e i ) . (18) In both cases the map S (cid:55)→ ( e k ) k =1 .. ( d +1) is measurable. The case (i) corresponds to the ’recurrent’ case where the path crosses zero and δ infinitely manytimes on [0 , ∞ ). Proof.
Set d = sup t ≥ D δt ( S ) ∈ N ∪ {∞} . Define ( θ + k , k ≥
1) as in (1). For k < d , set e k ( t ) = S ( t − θ + k − )1 [ θ + k − ,θ + k ) ( t ). Then it is easily verified, from the definition (1) of θ + k , that e k ∈ U δ andΛ( e k ) = θ + k − θ + k − . Measurability of the map S (cid:55)→ ( e k ) k ≥ follows from the measurability of the hittingtimes and the shift operator. To show uniqueness, we note that (17) implies that e k ( θ + k − + . ) = S | [ θ + k − ,θ + k ) so it is sufficient to show uniqueness of the sequence ( θ + k ) k ≥ . As D δt ( S ) < ∞ for each t >
0, the countableset { t > , ∆ D δt (cid:54) = 0 } is discrete and has a unique increasing ordering, which is given by ( θ + k ) k ≥ . Remark 3.5.
The above results decompose the path into one-sided δ -excursions i.e. with δ >
0. Onecan immediately obtain a similar decomposition for δ < − S . To obtain a decomposition in terms of two-sided δ -excursions, one can iterate these two results:first decompose S into δ -excursions, then decompose each δ -excursion into ( − δ ) − excursions. One mayfurther show that the resulting decomposition is independent of the order of these two operations.8 .3 Scenario analysis for mean-reversion strategies The drawdown [19] of a portfolio φ whose value at time t is V t ( φ ) is defined as∆( t ) = M t ( φ ) − V t ( φ ) where M t ( φ ) = max [0 ,t ] V t ( φ )is the running maximum.The decomposition of the path into δ − excursion given in Proposition 3.4 leads to simple expressionsfor the portfolio value, the maximum loss and the drawdown of the strategy: Proposition 3.6.
Along a path S with decomposition (16) ,(i) the gain V t ( φ + ) − V ( φ + ) = (cid:82) t φ + dS of the portfolio is given by V t ( φ + ) − V ( φ + ) = δ × D δt ( S ) + 1 [ τ + Dδt +1 ,θ + Dδt +1 ] (cid:16) δ − v D δt ( S )+1 ( t − τ + D δt +1 ) (cid:17) (19) (ii) the worst loss during [0 , t ] is given by max s ∈ [0 ,t ] (cid:0) V ( φ + ) − V s ( φ + ) (cid:1) = max k =0 ,...,D δt ( S ) { max [0 , ( t − τ + k +1 ) + ] ( v k +1 − ( k + 1) δ ) } . (20) (iii) the drawdown of φ + is given by ∆( t ) = max k =0 ,...,D δt ( S ) { max [0 , ( t − τ + k +1 ) + ] (( k + 1) δ − v k +1 ) }− δ × D δt ( S ) − [ τ + Dδt +1 ,θ + Dδt +1 ] ( t ) (cid:16) δ − v D δt ( S )+1 ( t − τ + D δt +1 ) (cid:17) . (21) Proof.
By definition of the portfolio φ + , we have V t ( φ + ) = V ( φ + ) − D δt ( S )+1 (cid:88) i =1 (cid:90) [ τ + i ,θ + i ) dS = D δt ( S ) (cid:88) i =1 ( S τ + i − S θ + i ) + ( S t ∧ τ + Dδt ( S )+1 − S t )thus V t ( φ + ) − V ( φ + ) = δ × D δt ( S ) + S t ∧ τ + Dδt ( S )+1 − S t . By the definition of v i this can be rewritten as V t ( φ + ) − V ( φ + ) = δ D δt ( S ) + 1 [ τ + Dδt +1 ,θ + Dδt +1 ] (cid:16) δ − v D δt ( S )+1 ( t − τ + D δt +1 ) (cid:17) , which then implies (20). Intuitively, decreasing the value of the threshold δ increases the number of level crossings and leads tomore transactions but with a lower profit per transaction. The exact behaviour of the trading strategyas δ → local time of the path at 0, which measures the time spent by the pathin a neighbourhood of zero [18].Let S ∈ C ([0 , ∞ ) , R ) and T >
0. We define the occupation measure of S by γ T ( A ) := (cid:90) T A ( S t ) dt, ∀ A ∈ B ( R ) . (22)9 200 400 600 800 1 , − . . . S t , V t ( φ + ) , . . ∆ ( t ) Figure 2: Decomposition of a path into δ -excursions with δ = 0 .
5. Top Left: decomposition into u i (blue) and v i (red). Top Right: Value of the portfolio φ + . Bottom: Drawdown ∆( t ).We will say that the path S admits a local time l T ( S, x ) if the measure γ T is absolutely continuous withrespect to Lebesgue measure on R , in which case we denote l T ( S, x ) := dγ T dx = lim ε → ε (cid:90) T [ x − ε,x + ε ] ( S t ) dt. The local time is characterized by the occupation time formula : (cid:90) T h ( S t ) dt = (cid:90) + ∞−∞ h ( x ) l T ( S, x ) dx, ∀ h ∈ C ( R , R ) . Intuitively, the local time l T ( S, x ) represents the time S spends at level x during [0 , T ]. We will beinterested in particular in the local time at 0, which we denote (cid:96) T ( S ) = l T ( S, T (cid:55)→ (cid:96) T ( S ) is increasing, which allows to define its right-continuous inverse, the inverselocal time at zero: ∀ l > , τ l = inf { t > , (cid:96) t ( S ) > l } . (23)We note that l (cid:55)→ τ l is an increasing c`adl`ag function of the variable l. The local time (cid:96) t ( S ) increases onthe set { t, S t = 0 } and is constant along any excursion from 0, so the discontinuities of τ correspond toexcursions of S , and jump intervals of τ correspond to the complement of the set where S visits 0: ∪ l> ( τ l − , τ l ) = { t ≥ , S t (cid:54) = 0 } . Thus the value l of local time along an excursion may be used as a natural index for labeling excursionsof S : the excursion at local time level l is given by e l ( t, S ) = (cid:40) S ( τ l − + t ) ( t ≤ τ l − τ l − ) , if τ l ( ω ) − τ l − ( ω ) > † if τ l ( ω ) = τ l − ( ω ) . (24)10oints of continuity of τ l , i.e. points at which τ l − = τ l correspond to ‘infinitesimal excursions’ whichmay arise if the path has non-zero local time at 0; we associate such excursions with a ‘cemetery’ state e l ( S ) = † . This defines an excursion process e : R + → E , = E , ∪ {†} . For a given set Γ ⊂ E , , we candefine the counting process, which counts excursions of S from 0 which lie in Γ, up to local time l : N l (Γ) := (cid:88) λ ≤ l Γ ( e λ ) . (25)Note that in general N l (Γ) can be infinite. We now establish an important connection between this excursion point process N and the decomposition into δ -excursions given by Proposition 3.4. Recall theset Γ δ of excursions from 0 to 0 which reach a level δ :Γ δ = (cid:110) f ∈ E , : max( f ) ≥ δ (cid:111) . Proposition 3.7.
Let S ∈ C ([0 , ∞ ) , R ) be a path with S = 0 which admits a local time (cid:96) t ( S ) at zero,with inverse τ is given by (1) . For δ > let D δt ( S ) be the number of δ − excursions of S on [0 , t ] , definedas in (17) . Then ∀ δ > , ∀ t > , N t (Γ δ ) < + ∞ , and ∀ t > , D δt ( S ) = N (cid:96) t ( S ) (Γ δ ) and ∀ l > , D δτ l ( S ) = N l (Γ δ ) . (26) Proof.
The condition N l (Γ δ ) < + ∞ is a consequence of the continuity of S . We will now establish aone-to-one correspondence between excursions e l ∈ Γ δ and intervals ( θ + i − , θ + i ) . As in Lemma 3.2, define i ≥ i -th δ − excursion:ˆ θ + i := sup { t < τ + i : S t = 0 } . To show that the two sets of intervals { (ˆ θ + i , θ + i ) } i ≥ and { ( τ l − , τ l ) } e l ∈ Γ δ coincide, we prove the followingtwo claims: • For each i ≥ l i ≥ , such that (ˆ θ + i , θ + i ) = ( τ l i − , τ l i ).Indeed, it is easy to see that on the interval (ˆ θ + i , θ + i ), S t >
0. Furthermore,ˆ e i ( t ) := S t +ˆ θ + i [0 , θ + i − ˆ θ + i ] ∈ Γ δ , since ˆ e i ( τ + i − ˆ θ + i ) ≥ δ. In particular (ˆ θ + i , θ + i ) is an interval of { t > , S t (cid:54) = 0 } , thus there existsunique l i such that (ˆ θ + i , θ + i ) = ( τ l i − , τ l i ) and e l i ∈ Γ δ . • Conversely, for every l ≥ e l ∈ Γ δ , there exists a unique index i ( l ) ≥ τ l − , τ l ) = (ˆ θ + i ( l ) , θ + i ( l ) ).Take the largest i = i ( l ) ≥ θ + i − ≤ τ l − then τ l − < θ i . Since on (ˆ θ + i , θ + i ), S t >
0, while S τ l − = 0 , we get that ˆ θ + i ≥ τ l − . The condition e l ∈ Γ δ implies that S reaches the level δ in ( τ l − , τ l ),by definition τ + i > ˆ θ + i is the first such time after θ i − , hence τ + i ∈ ( τ l − , τ l ). Since the intervals( τ l − , τ l ) and (ˆ θ + i , θ + i ) intersect, we conclude from the first claim that ( τ l − , τ l ) = (ˆ θ + i ( l ) , θ + i ( l ) ) (we alsouse the fact that the intervals { ( τ l − , τ l ) } l ≥ are disjoint).The correspondence between θ + i , i ≥ τ l , e l ∈ Γ δ , yields the result: D δt ( S ) = (cid:88) i ≥ θ + i ≤ t = (cid:88) i ≥ θ + i ( l ) ≤ t = (cid:88) τ l ≤ t Γ δ ( e l ) = (cid:88) l ≤ l t ( S ) Γ δ ( e l ) = N l t ( S ) (Γ δ ) . ehaviour of level-crossings as δ → . The behaviour of the above quantities as δ → δ is small, we account for the fact that trading takes placesonly at prices which are integer multiples of a ’tick’, i.e. only at times when S takes such values.Let δ n = 2 − n and introduce the partition π n defined by the hitting times of the grid δ n Z : t n := 0 , t nk +1 := inf (cid:8) t ≥ t nk : S t ∈ δ n Z \{ S t nk } (cid:9) . (27)Then sup π n | S ( t nk +1 ) − S ( t nk ) | → n → ∞ . We denote π = ( π n ) n ≥ . Following [12], we will say that S ∈ C ([0 , T ] , R ) has p-th order variation along π if there exists [ S ] pπ ∈ C ([0 , T ] , R + ) such that (cid:88) π n | S ( t nk +1 ∧ t ) − S ( t nk ∧ t ) | p → [ S ] pπ ( t ) . The smallest p ≥ S ] pπ (cid:54) = 0 then gives an index of ‘roughness’ for S along π . For example forBrownian paths p = 2 while for fractional Brownian motion with Hurst exponent H , p = 1 /H [5].For such a path, the number of down-crossings for levels close to zero is related to a slightly differentnotion of local time, defined in terms of a weighted occupation measure, weighted by the p-th ordervariation [12]: Definition 3.8 (Local time of order p [12]) . Let p ≥ and q ≥ . A continuous path S ∈ C ([0 , T ] , R ) has ( L q -)local time of order p along a sequence of partitions π = ( π n ) n ≥ of [0 , T ] if, for any t ∈ [0 , T ] ,the sequence of functions L π n ,pt ( S, . ) : x ∈ R (cid:55)→ L π n ,pt ( S, x ) := (cid:88) t nj ∈ π n (cid:20) S tnj ∧ t ,S tnj +1 ∧ t (cid:19) ( x ) (cid:12)(cid:12)(cid:12) S t nj +1 ∧ t − x (cid:12)(cid:12)(cid:12) p − converges in L q ( R ) to a limit L π, pt ( S, x ) ∈ L q ( R ) and the map t ∈ [0 , T ] (cid:55)→ L π, pt ( S, x ) ∈ L q ( R ) is weaklycontinuous. We call L π, p ( S, x ) the local time of order p of S at level x . L π, pt ( S, x ) measures the rate at which the path S accumulates p-th order variation around level x .Note that the local time of order p is non-zero only if S has non-zero p-th order variation along π i.e.[ S ] pπ >
0. If the convergence is uniform in ( t, x ) ∈ [0 , T ] × R , and the mapping ( x, t ) (cid:55)→ L π,pt ( S, x ) iscontinuous we call it the continuous local time of S [24].Note that the in the case of p = 2 the definitions in [4] and [12, 24] differ by a factor of 2; here weuse the latter notation. In the case p = 2 we will omit the index p in the notation; L π := L π, . Therelation between various notions of local time is discussed in [24].Following the arguments in [12, Lemma 3.4], one can establish a relation between the down-crossingsand up-crossings D δ n t ( S ) , U δ n t ( S ) of the interval [0 , δ n ]: for x ∈ [0 , δ n ] ,L π n ,pt ( S, x ) = D δ n t ( S ) | x | p − + U δ n t ( S ) | δ n − x | p − + O ( δ p − n ) . Since the numbers D δ n t ( S ) , U δ n t ( S ) can differ at most by one, we obtain that L π n t ( S ) = L π n t ( S,
0) = D δ n t ( S ) δ p − n + O ( δ p − n ) . If S has a continuous local time L π,p ( S, · ) along the sequence of Lebesgue partitions π , we conclude fromabove that lim n →∞ | δ n | p − D δ n t ( S ) = L π,pt ( S ) . The following proposition summarizes the behavior of the number of level crossings D δt ( S ) (representingthe number of trades) and the realized profit δD δt ( S ) as δ decreases to zero:12 roposition 3.9. Let δ n = 2 − n and p ≥ . Assume S ∈ C ([0 , T ] , R ) has a strictly positive local time L π,pt > of order p at zero along the sequence of partitions ( π n ) n ≥ defined by (27) . Then for any t ∈ (0 , T ] , as δ n → ,(i) if ≤ p < then δ n D δ n t ( S ) → .(ii) if p > then δ n D δ n t δ n → → ∞ , and D δ n t ( S ) δ n → ∼ L π,pt ( S ) δ p − n . (iii) if p = 2 then δ n D δ n t δ n → → L π, t ( S ) , i . e . D δ n t ( S ) δ n → ∼ L π, t ( S ) δ n . In particular, when p > δ should be chosen as small as possible, while for p ≤ δ ∗ ( S ) > δD δT ( S ).The assumptions of Proposition 3.9 are satisfied by typical sample paths of many classes of stochasticprocesses. Typical paths of semimartingales correspond to (iii), while paths of ’rough’ processes such asFractional Brownian motion with Hurst exponent H < / Example 3.10 (Continuous semimartingales) . Let S = M + A where M is a continuous martingaleand A is a continuous process with bounded variation (cid:82) T | dA t | on [0 , T ]. Denote by [ S ] = [ M ] thequadratic variation process of S . Then S admits a local time of order p = 2, which corresponds 1 / S at 0: L π, t ( S ) = lim ε → ε (cid:90) T [ − ε,ε ] ( S t ) d [ S ] t . Furthermore if for some q ≥ E (cid:18) [ M ] q/ T + ( (cid:90) T | dA t | ) q (cid:19) < ∞ , then it was shown by El Karoui [15] that t (cid:55)→ δ D δt is uniformly approximated in L q by L π, t ( S ) as δ → E (cid:18) sup ≤ t ≤ T (cid:12)(cid:12) δ D δt ( S ) − L π, t ( S ) (cid:12)(cid:12) q (cid:19) δ → → . Example 3.11 (Fractional Brownian motion) . Let B H be a fractional Brownian motion with Hurst pa-rameter H ∈ (0 , B H almost surely has a continuous local time L π, /H ( B H , · )of order p = 1 /H along the sequence of partitions π defined in (27), and L π, /Ht ( B H , x ) = (cid:96) t ( B H , x ) E (cid:104) | B H | H (cid:105) , where (cid:96) t ( B H , · ) is the occupation time density of B H . Denoting (cid:96) t ( B H ) = (cid:96) t ( B H , n →∞ | δ n | − HH D δ n t ( B H ) = (cid:96) t ( B H ) E (cid:104) | B H | H (cid:105)(cid:124) (cid:123)(cid:122) (cid:125) c H , so D δ n t ( B H ) δ n → ∼ c H (cid:96) t ( B H ) | δ n | − HH .
13e illustrate the relevance of Proposition 3.9 using a fractional Ornstein-Uhlenbeck process [9]: dS t = − λS t dt + γdB Ht , (28)where B H is a fractional Brownian motion with Hurst exponent H . Figure 3 shows, as a function of thethreshold δ , the number of δ − excursions estimated from values of S t on a discrete grid of N = 28 , (a) H = 0 .
7. (b) H = 0 .
5. (c) H = 0 . Figure 3: Behavior of D δT when δ → λ = 5 and γ = 0 .
1. Dotted line: asymptotic behavior δ H − H described in Proposition 3.9. We now consider the case where the trading signal S is described by a (one-dimensional) diffusionprocess, a situation often encountered in mathematical finance. Excursions of Markov processes werestudied in detail by Ito [21], who developed a beautiful description in terms of an infinite-dimensionalPoisson point process [6, 21, 36, 42]. We first recall some key results from Ito’s theory, then show howthey may be used to derive probabilistic properties of δ − excursions. Let us briefly recall some results from K. Ito’s excursion theory for Markov processes [21]. An excellentsurvey is given by Rogers [36]. For proofs and more detailed accounts we refer to [6, 21] or [37, Ch.XII].Let ( S t , t ≥
0) be a standard Markov process with continuous paths, whose state space is an interval I ⊂ R which includes zero. Then S has a local time (cid:96) t ( S ) at zero [22], defined as (cid:96) t ( S ) = lim ε → ε (cid:90) t [ − ε,ε ] ( S u ) du. We denote P ( t,x ) the law of ( S u , u ≥ t ) given S t = x , P a = P (0 ,a ) and P a the law of the process startingfrom S = a and stopped at T ( S ).We shall make the following assumption on S : Assumption 4.1. [0 , δ ] ⊂ I and P (cid:0) T δ ( S ) < ∞ (cid:1) = 1 , P δ (cid:0) T ( S ) < ∞ (cid:1) = 1 . (29)14hese assumptions are satisfied for a wide range of one-dimensional diffusion processes [22].The process t (cid:55)→ (cid:96) t ( S ) is increasing, and increases on S − ( { } ) = { t ≥ , S t = 0 } . We may defineits right-continuous inverse τ , the inverse local time at zero: ∀ λ > , τ λ = inf { t > , (cid:96) t ( S ) > λ } . (30)Then τ is an increasing process whose discontinuities correspond to the excursions of S away from 0: S − ( { } ) = { τ l , l ≥ } ∪ { τ l − , τ l − τ l − > } . During an excursion from 0, the local time is constant and τ undergoes a discontinuity whose size τ λ − τ λ − corresponds to the duration of the excursion. There is thus a one-to-one correspondencebetween excursions of S from 0 and the discontinuities of τ . Ito [21] defined the process of excursionsof S from
0, indexed by the local time at 0, as follows:
Definition 4.2 (Excursion process) . The excursion process of S is an E − valued process ( e λ , λ > defined as e λ ( t, ω ) = (cid:40) t ≤ τ λ ( ω ) − τ λ − ( ω ) S τ λ − ( ω )+ t ( ω ) if τ λ ( ω ) − τ λ − ( ω ) > † if τ λ ( ω ) = τ λ − ( ω ) . Here e λ designates the excursion along which the local time takes the value λ . Note that T ( e λ ) = τ λ − τ λ − .The following important result, due to Ito [21], characterizes the excursion process of a Markovprocess as as Poisson point process with values in E , : Theorem 4.3 (Ito’s excursion process [21]) . ( e λ , λ > is an F τ λ − Poisson point process: there existsa σ -finite measure ν on E such that, for any Γ ⊂ E with < ν (Γ) < ∞ , N λ (Γ) = (cid:88) l ≤ λ Γ ( e l ) (31) is a Poisson process with F τ λ − intensity ν (Γ) . ν is called the Ito excursion measure of S . For a proof and detailed discussion we refer to [6, 21, 37]. We have the following representation τ l ( ω ) = (cid:88) λ ≤ l T ( e λ ( ω )) , τ l − ( ω ) = (cid:88) λ
0) is a Poisson point process has manyinteresting implications, which we may exploit to derive a probabilistic description of δ − excursions.To explore the connection with δ -excursions, we will be interested in the case where Γ = Γ δ is theset of excursions which reach δ >
0, defined in (13):
Proposition 4.4.
Under Assumption 4.1, we have < ν (Γ δ ) < ∞ and P (cid:0) ∀ t ≥ , D δt ( S ) = N (cid:96) t ( S ) (Γ δ ) (cid:1) = 1 , P (cid:0) ∀ l > , N l (Γ δ ) = D δτ l ( S ) (cid:1) = 1 . (32) Furthermore: i) ( D δτ l ( S ) , l ≥ is a Poisson process with ( F τ l ) − intensity ν (Γ δ ) under P .(ii) ( D δt ( S ) , t ≥ is a renewal process under P : the durations of δ -excursions are independent andidentically distributed.Proof. Under Assumption 4.1 we can apply Proposition 3.7 to obtain ∀ t ≥ , D δt ( S ( ω )) = N (cid:96) t ( S ( ω ) ,a ) (Γ δ ) , and ∀ l > , N l (Γ δ ) = D δτ l ( S ( ω )) , for P − a.s. (i) then follows from Theorem 4.3 and P ( ∀ l > , D δτ l ( S ) = N l (Γ δ ) ) = 1. For (ii) we note that is alsoself-evident that the intervals θ + i +1 − θ + i are independent and identically distributed. For that note firstthat θ + i are stopping times, by strong Markov property of the law of process S on t > θ + i depends on thepast only through the information at time θ i . Since also S θ + i = 0 we conclude from the strong Markovproperty of S that θ + i +1 − θ + i is independent of θ + j +1 − θ + j , j < i and is distributed as θ +1 − θ +0 .As stated in Proposition 4.4, the counting process D δt is a renewal process i.e. the durations of δ − excursions are independent and identically distributed. The next result relates the distribution ofthese durations -through its Laplace transform- to the Ito excursion measure ν : Proposition 4.5.
Under Assumption 4.1, the duration of δ -excursions of S are independent randomvariables with moment generating function given by E [ e − λθ +1 ] = (cid:82) Γ δ ν ( df ) e − λ T ( f ) (cid:82) Γ δ ν ( df ) e − λ T ( f ) + (cid:82) E ν ( df )(1 − e − λ T ( f ) ) . (33) Proof.
We use the method of marked excursions described by Rogers & Williams [37, 36]. Let X λ bea Poisson process with intensity λ , independent from S , with jump times 0 < t < t < . . . where t i +1 − t i , i ≥ t = 0) are independent Exp( λ ) random variables. The discussion below followsthe approach described in [37, VI.49]. Let J := { g ∈ D ([0 , ∞ ) , N ) , ∆ t g ∈ { , }} be the space of c`adl`ag step functions with unit step size and m λ be the law of the process X λ . Weconsider the process ( S t , X t ) , t ≥ R × Z + and law ˜ P := P × m λ , which can beidentified with the coordinate process on ˜Ω := E × J. Let e l be the excursion of the process S at localtime l defined as in Definition 4.2. In addition, we define the increment function of the process X λ atlocal time l : η l ( t, ω ) = (cid:40) X λ ( τ l − + t ) ∧ τ l − X λτ l − if τ l − τ l − > † if τ l = τ l − . The map l (cid:55)→ ( e l , η l ) , l ≥ , defines a point process of marked excursions ˜ N on [0 , ∞ ) × ˜ E where˜ E := { ( f, g ) ∈ E × J : g ( t ) = g ( T ( f )) ∀ t ≥ T ( f ) } , defined by ˜ N l (Γ × B ) := (cid:88) λ ≤ l Γ × B ( e λ , η λ ) , for Γ × B ⊂ ˜ E . Then by [37, Theorem VI.49.2], ˜ N l is a Poisson point process with intensity measure˜ ν (Γ × B ) := (cid:90) Γ ν ( df ) m λ (cid:0) { g ∈ J : g ( · ∧ T ( f )) ∈ B } (cid:1) . (34)16e define the space of starred excursions as the set of marked excursions which contain a jump ofthe process X λ during its lifetime [0 , T ( f )] :˜ E ∗ := { ( f, g ) ∈ ˜ E : g ( ∞ ) > } . Then by formula (34) the counting measure for starred excursions is a Poisson process with intensity˜ ν ( ˜ E ∗ ) = (cid:90) E ν ( df )(1 − e − λ T ( f ) ) . Let ˜Γ δ be the set of marked excursions ( f, g ) ∈ ˜ E , with f ∈ Γ δ , and let ˜Γ ∗ δ = ˜Γ δ ∩ ˜ E ∗ be its subset ofstarred excursions, then ˜ ν (˜Γ δ ) = ˜ ν (Γ δ × J ) = ν (Γ δ )and ˜ ν (˜Γ ∗ δ ) = (cid:90) Γ δ ν ( df ) m λ (cid:0) { g ∈ J : g ( T ( f )) > } (cid:1) = (cid:90) Γ δ ν ( df )(1 − e − λ T ( f ) ) . We are now ready to prove (33). First, note that˜ P (cid:0) θ +1 < t (cid:1) = ˜ P (cid:18) First excursion in ˜Γ δ happensbefore the first starred excursion (cid:19) . (35)By independence of t from S , the LHS is exactly the moment generating function of θ +1 :˜ P (cid:0) θ +1 < t (cid:1) = E P [ e − λθ +1 ] . Meanwhile, by the property of the marked excursion point process˜ P (cid:16) the first excursion in Γ δ happensbefore the first ’starred’ excursion (cid:17) = ˜ ν (˜Γ δ \ ˜ E ∗ )˜ ν (˜Γ δ ∪ ˜ E ∗ ) = ˜ ν (˜Γ δ \ ˜ E ∗ )˜ ν (˜Γ δ \ ˜ E ∗ ) + ˜ ν ( ˜ E ∗ ) . We have ˜ ν (˜Γ δ \ ˜ E ∗ ) = ˜ ν (˜Γ δ ) − ˜ ν (˜Γ ∗ δ ) = ν (Γ δ ) − (cid:90) Γ δ ν ( df )(1 − e − λ T ( f ) ) = (cid:90) Γ δ ν ( df ) e − λ T ( f ) , thus the RHS of (35) is (cid:82) Γ δ ν ( df ) e − λ T ( f ) (cid:82) Γ δ ν ( df ) e − λ T ( f ) + (cid:82) E ν ( df )(1 − e − λ T ( f ) ) , hence the result.Various analytical descriptions have been proposed for Ito’s excursion measure ν [6, 45]; one which isparticularly relevant for the applications considered here is obtained by ’slicing’ the space of excursionsaccording to their maximum height M : ν ( df ) = (cid:90) ∞ F M ( dm ) Q m ( df )where F M is the distribution of the maximum height M = max( | f | ) of an excursion f ∈ E , and Q m isthe law of the excursion conditional on the maximum M = m . The distribution of many quantities ofinterest described in Section 3 only involve F M . In the case where S is Brownian motion, this is knownas the ’Williams decomposition’ [43] and has been studied in great detail [32, 35, 43, 45].17 xample 4.6 ( δ -excursions of Brownian motion) . Consider the case when S is a Brownian motion andlet ν + is the restriction of ν to the set of positive excursions. By Williams’ characterization of the Itoexcursion measure [43], F M ( dm ) = ν + (max( f ) ∈ dm ) = dm m and (cid:90) { f ∈E , , max( f )= m } ν + ( df ) e − λT ( f ) = (cid:32) √ λm sinh( √ λm ) (cid:33) . From this we obtain (cid:90) Γ δ ν ( df ) e − λ T ( f ) = (cid:90) ∞ δ (cid:32) √ λm sinh( √ λm ) (cid:33) dm m = √ λe √ λδ − . Similarly, we can compute (cid:90) E ν ( df )(1 − e − λ T ( f ) ) = √ λ. Substituting in formula (33), we obtain E [ e − λθ +1 ] = e − √ λδ , which corresponds to the Laplace transform of a L´evy distribution i.e. a one-side 1 / − stable distribution[28]. In particular, the duration ( θ + k +1 − θ + k ) of δ − excursions has infinite mean. δ -excursions One of the difficulties with Ito’s representation is the existence of infinitely many ‘infinitesimal’ excur-sions, which translate into the existence of a local time at zero and the infinite mass of Ito’s excursionmeasure ν ( . ).Using Assumption 4.1, together with the property ν (Γ δ ) < ∞ , we derive a simpler finite decompo-sition of the process in terms of δ -excursions, which is more amenable to applications: Theorem 4.7.
Let S be Markov process satisfying Assumption 4.1. There exists an IID sequence ( e k ) k ≥ of δ -excursions e k ∈ U δ such that P (cid:32) ∀ t ≥ , S t = (cid:88) k ≥ e k (cid:0) ( t − θ + k − ) (cid:1)(cid:33) = 1 (36) where θ +0 = 0 and θ + k = (cid:80) i ≤ k Λ( e i ) . The distribution Π δ of e k is a probability measure concentrated on U δ , given by the law of the first δ -excursion of S , i.e. the law of S ( . ∧ θ +1 ) under P : ∀ H ∈ C b ( E ) , (cid:90) E Π δ ( df ) H ( f ) = E (cid:2) H (cid:0) S ( . ∧ θ +1 ) (cid:1) (cid:3) . (37) Proof.
The existence and uniqueness of the decomposition is a consequence of Proposition 3.4 and themeasurability of the decomposition (17) applied to the sample paths of S . As θ + k are stopping times, thestrong Markov property of S implies that ( e k ) k ≥ are independently and identically distributed. We nowdescribe the law Π δ of e k in terms of the Ito excursion measure ν given in Theorem 4.3. By Proposition4.4, ν (Γ δ ) < ∞ . In Lemma 3.2, we constructed a measurable map f ∈ U δ (cid:55)→ ( T, g, γ f ) ∈ [0 , ∞ ) × E × Γ δ such that f = g ⊕ T γ f and g (0) = g ( T ) = 0 . U δ onto Γ δ , which projects any measurable set A ⊂ U δ , onto the set { γ f , f ∈ A } ⊂ Γ δ of their last excursions. This projection simply consists in mapping each path in U δ to its last excursion, which by Lemma 3.2 is always an element of Γ δ .The distribution Π δ of δ − excursions is related to the Ito excursion measure ν ( . ) through this pro-jection: for a measurable subset A ⊂ U δ , Π δ ( A ) = ν ( { γ f , f ∈ A } ) ν (Γ δ ) , (38)where γ f denotes the last excursion of f . It is then readily verified that Π δ is a probability measure on U δ , satisfying (37).This result may also be used as a device for simulating sample paths of the process. Rather thansimulating, as is often done, from transition probabilities of the process on a fixed time grid, if we areable to sample from Π δ then we may simulate S by sampling an IID sequence e k ∼ Π δ and constructingpaths of S by concatenation using (36). This leads to a more efficient computation of path-dependentquantities which have simple expressions in terms of excursions. Consider now the case where S is a diffusion process whose state space is some interval I ∈ R containing[0 , δ ], with infinitesimal generator G := a ( x )2 d dx + b ( x ) ddx , a ( . ) > , b ∈ L loc acting on C functions subject to appropriate boundary conditions. G can be written as G = 1 m ( x ) ddx s (cid:48) ( x ) ddx , s (cid:48) ( x ) = exp (cid:18) − (cid:90) x b ( y ) a ( y ) dy (cid:19) , m ( x ) = 2 s (cid:48) ( x ) a ( x ) . where m ( x ) is the speed measure of X and s (cid:48) ( x ) is the derivative of the scale function s ( x ) [22]. Thescale function s is defined up to an additive constant and we will use the normalization s (0) = 0 in thesequel. For Brownian motion, we have m ( x ) = 2 and s ( x ) = x . The Laplace transform of the hittingtime distribution may then be expressed in terms of m and s [7, 22]: ∀ ( x, y ) ∈ I , ∀ λ ≥ , E x (cid:2) e − λT y ( S ) (cid:3) = (cid:40) Φ λ, − ( x ) / Φ λ, − ( y ) , if x < y, Φ λ, + ( x ) / Φ λ, + ( y ) , if x > y, where Φ λ, − (resp. Φ λ, + ) is the unique increasing (resp. decreasing), up to multiplicative constant factors,non-negative solution of the differential equation G Φ = λ Φ . Using these results, we can now derive various quantities of interest related to the profit and loss of thestrategy φ + defined in (4).Recall that ( τ + k − θ + k − ), θ + k − τ + k and ( θ + k − θ + k − ) denote, respectively, the duration of the IIDwaiting period, duration of the IID holding period and the lifetime of the IID δ -excursion of S . Let F θ ( t ) = P ( θ − k θ + k − ≤ t ) = P δ ( T ( S ) ≤ t ) be the distribution of the duration of holding period.19 Distribution of holding and waiting periods:
The distributions of ( τ + k − θ + k − ), θ + k − τ + k and( θ + k − θ + k − ) may be characterized through their Laplace transforms L τ ( λ ) := E [ e − λ ( τ + k − θ + k − ) ] = Φ λ, − (0)Φ λ, − ( δ ) , E δ [ e − λ ( θ + k − τ + k ) ] = E δ [ e − λT ( S ) ] = Φ λ, + ( δ )Φ λ, + (0) . (39)By the strong Markov property, these two variables are independent so L θ ( λ ) := (cid:90) ∞ e − λs F θ ( ds ) = E [ e − λ ( θ + k − θ + k − ) ] = Φ λ, − (0)Φ λ, − ( δ ) × Φ λ, + ( δ )Φ λ, + (0) . (40) • Expected number of trades : Recall that D δt ( S ) is the number of transactions on [0 , t ]. Theexpected number n δ ( t ) = E [ D δt ( S )] of transactions up to t is related to F θ via the renewal equation [11]: n δ ( t ) = F θ ( t ) + (cid:90) t n δ ( t − s ) F θ ( ds ) , i . e . n δ = F θ + n δ (cid:63) F θ . (41)From the renewal equation for the Laplace transform of n δ , we obtain the Laplace transform of n δ using (33): L [ n δ ]( λ ) := (cid:90) ∞ e − λ t n δ ( t ) dt = L θ ( λ )1 − L θ ( λ ) • Maximum loss per trade cycle and stop-loss probabilities:
As a consequence of the Markovproperty, the maximum loss during the holding period [ τ + k , θ + k ] is independent and identicallydistributed across trading periods k = 1 , , .. with distribution given by P (cid:32) max t ∈ [ τ + k ,θ + k ] (cid:16) V τ + k ( φ + ) − V t ( φ + ) (cid:17) ≥ x (cid:33) = P δ ( T x + δ ( S ) < T ( S )) = s ( δ ) s ( x + δ ) . (42)This quantity also represents the probability of hitting an exposure limit M and may be used forsetting such a limit. For example an exposure limit M q chosen such that the stop-loss is triggeredwith probability 0 < q < M q = s − (cid:18) s ( δ ) q (cid:19) − δ • Expected maximum loss per trade cycle:
The expected maximum loss for the the strategy(6) with stop-loss at level M is given by E (cid:34) max t ∈ [ τ + k ,κ + k ] (cid:16) V τ + k ( φ + M ) − V t ( φ + M ) (cid:17)(cid:35) = E δ (cid:20) max t ∈ [0 ,T ( S ) ∧ T M + δ ( S )] ( S t − δ ) (cid:21) = (cid:90) M s ( δ ) dxs ( δ + x ) . • Distribution of realized profit : Let π Mk be the realized profit of the strategy φ + M with stop-losslevel M over a trading cycle [ θ + k − , θ + k ]. Then ( π Mk , k ≥
1) are IID Bernoulli variables with P ( π Mk = − M ) = s ( δ ) s ( M + δ ) = 1 − P ( π Mk = δ ) . This leads to a binomial profit/loss distribution after n trade cycles. For a fixed horizon t , theexpected realized profit is given by E D δt ( S ) (cid:88) k =1 π Mk = n δ ( t ) (cid:18) δ − ( M + δ ) s ( δ ) s ( M + δ ) (cid:19) , (43)where n δ is given by (41). 20o compute the Laplace transform of the distribution of the portfolio value V t ( φ + ), let us introduce U ( t, λ, δ ) := E (cid:104) e − λ ( S ( t ∧ τ +1 ) − S ( t ∧ θ +1 ) ) (cid:105) = (cid:90) E Π δ ( df ) e − λ ( f ( t ) − f ( t ∧ T ( f ))) ,U ( t, λ, δ ) := E δ (cid:104) e λ S ( t ∧ T ( S )) (cid:105) , ˜ U ( z, λ, δ ) = (cid:90) ∞ (cid:2) e − z t U ( t, λ, δ ) dt. (cid:3) U is the solution of the following boundary value problem: (cid:40) ∂∂t U ( t, λ, x ) = G x U ( t, λ, x ) , x ∈ R \ { } ,U (0 , λ, x ) = e λx , ∀ x ∈ R ; U ( t, λ,
0) = 1 . (44) Proposition 4.8.
Denote by H δ ( t, λ ) the Laplace transform of the distribution of V t ( φ + ) : H δ ( t, λ ) := E (cid:104) e − λ V t ( φ + ) (cid:105) . Then under Assumption 4.1, we have H δ ( t, λ ) = e − λδ (cid:0) H δ ( · , λ ) ∗ F θ (cid:1) ( t ) + U ( t, λ, δ ) − e − λδ = e − λδ (cid:0) H δ ( · , λ ) (cid:63) F θ (cid:1) ( t ) + e − λδ ( U ( · , λ, δ ) (cid:63) F τ ) ( t ) − e − λδ . In particular the Laplace transform of H in the time variable is given by: L (cid:2) H δ ( · , λ ) (cid:3) ( z ) = (cid:90) ∞ H δ ( t, λ ) e − zt dt = ˜ U ( z, λ, δ ) L τ ( z ) − /ze λδ − L θ ( z ) . (45)For a proof see Appendix A.1. In this section we provide some explicit calculations for two classical models: Brownian motion and theOrnstein-Uhlenbeck process. δ -excursions Brownian excursions have been extensively studied starting with L´evy [29], and a great deal of analyticalresults are available, see e.g. [32] or [34, Ch. XII].The case where the signal S follows Brownian motion corresponds to no mean reversion i.e. a case ofmodel mis-specification for mean-reversion strategies based on the signal S . As the following propositionshows, the strategy φ + has a heavy-tailed loss distribution and infinite expected holding periods beforerealized profits materialize: Proposition 5.1 ( δ -excursions of Brownian motion.) . Let S = σB where B is a standard Brownianmotion. Then(i) The duration θ + k +1 − θ + k of δ -excursions are IID variables with θ + k +1 − θ + k (law) = δ σ (cid:18) Z + 1 Z (cid:48) (cid:19) , where Z, Z (cid:48) are independent standard Gaussian variables. ii) The distribution of the waiting time τ + k − θ + k − is given by P ( τ + k − θ + k − > t ) = P ( T δ ( S ) > t ) = 1 − (cid:18) δσ √ t (cid:19) where Φ is the standard normal distribution.(iii) The worst loss during each holding period [ θ + k , τ + k ] has a Pareto distribution with tail index 1: P (cid:32) max t ∈ [ τ + k ,θ + k ] V τ + k ( φ + ) − V t ( φ + ) ≥ M (cid:33) = δM + δ . In particular, the expected maximum loss for a strategy with exposure limit M is δ log(1 + M/δ ) ,which goes to infinity as the exposure limit is removed ( M → ∞ ).(iv) The distribution of the portfolio value V t ( φ + ) satisfies (cid:90) ∞ E (cid:104) e − λV t ( φ + ) (cid:105) e − zt dt = σ λ e − √ zσ δ − ze λδ e −√ zσ δ − ( σ λ − z ) z ( σ λ − z ) (cid:16) e λδ − e − √ zσ δ (cid:17) , z > σ λ . The proof is given in Appendix A.2. These results provide insights for the assessment of modelrisk in mean-reversion strategies: it shows that when mean-reversion is absent, strategies based on thisassumption lead to heavy-tailed loss distributions, even if underlying asset returns are not heavy tailed.This example also shows that the distribution of returns for the underlying asset (in this case: Gaussian)does not provide the right benchmark for understanding the risk of dynamic trading strategies, whichrelate to the height and duration of the excursions (in this case, both heavy-tailed with infinite mean!).
The most widely used example of mean-reverting process in finance is the Ornstein-Uhlenbeck (OU)process, defined as the unique solution ( S t ) t ≥ to the stochastic differential equation dS t = γdB t + α ( µ − S t ) dt, i . e , S t = e − αt (cid:18) S + γ (cid:90) t e αs dB s (cid:19) + µ (1 − e − αt ) , (46)where α ∈ R and γ ∈ R and B is a standard Brownian motion. The stationary distribution is N ( µ, γ α ).We denote σ ∞ = γ α the stationary variance.Distributional properties of excursions for the OU process have been studied in detail by Salminenet al [39]. We now use these results to study δ -excursions of the OU process.An application of the formulas in Section 4 yields: Proposition 5.2 (Property of δ -excursions for OU Process) . Let S be the OU process defined in (46) .Then the duration ( θ + k − θ + k − ) of the δ -excursion satisfies E [ e − λ ( θ + k − θ + k − ) ] = Φ λ, − (0)Φ λ, − ( δ ) × Φ λ, + ( δ )Φ λ, + (0) , where Φ λ, − ( x ) = (cid:90) ∞ u λα − exp (cid:18)(cid:114) αγ ( x − µ ) u − u (cid:19) du, Φ λ, + ( x ) = (cid:90) ∞ u λα − exp (cid:18)(cid:114) αγ ( µ − x ) u − u (cid:19) du. olding periods ( θ + k − τ + k ) are independent variables; for µ = 0 their probability density is given by f ( t ) = δγ √ π exp (cid:18) − δ αe − αt γ sinh( αt ) + αt (cid:19) (cid:18) α sinh( αt ) (cid:19) / (47) The distribution of the worst loss during the k th holding period [ τ + k , θ + k ] is given by P (cid:32) max t ∈ [ τ + k ,θ + k ] V ( φ + ) − V t ( φ + ) > M (cid:33) = s ( δ ) s ( M + δ ) , where s ( x ) = (cid:90) x exp (cid:18) αγ y − αµγ y (cid:19) dy. and E (cid:34) max t ∈ [ τ + k ,θ + k ] V ( φ + ) − V t ( φ + ) (cid:35) = (cid:90) ∞ s ( δ ) s ( δ + x ) dx. (48) Proof. ( θ + k − τ + k ) represents the hitting time of level 0 starting from δ for the OU process. The probabilitydensity of this hitting time is given [41] by (47). The scale function [22] of the OU Process (46) is givenby s ( x ) = (cid:90) x exp (cid:18) αγ y − αµγ y (cid:19) dy. Then for any
M > P δ (cid:18) max t ∈ [0 ,T ( S )] S t ≥ M + δ (cid:19) = P δ ( T M + δ ( S ) < T ( S )) = s ( δ ) s ( M + δ ) . These results allow to compute various quantities of interest for the trading strategy φ + as a functionof model parameters; these are displayed in Figure 4. In most empirical studies the threshold δ is chosenclose to the standard deviation of the signal [17]: δ (cid:39) σ ∞ . We observe that the expected maximum lossper trade cycle has a maximum around δ (cid:39) σ ∞ (Figure 4a), while the expected profit is maximized ata lower threshold δ (cid:39) . σ ∞ (Figure 4c). This implies that the choice δ = σ ∞ , may not achieve anoptimal risk-return tradeoff.One may represent this risk-return trade-off in terms of an ‘efficient frontier’. Figure 4d shows therealized profit vs the expected maximum loss for various levels of the threshold δ . The profitability of thestrategy strongly depends on the mean-reversion rate α , as expected. The maximum profit is obtainedfor δ (cid:39) . σ ∞ but the choice δ = σ ∞ is never optimal. These results are consistent with previous studiesusing the OU model [27].As observed in Figure 4b, the expected worst loss diverges to infinity as the mean reversion rate α →
0, which corresponds to the Brownian case (see Proposition 5.1). This indicates a significantdownside risk for such strategies if mean reversion is ‘slow’ i.e. if α is small. δ -excursions So far we have been using excursions as a way to dissect sample path properties of a process. Howeverone can use this entire apparatus as a method for constructing stochastic models, using excursions asbuilding blocks. This approach was first carried out by Ito, in his construction of a ’recurrent extension’of a Markov process defined up to its first return to zero [21] and explored in depth by Salisbury [38].Construction of regenerative processes by concatenation of independent excursions has been studied byLambert and Simatos [26] and Yano [44]. Our δ − excursion concept is related to, but slightly different23 a) Expected worst loss as a function of δ/σ ∞ . (b) Expected worst loss as a function ofmean-reversion parameter α ( δ = 0 . δ E [ D δT ] as a function of δ/σ ∞ . (d) Efficient frontier (Star: δ = σ ∞ ). Figure 4: Distributional properties of the trading strategy (4) when the trading signal is an OU processwith γ = 0 .
1. (No stop-loss.) (a) Realized profit per cycle; M = δ = 0 .
1. (b) Efficient frontier with various choice of M ∈ [0 . σ ∞ , σ ∞ ]. ( α = 0 . σ ∞ = 0 . Figure 5: Distributional properties of the trading strategy (6) with stop-loss M and when the tradingsignal is an OU process with γ = 0 . δ may be seen as ’big excursions’ in the senseof [26] and relate to δ − excursions through the last exit decomposition, as noted in Lemma 3.2.We present in this section a variation on this theme, which allows to construct processes by con-catenation of a sequence of δ -excursions with desired features, and use this approach to provide newconstructions as well as a non-parametric approach to scenario simulation based on excursions. Unlike24to’s original synthesis theorem, our construction leads to Markov processes on an enlarged state space.Compared with [26, 44], we retain a discrete construction and are able to incorporate dependence in thesequence, as shown in the examples below, without losing analytical tractability. δ -excursions We now present a converse to Theorem 4.7 which allows to construct a stochastic process by pastingtogether • excursions from 0 to δ of a Markov process S + , with • excursions from δ to 0 of a Markov process S − . Theorem 6.1 (Markovian concatenation of δ -excursions) . Let I ⊂ R be an interval and δ > . Let ( S + , P x + ) and ( S − , P x − ) be regular Markov processes with state space I satisfying P ( S +0 = 0) = P − ( S − = 0) = 1 , P (0 < T δ ( S + ) < ∞ ) = 1 , P δ − (0 < T ( S − ) < ∞ ) = 1 . Consider the process e defined by concatenating S + and S − at T δ ( S + ) e = S + ⊕ T δ ( S + ) S − , S + ∼ P , S − ∼ P δ − (49) Then the law of e defines a probability measure Π δ on U δ . Let ( e k ) k ≥ be an IID sequence with law Π δ and ∆ = 0 , ∆ k = k (cid:88) i =1 Λ( e i ) , X t = (cid:88) k ≥ e k (( t − ∆ k − )) . (50) Then X is unique in law and its δ − excursions are independent with law Π δ . Π δ is the law of X ( . ∧ ∆ ) and (50) is the decomposition of X into δ − excursions.Proof. The proof consists in constructing an auxiliary Markov process Y such that the δ -excursions of X corresponds to the (classical) excursions of Y away from a certain point.Intuitively, to describe the transitions of the process X , in addition to the current position x ∈ I weneed to know whether we are in an excursion of type (cid:48) + (cid:48) or (cid:48) − (cid:48) . We therefore construct a Markovianlifting of X i.e. a Markov process Y = ( Y , Y ) on an enlarged state space where the second componentis in { + , −} and show that X = Y satisfies the requirement of the theorem. Let E = (( I ∩ ( −∞ , δ ] ) × { + } ) (cid:124) (cid:123)(cid:122) (cid:125) E + ∪ (([0 , ∞ ) ∩ I ) × {−} ) (cid:124) (cid:123)(cid:122) (cid:125) E − ⊂ I × { + , −} . be state space E shown in Figure 3: E + corresponds to the region shaded in blue, while E − correspondsto the region shaded in green.One method of proof is to consider e as a Markov process on E absorbed at (0 , − ) and construct Y as a recurrent extension of e defined by (49). Consider the process (cid:98) Y defined on [0 , Λ( e )] as (cid:98) Y (0) = (0 , − ) , (cid:98) Y ( t ) = ( e ( t ) , +) 0 < t ≤ T δ ( S + ) , (cid:98) Y ( t ) = ( e ( t ) , − ) t > T δ ( S + )as an E − valued Markov process absorbed at (0 , − ). Ito’s recurrent extension theorem [21, Theorem6.1] then implies the existence of a unique Markov process ( Y, P Y ) with state space E such that the lawof ( Y ( . ∧ T ∂E )) coincides with that of (cid:98) Y . From the assumptions on the hitting times of S + , S − , this25orresponds to the ’discrete visiting case’ of Ito’s construction [21, Sec. 6] so Y is a concatenation ofIID copies of ˆ Y . Let us now describe the Markov process Y through its infinitesimal generator. Denoteby C ( I ) = { f ∈ C ( I ) , ∂f ∈ C ( I ) , ∂ f ∈ C ( I ) } . Let L + (resp. L − ) be the infinitesimal generator of S + (resp. S − ) on C ( I ), in the sense that ( S ± , P ± ) is the unique solution of the martingale problem for L ± . For f : E (cid:55)→ R , denote by f | E + (resp. f | E − ) its restriction to E + (resp. E − ). We define D L = (cid:110) f : E → R , f | E + ( ., +) = f + ∈ C ( I ∩ ( −∞ , δ ] ) ,f | E − ( ., − ) = f − ∈ C ([0 , ∞ ) ∩ I ) , f + ( δ ) = f − ( δ ) , f − (0) = f + (0) (cid:111) . (51)Define the operator L : D L → C ( E ) by Lf ( x, +) = L + f + ( x ) , ∀ x ∈ I ∩ ( −∞ , δ ) Lf ( x, − ) = L − f − ( x ) , ∀ x ∈ I ∩ (0 , ∞ ) (52) Lf ( δ, +) = L − f − ( δ ) , Lf (0 , − ) = L + f + (0) . We shall now show that ( Y, P Y ) is the unique solution to the martingale problem for ( L, (0 , +)) onthe canonical space D ([0 , ∞ ) , E ) [16, Ch.4, Sec. 5]. Denote by S the canonical process and let f ∈D L . P x + (resp. P x − )) is the unique solution to the martingale problem for ( L + , x ) (resp. ( L − , x )) on D ([0 , ∞ ) , I ), so using the notation in (52), M + t = f + ( S t ) − (cid:82) t L + f + ( S u ) du is a P + − martingale, and M − t = f − ( S t ) − (cid:82) t L − f − ( S u ) du is a P − − martingale. Let us now show that M t = f ( Y t ) − (cid:82) t Lf ( Y u ) du isa P Y − martingale. M t = D δt ( S ) (cid:88) i =1 (cid:34) f ( Y τ + i ) − f ( Y θ + i − ) − (cid:90) τ + i θ + i − Lf ( Y u ) du + f ( Y θ + i ) − f ( Y τ + i ) − (cid:90) θ + i τ + i Lf ( Y u ) du (cid:35) + f ( Y τ + Dδt ( S )+1 ∧ t ) − f ( Y θ + Dδt ( S ) ∧ t ) − (cid:90) τ + Dδt ( S )+1 ∧ tθ + Dδt ( S ) ∧ t Lf ( Y u ) du + f ( Y θ + Dδt ( S )+1 ∧ t ) − f ( Y τ + Dδt ( S )+1 ∧ t ) − (cid:90) θ + Dδt ( S )+1 ∧ tτ + Dδt ( S )+1 ∧ t Lf ( Y u ) du = D δt ( S ) (cid:88) i =1 (cid:16) M + τ + i − M + θ + i − + f + ( δ ) − f − ( δ ) (cid:17) + D δt ( S ) (cid:88) i =1 (cid:16) M + θ + i − M + τ + i + f − (0) − f + (0) (cid:17) + M + τ + Dδt ( S )+1 ∧ t − M + θ + Dδt ( S ) ∧ t + 1 t ≥ τ + Dδt ( S )+1 ( f + ( δ ) − f − ( δ )) + (cid:18) M − θ + Dδt ( S )+1 ∧ t − M − τ + Dδt ( S )+1 ∧ t (cid:19) where θ + k , τ + k are the stopping times introduced in (1). The boundary terms f − (0) − f + (0) and f + ( δ ) − f − ( δ ) are zero by definition of D L , and all other terms are martingale differences under P Y , so M is a P Y -martingale. Thus ( Y, P Y ) is indeed a solution of the martingale problem for ( L, (0 , +)) . The behavior of the Markovian lifting Y can be described from the infinitesimal generator (52): while Y = +, Y t evolves according to the transition probabilities of S + up to its first hitting time of δ , atwhich point Y jumps to − . Thereafter, Y evolves according to the transition probabilities of S − untilit’s next hitting time of zero 0 at which point Y jumps back to +, and so on.26 > δ ( δ, +) ( δ, − ) x ∈ (0 , δ ) x = 0 (0 , − ) x < − Figure 6: State space E of the Markovian lifting of the process constructed in Theorem 6.1.Though this construction shares at first glance some features with the Ito’s synthesis theorem [21, 38],there are some key differences. When δ = 0 we recover Ito’s construction: X is then the recurrentextension of ( S + ( . ∧ T ( S + )) , P ) . However, for δ > X itself is not a Markov process, but the projection of a Markov process Y on anenlarged state space, which may be thought of as a ’regime-switching’ model with two regimes { + , −} .The process X • behaves like ( S + , , P ) during its excursions from 0 to δ (+ regime): for measurable A ⊂ R ∀ h > , P ( X t + h ∈ A | X t = 0 , T δt ( X ) > h ) = P ( S + h ∈ A ); • behaves like ( S − , P δ − ) during its excursions from δ to 0 ( − regime): for measurable A ⊂ R ∀ h > , P ( X t + h ∈ A | X t = δ, T t ( X ) > h ) = P δ − ( S − h ∈ A ) . Also, unlike the Ito synthesis theorem, the process is constructed as a discrete concatenation: underAssumption 4.1 we have P ( ∀ k ≥ , ∆ k +1 > ∆ k ) = 1 and local time is not involved in this construction.Note that any δ > δ may have a naturalinterpretation.Theorem 6.1 provides a systematic and tractable way of constructing stochastic processes whoseexcursions have some desired properties. We now provide an example of how this result may be used toconstruct stochastic processes with asymmetric upward and downward excursions. Example 6.2 (Asymmetric OU process) . Recall the OU process defined in (46), which is characterizedby parameter ( α, µ, γ ). Denote by P x + the law of the OU process with parameter ( α + , µ + , γ + ) and P x − thelaw of the OU process with ( α − , µ − , γ − ). We can construct an asymmetric OU-process using asymmetric δ -excursions with formula (49). Figure 7 shows an example of sample path of X .27 20 40 60 80 100 120 140 160 180 200 − − . . S t Figure 7: Sample path of an asymmetric OU process with δ = 0 .
5. Blue segments represent excursionsfrom 0 to δ , generated with ( α + , µ + , γ + ) = (0 . , , . δ to 0, generated with ( α − , µ − , γ − ) = (0 . , , . N = 10000 generated with parameters ( α + , µ + , γ + ) = (0 . , , . α − , µ − , γ − ) = (0 . , , .
1) and δ = 0 .
5, estimating an OU process yields γ − < (cid:98) γ = 0 . < γ + , and α − < (cid:98) α = 0 . < α + . We thus observe that the OU model either overestimates or underestimates the volatility and mean-reversion in each regime.
The above examples use as building blocks parametric models of Markov processes to construct othermodels with desirable excursion properties. More generally, Proposition 3.3 may be used as a non-parametric method for simulating paths whose excursions match those observed in data.Consider a data set consisting of observations on the path ( S t , t ∈ [0 , T ]) of a P − Markov processsatisfying Assumption 4.1 for some δ > S into a sequence ( e k ) k =1 ..D δT of δ − excursionsas in (17). By Theorem 4.7, e k are IID variables with values in U δ , whose law we denote Π δ . Denotingby (cid:15) x a unit point mass at x , Π T = 1 D δT ( S ) D δT ( S ) (cid:88) k =1 (cid:15) e k defines a probability measure on U δ which we call the empirical δ -excursion measure . Under Assumption4.1, D δT ( S ) → ∞ as T → ∞ , so by the law of large numbers the empirical excursion measure Π T providesa good approximation of Π δ for large T and for any Glivenko-Cantelli class F of functions [40] on U δ ,representing properties of δ -excursions, we have ∀ H ∈ F , (cid:90) H ( f )Π T ( df ) T →∞ → (cid:90) H ( f )Π δ ( df ) . The point is that it is fairly easy to simulate samples from Π T and evaluate (cid:82) F ( f )Π T ( df ): this may bedone by randomly resampling from the empirical sequence of excursions ( e k ). This leads to an approachfor non-parametric scenario simulation based on δ − excursions:28 on-parametric scenario simulation by pasting of excursions.Input data : sample path ( S t , t ∈ [0 , T ]) • Decompose ( S t , t ∈ [0 , T ]) into δ − excursions e , ...e N ∈ U δ using Proposition3.4. • Generate an IID sequence of integers ( k ( ω ) , k ( ω ) , .. ) where k i ∼ UNIF( { , , .., N } ). • Construct a path X ( ω ) as in (17) by concatenating the excursions in the order given by( k i , i ≥ X t ( ω ) = (cid:88) i ≥ e k i ( ω ) (( t − θ + i − ) where θ +0 = 0 , θ + i = i (cid:88) j =1 Λ( e k i ) . Output : simulated sample path X Table 1: Non-parametric scenario simulation by pasting of excursions.If N = D δT ( S ) is large, the paths X generated in this way have δ − excursions whose properties mimicthose of S .We illustrate the flexibility of this approach in an example based on financial data. Example 6.3 (Pairs trading) . Pairs trading [33, 17] is a trading strategy based on identifying a sta-tionary linear combination of two stock prices and using this linear combination as a trading signalfor generating buy/sell transactions in the pair. In most applications the signal is then modeled as anAR(1)/ Ornstein-Uhlenbeck process [27, 33].As an example, we use second-by-second NYSE price records of CocaCola (KO) and PepsiCola (PEP)shares during trading hours 10:00AM-4:00PM for the period 07/01/2013 - 07/01/2020 to construct apair-trading signal. Denote PEP( t ) (resp. CO( t )) the mid-price of Pepsi (resp. Coca-Cola). The signalis constructed as S t := PEP( t ) − a t CO( t ) where the coefficient a t is piece-wise constant, updated oneach trading day by an ordinary least square regression of PEP( · ) on CO( · ) over the previous 5 days.To assess the roughness of the signal S , we analyze the number of level crossings as a function of δ and apply Proposition 3.9. Recall that log( D δt ( S )) ∼ − ( p −
1) log( δ ) + constant as δ →
0, where p measure the roughness of the path. We estimate the exponent p by linear regression of log( D δt ( S )) onlog( δ ). As shown in Figure 8a, the estimated exponent of the trading signal S is around p = 1 .
85, whichimplies that the path is slightly smoother than the Brownian motion (for which p = 2).There are two types of crossings with different time-scales along the path: crossings due to the meanreverting phenomenon on a longer time-scale and crossings with small magnitudes due to the roughnessof the path once the signals revert to level 0. The crossings of the first type could be captured by all δ with appropriate choices. Crossings of the second type show up as δ →
0. Empirical estimates seemto indicate a non-zero limit of the realized profit δD δT as δ →
0. This is consistent with the result inProposition 3.9 for p = 2, indicating that it is not use small thresholds δ for trading this pair (see Figure8b). The realized profit is maximized at δ (cid:39) . σ , which is quite different from the OU model, whererealized profit is maximized for δ < σ . 29 a) Number of level crossings D δT . (Orange dashedline: δ p − with p = 1 .
85; Green dotted line: asymp-totics for p = 2 . T = 1 day). (b) Realized profit δD δT ( T = 1 day). Figure 8: Number of crossings D δT and realized profit δD δT as a function of threshold δ .Recall that the trading strategy φ + consists in shorting the pair when S t crosses the threshold δ from below and unwinds the position when S t returns to 0. Denote by σ the standard deviation of S t .In Figure 9, we provide the empirical distributions of the durations for waiting period waiting period( τ + k − θ + k − ) and holding period ( θ + k − τ + k ) and the maximum loss during the holding period, when δ = σ is the intraday standard deviation of the signal, a common choice for mean-reversion strategies [33].As seen from the semi-logarithmic plots in Figures 9b and 9c, the durations of the holding periodand the waiting period are approximately exponentially distributed. The maximum loss has a Paretotail with exponent k = 1, which is very heavy tailed and indicates infinite mean and variance, as shownby the log-log plot in Figure 9d. This combination of a Pareto tail for the excursion height and anexponential duration for δ − excursions corresponds neither to the Brownian case nor to the case of theOrnstein-Uhlenbeck process.Estimating an Ornstein-Uhlenbeck process as in (46) using the method of moments yields the fol-lowing parameter estimates (time is measured in seconds): (cid:98) α = 1 . × − , (cid:98) µ = − . , and (cid:98) γ = 4 . × − . (53)The corresponding distributions for the duration of the holding period, the waiting period and the worstloss during the holding period are displayed ( green dotted lines) alongside the empirical distributionsof these quantities in Figure 9. The discrepancy between the green dotted lines and the blue solid linesin Figures 9b, 9c, and 9d illustrates that the distributions computed using the OU model give a poorapproximation of the corresponding empirical distributions, leading to an inaccurate representation ofthe risk and return profile of the strategy. This is yet another indication of the risk of model mis-specification in such mean-reversion strategies. 30 a) Coca-Pepsi pair trading signal: 07/16/2013. (b) Rank-frequency plot for the waiting period ( τ + k − θ + k − ), semi-logarithmic scale.(c) Rank-frequency plot for the holding period ( θ + k − τ + k ), semi-logarithmic scale. (d) Rank-frequency plot for maximum loss duringthe holding period (log-log scale). Orange dotted line:Pareto distribution with exponent k = 1 . Figure 9: Coca-Pepsi pair trading signal KO( t ) − a t PEP( t ) at one-second frequency, for δ = σ (2007-2020). Data (blue) vs OU model (dotted).We tackle the shortcomings of the OU model by using our non-parametric scenario simulation ap-proach to evaluate the quantities of interest as described in Table 1: we decompose the empirical pathinto δ − excursions and generate scenarios by random pasting of such empirical δ − excursions. Figure 10shows a typical sample path generated in this manner, for δ = σ . By construction, paths generatedin this way retain the roughness properties of the observe path as well as the empirical distribution ofheights and durations of δ − excursions. 31igure 10: Example of sample path generated using non-parametric method, using the pairs tradingsignal described in Section 6.3. References [1]
C. Alexander , Optimal hedging using cointegration , Philosophical Transactions of the RoyalSociety of London. Series A, 357 (1999), pp. 2039–2058.[2]
C. Alexander and A. Dimitriu , Indexing and statistical arbitrage , The Journal of PortfolioManagement, 31 (2005), pp. 50–63.[3]
M. Avellaneda and J.-H. Lee , Statistical arbitrage in the US equities market , QuantitativeFinance, 10 (2010), pp. 761–782.[4]
J. Bertoin , Temps locaux et int´egration stochastique pour les processus de Dirichlet , in S´eminairede Probabilit´es, XXI, vol. 1247 of Lecture Notes in Math., Springer, Berlin, 1987, pp. 191–205.[5]
F. Biagini, Y. Hu, B. Øksendal, and T. Zhang , Stochastic calculus for fractional Brownianmotion and applications , Springer-Verlag, 2008.[6]
R. M. Blumenthal , Excursions of Markov processes , Springer, 2012.[7]
A. N. Borodin and P. Salminen , Handbook of Brownian motion-facts and formulae , Birkh¨auser,2012.[8]
N. Cai, N. Chen, and X. Wan , Occupation times of jump-diffusion processes with double expo-nential jumps and the pricing of options , Mathematics of Operations Research, 35 (2010), pp. 412–437.[9]
P. Cheridito, H. Kawaguchi, and M. Maejima , Fractional Ornstein-Uhlenbeck processes ,Electron. J. Probab., 8 (2003), p. 14 p.[10]
M. Chesney, M. Jeanblanc-Picqu´e, and M. Yor , Brownian excursions and parisian barrieroptions , Advances in Applied Probability, (1997), pp. 165–184.3211]
E. Cinlar , Markov renewal theory , Advances in Applied Probability, 1 (1969), pp. 123–187.[12]
R. Cont and N. Perkowski , Pathwise integration and change of variable formulas for continuouspaths with arbitrary regularity , Trans. Amer. Math. Soc. Ser. B, 6 (2019), pp. 161–186.[13]
A. Dassios and S. Wu , Perturbed brownian motion and its application to parisian option pricing ,Finance and Stochastics, 14 (2010), pp. 473–494.[14]
E. B. Dynkin , Wanderings of a Markov process , Theory of Probability & Its Applications, 16(1971), pp. 401–428.[15]
N. El Karoui , Sur les mont´ees des semi-martingales , in Temps locaux, no. 52-53 in Ast´erisque,Soci´et´e math´ematique de France, 1978, pp. 63–72.[16]
S. N. Ethier and T. G. Kurtz , Markov processes: characterization and convergence , JohnWiley & Sons, 1986.[17]
E. Gatev, W. N. Goetzmann, and K. G. Rouwenhorst , Pairs trading: Performance of arelative-value arbitrage rule , The Review of Financial Studies, 19 (2006), pp. 797–827.[18]
D. Geman and J. Horowitz , Occupation densities , Ann. Probab., 8 (1980), pp. 1–67.[19]
S. J. Grossman and Z. Zhou , Optimal investment strategies for controlling drawdowns , Math-ematical Finance, 3 (1993), pp. 241–276.[20]
S. Hogan, R. Jarrow, M. Teo, and M. Warachka , Testing market efficiency using statisticalarbitrage with applications to momentum and value strategies , Journal of Financial Economics, 73(2004), pp. 525 – 565.[21]
K. Ito , Poisson point processes attached to Markov processes , in Proceedings of the Sixth BerkeleySymposium on Mathematical Statistics and Probability, vol. 3, University of California Press, 1972,pp. 225–239.[22]
K. Itˆo and H. P. McKean , Diffusion processes and their sample paths , Springer, 1965.[23]
A. Johansen and D. Sornette , Large stock market price drawdowns are outliers , Journal ofRisk, 4 (2002), pp. 69–110.[24]
D. Kim , Local times for continuous paths of arbitrary regularity , 2019.[25]
P. Lakner, J. Reed, and F. Simatos , Scaling limit of a limit order book model via the regen-erative characterization of l´evy trees , Stochastic Systems, 7 (2017), pp. 342–373.[26]
A. Lambert and F. Simatos , The weak convergence of regenerative processes using some excur-sion path decompositions , Ann. Inst. H. Poincar´e Probab. Statist., 50 (2014), pp. 492–511.[27]
T. Leung and X. Li , Optimal mean reversion trading: Mathematical analysis and practical ap-plications , World Scientific, 2015.[28]
P. L´evy , Sur certains processus stochastiques homog`enes , Compositio Mathematica, 7 (1940),pp. 283–339.[29]
P. L´evy , Processus stochastiques et mouvement Brownien , Gauthier Villars, 1948.3330]
M. R. Pistorius , An excursion-theoretical approach to some boundary crossing problems and theSkorokhod embedding for reflected L´evy processes , in S´eminaire de Probabilit´es XL, Springer, 2007,pp. 287–307.[31]
J. Pitman and M. Yor , Hitting, occupation and inverse local times of one-dimensional diffusions:martingale and excursion approaches , Bernoulli, 9 (2003), pp. 1–24.[32]
J. Pitman and M. Yor , Itˆo’s excursion theory and its applications , Japanese Journal of mathe-matics, 2 (2007), pp. 83–96.[33]
H. Rad, R. K. Y. Low, and R. Faff , The profitability of pairs trading strategies: distance,cointegration and copula methods , Quantitative Finance, 16 (2016), pp. 1541–1558.[34]
D. Revuz and M. Yor , Continuous martingales and Brownian motion , Springer-Verlag, Berlin,1999.[35]
L. Rogers , Williams’ characterisation of the Brownian excursion law: proof and applications , inS´eminaire de Probabilit´es XV 1979/80, Springer, 1981, pp. 227–250.[36]
L. C. G. Rogers , A guided tour through excursions , Bulletin of the London mathematical Society,21 (1989), pp. 305–341.[37]
L. C. G. Rogers and D. Williams , Diffusions, Markov processes and martingales. Volume 2:Itˆo calculus , vol. 2, Cambridge University Press, 2000.[38]
T. S. Salisbury , Construction of right processes from excursions , Probability theory and relatedfields, 73 (1986), pp. 351–367.[39]
P. Salminen, P. Vallois, and M. Yor , On the excursion theory for linear diffusions , JapaneseJournal of Mathematics, 2 (2007), pp. 97–127.[40]
M. Talagrand , The Glivenko-Cantelli problem , Ann. Probab., 15 (1987), pp. 837–870.[41]
M. C. Wang and G. E. Uhlenbeck , On the theory of the Brownian motion ii , Reviews ofmodern physics, 17 (1945), p. 323.[42]
S. Watanabe , Ito’s theory of excursion point processes and its developments , Stochastic Processesand their Applications, 120 (2010), pp. 653 – 677.[43]
D. Williams , Path Decomposition and Continuity of Local Time for One-Dimensional Diffusions,I , Proceedings of the London Mathematical Society, s3-28 (1974), pp. 738–768.[44]
K. Yano , Functional limit theorems for processes pieced together from excursions , Journal of theMathematical Society of Japan, 67 (2015), pp. 1859–1890.[45]
J.-Y. Yen and M. Yor , Local times and excursion theory for Brownian motion , vol. 2088,Springer, 2013.[46]
H. Zhang , Occupation times, drawdowns, and drawups for one-dimensional regular diffusions ,Adv. in Appl. Probab., 47 (2015), pp. 210–230.34
Technical proofs
A.1 Proof of Proposition 4.8
The strong Markov property of S implies that V t ( φ + ) = (cid:88) i ≥ S t ∧ τ + i − S t ∧ θ + i ( law ) = S t ∧ τ +1 − S t ∧ θ +1 + V ( θ +1 ) + ( φ + )Depending on whether θ +1 < t or > t , we have either S t ∧ τ +1 − S t ∧ θ +1 = δ or V ( t − θ +1 ) + ( φ + ) = 0, so e − λV t ( φ + ) ( law ) = e − λ (cid:18) S t ∧ τ +1 − S t ∧ θ +1 (cid:19) − λV ( t − θ +1 )+ ( φ + ) = e − λδ − λV ( t − θ +1 )+ ( φ + ) + e − λ (cid:18) S t ∧ τ +1 − S t ∧ θ +1 (cid:19) − e − λδ . Hence taking the expectation, we obtain H δ ( t, λ ) = e − λδ (cid:0) H δ ( · , λ ) (cid:63) F θ ( ds ) (cid:1) ( t ) + U ( t, λ, δ ) − e − λδ . It remains to note that, since for given τ +1 ; S ( t ∧ θ +1 ) − S ( t ∧ τ +1 ) has the same law under P as S (( t − τ +1 ) + ∧ T ) − δ under P δ , then U ( t, λ, δ ) = E (cid:104) E (cid:104) e − λ ( S ( t ∧ τ +1 ) − S ( t ∧ θ +1 ) ) (cid:12)(cid:12)(cid:12) F τ +1 (cid:105)(cid:105) = E (cid:2) e − λδ U (( t − τ +1 ) + , λ, δ ) (cid:3) = e − λδ ( U ( · , λ, δ ) (cid:63) F τ ( ds )) ( t ) . Plugging this in the previous identity, we get H δ ( t, λ ) = e − λδ (cid:0) H δ ( · , λ ) (cid:63) F θ (cid:1) ( t ) + e − λδ ( U ( · , λ, δ ) (cid:63) F τ ) ( t ) − e − λδ . Equation (45) follows by applying Laplace transform in t and solving for L (cid:2) H δ ( · , λ ) (cid:3) ( · ). A.2 Proof of Proposition 5.1
First we note that ( θ + i +1 − τ + i +1 ) (law) = ( τ + i +1 − θ + i ) (law) = T δ , and P ( T δ ( S ) > t ) = P (cid:18) max u ∈ [0 ,t ] S u < δ (cid:19) = P ( | S t | < δ ) = P ( √ t | S | < δ ) = 1 − (cid:18) δσ √ t (cid:19) where Φ is the standard normal distribution. So θ + i +1 − θ + i = ( θ + i +1 − τ + i +1 ) + ( τ + i +1 − θ + i ) (Law) = δ σ (cid:18) Z + 1 Z (cid:48) (cid:19) , where Z, Z (cid:48) are independent N (0 , M > P (cid:32) max t ∈ [ τ +1 ,θ +1 ] V ( φ + ) − V t ( φ + ) ≥ M (cid:33) = P δ ( T M + δ ( S ) < T ( S )) = δM + δ . which then implies that the expected maximum loss is infinite. To show (iv) note that when G x = σ ∂ x one can solve boundary value problem (44) to obtain U ( t, λ, x ) = 1 + 1 σ √ πt (cid:90) ∞ (cid:18) e − ( x − y )22 σ t − e − ( x + y )22 σ t (cid:19) ( e λy − dy = e λ x + p ( λ ) t Φ (cid:18) x + λσ tσ √ t (cid:19) − e − λ x + p ( λ ) t Φ (cid:18) − x + λσ tσ √ t (cid:19) + 2Φ (cid:18) − xσ √ t (cid:19) . p ( λ ) := σ λ and Φ is the standard normal distribution. Hence U ( t, λ, x ) grows like e p ( λ ) t as t → + ∞ so the Laplace transform ˜ U ( z, λ, x ) is well-defined for z > p ( λ ). Taking the Laplace transformin the PDE for U , we obtain˜ U ( z, λ, x ) = p ( λ ) z ( p ( λ ) − z ) Φ z, + ( x )Φ z, + (0) − e λx p ( λ ) − z , z > p ( λ ) . Using (40) and formula (45) in Proposition 4.8, we obtain L (cid:2) H δ ( · , λ ) (cid:3) ( z ) = p ( λ ) z ( p ( λ ) − z ) L θ ( z ) − e λδ p ( λ ) − z L τ ( z ) − /ze λδ − L θ ( z ) , z > p ( λ ) . In this case we have explicit forms for Φ z, ± ( x ) = c e λ ∓ x , where λ ± := ± √ zσ . Thus we have L τ ( z ) = e − √ zσ δ , L θ ( z ) = e − √ zσ δ . Therefore, L (cid:2) H δ ( · , λ ) (cid:3) ( z ) = p ( λ ) e − √ zσ δ − ze λδ e −√ zσ δ − ( p ( λ ) − z ) z ( p ( λ ) − z ) (cid:16) e λδ − e − √ zσ δ (cid:17) , z > p ( λ ) . Hence (cid:90) ∞ E (cid:104) e − λV t ( φ + ) (cid:105) e − zt dt = σ λ e − √ zσ δ − ze λδ e −√ zσ δ − ( σ λ − z ) z ( σ λ − z ) (cid:16) e λδ − e − √ zσ δ (cid:17) , for z > σ λ ..