When to sell an asset amid anxiety about drawdowns
aarXiv: math.PR/00000000
When to sell an asset amid anxiety about drawdowns
Neofytos Rodosthenous ∗ School of Mathematical Sciences, Queen Mary University of LondonMile End Road, London E1 4NS, UKe-mail: [email protected]
Hongzhong Zhang
Department of IEOR, Columbia University500W 120th Street, New York, NY 10027, USA
Abstract:
We consider risk averse investors with different levels of anxiety about asset price draw-downs. The latter is defined as the distance of the current price away from its best performance sinceinception. These drawdowns can increase either continuously or by jumps, and will contribute towardsthe investor’s overall impatience when breaching the investor’s private tolerance level. We investigatethe unusual reactions of investors when aiming to sell an asset under such adverse market conditions.Mathematically, we study the optimal stopping of the utility of an asset sale with a random discount-ing that captures the investor’s overall impatience. The random discounting is given by the cumulativeamount of time spent by the drawdowns in an undesirable high region, fine tuned by the investor’spersonal tolerance and anxiety about drawdowns. We prove that in addition to the traditional take-profit sales, the real-life employed stop-loss orders and trailing stops may become part of the optimalselling strategy, depending on different personal characteristics. This paper thus provides insights onthe effect of anxiety and its distinction with traditional risk aversion on decision making.
MSC 2010 subject classifications:
Keywords and phrases:
Drawdown, L´evy process, optimal stopping, Omega clock, random discountrate, stop-loss orders, trailing stops.
1. Introduction
There are many economic, financial and psychological (behavioural) reasons that drive investors to panicwhen their asset prices experience a relatively large fall or have a relatively low value for a significant amountof time. This often results in the assets being sold at a lower than anticipated price. A common scenario,that leads investors to such decisions, is financial markets acting contrary to an investor’s expectation fora lot longer than their investment capital can hold out. Another scenario is when asset prices remain atlow levels, supervisors of traders grow impatient and ask “How much longer do we have to carry this tradebefore it profits?”. One can also consider the case of Commodity Trading Advisors who determine variousrules for the magnitude and duration of their client accounts’ drawdowns. These can be accounts that areshut down either when certain drawdowns are breached or after small long-lasting drawdowns (see, e.g. [4]for more details). In all these examples, investors are driven to liquidate assets at undesirable low prices.This phenomenon of “selling low” is also evident in financial markets through the extensive use of tradi-tional stop-loss and trailing stop orders placed by investors. Stop-loss orders are usually placed to minimiserisks associated with trading accounts, to bound losses or to protect profits, and they have a fixed value. Trailing stops serve a similar purpose, but contrary to stop-loss orders, they automatically follow price move-ments, e.g. the asset’s best historical performance. Even though such strategies sell a “losing” investmentwithout guaranteeing that this is better than holding onto the assets, they are frequently used in practicemainly due to investors’ anxiety of incurring further losses. Besides their aforementioned purpose, placingtrailing stop orders has the additional benefit of allowing investors to focus on multiple open positions atthe same time, thanks to their special self-adjustment feature (see [11] for a study of trailing stop strategies,their optimal value, duration and distribution of gains). Recently, the use of these strategies was also studiedin various mathematical frameworks. For instance, the optimal combination of an up-crossing target priceand a stop-loss, chosen from a specific given set, was studied in [36] under a switching geometric Brownian ∗ Corresponding author. See [10] for a recent empirical study stressing the importance of stop-loss orders and their usefulness for reducing investors’disposition effect. 1 imsart-generic ver. 2014/10/16 file: DDOmega_2020-05-19.tex date: June 2, 2020 a r X i v : . [ q -f i n . M F ] M a y odosthenous and Zhang/Selling amid anxiety about drawdowns motion model. An investigation in [17] under a drifted Brownian motion showed that the possibility of anegative drift also suggests the use of combinations of such strategies. Moreover, buying and selling strategieswith exogenous trailing stops were considered in conjunction with take-profit orders in [22] under diffusionmodels. In practice, such stop-loss and trailing stop orders are available for use on stock, option and futuresexchanges.Although trailing stops and stop-loss orders have been widely used, there is no quantitative model thatcan explain the rationale behind such practices. In all aforementioned studies [36, 17, 22, 37], the use oftrailing stops is exogenously imposed. On the other hand, Russian options and their extensions (see e.g. [34])involve the use of trailing stops, but as in regret theory, their objective is to protect against drawdowns, andis not concerned about the utility realised from an asset sale. The purpose of this paper is thus to providea framework that rationalises the use of these types of orders from the perspective of selling an asset. Inparticular, we neither impose any exogenous hard constraints on the set of selling strategies, nor do we havea reward that involves the running maximum of the asset’s price. In such case, a traditional take-profit saleis usually optimal. However, we demonstrate in the present paper that, the use of stop-loss and trailingstop strategies naturally arises from the growing “anxiety” of the investor, as the asset price’s performanceremains at undesirable levels. A decision making process affected by (asset price) path-dependence can alsobe found, via the so-called history-dependent risk aversion models, in [7] and references therein. Contraryto our work, the path-dependence affects directly the risk aversion in these models, e.g. it is incorporatedin a state-dependent risk aversion coefficient ρ of utility functions as in (1.4). Our model can be seen asan expansion of this class of path-dependent risk aversion studies in the novel direction that we rigorouslypresent in the sequel.To fix ideas, consider a financial market with a risky asset whose price process e X is modelled by aspectrally negative L´evy process X = ( X t ) t ≥ on a filtered probability space (Ω , F , F = ( F t ) t ≥ , P ). Empiricalevidence suggests that such a financial market model, allowing for negative asset price jumps, is appropriatein various settings, such as equity, fixed income and credit risk (see [3] and [26] among others). In order tomodel the aforementioned anxiety of investors, consider also the best performance of the asset e X , where X = ( X t ) t ≥ is the running maximum process associated with X , given by X t = s ∨ sup u ∈ [0 ,t ] X u . Namely,the best performance until time t is the maximum between the highest price of the asset during the timeinterval [0 , t ] and the constant s ∈ R . The latter represents the “starting maximum” of the asset price andcan be interpreted as the highest asset price over some previous time period ( − t , t > q >
0, we constructan (impatience) Omega clock, which measures the amount of time that X is below its running maximum X by a pre-specified level c > (cid:36) ct := q (cid:90) t { X u
0, by U ( u ) := (cid:40) u − ρ − − ρ , if ρ ∈ [0 , , log( u ) , if ρ = 1 . (1.4) imsart-generic ver. 2014/10/16 file: DDOmega_2020-05-19.tex date: June 2, 2020 odosthenous and Zhang/Selling amid anxiety about drawdowns Note that, when ρ = 0, by scaling K · U ( uK ) = u − K we also obtain the revenue from selling the asset withprice u = e X net of the transaction cost K > q implies a strong penalisation of each unit of time, that the asset price’sdrawdown breaches the investor’s tolerance level c . This will naturally demand selling the asset sooner,and in some occasions, this sale may even happen with a loss, if its price does not go up quickly. Indeed,our analysis captures this phenomenon by proving that traditional stop-loss type strategies are optimaleither when investors have a high tolerance level c for drawdowns, but severe anxiety when these assetprice drawdowns occur, see Theorem 2.4 (also Figure 3); or when investors have severe anxiety and a lowdrawdown tolerance, see Theorem 2.6 (also Figures 4). Furthermore, in the latter class of investors withsevere anxiety even about small drawdowns, we prove that trailing stop type strategies also become partof the optimal strategy, see Theorem 2.6 (also Figures 4–6). Overall, through the study of problem (1.2),this paper manages to answer the following (qualitative and quantitative) question: “what are the individualtolerance and anxiety characteristics of an investor that may result in an optimal use of trailing stop andstop-loss type strategies?” . Our results on the optimal selling strategy are robust with respect to the choice ofutility function. Their qualitative nature remains the same and they only change quantitatively when tuningthe risk aversion coefficient ρ . On the other hand, irrespective of the risk aversion of investors, the absenceof severe anxiety always produces take-profit selling strategies (see Section 2.1). Thus, our novel results aremainly driven by anxiety, a risk factor which is not captured by traditional risk aversion.Drawdown is widely used as a path-dependent risk indicator (see, e.g. [37]) and has been often used as aconstraint for portfolio optimisation. The growth optimal portfolio under exogenous drawdown constraintsis explicitly constructed for diffusion models in [12, 5]. Their strategy entails continuous buying and sellingof risky assets, in order to meet the hard constraint on drawdowns. In particular, the portfolio will be 100%invested in the risk-free asset once the drawdown reaches their pre-specified tolerance level. In contrast, ourframework only imposes a soft drawdown constraint, in that breaching the investor’s drawdown tolerancedoes not automatically trigger a sale of the risky asset. Additionally, our investment is irreversible andindivisible, thus the investor’s objective is to find the optimal timing of a sale instead of the continuousrebalancing of the holding position.The irreversible sale of a real asset which is indivisible, at a time chosen by the investor, is a classicaltopic in the optimal stopping literature. Optimal timing of an asset sale such that an expected utility ismaximised was studied in [14] under an exponential utility and in [15] under a CRRA utility (see also [16])in a diffusion framework. The problem under a risk neutral utility, which is simply given by the revenue fromthe sale, namely the value of the asset e X τ net of the transaction cost K at the selling time τ , was studied in[27] in a general L´evy model. The optimal sale of an asset under a utility given by the asset price scaled byits running maximum was studied in [8] in a geometric Brownian motion model. All aforementioned studiesare performed either with a constant discount rate or without discounting. The study of the above optimalstopping problem with a random (stochastic path-dependent) discount rate and exponential L´evy asset pricemodels was developed in [35]. There, a risk neutral utility U (e x ) = e x − K was considered, in a simplifiedversion of the problem formulated in (1.2), namely v ( x ; y ) := sup τ ∈T E x [e − A yτ U (e X τ ) { τ< ∞} ] , ∀ x, y ∈ R , (1.5)where instead of the original Omega clock (1.1), a “level Omega clock” was used to define A y , given by theoccupation time A yt = rt + q (cid:90) t { X u 2. Model and main results We consider a filtered probability space (Ω , F , F = ( F t ) t ≥ , P ) on which we define the logarithm of the assetprice (log price) X = ( X t ) t ≥ to be a spectrally negative L´evy process. Here F is the augmented naturalfiltration of X . We denote by ( µ, σ , Π) the L´evy triplet of X , and by ψ its Laplace exponent, i.e. ψ ( β ) := log E [e βX ] = µβ + 12 σ β + (cid:90) −∞ (cid:0) e βx − − βx { x> − } (cid:1) Π(d x ) , for every β ∈ H + ≡ { z ∈ C : (cid:60) z ≥ } . Here, the L´evy measure Π(d x ) is supported on ( −∞ , 0) satisfying (cid:82) −∞ (1 ∧ x )Π(d x ) < ∞ , is atom-less and has a (weakly) monotone density π ( · ), i.e. π ( − x ) is non-increasingin x > −∞ , − x ) = (cid:90) − x −∞ π ( u )d u, ∀ x > . This condition is weaker than the standard one of the tail measure Π( −∞ , − x ) either having a completelymonotone density or being log-convex. Examples of processes that satisfy the condition include spectrallynegative α -stable process, spectrally negative CGMY model and spectrally negative hyper-exponential model.We assume that the discount rate r > ψ (1), which is equivalent to the discounted asset price (e − rt + X t ) t ≥ being a super-martingale. For any given r ≥ 0, the equation ψ ( β ) = r has at least one positive solution, andwe denote the largest one by Φ( r ). Notice that r > ψ (1) implies that Φ( r ) > 1. Finally, the r -scale function W ( r ) : R (cid:55)→ [0 , ∞ ) is a function vanishing on ( −∞ , , ∞ ), with a Laplace transform givenby (cid:90) ∞ e − βx W ( r ) ( x )d x = 1 ψ ( β ) − r , for β > Φ( r ) . We assume that W ( r ) ( · ) ∈ C (0 , ∞ ) for all r ≥ 0, which is guaranteed if σ > The r -scale function W ( r ) ( · ) is closely related to the first passage times of the spectrally negative However, σ > W ( r ) ( · ) ∈ C (0 , ∞ ). For instance, a spectrally negative α -stable processwith α ∈ (1 , 2) satisfies this condition without a Gaussian component. imsart-generic ver. 2014/10/16 file: DDOmega_2020-05-19.tex date: June 2, 2020 odosthenous and Zhang/Selling amid anxiety about drawdowns L´evy process X , which are defined by T ± x := inf { t ≥ X t ≷ x } , x ∈ R . In Appendix A we list a handful of useful properties and identities of r -scale functions. As seen in [35], the optimal selling strategy, for problem (1.5) with a risk neutral utility, is largely affected bythe investor’s own anxiety rate q . To extend the arguments there to a general CRRA utility and eventuallyto our objective (1.2), we introduce the concept of mild anxiety and severe anxiety , after conducting somepreliminary analysis of problem (1.5).We focus on the case of ρ ∈ [0 , ρ ↑ q = 0. In this case, we always discount at rate r > 0. Given that z (cid:55)→ U (e z ) − r ) dd z U (e z ) = e (1 − ρ ) z (cid:18) − ρ − r ) (cid:19) − − ρ (2.1)is strictly increasing over R , we know from [25, Theorem 2.2] (see also footnote 4) that problem (1.5) is solvedby a take-profit (up-crossing) selling strategy when the log price reaches the target − ρ log( Φ( r )Φ( r ) − ρ ) > . With anxiety, i.e. q > 0, we consider the function g ( · ) given by g ( x ) = e (1 − ρ ) x (cid:18) − ρ − x ) (cid:19) with Λ( x ) := dd x log I ( r,q ) ( x ) , (2.2)where I ( r,q ) ( · ) is defined by I ( r,q ) ( x ) := (cid:90) ∞ e − Φ( r + q ) u W ( r ) ( u + x )d u, ∀ x ∈ R . (2.3)The function g ( · ) is continuous everywhere with one possible discontinuity at 0. It is known from [35, Lemma4.2] that Λ( · ) is strictly decreasing over R + , with limits Λ(0+) ≤ Φ( r + q ) and Λ( ∞ ) = Φ( r ). Moreover,following similar analysis to [35], we know that g ( · ) is strictly increasing over ( −∞ , 0) with a lower limit g ( −∞ ) = 0, and is ultimately increasing over [ u, ∞ ) for u ≥ g ( ∞ ) = ∞ . Thus, one can unambiguously define (the largest local minimum of g ) u := inf { u ∈ R : g ( · ) is non-decreasing over [ u, ∞ ) } . Notably, constant u is either non-negative, or −∞ .In case u = −∞ , the function g ( · ) is non-decreasing over R , so for any fixed y ∈ R , the mapping z (cid:55)→ U (e z ) − z − y ) dd z U (e z ) = e (1 − ρ ) y g ( z − y ) − − ρ , ∀ z ∈ R (2.4)is also non-decreasing. By [25, Theorem 2.2] and Lemma A.2 (see also footnote 4), we immediately knowthat (1.5) is also solved by a take-profit selling strategy, regardless of the risk tolerance level y . In this case,the selling target log price is given by z (cid:63) ( y ) := y + inf (cid:110) u > u : g ( u ) > e − (1 − ρ ) y / (1 − ρ ) (cid:111) . (2.5) Notice also that the assumption r > ψ (1) implies0 ≤ lim z →∞ E x [e − A yT + z U ( X T + z ) { T + z < ∞} ] ≤ lim z →∞ E x [e − rT + z U ( X T + z ) { T + z < ∞} ] = lim z →∞ U ( z )e − Φ( r ) z = 0 . The definition of z (cid:63) ( y ) holds for any value of u . imsart-generic ver. 2014/10/16 file: DDOmega_2020-05-19.tex date: June 2, 2020 odosthenous and Zhang/Selling amid anxiety about drawdowns q - - - Fig 1: Plot of H (cid:63) as a function of q . Here, we consider a risk neutral investor (i.e. ρ = 0) with a discountrate r = 0 . 18 and the Laplace exponent used is ψ ( β ) = 0 . β + 0 . β − . ββ +4 ) (so the jump distributionis exponential).Equivalently, z (cid:63) ( y ) is the largest root over ( y + u, ∞ ), which is simply R when u = −∞ , to equation g ( z − y ) = 11 − ρ e − (1 − ρ ) y . (2.6)Because of the qualitative similarity of this type of optimal strategy with that under no anxiety (i.e. q = 0),we henceforth refer to the case of u = −∞ as the case of mild anxiety .The remaining case, when u ≥ 0, is referred to as the case of severe anxiety . In this case, contrary to theprevious mild one, an investor may choose an additional stop-loss type strategy to “cut the loss” dependingon the risk tolerance level y , which is a phenomenon already documented in [35] under a risk neutral utility .Furthermore, the representation of candidate threshold z (cid:63) ( y ) as the largest root to (2.6) does not alwayshold under severe anxiety. Specifically, only if y ≤ y := − − ρ log (cid:0) (1 − ρ ) g ( u ) (cid:1) , (2.7)can we identify z (cid:63) ( y ) as the largest root over ( y + u, ∞ ) to (2.6). If y > y , z (cid:63) ( y ) of (2.5) is simply equal to y + u .We close this subsection by providing a convenient criterion that distinguishes severe from mild anxiety:when the tail jump measure of the L´evy process, denoted by Π( x ) := Π( −∞ , − x ) (for x > u = −∞ holds ⇔ H (cid:63) := (Φ( r + q ) − ρ )(Φ( r + q ) − qW ( r ) (0)) − qW ( r ) (cid:48) (0+) ≥ . (2.8)Figure 1 plots H (cid:63) as a function of q , illustrating the relationship between mild anxiety ( u = −∞ ) and small q , as well as severe anxiety ( u ≥ 0) and large q . In this section we present our main result, the value function and the optimal selling strategy for problem(1.2), when the investor has either mild (i.e. u = −∞ ) or severe (i.e. u ≥ 0) anxiety.To begin, we note that the optimal stopping of reward U (e X ) with a constant discounting rate r + q , issolved by a take-profit selling strategy with target log price b = 11 − ρ log (cid:16) Φ( r + q )Φ( r + q ) − ρ (cid:17) > , (2.9) We shall see that such distinction still exists for our generalised problem (1.5) (see Section 4) and our main objective (1.2)(see main results in Section 2.2 below). imsart-generic ver. 2014/10/16 file: DDOmega_2020-05-19.tex date: June 2, 2020 odosthenous and Zhang/Selling amid anxiety about drawdowns Fig 2: Illustration of the optimal stopping region under mild anxiety ( u = −∞ ). Model parameters: c =0 . , r = 0 . , q = 0 . 003 and Laplace exponent used: ψ ( β ) = 0 . β +0 . β − . ββ +4 ). The investor solvesa risk-neutral sale with transaction cost K = 10. Here, we have H (cid:63) = 0 . , b = 4 . z c = 4 . , 3) in the figure.and the value function is given by v ( x ) := sup τ ∈T E x [e − ( r + q ) τ U (e X τ ) { τ< ∞} ] = { x ≤ b } e Φ( r + q )( x − b ) U (e b ) + { x>b } U (e x ) . (2.10)Function v ( · ) is smooth everywhere off the set { b } , and is continuously differentiable over R . These resultsfollow from [25, Theorem 2.2] together with the increasing property of mapping (2.1) as we replace discountrate r by r + q .Define also the log price threshold z c := 11 − ρ log (cid:18) Λ( c )Λ( c ) − ρ (cid:19) . (2.11)By the monotonicity of Λ( · ) over R + , we know that z c > b . In the first result of this section, we present the properties of two types of take-profit sale targets e z (cid:63) ( · ) from (2.5) or e z c from (2.11), which can be attained either before or after the asset price improves its bestperformance e s , respectively. Lemma 2.1. If u = −∞ , then(i). the function z (cid:63) ( · ) of (2.5) is continuous and strictly decreasing over ( −∞ , b ] , and z (cid:63) ( y ) ≡ b for all y ≥ b ;(ii). the log price z c from (2.11) satisfies z (cid:63) ( z c − c ) = z c and z c < b + c . The optimality of the above selling strategies is given in the following result and is proved in Section5.1 via the use of variational inequalities (see the beginning of Section 5) and the results obtained in thesubsequent Sections 3 and 4.1. imsart-generic ver. 2014/10/16 file: DDOmega_2020-05-19.tex date: June 2, 2020 odosthenous and Zhang/Selling amid anxiety about drawdowns Theorem 2.2. For an investor with mild anxiety (i.e. u = −∞ ), the value function for problem (1.2) isgiven by V ( x, s ; c ) = e − Λ( c )( z c − s ) I ( r,q ) ( x − s + c ) I ( r,q ) ( c ) U (e z c ) , if s < z c ; { x 0, the structure of the optimal selling strategy for problem (1.2) changes as theinvestor’s tolerance level c varies. In order to define the critical regions of c -values, we firstly need to specifytwo values (cid:98) y and (cid:101) y that are closely related to variational inequalities and martingale methods associated tothe optimality of take-profit selling strategies. To formalise the following results, consider the function χ ( x ) := r − ρ − r − ψ (1 − ρ )1 − ρ e (1 − ρ ) x + (cid:90) b − x −∞ (cid:0) v ( x + w ) − U (e x + w ) (cid:1) Π(d w ) , ∀ x ≥ b. (2.13)Following similar analysis to [35], one can show that χ ( · ) is continuous and strictly decreasing over [ b, ∞ ),and satisfies χ ( ∞ ) = −∞ . Thus, we can define (cid:98) y := inf { y ≥ b : χ ( y ) ≤ } . (2.14)Moreover, we define another critical log price threshold (cid:101) y , given by (cid:101) y := inf (cid:26) y ≤ y : sup x For an investor with severe anxiety (i.e. u ≥ ) and a high drawdown tolerance c ≥ (cid:101) c , thevalue function for problem (1.2) is given by V ( x, s ; c ) = e − Λ( c )( z c − s ) I ( r,q ) ( x − s + c ) I ( r,q ) ( c ) U (e z c ) , if s < z c ; { x 0) and high drawdowntolerance ( c ≥ (cid:101) c ). Model parameters: c = 1 . , r = 0 . , q = 1 and Laplace exponent used: ψ ( β ) =0 . β + 0 . β − . ββ +4 ). The investor solves a risk-neutral sale with transaction cost K = 10. Herewe have H (cid:63) = − . , b = 2 . , (cid:101) c = 1 . , (cid:101) y = 2 . , (cid:98) y = 4 . z c = 4 . , 2) in the figure. Lemma 2.5. Let ∆( · , a ; y ) and a (cid:63) ( · ) be defined by (2.17) – (2.18) and f ( x ) := (cid:82) b − x −∞ ( v ( x + w ) − U (e x + w ))Π(d w ) .Then, there exists a unique solution a ( · ) to the first order non-linear ODE a (cid:48) ( s ) = qW ( r ) ( c ) W ( r,q ) ( s, a ( s ); s − c ) (1 − ρ ) (cid:0) v ( s − c ) + ∆( s − c, a ( s ); s − c ) (cid:1) ( r + q − ψ (1 − ρ ))e (1 − ρ ) a ( s ) − ( r + q ) − (1 − ρ ) f ( a ( s )) , ∀ s ≤ b (cid:63) ( y c ) ,a ( b (cid:63) ( y c )) = a (cid:63) ( y c ) , (2.22) which can be extended smoothly for s ≤ b (cid:63) ( y c ) as long as a ( s ) ≥ b . Moreover, there exists a unique s c ∈ ( b + c, b (cid:63) ( y c )) such that a ( s c ) = b and we have a (cid:48) ( s ) > , a ( s ) < s − c, ∆( x, a ( s ); s − c ) > , ∀ ( x, s ) ∈ O + s.t. a ( s ) < x ≤ s − c and s c ≤ s ≤ b (cid:63) ( y c ) . The optimality of the selling strategies presented in Lemma 2.3 together with the aforementioned trailingstop type sale target, is given below for this class of investors. This is proved in Section 5.2 via the use ofvariational inequalities (see the beginning of Section 5) and the results obtained in the subsequent Sections3 and 4.2. Theorem 2.6. For an investor with severe anxiety (i.e. u ≥ ) and a low drawdown tolerance c < (cid:101) c , thevalue function for problem (1.2) is given by V ( x, s ; c ) = e − Λ( c )( s c − s ) I ( r,q ) ( x − s + c ) I ( r,q ) ( c ) V ( s c , s c ; c ) , if s < s c ; v ( x ) + ∆( x, a ( s ); s − c ) { x ≥ a ( s ) } , if s c ≤ s < b (cid:63) ( y c ); v ( x ) + { a (cid:63) ( s − c ) 0) and low drawdown tolerance( c < (cid:101) c ). Model parameters: c = 0 . , r = 0 . , q = 1 and Laplace exponent used: ψ ( β ) = 0 . β +0 . β − . ββ +4 ). The investor solves a risk-neutral sale with transaction cost K = 10. Here we have H (cid:63) = − . , b = 2 . , (cid:101) c = 1 . , (cid:98) y = 4 . , s c = 3 . b (cid:63) ( y c ) = 4 . , 2) in the figure. The optimal selling region (see Figure 4 for an illustration) is S c = S c ∪ S c ∪ S c , where S c = { ( x, s ) ∈ O + : b ≤ x ≤ a ( s ) and s c ≤ s < b (cid:63) ( y c ) } , S c = { ( x, s ) ∈ O + : b ≤ x ≤ a (cid:63) ( s − c ) or x ≥ b (cid:63) ( s − c ) , and b (cid:63) ( y c ) ≤ s < (cid:98) y + c } , S c = { ( x, s ) ∈ O + : x ≥ b and s ≥ (cid:98) y + c } . Contrary to Theorems 2.2 and 2.4, we observe that the regions of the value function communicate inTheorem 2.6. If an investor with severe anxiety has a low tolerance for drawdowns, then the optimal sellingstrategy may involve some holding period of no trade, and some period when a fixed take-profit target is setup together with a sequence of protective, adaptive stop-loss orders. To be more precise:(i) When the starting maximum log price s is lower than s c , the investor should hold onto the asset untilits log price reaches s c (no trade region).(ii) If the maximum log price X is at least s c , but lower than b (cid:63) ( y c ), the investor should set up a take-profittarget log price b (cid:63) ( y c ), while also consider to (optimally) sell the asset when the log price X jumpsdown to the interval [ b, a ( X )]. The latter strategy is precisely a generalised trailing stop, where thestochastic floor increases along with the running maximum X (see also [22]).(iii) If the trailing stop order is not activated as X continuously increases towards the take-profit target logprice b (cid:63) ( y c ), it will be optimal to sell the asset once X reaches b (cid:63) ( y c ).(iv) As an independent case, not communicating with the aforementioned ones (i)–(iii), when the startingmaximum log price s is higher than b (cid:63) ( y c ), we obtain similar economic insights as in the severe anxietywith high drawdown tolerance, since the optimal selling strategy is realised either at a traditionaltake-profit sale, before a new maximum is established, or at a stop-loss type order. Remark 2.7. It is worth mentioning that, when setting a trailing stop type order, there is a possibilitythat the asset log price jumps downwards from the interval ( a ( s ) , s ] to the interval ( −∞ , b ) . In this case, weproved that it is optimal for the investor not to sell the asset, but rather be patient by waiting until T + b to sell(this is also true for all aforementioned cases under severe anxiety that require the use of a stop-loss typestrategy). This reflects the additional protection sought by investors in financial markets through a trailing imsart-generic ver. 2014/10/16 file: DDOmega_2020-05-19.tex date: June 2, 2020 odosthenous and Zhang/Selling amid anxiety about drawdowns stopping region a t X t - cX t Fig 5: A simulated sample path under parameters: c = 0 . r = 0 . , q = 1, Laplace exponent ψ ( β ) =0 . β + 0 . β − . ββ +4 ), and initial value X = X = 2. The investor solves a risk-neutral sale withtransaction cost K = 10. In the figure, a t = a ( X t ) is the trailing stop threshold, which is set up once X reaches s c = 3 . b (cid:63) ( y c ) = 4 . . 949 and t = 6 . X t a t stopping region X t - c Fig 6: A simulated sample path under parameters: c = 0 . r = 0 . , q = 1, Laplace exponent ψ ( β ) =0 . β + 0 . β − . ββ +4 ), and initial value X = X = 2. The investor solves a risk-neutral sale withtransaction cost K = 10. In the figure, a t = a ( X t ) is the trailing stop threshold, which is set up once X reaches s c = 3 . b (cid:63) ( y c ) = 4 . t = 11 . imsart-generic ver. 2014/10/16 file: DDOmega_2020-05-19.tex date: June 2, 2020 odosthenous and Zhang/Selling amid anxiety about drawdowns stop (resp., stop-loss) with a limit. The purpose is to secure a price only when the asset price experiences adrawdown from its peak that crosses the trailing stop (resp., stop-loss) threshold but not the limit. We canfurther interpret this result as missing out on selling the asset after a relatively big price jump, since theasset has already lost enough value that it “costs” nothing to wait for some more time until the log pricerecovers to b . In order to illustrate the optimal selling strategy proposed in Theorem 2.6, we present two numericalcase studies for identical assets: In Figure 5, the asset is optimally sold at time t = 6 . a ( X t ) = 2 . x = 2; in Figure 6, the trailing stop is never activated (even though itis set up at some point), so the investor sells the asset at the take-profit threshold b (cid:63) ( y c ) = 4 . Remark 2.8. We comment here that the rationale behind the consideration of the ODE (2.22) is the im-position of Neumann condition on the value of the optimal selling strategy at the diagonal ∂ O + of thetwo-dimensional state space. Different from the majority of existing literature on optimal stopping problemsinvolving the maximum process, we do have a boundary condition at s = b (cid:63) ( y c ) for the ODE, which helps usto obtain a unique solution a ( · ) as the candidate down-crossing sell order. When such a boundary conditionis not available, an appropriate (unique) candidate must be chosen from the set of infinitely many solutionsof the ODE, by relying on various different methods; e.g. using the transversality condition (see [13], [34],among others), or the maximality principle from [32] (see [28] for diffusion models, [21] for L´evy models,among others). The remaining Sections 3–5 are devoted to proving the main results of the paper that have been presentedin this section. Given that the structure of the stopping region is explicitly determined by model parameters via equationsof differentiable functions, we know that the optimal stopping boundaries are continuous in c and q . It iseasy to see that the set of optimal selling regions shrinks with the investor’s tolerance level c for asset pricedrawdowns and increases with the anxiety rate q about these drawdowns. Namely, when c decreases or when q increases, investors should become more proactive and (optimally) sell their asset at lower profit-takingand/or higher stop-loss/trailing stop targets. These results can also be proved independently of the theorydeveloped in Section 2.2, directly via the expression of the value function in (1.2). In particular, the resultfor c is given in Proposition 3.1 in Section 3.1 below, while one can similarly prove the monotonicity withrespect to q from (1.3) and (1.1), by observing that q (cid:55)→ R ct ( q ) is non-decreasing. We can further prove that the set of optimal selling strategies for investors with mild anxiety (cf. Theorem2.2 in Section 2.2.1) is strictly increasing with the investors’ risk aversion coefficient ρ . In particular, we provein Lemma B.1 that the mappings ρ (cid:55)→ b ( ρ ), ρ (cid:55)→ z (cid:63) ( y ; ρ ) and ρ (cid:55)→ z c ( ρ ) are all strictly decreasing, wheneverthey are defined (cf. Section 2.2). Therefore, more risk averse investors should be more proactive and(optimally) sell their asset at lower profit-taking targets.Our analytical results in Section 2.2 allow also for a numerical study of comparative statics with respectto general model parameter configurations, including cases of severe anxiety. To set up a numerical studyfor comparative statics of the risk-aversion coefficient ρ , volatility σ and jump distribution intensity η , weconsider an asset with log price process given by a compound Poisson jump process plus a Brownian motionwith drift. Namely, we consider an asset price model with Laplace exponent ψ ( β ) = 0 . β + 12 σ β − . ββ + η (2.24)within seven model parameter configurations, as shown in Table 1 below.We illustrate in Figure 7 the optimal stopping regions for investors with severe anxiety and low drawdowntolerance (cf. Theorem 2.6), when perturbing their risk aversion coefficient ρ , the log price’s volatility σ , or Here, we used the notation R ct = R ct ( q ) to stress the dependence of the Omega clock on the parameter q . Here, we used the notation b = b ( ρ ), z (cid:63) ( y ) = z (cid:63) ( y ; ρ ) and z c = z c ( ρ ) to stress the dependence of these take-profit sellingtargets on the parameter ρ . imsart-generic ver. 2014/10/16 file: DDOmega_2020-05-19.tex date: June 2, 2020 odosthenous and Zhang/Selling amid anxiety about drawdowns σ η r q ρ c Benchmark 0.2 4 0.18 1 0.25 0.3568Smaller ρ ρ . σ . σ . η η Table 1 Configurations of model parameters for comparative statics Configuration b b (cid:63) ( y c ) s c b (cid:63) ( (cid:98) y )Benchmark 0.2277 1.2793 0.7331 1.5593Smaller ρ ρ σ σ η η Table 2 Key selling thresholds for the problem configurations in Table 1 the intensity η of the jumps’ exponential distribution according to the configurations in Table 1. The keyselling thresholds for each case are also shown in Table 2 complementing the illustrations in Figure 7.One can observe from Figure 7 and Table 2 that: ( a ) the set of selling strategies appears to grow (resp.,shrink) with respect to ρ (resp., σ and η ) also in the case of severe anxiety; ( b ) the changes in the trailingstop a ( s ), stop-loss a (cid:63) ( s ) and take-profit b targets, whenever they exist for each fixed s > 0, are relativelyminor under all parameters examined, compared to the resulting changes in the take-profit target b (cid:63) ( y c ),which are much more significant. In summary,1. for both degrees of anxiety, the more risk averse investors tend to be more proactive and (optimally)sell their asset at lower profit-taking and/or higher stop-loss (or trailing stop) targets.2. in cases of severe anxiety, the more volatile the asset price, the longer investors wait before (optimally)selling their asset, as it is widely understood that options tend to gain value when volatility increases;3. in cases of severe anxiety, the higher the intensity of the exponential distribution, the smaller the sizeof negative jumps tends to be, hence the investors wait longer before (optimally) selling their asset. 3. Bounds for the value function V ( x, s ; c ) Fixing any tolerance level c > 0, we denote by S c the “stopping region” of log prices. This contains all thestates of price and maximum price at which the investor should (optimally) sell the asset. By the generaltheory of optimal stopping for Markov processes (see, e.g. [33, Ch. I, Sec. 2.2]), we define S c := { ( x, s ) ∈ O + : V ( x, s ; c ) − U (e x ) = 0 } . As the first step of determining the value function V ( · , · ; c ) of (1.2), we derive a lower and an upper bound.We also provide bounds for the stopping region S c , which will be useful in proofs in later sections. V ( x, s ; c ) and upper bound for S c For any c (cid:48) > c > 0, it is easily seen from (1.3) and (1.1) that R c (cid:48) t ≤ R ct ≤ R t ≤ ( r + q ) t holds for all t ≥ v ( x ), the optimal expected value of utility U (e X ) discounted at rate r + q > Proposition 3.1. For any c (cid:48) > c > , v ( x ) ≤ V ( x, s ; c ) ≤ V ( x, s ; c (cid:48) ) , ∀ ( x, s ) ∈ O + , imsart-generic ver. 2014/10/16 file: DDOmega_2020-05-19.tex date: June 2, 2020 odosthenous and Zhang/Selling amid anxiety about drawdowns and S c ⊆ [ b, ∞ ) ∩ O + . Proof. For any fixed ( x, s ) ∈ O + , c (cid:48) > c > τ ∈ T , we have on the event { τ < ∞} that, the following inequalitiese − ( r + q ) τ U (e X τ ) ≤ e − R cτ U (e X τ ) ≤ e − R c (cid:48) τ U (e X τ )hold true ( P x,s -a.s.). Taking expectations E x,s and the suprema over all stopping times τ , we obtain inview of the definition (2.10) of v that v ( x ) ≤ V ( x, s ; c ) ≤ V ( x, s ; c (cid:48) ) . The bound for S c follows immediately.As a result of Proposition 3.1, we may equivalently express our objective (1.2) as V ( x, s ; c ) = sup τ ∈T E x,s [e − R cτ v ( X τ ) { τ< ∞} ] , ∀ ( x, s ) ∈ O + . (3.1)This will be a more convenient form of our problem in some parts of the forthcoming analysis (cf. Section5.2). V ( x, s ; c ) and lower bound for S c Consider the value function v ( x ; y ) of (1.5), i.e. optimal expected value of utility U (e X ) with discounting A y . We define the optimal stopping region of problem (1.5), (see, e.g. [33, Ch. I, Sec. 2.2]) by D y := { x ∈ R : v ( x ; y ) − U (e x ) = 0 } , ∀ y ∈ R . (3.2)Problem (1.5) can be considered as a simplified version of the original problem (1.2), when the discountfactor does not update with the running maximum X . Specifically, for any fixed c > x, s ) ∈ O + ,the continuous additive functional A s − c is almost surely dominated from above by R c under P x,s . Thus, thevalue function of (1.2) is always bounded from above by the value of (1.5) when y = s − c . Proposition 3.2. For any fixed c ≥ , we have V ( x, s ; c ) ≤ v ( x ; s − c ) , ∀ ( x, s ) ∈ O + . (3.3) Then, the optimal stopping region D y defined by (3.2) satisfies ( D s − c × { s } ) ∩ O + ⊆ S c . (3.4) Moreover, the equalities in (3.3) and (3.4) hold if and only if [ s, ∞ ) ⊆ D s − c .Proof. We only need to prove that the equality in (3.3) holds if and only if [ s, ∞ ) ⊆ D s − c . But the lattercondition means that it is optimal to stop in problem (1.5) before the asset log price X reaches s . This isequivalent to the running maximum X remaining constant, equal to s , in which case the value of the samestrategy for problem (1.2) under P x,s is the same as v ( x ; s − c ). Therefore, the optimal value V ( x, s ; c ) forproblem (1.2) is no less than v ( x ; s − c ). This completes the proof.In the next section, we focus on solving for the upper bound v ( x ; y ). Note that the s -component is actually redundant when the objective function does not involve the running maximum. imsart-generic ver. 2014/10/16 file: DDOmega_2020-05-19.tex date: June 2, 2020 odosthenous and Zhang/Selling amid anxiety about drawdowns 4. Cracking problem (1.5) with value function v ( x ; y ) This section is concerned with the study of problem (1.5), which servers as a cornerstone in the analysisof problem (1.2). Recall that, the case of risk neutral utility has already been treated in [35, Theorem 2.4,Theorem 2.5]. Proposition 4.1. The value function of (1.5) satisfies the following properties:(i) v ( x ; y ) is strictly increasing and continuous in x over R , and is non-increasing and continuous in y over R ;(ii) if there exists a constant a ∈ D y ∩ ( −∞ , y ] , then y ≥ b and [ b, a ] ⊂ D y ;(iii) the optimal stopping region D y is a union of disjoint closed intervals and there is at most one componentthat lies in ( y, ∞ ) .Proof. For any ρ ∈ [0 , u − ρ − − ρ (cid:12)(cid:12)(cid:12)(cid:12) u =1 = 0 , ∂∂u (cid:12)(cid:12)(cid:12)(cid:12) u =1 (cid:18) u − ρ − − ρ (cid:19) = 1 , and ∂∂ρ ∂∂u (cid:18) u − ρ − − ρ (cid:19) = − u − ρ log u < ∀ u > , that (cid:18) u − ρ − − ρ (cid:19) + ≤ ( u − + , ∀ u ∈ R + . Hence, by the dominated convergence theorem, one can repeat the steps used in the proof of [35, Proposition3.1] to prove ( i ) and ( ii ). The claim in the first half of ( iii ) follows from the fact that v ( x ; y ) is continuous in x over R ; the second half of the claim in ( iii ) can be proved in the same way as in [35, Proposition A.1].We recall that, if u = −∞ or u ≥ y < y , the candidate up-crossing selling threshold z (cid:63) ( y ) of(2.5), is actually the largest root to (2.6). By the monotone property of g we know that z (cid:63) ( y ) − y is strictlydecreasing. Moreover, Proposition 4.1(i) also implies that if D y = [ z (cid:63) ( y ) , ∞ ) for all y in some interval I withnonempty interior, then z (cid:63) ( y ) is continuous and non-increasing in y over I . Proposition 4.1(ii)–(iii) implythat there are three possibilities for the stopping region: (I) D y does not include b , or (II) D y = [ b, ∞ ), or(III) D y = [ b, a ] ∩ [ b, ∞ ) for some b ≤ a < y < b .In what follows, we study the problem (1.5) separately in the two cases of mild (i.e. u = −∞ ) and severe(i.e. u ≥ 0) anxiety (see Section 2.1). The following results generalise [35, Theorem 2.4, Theorem 2.5] to thecase of risk averse investors. Theorem 4.2. For an investor with mild anxiety (i.e. u = −∞ ), the optimal stopping region and the valuefunction for problem (1.5) are given by D y = [ z (cid:63) ( y ) , ∞ ) , and v ( x ; y ) = { x 0. Also, Proposition 4.1 and succeedingdiscussions imply the continuity and non-increasing property of z (cid:63) ( y ) over R .To prove the strictly decreasing property of z (cid:63) ( y ) over ( −∞ , b ], suppose that there exist y < y < b suchthat z (cid:63) ( y ) = z (cid:63) ( y ) ≥ b , aiming for a contradiction. Then, we must have v ( x ; y ) ≡ v ( x ; y ) for all x ∈ R .However, it is easily seen that A y T + z(cid:63) ( y < A y T + z(cid:63) ( y , P x − a.s. for any x > y ⇒ v ( x ; y ) > v ( x ; y ) for any x > y ,which is a contradiction. Hence, z (cid:63) ( y ) must be strictly decreasing over ( −∞ , b ]. imsart-generic ver. 2014/10/16 file: DDOmega_2020-05-19.tex date: June 2, 2020 odosthenous and Zhang/Selling amid anxiety about drawdowns Theorem 4.3. For an investor with severe anxiety (i.e. u ≥ ), we have b < (cid:101) y < y and (cid:101) y < (cid:98) y . The optimalstopping region and the value function for problem (1.5) are given as follows: (a) if y < (cid:101) y , then D y = [ z (cid:63) ( y ) , ∞ ) , while if y = (cid:101) y , then D y = { b } ∪ [ z (cid:63) ( y ) , ∞ ) . The value function v ( x ; y ) is given by (4.1) ; (b) if (cid:101) y < y < (cid:98) y , then D y = [ b, a (cid:63) ( y )] ∪ [ b (cid:63) ( y ) , ∞ ) , where a (cid:63) ( y ) and b (cid:63) ( y ) are defined in (2.17) , and v ( x ; y ) = v ( x ) + { a (cid:63) ( y ) 0, we have g ( u − (cid:15) ) > e ( ρ − y / (1 − ρ ), so ∂∂x D ( x ; y ) x = z (cid:63) ( y ) − (cid:15) < 0. This implies that D ( x ; y ) > D ( z (cid:63) ( y ); y ) = 1for all x in a sufficiently small left neighborhood of z (cid:63) ( y ). Hence, we conclude that (cid:101) y < y . Proof of part (a) . By the construction of (cid:101) y , we know that for any fixed y < (cid:101) y , D ( x ; y ) < , ∀ x < z (cid:63) ( y ) , which implies that the selling strategy T + z ∗ ( y ) is optimal for all y < (cid:101) y .If y = (cid:101) y , we may also conclude that the candidate value function v ( · ; (cid:101) y ) of (4.1) is the true value function.Moreover, by the above properties of (cid:101) y , we know that there exists a point x < u + (cid:101) y satisfying D ( x ; (cid:101) y ) = sup x b ,where L is the infinitesimal generator of X . Precisely, for all functions F ( · ) ∈ C ( R ), L F is given by L F ( x ) = 12 σ F (cid:48)(cid:48) ( x ) + µF (cid:48) ( x ) + (cid:90) −∞ ( F ( x + z ) − F ( x ) − { z> − } zF (cid:48) ( x ))Π(d z ) . Given that ( L − r ) v ( x ) = qv ( x ) > x < b , we know that (cid:98) y defined by (2.14) is the smallest y -valuesuch that v ( · ) is super-harmonic with respect to the discount rate r + q { x 5. Proofs of the main results In this section, we shall prove that the solution to the problem (1.2), takes the forms presented in Theorems2.2, 2.4 and 2.6. Our main verification approach is through the Hamilton-Jacobi-Bellman equation. Specif-ically, suppose that we can find a function w : O + → (0 , ∞ ) in C , ( O + ) ∩ C , ( O + \{ ( θ , s ) , . . . , ( θ k , s ) } )(resp., C , ( O + )) if X has paths of unbounded (resp., bounded) variation, for some θ , . . . , θ k ∈ R , such that w ( x, s ) ≥ U (e x ) and is super-harmonic. That is, w satisfies the variational inequalitymax (cid:8) ( L − r − q { x The first claim is already proved in Theorem 4.2, while the second claim followsstraightforwardly by definition (2.11) and (2.5)–(2.6). Proof of Theorem 2.2. Proof of part for s ≥ z c . We know from Theorem 4.2 under the case of u = −∞ , thatthe optimal stopping region D y for problem (1.5) is the half-line [ z (cid:63) ( y ) , ∞ ). On the other hand, Proposition3.2 asserts that if [ s, ∞ ) ⊆ D s − c ≡ [ z (cid:63) ( s − c ) , ∞ ) , then V ( x, s ; c ) = v ( x ; s − c ) . It is thus natural to consider the critical s -value, such that D s − c = [ s, ∞ ), namely, z (cid:63) ( s − c ) = s ⇔ m ( s − c ) = c , (5.4) imsart-generic ver. 2014/10/16 file: DDOmega_2020-05-19.tex date: June 2, 2020 odosthenous and Zhang/Selling amid anxiety about drawdowns where m ( · ) is defined by (5.3). By the construction of m ( · ) we know that ∃ ! y c ∈ ( −∞ , b ) that solves m ( y c ) = c . In fact, using (2.2) and (2.5), one can obtain an explicit expression for y c : y c = 11 − ρ log (cid:18) Λ( c )Λ( c ) − ρ (cid:19) − c. (5.5)Therefore, we can define z c as the s -value determined by (5.4), which is given by (2.11). Taking into accountthe definition and expression of z c in (2.11), we observe from the monotonicity of D y in y (cf. Theorem 4.2)that if s ≥ z c (or equivalently, s − c ≥ y c ) , then [ s, ∞ ) ⊆ [ z c , ∞ ) ≡ D y c ⊆ D s − c . Hence by Proposition 3.2, we know that V ( x ; s ; c ) ≡ v ( x ; s − c ) for any ( x, s ) ∈ O + such that s ≥ z c , where v admits the expression (4.1). The result follows by finally observing that, for all x ∈ [ z (cid:63) ( s − c ) , s ], we have ∂∂s (cid:12)(cid:12)(cid:12)(cid:12) x = s V ( x, s ; c ) = ∂∂s (cid:12)(cid:12)(cid:12)(cid:12) x = s U (e x ) = 0 , ∀ s > z c ⇒ V ( · , · ; s ) satisfies (5.2) for all s > z c . Proof of part for s < z c . For the remaining case, we prove that it is optimal to wait until the process( X, X ) reaches the point ( z c , z c ), i.e. to sell at T + z c . The value of such strategy, denoted by V ( x, s ; c ), is givenby V ( x, s ; c ) := E x,s (cid:104) exp( − R cT + zc ) U (cid:0) exp( X T + zc ) (cid:1) { T + zc < ∞} (cid:105) = E x,s (cid:104) exp( − R cT + s ) V ( s, s ; c ) { T + s < ∞} (cid:105) (5.6)= E x,s (cid:104) exp( − A s − cT + s ) V ( s, s ; c ) { T + s < ∞} (cid:105) (A.5) = I ( r,q ) ( x − s + c ) I ( r,q ) ( c ) V ( s, s ; c ) , for all x < s < z c , thanks to the strong Markov property of ( X, X ). Using the Neumann boundary condition ∂∂s | x = s V ( x ; s ; c ) =0, as we expect the process ( X, X ) to reflect at the diagonal of the state space ∂ O + until it reaches the point( z c , z c ), we obtain from (5.6) that V ( x, s ; c ) = e − Λ( c )( z c − s ) I ( r,q ) ( x − s + c ) I ( r,q ) ( c ) U (e z c ) , ∀ ( x, s ) ∈ O + s.t. s < z c . (5.7)Notice that, the positive function given by the right-hand side of (2.12) is precisely V ( x, s ; c ). Then byconstruction, it is easily seen that the mapping ( x, s ) (cid:55)→ V ( x, s ; c ) is continuous over O + , x (cid:55)→ V ( x, s ; c ) is C over ( −∞ , s ) for each fixed s , and it satisfies the Neumann condition (5.2) for all s < z c . In addition, onecan show that the Neumann condition also holds at s = z c by respectively computing the left and the rightderivative with respect to s . Hence, it remains to show that V solves the variational inequality (5.1) for all( x, s ) ∈ O + \{ ( s, s ) } .We first prove that the inequalities involving the infinitesimal generator hold. To this end, we fix x < s ≤ z c and by using the definition of V from (5.6) and similar arguments to Section 4 of [2], we know that( L − ( r + q { x Part ( i ), the detailed constructions of y , (cid:101) y and (cid:98) y and the inequalities in part ( ii ), aswell as part ( iv ) are already proved in Theorem 4.3. Part ( iii ) follows straightforwardly from definitions(2.11), (2.15) and (2.5)–(2.6). Proof of Theorem 2.4. Construction of (cid:101) c from (2.16) . Recall from Theorem 4.3, under u ≥ 0, that the opti-mal strategy for the simplified problem (1.5) changes qualitatively when the value y is less or greater thanthe key level (cid:101) y . Naturally, one may consider the tolerance level (cid:101) c associated with (cid:101) y through the definition (cid:101) c := m ( (cid:101) y ) , where m ( · ) is defined by (5.3) , (5.12)which is equivalent to the definition in (2.16). By the construction of m ( · ) we know that (cid:101) c is uniquely definedand positive. Proof of part for s ≥ z c . Following the same reasoning as in Section 5.1, we consider the critical s -valuesatisfying the properties in (5.4) and thus solving m ( s − c ) = c . By the construction of m ( · ), ∃ ! y c ∈ ( −∞ , (cid:98) y ] that solves m ( y ) = c . (5.13)However, in this case, we can further conclude from the monotonicity of m ( · ), the assumption c ≥ (cid:101) c , thedefinition (5.12) of (cid:101) c and the fact that (cid:101) y < (cid:98) y that m ( y c ) = c ≥ (cid:101) c = m ( (cid:101) y ) ⇔ y c ≤ (cid:101) y. imsart-generic ver. 2014/10/16 file: DDOmega_2020-05-19.tex date: June 2, 2020 odosthenous and Zhang/Selling amid anxiety about drawdowns Thus, by Theorem 4.3.(a), we can define the critical s -value as z c := y c + c with y c given explicitly by (5.5),yielding that z c is indeed given by (2.11) in this case as well. Moreover, recall from Theorem 4.3.(a) that,the optimal stopping region D y c for problem (1.5) is given by D y c = (cid:40) [ z c , ∞ ) , if y c < (cid:101) y ⇔ c > (cid:101) c, { b } ∪ [ z c , ∞ ) , if y c = (cid:101) y ⇔ c = (cid:101) c (where z c = z (cid:101) c = z (cid:63) ( (cid:101) y )) . In both cases, using the monotonicity of D y in y (cf. Theorem 4.3), it is straightforward to see thatif s ≥ z c , then [ s, ∞ ) ⊆ [ z c , ∞ ) ⊆ D y c ⊆ D s − c . Hence, by Proposition 3.2 we know that V ( x, s ; c ) ≡ v ( x ; s − c ) for any ( x, s ) ∈ O + such that s ≥ z c , where v admits the expressions in Theorem 4.3 for y = s − c . The result follows by the straightforward observationthat the Neumann condition (5.2) holds for all s ≥ z c (see e.g. Section 5.1 for similar arguments). Proof of part for s < z c . For the remaining case, we prove that it is optimal to wait until the process( X, X ) reaches the point ( z c , z c ). Notice that the positive function given by the right-hand side of (2.20)identifies with V ( x, s ; c ) from (5.6) (see also (5.7)), and the proof follows similar arguments as the ones in theproof for Theorem 2.2. Therefore, the only non-trivial task, before establishing the optimality of V , is to provethe dominance of V over the intrinsic value U (e x ). We examine below the ratio R ( x, s ; c ) = U (e x ) /V ( x, s ; c )for s < z c .When u > 0, we know that g ( · ) is non-monotone anymore, hence we cannot draw any conclusions fromthe partial derivative (5.9) of R ( x, s ; c ). To this end, we employ a different technique than in the proof ofTheorem 2.2. We begin by noticing that for s < z c , we have by (2.11) and (5.5) that s − c < z c − c = y c ,thus m ( s − c ) ≥ m ( y c ) = c ≥ (cid:101) c = m ( (cid:101) y ) ⇔ s − c ≤ y c ≤ (cid:101) y. (5.14)Therefore, in light of Theorem 4.3.(a), we have z (cid:63) ( s − c ) ≥ z (cid:63) ( y c ) = z c ≥ z (cid:63) ( (cid:101) y ) > b. Then, we take the partial derivative of the expression (5.7) of V ( x, s ; c ) with respect to s , for x ≤ s < z c , ∂∂s V ( x, s ; c ) = (cid:40) V ( x, s ; c ) (Λ( c ) − Φ( r + q )) , for x ∈ ( −∞ , s − c ) ,V ( x, s ; c ) (Λ( c ) − Λ( x − s + c )) , for x ∈ [ s − c, s ) , (cid:41) > , ∀ x < s , due to the fact that Λ( · ) is strictly decreasing on R + (cf. [35, Lemma 4.2]). We therefore have V ( x, s ; c ) ≥ lim s ↑ z c V ( x, s ; c ) = I ( r,q ) ( x − z c + c ) I ( r,q ) ( c ) U (e z c ) (2.11) , (5.5) = I ( r,q ) ( x − y c ) I ( r,q ) ( z c − y c ) U (e z c ) . In light of (5.14) and Theorem 4.3.(a) for y = s − c ≤ y c ≤ (cid:101) y , we know from the above and (4.1) that V ( x, s ; c ) ≥ I ( r,q ) ( x − y c ) I ( r,q ) ( z c − y c ) U (e z c ) = v ( x ; y c ) ≥ U (e x ) , ∀ ( x, s ) ∈ O + s.t. s < z c . Finally, we comment that the only possibility for V ( x, s ; c ) = U (e x ) is realised when the two inequalitiesabove are equalities. In particular, this can only occur when either x = s → z c , or when s → z c = z (cid:63) ( (cid:101) y ) and x = b (see Theorem 4.3.(a) for y = s − c = y c = (cid:101) y ). Proof of Lemma 2.5. The existence and the uniqueness of the solution follow from classical results for non-linear ODEs. To show the other properties, let θ ( s ) = s − a ( s ), and let F ( s, a ) be the slope field of (2.22),i.e. F ( s, a ) = qW ( r ) ( c ) W ( r,q ) ( s, a ; s − c ) (1 − ρ ) (cid:0) v ( s − c ) + ∆( s − c, a ( s ); s − c ) (cid:1) ( r + q − ψ (1 − ρ ))e (1 − ρ ) a ( s ) − ( r + q ) − (1 − ρ ) f ( a ( s ))Then θ ( · ) satisfies the ordinary differential equation θ (cid:48) ( s ) = 1 − F ( s, s − θ ( s )) . imsart-generic ver. 2014/10/16 file: DDOmega_2020-05-19.tex date: June 2, 2020 odosthenous and Zhang/Selling amid anxiety about drawdowns By equation (A.4), we see that for a = y = s − c , which occurs whenever θ ( s ) = c , for some s ≥ b + c , we get F ( s, s − c ) = q (1 − ρ ) U (e s − c )( r + q − ψ (1 − ρ ))e (1 − ρ ) ( s − c ) − ( r + q ) − (1 − ρ ) f ( s − c ) , since ∆( a, a ; a ) = 0 by its definition (2.18). Hence, for such values of s , we have1 − F ( s, s − c ) = ( r − ψ (1 − ρ ))e (1 − ρ ) ( s − c ) − r − (1 − ρ ) f ( s − c )( r + q − ψ (1 − ρ ))e (1 − ρ ) ( s − c ) − ( r + q ) − (1 − ρ ) f ( s − c ) (5.15)However, from (2.21) we have b (cid:63) ( y c ) − y c = c , thus we get in this case that b ≤ s − c ≤ b (cid:63) ( y c ) − c = y c < (cid:98) y .Therefore, the inequality χ ( x ) > x < (cid:98) y (see (2.14)) yields that( r − ψ (1 − ρ ))e (1 − ρ ) ( s − c ) − r − (1 − ρ ) f ( s − c ) < . (5.16)On the other hand, following the proof of [35, Lemma 4.9] one can show that f ( · ) is decreasing and continuousover [ b, ∞ ), with limit f ( b ) = e (1 − ρ ) b − ρ ( r + q − ψ (1 − ρ )) − r + q − ρ − σ Φ( r + q ) . (5.17)Therefore,( r + q − ψ (1 − ρ ))e (1 − ρ ) ( s − c ) − ( r + q ) − (1 − ρ ) f ( s − c ) (5.18) > ( r + q − ψ (1 − ρ ))e (1 − ρ ) b − ( r + q ) − (1 − ρ ) f ( b ) = 12 (1 − ρ ) σ Φ( r + q ) ≥ , for all s > b + c, where the last equality follows from (5.17). From (5.15), (5.16) and (5.18) we know thatfor all s ≤ z c such that θ ( s ) = c , we have θ (cid:48) ( s ) = 1 − F ( s, s − θ ( s )) < s ∈ [ b + c, b (cid:63) ( y c )) such that θ ( s ) = c , then θ ( s ) < c for all s ∈ ( s , b (cid:63) ( y c )]. However,notice that θ ( z c ) = θ ( b (cid:63) ( y c )) = b (cid:63) ( y c ) − a ( b (cid:63) ( y c )) = b (cid:63) ( y c ) − a (cid:63) ( y c ) > b (cid:63) ( y c ) − y c = c , which is a contradiction. Therefore, such an s cannot exist and the only possibility is θ ( s ) > c , i.e. a ( s ) < s − c ,as long as s ∈ [ b + c, b (cid:63) ( y c )) and a ( s ) is well-defined.In the final part of the proof, we use the aforementioned property of a ( s ) < s − c , in order to examinethe behaviour of a ( · ) when a ( s ) > b , which also implies that s − c > b . Then, for any s fixed and all x ∈ ( a ( s ) , s − c ], v ( x ) + ∆( x, a ( s ); s − c ) = U (e x ) + ∆( x, a ( s ); s − c ) > U (e x ) > U (e a ( s ) ) > U (e b ) > . Combining all the above with the probabilistic meaning of W ( r,q ) ( s, a ( s ); s − c ) in Lemma A.1, we concludethat a (cid:48) ( s ) = F ( s, a ( s )) > s > b + c , as long as a ( s ) > b . We now define s c := sup { s < b (cid:63) ( y c ) : a ( s ) ≤ b } . Notice that a ( b (cid:63) ( y c )) = a (cid:63) ( y c ) > a (cid:63) ( (cid:101) y ) = b = a ( s c ). Thus, from the monotonicity of a ( · ) we know that s c < b (cid:63) ( y c ). We complete the proof by showing that s c > b + c . Arguing by contradiction, we suppose that s c ≤ b + c , which implies that a ( s ) is well-defined at s = b + c and a ( b + c ) ≥ b . However, it follows from theestablished fact a ( s ) < s − c , that a ( b + c ) < b + c − c = b , which is indeed a contradiction. Proof of Theorem 2.6. Let (cid:101) c be the level defined by (2.16). Proof of part for s ≥ b (cid:63) ( y c ) . By the construction of m ( · ) and the fact that (cid:101) y < (cid:98) y , we conclude that thecritical value y c from (5.13), satisfies m ( y c ) = c < (cid:101) c = m ( (cid:101) y ) ⇔ y c > (cid:101) y ⇔ y c ∈ ( (cid:101) y, (cid:98) y ) . imsart-generic ver. 2014/10/16 file: DDOmega_2020-05-19.tex date: June 2, 2020 odosthenous and Zhang/Selling amid anxiety about drawdowns Using the definition (5.3) of m ( · ), we know that the unique value y c satisfies (2.21). Recall from Theorem4.3.(b) that, for y c ∈ ( (cid:101) y, (cid:98) y ), the optimal stopping region D y c for problem (1.5) is given by D y c = [ b, a (cid:63) ( y c )] ∪ [ b (cid:63) ( y c ) , ∞ ) . (5.19)By the monotonicity of D y in y (cf. Theorem 4.3), we have that,if s ≥ b (cid:63) ( y c ) , then [ s, ∞ ) ⊆ [ b (cid:63) ( y c ) , ∞ ) ⊆ D y c ⊆ D s − c . Hence, by Proposition 3.2 we know that V ( x, s ; c ) ≡ v ( x ; s − c ) for any ( x, s ) ∈ O + such that s ≥ b (cid:63) ( y c ), where v admits the expressions in Theorem 4.3 for y = s − c . The result follows by the straightforward observationthat the Neumann condition (5.2) holds for all s ≥ b (cid:63) ( y c ) (see e.g. Section 5.1 for similar arguments). Proof of part for s < b (cid:63) ( y c ) . We now focus on the remaining case that s < b (cid:63) ( y c ). As in Theorems 2.2and 2.4, we prove that the take-profit selling strategy T + b (cid:63) ( y c ) still constitutes part of the optimal strategyif s < b (cid:63) ( y c ). However, we shall prove that the optimal selling strategy is also partially given by a trailingstop type order, which requires selling the asset if its log price drops below some moving threshold a ( s ) thatdepends on the running best performance s of the asset log price.To be more precise, we define the value (cid:101) V of the two-sided exit strategy from an interval ( a, b (cid:63) ( y c )) bythe asset log price process X , where a = a ( s ) is such that a < s − c for a fixed s ≤ b (cid:63) ( y c ). In view of theequivalent expression in (3.1) of our original problem (1.2), the value of this strategy is given by (cid:101) V ( x, s ; c, a ) := E x,s (cid:20) exp( − R cT − a ∧ T + b(cid:63) ( yc ) ) v ( X T − a ∧ T + b(cid:63) ( yc ) ) (cid:21) . (5.20)We now derive a useful renewal equation satisfied by (cid:101) V . Since the second component of the process ( X, X )is constant up to time T + s , we know that R ct = A s − ct for all t ≤ T + s ( P x,s -a.s.), so we can rewrite (5.20) inthe form (cid:101) V ( x, s ; c, a ) = E x,s (cid:104) exp( − A s − cT − a ) v ( X T − a ) { T − a 0, so by imposing smooth fit at x = a , i.e. ∂∂x (cid:101) V ( a, s ; c, a ) = v (cid:48) ( a ), we again obtain (5.22).In both cases, we obtain from (5.21) and (5.22) that the threshold a = a ( s ) (if it exists) and function (cid:101) V ( · , s ; c, a ( s )) satisfy (cid:101) V ( x, s ; c, a ( s )) = v ( x ) + ∆( x, a ( s ); s − c ) ≡ v ( x ) + (cid:90) s − ca ( s ) W ( r,q ) ( x, w ; s − c ) · [ q v ( w ) − χ ( w )]d w − (cid:90) x ∨ ( s − c ) s − c W ( r ) ( x − w ) · χ ( w )d w. (5.23)Then by the above construction, it is easily seen that the mapping ( x, s ) (cid:55)→ (cid:101) V ( x, s ; c, a ( s )) is continuous over O + , and x (cid:55)→ (cid:101) V ( x, s ; c, a ( s )) is C over ( −∞ , s ) for each fixed s in the unbounded variation case. Also, bystraightforward calculations, the function (cid:101) V ( · , · ; c, a ( s )) from (5.23) (defined in (5.20)) satisfies the Neumanncondition (5.2) if and only if a ( · ) solves the ODE (2.22), where the boundary condition follows from thestructure of the optimal selling region (5.19) when s = b (cid:63) ( y c ). Given that { ( x, s ) ∈ O + : x < b } is alwayspart of the continuation region O + \S c of problem (1.2) (cf. Proposition 3.1), the candidate optimal threshold a ( s ) must satisfy a ( s ) ≥ b , which is the final condition imposed in Lemma 2.5. Notice that, Lemma 2.5 thenimplies that the function a ( · ) is strictly increasing and there exists a unique value s c < b (cid:63) ( y c ), such that a ( s c ) = b . The above function (cid:101) V is precisely the positive function from the right-hand side of (2.23) for s c ≤ s < b (cid:63) ( y c ). Proof of sub-part for s < s c . In light of the above, the selling strategy in (5.20) (see also (5.23)) is acandidate only for s c ≤ s < b (cid:63) ( y c ), while for all s < s c , it is optimal to simply wait until the asset log priceincreases to s c and then follow the optimal strategy V ( s c , s c ; c ). Thus, the expression of the value functionin (2.23) for all s < s c follows from similar arguments to the ones leading to (5.6)–(5.7) and their optimalityin the proof of Theorem 2.4, as soon as we prove the optimality of (cid:101) V ( x, s ; c, a ( s )) for all s c ≤ s < b (cid:63) ( y c ). Proof of sub-part for s c ≤ s < b (cid:63) ( y c ) . Therefore, in order to complete the proof, it suffices to show that (cid:101) V satisfies the variational inequality (5.1) for all ( x, s ) ∈ O + \{ ( s, s ) } such that s c ≤ s < b (cid:63) ( y c ).Firstly, it is seen by construction and the definition (2.18) of ∆, that the inequalities involving theinfinitesimal generator can be straightforwardly verified, since( L − ( r + q { x 0. We therefore see thatthe mapping s (cid:55)→ (cid:101) V ( x, s ; c, a ( s )) is strictly decreasing on [ s c , b (cid:63) ( y c )) for any fixed x ∈ ( s − c, s ]. (5.24)We then argue by contradiction, assuming that there exists a pair ( x , s ) for x ∈ ( s − c, s ] and s ∈ [ s c , b (cid:63) ( y c )), such that (cid:101) V ( x , s ; c, a ( s )) ≤ U (e x ). We arrive to a contradiction in both scenarios of x ∈ ( y c , s ] and x ∈ ( s − c, y c ]. In particular, on one hand, if y c < x ≤ s < b (cid:63) ( y c ), then (5.24) yields that U (e x ) ≥ (cid:101) V ( x , s ; c, a ( s )) > (cid:101) V ( x , b (cid:63) ( y c ); c, a ( b (cid:63) ( y c )) (2.22) = v ( x ) + ∆( x , a (cid:63) ( y c ); y c ) . However, combining the above with Theorem 4.3.(b) for y = y c > (cid:101) y , we get the contradiction U (e x ) > v ( x ) + ∆( x , a (cid:63) ( y c ); y c ) (4.2) = v ( x ; y c ) ≥ U (e x ) . On the other hand, if s − c < x ≤ y c , then we define s := x + c ∈ ( s , b (cid:63) ( y c )] and use again (5.24) to getthe contradiction U (e x ) ≥ (cid:101) V ( x , s ; c, a ( s )) > (cid:101) V ( x , s ; c, a ( s )) = v ( x ) + ∆( x , a ( s ); x ) ≥ U (e x ) , where the last inequality follows from Lemma 2.5. In summary we conclude that (cid:101) V ( x, s ; c, a ( s )) ≡ v ( x ) + ∆( x, a ( s ); s − c ) > U (e x ) , ∀ ( x, s ) ∈ { ( x, s ) ∈ O + : a ( s ) < x, s c ≤ s < b (cid:63) ( y c ) } , which completes the proof. Appendix A: Preliminaries on scale functions A well-known fluctuation identity of spectrally negative L´evy processes (see e.g. [19, Theorem 8.1]) is given,for r ≥ x ∈ [ a, b ], by E x [e − rT + b { T + b For any r ≥ , q > , and x ≤ b with a ≤ y ≤ b , we have E x [exp( − A yT + b ) { T + b Lemma A.2. For any x < b , we have E x (cid:2) exp( − A yT + b ) (cid:3) = I ( r,q ) ( x − y ) I ( r,q ) ( b − y ) , where I ( r,q ) ( · ) is given by (2.3) . (A.5)Finally, the following lemma gives the behaviour of scale functions at 0+ and ∞ ; see, e.g., [18, Lemmata3.1, 3.2, 3.3], and [9, (3.13)]. Lemma A.3. For any r > , W ( r ) (0) = (cid:26) , unbounded variation , γ , bounded variation , W ( r ) (cid:48) (0+) = σ , if σ > , ∞ , if σ = 0 and Π( −∞ , 0) = ∞ , r +Π( −∞ , γ , if σ = 0 and Π( −∞ , < ∞ , Appendix B: Technical resultsLemma B.1. The take-profit selling targets b , z c and z (cid:63) ( y ) , defined, respectively, by (2.9) , (2.11) and (2.5) for all relevant y -values in Theorems 2.2 and 2.4, are strictly decreasing functions of the risk aversioncoefficient ρ ∈ [0 , .Proof. Suppose that 0 ≤ ρ < ρ < 1. Denote by z (cid:63) ( y ), b and z (cid:63) ( y ), b the selling strategies defined by(2.5), (2.9), under ρ and ρ , respectively.We can firstly observe from the definitions (2.9) and (2.11) that both b = b ( ρ ) and z c = z c ( ρ ) are strictlydecreasing in ρ ∈ [0 , 1) independently of the case under consideration (cf. Section 2.2). Taking into account (from above) that b > b holds, we can straightforwardly conclude from Theorems2.2 and 2.4 that z (cid:63) ( y ) ≡ b > b ≡ z (cid:63) ( y ) for all y ≥ b . In view of the inequalities ∞ > z (cid:63) ( −∞ ) = log Φ( r ) − log(Φ( r ) − ρ )1 − ρ > log Φ( r ) − log(Φ( r ) − ρ )1 − ρ = z (cid:63) ( −∞ ) > z (cid:63) ( y ) > z (cid:63) ( y ) for all y < b . To prove this by contradiction, we assume that there exists y < b such that z (cid:63) ( y ) = z (cid:63) ( y ) = z (this suffices as z (cid:63)i ( y ) , i = 1 , 2, is continuous in y ). Given that z (cid:63) ( y ) solves (2.6),it follows that 1 = e (1 − ρ ) z (cid:18) − − ρ Λ( z − y ) (cid:19) = e (1 − ρ ) z (cid:18) − − ρ Λ( z − y ) (cid:19) . However, firstly note that the function ρ (cid:55)→ e (1 − ρ ) z (1 − − ρ Λ( z − y ) ) has fixed convexity in [0 , z, y , observe that ρ = 1 is already a solution. Hence, having two more distinct roots ρ , ρ ∈ [0 , z (cid:63) ( y ) (cid:54) = z (cid:63) ( y ) for all y < b . Combining theabove results, we can conclude that we eventually have z (cid:63) ( y ) > z (cid:63) ( y ) for all y in their common domainindependently of the case under consideration (cf. Theorems 2.2 and 2.4 in Section 2.2). Acknowledgments Neofytos Rodosthenous gratefully acknowledges support from EPSRC Grant Number EP/P017193/1. References [1] Alili, L. and Kyprianou, A. 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SIAM Journal on Control and Optimizatoin Zhang, H. (2018). Stochastic Drawdowns . World Scientific. imsart-generic ver. 2014/10/16 file: DDOmega_2020-05-19.tex date: June 2, 2020 odosthenous and Zhang/Selling amid anxiety about drawdowns ρ (b) Benchmark (c) Larger ρ (d) Smaller σ (e) Benchmark (f) Larger σ (g) Smaller η (h) Benchmark (i) Larger η Fig 7: Stopping regions for the problem configurations in Table 1 (see also Table 2 for the values of keythresholds). imsart-generic ver. 2014/10/16 file: DDOmega_2020-05-19.tex date: June 2, 2020 odosthenous and Zhang/Selling amid anxiety about drawdowns y (a) b (cid:63) ( y ) and a (cid:63) ( y ) y (b) m ( y ) Fig 8: Using a compound Poisson plus a drifted Brownian motion model with H (cid:63) < u > b (cid:63) ( y ) (in dashed line) and a (cid:63) ( y ) defined in Theorem 4.3 infigure (a); In figure (b), we plot the function m ( y ) defined in (5.3). Both b (cid:63) ( y ) − a (cid:63) ( y ) = 0 and m ( y ) = 0occur at y = (cid:98) y .. m ( y ) := (cid:40) z (cid:63) ( y ) − y, ∀ y ≤ b, if u = −∞ ; z (cid:63) ( y ) { y ≤ (cid:101) y } + b (cid:63) ( y ) { (cid:101) y e Λ( c )( z c − s ) Λ( c ) U (e z c ) (cid:18) − ρ − e (1 − ρ ) z c (cid:18) − ρ − c ) (cid:19)(cid:19) = 0 , ∀ s < z c . In view of R ( z c , z c ; c ) = 1, which follows straightforwardly from (5.10) for s = z c , we thus obtain U (e x ) V ( x, s ; c ) = R ( x, s ; c ) ≤ R ( s, s ; c ) < R ( z c , z c ; c ) = 1 , ∀ x ≤ s < z c , which implies that V ( x, s ; c ) ≥ U (e x ) , ∀ x ≤ s ≤ z c . (5.11)Combining (5.8) and (5.11), we conclude that the variational inequality (5.1) is satisfied and consequentlywe prove the optimality of V ( x, s ; c ) for all s ≤ z c . Proof of Lemma 2.3. U (e x ) , ∀ a ( s ) < x ≤ s, s c ≤ s < b (cid:63) ( y c ) . We know from Lemma 2.5 that (cid:101) V ( x, s ; c, a ( s )) = v ( x ) + ∆( x, a ( s ); s − c ) > U (e x ) , for all x ∈ ( a ( s ) , s − c ] (cid:40) [ b, ∞ ) . We therefore focus on any fixed x ∈ ( s − c, s ], for which we have ∂∂s (cid:101) V ( x, s ; c, a ( s )) = ∂∂s ∆( x, a ( s ); s − c ),where ∂∂s ∆( x, a ( s ); s − c ) = a (cid:48) ( s ) W ( r,q ) ( x, a ( s ); s − c ) ( L − r − q ) v ( a ( s )) + qW ( r ) ( x − s + c ) (cid:101) V ( s − c, s ; c, a ( s )) . Using the expression of (cid:101) V ( s − c, s ; c, a ( s )) from (5.23) and the ODE (2.22) solved by a ( · ), we get (cid:101) V ( s − c, s ; c, a ( s )) = − a (cid:48) ( s ) W ( r,q ) ( s, a ( s ); s − c ) qW ( r ) ( c ) ( L − r − q ) v ( a ( s )) . imsart-generic ver. 2014/10/16 file: DDOmega_2020-05-19.tex date: June 2, 2020 odosthenous and Zhang/Selling amid anxiety about drawdowns Combining all of the above, we obtain that, for all x ∈ ( s − c, s ] and s ∈ ( s c , b (cid:63) ( y c )), ∂∂s (cid:101) V ( x, s ; c, a ( s ))= a (cid:48) ( s ) W ( r,q ) ( s, a ( s ); s − c ) (cid:18) W ( r,q ) ( x, a ( s ); s − c ) W ( r,q ) ( s, a ( s ); s − c ) − W ( r ) ( x − s + c ) W ( r ) ( c ) (cid:19) ( L − r − q ) v ( a ( s ))= a (cid:48) ( s ) W ( r,q ) ( s, a ( s ); s − c ) (cid:18) E x [exp( − A s − cT + s ) { T + s