The importance of dynamic risk constraints for limited liability operators
aa r X i v : . [ q -f i n . P M ] N ov THE IMPORTANCE OF DYNAMIC RISK CONSTRAINTS FOR LIMITEDLIABILITY OPERATORS
JOHN ARMSTRONG, DAMIANO BRIGO, AND ALEX S.L. TSE
Abstract.
Previous literature shows that prevalent risk measures such as Value at Risk or ExpectedShortfall are ineffective to curb excessive risk-taking by a tail-risk-seeking trader with S-shaped utilityfunction in the context of portfolio optimisation. However, these conclusions hold only when the con-straints are static in the sense that the risk measure is just applied to the terminal portfolio value. Inthis paper, we consider a portfolio optimisation problem featuring S-shaped utility and a dynamic riskconstraint which is imposed throughout the entire trading horizon. Provided that the risk control policyis sufficiently strict relative to the asset performance, the trader’s portfolio strategies and the resultingmaximal expected utility can be effectively constrained by a dynamic risk measure. Finally, we arguethat dynamic risk constraints might still be ineffective if the trader has access to a derivatives market. Introduction
Portfolio optimisations are typically formulated as an expected utility maximisation problem faced bya risk averse agent with concave utility function. However, a simple concave function may not be sufficientto model agents’ preferences in an actual trading environment. For example, the limited-liability featureof a financial institution as well as standard remuneration scheme tend to create incentive distortion wherea successful trader can share the profits via bonuses but a failed trader can simply walk away withoutpunishment. Thus gains and losses can be perceived very differently by an agent leading to deviationfrom a concave utility function. See for example Carpenter (2000) and Bichuch and Sturm (2014). Ata psychological level, the seminal work of Kahneman and Tversky (1979) and many of the other follow-up studies reveal that individuals are risk averse over positive outcomes but risk seeking over negativeoutcomes. These stylised preferences can be better captured by an S-shaped utility function which isconcave on gains and convex on losses.This paper concerns the risk-taking behaviours of “tail-risk-seeking traders” who do not care muchabout extreme losses and hence their utility function is S-shaped. It is of great regulatory interests tounderstand how the trading activities of a tail-risk-seeking trader can be controlled by standard riskmeasures. A surprising result has been reported in a recent paper of Armstrong and Brigo (2019) thatValue at Risk (VaR) and Expected Shortfall (ES) are totally ineffective to curb the risk-taking behavioursof tail-risk-seeking traders. They consider a portfolio optimisation problem under S-shaped utility functionand find that the value function of the trader remains the same upon imposing a static VaR/ES constraint
Date : November 9, 2020. on the terminal portfolio value. In other words, neither VaR nor ES can alter the maximal expected utilityattained by a tail-risk-seeking trader compared to the benchmark case without any risk constraint. Thiscasts doubt over the usefulness of prevalent risk management protocols to combat excessive risk-takingby traders with more realistic preferences. An earlier restricted version of the same result, focusingonly on a Black-Scholes option market, is in Armstrong and Brigo (2018). A further related result inArmstrong and Brigo (2020) introduces the notion of ρ -arbitrage for a coherent risk measure ρ . Positivehomogeneity of the measure ρ is the key property that is used to reach the result. A risk measure ρ isdefined to be ineffective if a static risk constraint based on that measure cannot lower the expected utility ofa limited liability trader. A ρ –arbitrage is defined as a portfolio payoff with non-positive price, non-positiverisk as measured by ρ but with strictly positive probability of being strictly positive. The ineffectivenessof static risk constraints based on the coherent risk measure ρ is shown to be equivalent to the existence ofa ρ –arbitrage. Again, the emphasis for us, in this paper, is that also in Armstrong and Brigo (2018) andArmstrong and Brigo (2020) the risk constraints are static and that the situation becomes very differentwith dynamic risk constraints.Indeed, in view of the above negative result, we explore a simple remedy which resurrects VaR/ES asa tool to risk manage tail-risk-seeking traders: the risk measure is imposed dynamically throughout theentire trading horizon. At each point of time given the current assets holding in place, the portfolio riskexposure is computed by projecting the distribution of the portfolio return over an evaluation windowunder the assumption that the assets holding remains unchanged. There are several advantages with sucha dynamic risk constraint. First, this risk management approach is more consistent with the industrialpractice where the risk exposure of the trader’s positions is typically reported and monitored at leastdaily. Second, imposing a static risk constraint on the terminal portfolio value only usually leads to atime-inconsistent optimisation problem where the optimal strategy solved at a future time point may notbe consistent with the one derived in the past. This results in difficulty with interpreting the notion ofoptimality, and one has to make further assumptions (such as whether the agent can pre-commit to theoptimal strategy derived at time-zero) to pin down a unique prediction of the trader’s action. The ideaof time-inconsistency in dynamic optimisation problems can be dated back to Strotz (1955).Our main contribution is to show that a dynamic VaR or ES constraint can indeed constrain a tail-risk-seeking trader, in the sense that the maximal expected utility attained can be reduced provided thatthe risk control policy is sufficiently strict relative to the Sharpe ratio of the risky asset. The differencebetween a static and a dynamic risk constraint is drastic both mathematically and economically. Ina complete market, any arbitrary payoff can be synthesised by dynamic replication. As a result, theproblem of solving for the optimal trading strategy is equivalent to finding a utility-maximising payoffwhose no-arbitrage price is equal to the initial wealth available. This duality principle which converts adynamic stochastic control problem into a static optimisation has been widely adopted to solve portfolio HE IMPORTANCE OF DYNAMIC RISK CONSTRAINTS FOR LIMITED LIABILITY OPERATORS 3 optimisation problems. A static risk measure applied to the terminal portfolio value only restricts theclass of the admissible payoffs. Armstrong and Brigo (2019) show that one can construct a sequence ofdigital options which pay a small positive amount most of the time but incur an extreme loss with atiny probability, and that these payoffs can be carefully engineered to satisfy any given VaR/ES limit.The resulting expected utilities will converge to the same utility level associated with an unconstrainedproblem.The conclusion changes significantly when the risk constraint is applied dynamically instead. To complywith the given risk limit at each time point, the notional invested in the underlying assets has to be cappedif the risk policy is sufficiently strict. Thus a dynamic risk constraint now has first-order impact on theadmissible trading strategies. The usual duality approach no longer works because the restriction on thetrading strategies from the outset precludes dynamic replication of a claim. We therefore have to resortto the primal HJB equation approach to solve the portfolio optimisation problem. Although a close-formsolution is not available in general, we can nonetheless deduce the analytical conditions on the modelparameters under which a dynamic VaR/ES constraint becomes effective. In a special case where theexcess return of the asset is zero, we can provide a finer characterisation of the optimal trading strategy.Our results show that a dynamic risk constraint can be effective against a “delta-one trader” who canonly invest in the underlying risky assets. What will happen if the trader can access derivatives trading aswell? In the context of utility maximisation under market completeness, there is no economic differencebetween delta-one and derivatives trading since any payoff can be replicated by dynamic trading in theunderlying assets. We argue, however, that a dynamic risk constraint such as ES will become ineffectiveagain if derivatives trading is allowed. The key idea is that a derivatives trader can exploit dynamicrebalancing to continuously roll-over some risky digital options to ensure the risk constraint is satisfied atall time while generating an arbitrarily high level of utility.We conclude the introduction by discussing some related work. A vast literature on continuous-timeportfolio optimisation has emerged since Merton (1969, 1971). One natural extension of the originalMerton model is to incorporate additional constraints in form of a risk functional applied to the terminalportfolio value. Examples of the extra constraints include VaR (Basak and Shapiro (2001)), expectedloss and other similar shortfall-style measures (Gabih et al. (2005)), probability of outperforming a givenbenchmark (Boyle and Tian (2007)) and utility-based shortfall risk (Gundel and Weber (2008)). In thesepapers, the combination of concave utility function and static risk constraint facilitates the use of the dualapproach to solve the underlying optimisation problems.There has been a recent strand of literature focusing on dynamic risk constraints. Yiu (2004), Cuoco et al.(2008), Akume et al. (2010) consider similar portfolio optimisation problems with VaR/ES constraintsunder different modelling setups. The optimal trading strategy behaves very differently when a staticconstraint is replaced by a dynamic one. For example, Basak and Shapiro (2001) show that a static VaR
HE IMPORTANCE OF DYNAMIC RISK CONSTRAINTS FOR LIMITED LIABILITY OPERATORS 4 constraint may induce the trader to take more risk (relative to the unconstrained case) in the bad stateof the world, whereas Cuoco et al. (2008) show that if the VaR constraint is applied dynamically thenthe optimal risk exposure can be unanimously reduced. HJB equation formulation has to be used whensolving the problem with dynamic risk constraints. All the papers cited above work with a concave utilityfunction, and thus the problem is still relatively standard to yield analytical and numerical progress.S-shaped utility maximisation has received a lot of attention in the context of behavioural economicsand convex incentive scheme. Despite the non-standard shape of the underlying utility function, dualitymethod can still be suitably adapted to solve the optimisation problems. See for example Berkelaar et al.(2004), Reichlin (2013), Bichuch and Sturm (2014) and the references therein. Papers on dynamic port-folio optimisation which simultaneously feature S-shaped utility as well as VaR/ES constraint includeArmstrong and Brigo (2019), Guan and Liang (2016) and Dong and Zheng (2020). But again, the con-straints are static in nature which are only imposed at the terminal time point. Our work fills the gap inthe literature by considering S-shaped utility function and dynamic risk constraint in conjunction. In thesame spirit that Cuoco et al. (2008) is the dynamic version of Basak and Shapiro (2001) under concaveutility function, our work can be viewed as the dynamic version of Armstrong and Brigo (2019) underS-shaped utility to give insights on the new economic phenomena when a more realistic risk managementapproach is adopted.The rest of the paper is organised as follows. Section 2 gives an overview of the modelling framework.The main results of the paper are stated in Section 3 with some numerical illustrations. A special casethat the excess return of the asset being zero is analysed in details in Section 4. We briefly discuss inSection 5 how the results will change if the trader can access a derivatives market. Section 6 concludes.Miscellaneous technical materials are deferred to the appendix.2.
Modelling setup
The economy.
For simplicity of exposition, in the main body of this paper we consider a standardBlack-Scholes economy with a riskfree bond and one risky asset only. Extension to the multi-asset setupis discussed in the Appendix C.Fix a terminal horizon
T >
0. Let (Ω , F , {F t } ≤ t ≤ T , P ) be a filtered probability space satisfying theusual conditions which supports a one-dimensional Brownian motion B = ( B t ) t ≥ . The risky asset hasprice process S = ( S t ) t ≥ following a geometric Brownian motion dS t S t = µdt + σdB t with drift µ and volatility σ >
0, and the riskfree bond has a constant interest rate of r . A trader investsin the two assets dynamically where an amount of Π t is invested in the risky asset at time t . The portfoliostrategy Π = (Π t ) t ≥ is said to be admissible if it is adapted and R T Π t dt < ∞ almost surely. The set of HE IMPORTANCE OF DYNAMIC RISK CONSTRAINTS FOR LIMITED LIABILITY OPERATORS 5 admissible portfolio strategies is denoted by A . The portfolio value process X = ( X t ) t ≥ then evolves as dX t = Π t S t dS t + r ( X t − Π t ) dt = rX t dt + Π t [( µ − r ) dt + σdB t ] , X = x , (1)where x is an exogenously given initial capital of the trader.2.2. Dynamic risk constraints.
Suppose r = 0 for the moment. The dynamics (1) can be rewritten as dX t = κ ( θ t − X t ) dt + σ Π t dB t with κ := − r and θ t := − ( µ − r )Π t r . This is an Ornstein–Uhlenbeck process and thus X t +∆ = e − κ ∆ X t + κ Z t +∆ t e − κ ( t +∆ − s ) θ s ds + σ Z t +∆ t e − κ ( t +∆ − s ) Π s dB s for any t and ∆ >
0. We then deduce X t +∆ − e r ∆ X t = ( µ − r ) Z t +∆ t e r ( t +∆ − s ) Π s ds + σ Z t +∆ t e r ( t +∆ − s ) Π s dB s (2)which could be interpreted as the (numeraire-adjusted) portfolio gain/loss over the time horizon [ t, t + ∆].At each instant of time t , a risk manager assesses the risk associated with the portfolio return givenby (2). Since the risk manager typically does not have the knowledge of the trader’s portfolio strategybeyond the current time t , he assumes the portfolio strategy Π will be held fixed over the risk evaluationwindow [ t, t + ∆]. Then the time- t estimated random variable of portfolio loss over [ t, t + ∆], denoted by L t , is given by − L t := ( µ − r )Π t Z t +∆ t e r ( t +∆ − s ) ds + σ Π t Z t +∆ t e r ( t +∆ − s ) dB s = ( µ − r )( e r ∆ − r Π t + σ Π t Z t +∆ t e r ( t +∆ − s ) dB s and as such L t is normally distributed with mean and variance of E [ L t ] = − ( µ − r )( e r ∆ − r Π t , Var( L t ) = ( e r ∆ − σ r Π t . (3)The special case of r = 0 can be recovered by considering the appropriate limits in (3). Remark . In the literature, there are multiple ways to estimate the projected distribution of portfoliogain/loss. Our approach is based on Yiu (2004) where the notional invested in the risky asset Π t is assumedto be fixed by the risk manager. Alternatively, the risk manager can also assume the proportion of capitalinvested in the risky asset Π t /X t is fixed - this assumption is adopted for example by Cuoco et al. (2008).The latter approach leads to a more difficult mathematical problem in general because the projecteddistribution will then also depend on the current portfolio value X t . The question about which approachis more superior depends on the risk management practice adopted at a particular institution. Anothervery plausible approach is to assume the quantity of the assets n t := Π t /S t to be fixed (this could be HE IMPORTANCE OF DYNAMIC RISK CONSTRAINTS FOR LIMITED LIABILITY OPERATORS 6 more relevant in the context of equity trading where stock and future positions are typically recorded interms of quantity rather than notional). Then starting from (2) we can deduce that − L t := ( µ − r ) n t Z t +∆ t e r ( t +∆ − s ) S u du + σn t Z t +∆ t e r ( t +∆ − u ) S u dB u = n t Z t +∆ t e r ( t +∆ − u ) (( µ − r ) S u du + σS u dB u )= n t e r ( t +∆) Z t +∆ t e − ru ( dS u − rS u du )= n t e r ( t +∆) Z t +∆ t d ( e − ru S u )= n t ( S t +∆ − e r ∆ S t )= Π t (cid:20) e (cid:16) µ − σ (cid:17) ( t +∆)+ σ ( B t +∆ − B t ) − e r ∆ (cid:21) which only depends on the current state via Π t = n t S t . This is qualitatively very similar to the approachused by Yiu (2004) and us, except that L t is now linked to some log-normal random variable.A dynamic risk constraint is imposed such that ρ ( L t ) ≤ R for all t ∈ [0 , T ). Here ρ ( · ) is some riskmeasure and R > α (with α < .
5) such that ρ ( L t ) = VaR α ( L t ) := sup { x ∈ R : P ( L t ≥ x ) > α } ,then using the Gaussian property of L t and (3) the constraint can be specialised to − ( µ − r )( e r ∆ − r Π t − σ r e r ∆ − r Φ − ( α ) | Π t | ≤ R where Φ denotes the cumulative distribution function (cdf) of a N (0 ,
1) random variable. We define theset K VaR := ( π ∈ R : − ( µ − r )( e r ∆ − r π − σ r e r ∆ − r Φ − ( α ) | π | ≤ R ) (4)such that compliance with the dynamic VaR constraint at time t is equivalent to Π t ∈ K VaR .Similarly, if the risk measure is taken as ES with confidence level α such that ρ ( L t ) = E [ L t | L t ≥ VaR α ( L t )], then the constraint becomes − ( µ − r )( e r ∆ − r Π t + σ r e r ∆ − r φ (Φ − ( α )) α | Π t | ≤ R with φ ( · ) being the probability density function (pdf) of a N (0 ,
1) random variable. We then define theset K ES := ( π ∈ R : − ( µ − r )( e r ∆ − r π + σ r e r ∆ − r φ (Φ − ( α )) α | π | ≤ R ) (5)where we require Π t ∈ K ES for all t in order to satisfy the dynamic ES constraint.It turns out that the nature of the sets K VaR and K ES crucially depends on the Sharpe ratio of therisky asset µ − rσ , as the following lemma shows. HE IMPORTANCE OF DYNAMIC RISK CONSTRAINTS FOR LIMITED LIABILITY OPERATORS 7
Lemma 1.
Define the constants M VaR := − r e r ∆ − r re r ∆ − − ( α ) > , M ES := r e r ∆ − r re r ∆ − φ (Φ − ( α )) α > . (6) Then for i ∈ { VaR , ES } , the sets K i defined in (4) and (5) have the following properties:(1) If µ − rσ ≥ M i , there exists −∞ < k i < such that K i = [ k i , ∞ ) ;(2) If | µ − rσ | < M i , there exists −∞ < k i < < k i < ∞ such that K i = [ k i , k i ] ;(3) If µ − rσ ≤ − M i , there exists < k i < ∞ such that K i = ( −∞ , k i ] .Moreover, k VaR = − R − σ q e r ∆ − r Φ − ( α ) + ( µ − r )( e r ∆ − r , k VaR = R − σ q e r ∆ − r Φ − ( α ) − ( µ − r )( e r ∆ − r ; k ES = − Rσ q e r ∆ − r φ (Φ − ( α )) α + ( µ − r )( e r ∆ − r , k ES = Rσ q e r ∆ − r φ (Φ − ( α )) α − ( µ − r )( e r ∆ − r . Proof.
This is a simple exercise of analysing the piecewise linear function arising in the definition of K VaR and K ES . (cid:3) The constants M i defined in (6) encapsulate the risk management parameters α and ∆. Unless thequality of the investment asset is very good (measured by the magnitude of its Sharpe ratio) relative to M i , a dynamic VaR or ES constraint will result in a restriction that Π t needs to take value in a boundedset, i.e. a delta limit restriction where both the long and short position in the underlying asset cannotexceed certain notional levels given by k i and k i . It is also not hard to see that M i is decreasing in both α and ∆. Hence a small confidence level of the VaR/ES constraint or a tight risk evaluation window willmore likely lead to a bounded investment set K i . Provided that k i and k i exist, one can also easily checkthat | k i | and k i are both decreasing in σ and increasing in R and α . Hence a high asset volatility, low risklimit or tight confidence level of the VaR/ES measure will result in small absolute delta notional limit.2.3. Trader’s utility function and optimisation problem.
We assume that the trading decision ismade by a “tail-risk-seeking trader” who is insensitive towards extreme losses. His utility function U ( · )is S-shaped and his goal is to maximise the expected utility of the terminal portfolio value. The onlyassumption required over U is the following. Assumption 1.
The utility function U : R → R is a continuous, increasing and concave (resp. convex)function on x > (resp. x < ) with U (0) = 0 and lim x →−∞ U ( x ) x = 0 . In particular, the trader is locally risk averse over the domain of gains but locally risk seeking overthe domain of losses. Moreover, the assumption on the left-tail behaviour of the utility function furthersuggests that the trader is tail-risk-seeking in that the “dis-utility” due to extreme losses has a sub-lineargrowth. We do not require U ( x ) to be differentiable. This allows us to consider for example the piecewise HE IMPORTANCE OF DYNAMIC RISK CONSTRAINTS FOR LIMITED LIABILITY OPERATORS 8 power utility function of Kahneman and Tversky (1979) which is not differentiable at x = 0, or an optionpayoff function which may contain kinks.Mathematically, the underlying optimisation problem is V ( t, x ) := sup Π ∈A ( K ) E ( t,x ) [ U ( X Π T )] (7)where X = X Π has dynamics described by (1), and A ( K ) is the admissible set of the portfolio strategiesunder a given dynamic risk constraint in form of A ( K ) := { Π ∈ A : Π( t, ω ) ∈ K L ⊗ P -a.e. ( t, ω ) } with K ⊆ R being some given set and L is Lebesgue measure. For example, if the risk constraint is absentwe simply take K = K := R and then A ( K ) = A . If a dynamic VaR constraint is in place, we set K = K VaR as defined in (4). Likewise a choice of K = K ES given by (5) corresponds to a dynamic ESconstraint. Remark . Portfolio optimisation problem in form of (7) with U being a strictly concave, twice-differentiablefunction is studied by Cvitani´c and Karatzas (1992). Their results cannot be applied to our setup be-cause our utility function is S-shaped. Dong and Zheng (2019) consider a version of the problem withS-shaped utility and short-selling restrictions. Their solution method is based a concavification argumentin conjunction with the results by Bian et al. (2011) which cover non-smooth utility function but onlyunder the assumption that the set K is in form of a convex cone. For our model, Lemma 1 suggests thatthe set K under VaR/ES constraint cannot be a convex cone. Thus we cannot apply their approaches tosolve our problem.Let V i ( t, x ) be the value function of problem (7) under K = K i with i ∈ { , VaR , ES } denoting thelabel identifying which dynamic risk measure is being adopted (i.e no risk constraint at all, Value at Riskand Expected Shortfall). We first state a benchmark result based on Armstrong and Brigo (2019). Proposition 1 (Theorem 4.1 of Armstrong and Brigo (2019)) . The value function of the unconstrainedportfolio optimisation problem is V ( t, x ) = sup s U ( s ) .Sketch of proof. Without loss of generality we just need to prove the result at t = 0. By standard dualityargument (see for example Karatzas et al. (1987)), the portfolio optimisation problem (7) without anyadditional risk constraint is equivalent to solving V (0 , x ) = sup X T ∈F T E [ U ( X T )] E [ ξ T X T ] ≤ x where ξ T := exp "(cid:18) − µ − rσ (cid:19) B T − r + 12 (cid:18) µ − rσ (cid:19) ! T HE IMPORTANCE OF DYNAMIC RISK CONSTRAINTS FOR LIMITED LIABILITY OPERATORS 9 is the pricing kernel in the Black-Scholes economy. Now consider a digital payoff in form of X T = − b P ( ξ T > k ) ( ξ T >k ) + b ( ξ T ≤ k ) (8)for b > k >
0. The budget constraint can be written as − b P ( ξ T > k ) E [ ξ T ( ξ T >k ) ] + b E [ ξ T ( ξ T ≤ k ) ] ≤ x ⇐⇒ − b ≤ P ( ξ T > k ) E [ ξ T ( ξ T >k ) ] (cid:2) x − b E [ ξ T ( ξ T ≤ k ) ] (cid:3) . If ξ T is unbounded from the above (which is the case in the Black-Scholes model), then lim k →∞ P ( ξ T >k ) E [ ξ T ( ξT >k ) ] =0. In turn for any b > k such that the budget constraint issatisfied. The value function must be no less than the expected utility attained by this payoff structure,i.e. V (0 , x ) ≥ U (cid:18) − b P ( ξ T > k ) (cid:19) P ( ξ T > k ) + U ( b ) P ( ξ T < k ) . Under Assumption 1, lim x →−∞ U ( x ) x = 0. Thus on sending k → ∞ we deduce V (0 , x ) ≥ U ( b ). Theresult follows since b > (cid:3) Without any risk constraint in place, the tail-risk-seeking trader can attain any arbitrarily high utilityby replicating a sequence of digital options which pay a positive amount with a large probability butincur an extremely disastrous loss with very small probability. Armstrong and Brigo (2019) show thatthis result does not change even if a static VaR/ES constraint is imposed on the terminal portfolio value,in the sense that the trader can still manipulate the digital structure to attain an arbitrarily high utilitylevel while satisfying the additional constraints.We are interested in studying whether such conclusion will change if we adopt a dynamic risk constraintinstead. With the unconstrained optimisation problem as our benchmark, we first give below a formaldefinition of the effectiveness of a dynamic risk constraint.
Definition 1.
A dynamic risk constraint i ∈ { VaR , ES } is said to be effective if for each t < T thereexists x such that V i ( t, x ) < V ( t, x ) = sup s U ( s ) . The notion of effectiveness in Definition 1 may appear to be somewhat weak as we do not insistthat the trader’s expected utility have to be strictly reduced at all states ( t, x ). Indeed for a generalutility function, we cannot expect V i ( t, x ) < sup s U ( s ) for all ( t, x ). For example, consider a call spreadpayoff U ( x ) = ( x + 1) + − ( x − + − t = 0 for all t is anadmissible strategy under a given dynamic risk constraint i and interest rate is non-negative, we alwayshave V i ( t, x ) = 1 = sup s U ( s ) for all t < T and x ≥ Main results
We first give a useful proposition which is the building block of the main results in this paper.
HE IMPORTANCE OF DYNAMIC RISK CONSTRAINTS FOR LIMITED LIABILITY OPERATORS 10
Proposition 2.
For the optimisation problem (7) , if the set K is bounded then for every t < T thereexists x such that V ( t, x ) < sup s U ( s ) .Proof. Since U ( x ) ≤ x ≤ U ( x ) is concave on x >
0, for any constant
C > m > U ( x ) ≤ ¯ U ( x ) := mx + + C for all x . Then J ( t, x ; Π) := E ( t,x ) [ U ( X Π T )] ≤ E ( t,x ) [ ¯ U ( X Π T )] . Since K is bounded, there exits b ∈ (0 , ∞ ) such that K ⊆ [ − b, b ] =: ¯ K . Then V ( t, x ) = sup Π t ∈A ( K ) J ( t, x ; Π) ≤ sup Π t ∈A ( K ) E ( t,x ) [ ¯ U ( X Π T )] ≤ sup Π t ∈A ( ¯ K ) E ( t,x ) [ ¯ U ( X Π T )] =: ¯ V ( t, x ) . (9)We can now derive the expression of ¯ V ( t, x ) as the value function of a stochastic control problem withpayoff function ¯ U which is increasing and convex. Formally, we expect ¯ V ( t, x ) to be the (viscosity) solutionof the HJB equation − V t − sup π ∈ [ − b,b ] n [( µ − r ) π + rx ] V x + σ V xx π o = 0 , t < T ; V ( t, x ) = ¯ U ( x ) , t = T. (10)Suppose µ ≥ r and recall that ¯ U is convex. Since the dynamics of the portfolio process is dX t =[ rX t + ( µ − r )Π t ] dt + σ Π t dB t where its drift and volatility are both increasing in Π t , we expect theoptimal strategy is to choose the largest possible value of Π t within the bounded set ¯ K . Hence thecandidate optimal control for problem (9) is Π ∗ t = b < ∞ . The corresponding candidate value function isthus w ( t, x ) := E ( t,x ) [ U ( X Π ∗ T )]and the wealth process under the candidate optimal control is dX ∗ t = rX ∗ t dt + b [( µ − r ) dt + σdB t ] . Then X ∗ s = xe r ( s − t ) + b ( µ − r ) r ( e r ( s − t ) −
1) + bσ Z st e r ( s − u ) dB u for s ≥ t and X ∗ t = x, such that X ∗ T is normally distributed with mean xe r ( T − t ) + b ( µ − r ) r ( e r ( T − t ) −
1) and variance b σ r ( e r ( T − t ) − w ( t, x ) = E [ C + m ( X ∗ T ) + ]= C + m h xe r ( T − t ) + b ( µ − r ) r ( e r ( T − t ) − i Φ xe r ( T − t ) + b ( µ − r ) r ( e r ( T − t ) − bσ q e r ( T − t ) − r HE IMPORTANCE OF DYNAMIC RISK CONSTRAINTS FOR LIMITED LIABILITY OPERATORS 11 + mbσ r e r ( T − t ) − r φ xe r ( T − t ) + b ( µ − r ) r ( e r ( T − t ) − bσ q e r ( T − t ) − r (11)where Φ and φ are the cdf and pdf of a standard N (0 ,
1) random variable respectively. w ( t, x ) is indeed C × on [0 , T ) × R , and is increasing convex in x . It can be easily shown that w is a solution to the HJBequation (10). Standard verification arguments then lead to the conclusion that ¯ V ( t, x ) = w ( t, x ). Finally,for each fixed t we have ¯ V ( t, x ) = w ( t, x ) → C as x ↓ −∞ . But the constant C > V ( t, x ) ≤ ¯ V ( t, x ), the desired result follows if we choose C ∈ (0 , sup s U ( s )).The case of µ < r can be handled similarly except that the optimal control will become Π ∗ t = − b instead. (cid:3) The implication of Proposition 2 is that a delta notional limit on the risky asset alone is sufficient toconstrain a tail-risk-seeking trader. For an unconstrained problem, as discussed in the proof of Proposition1 one can attain an arbitrarily high utility level by replicating some digital options. But it is known thatthe delta of a digital option can be unboundedly large when the time to maturity becomes short and theunderlying stock price is near the strike. Hence a trader cannot replicate a digital option and hold theposition until maturity while complying the dynamic risk constraint with certainty. In practice, a tradingdesk with a substantial at-the-money digital option position with short maturity will often be requestedto wind-down the trade to reduce the pin risk.Next we state the main theorem of this paper which provides a precise condition under which a dynamicVaR/ES constraint can effectively restrict a rough trader.
Theorem 1.
Recall the constants M VaR and M ES introduced in (6) . A dynamic Value at Risk constraintis effective if and only if | µ − rσ | < M VaR . A dynamic Expected Shortfall constraint is effective if and onlyif | µ − rσ | < M ES .Proof. In view of Lemma 1 and Proposition 2 we only need to prove the “only if” part of the theorem.In the proof of Proposition 1, a utility level of sup s U ( s ) can be attained by replicating a sequence ofpayoffs in form of X T = a ( ξ T >k ) + b ( ξ T ≤ k ) with a < < b where ξ T is the pricing kernel in the Black-Scholes economy. But ξ T = exp (cid:20) − µ − rσ B T − (cid:18) r + 12 λ (cid:19) T (cid:21) ∝ S − µ − rσ T . Hence X T is increasing (resp. decreasing) in S T if µ > r (resp. µ < r ).Suppose µ − rσ ≥ M i > i ∈ { VaR , ES } . Then by Lemma 1 the admissible set is in form of[ k i , ∞ ) where k i ∈ ( −∞ , µ > r , if we view X T as a contingent claim written on therisky asset, the payoff X T = X ( S T ) is an increasing function and thus the option must have non-negative HE IMPORTANCE OF DYNAMIC RISK CONSTRAINTS FOR LIMITED LIABILITY OPERATORS 12 delta for all ( t, x ). Hence only long position is ever required to replicate this claim. The sequence ofstrategies replicating the digital options which yield a utility level of sup s U ( s ) must also belong to A ( K i )as well. In this case, the dynamic risk i constraint is not effective. Similar results hold for the case of µ − rσ ≤ − M i < (cid:3) A dynamic risk constraint i ∈ { VaR , ES } restricts a tail-risk-seeking trader if and only if the (magnitudeof) Sharpe ratio is smaller than the constant M i . Surprisingly, from the definition of M i in (6) we seethat it does not depend on the risk limit level R at all but only the evaluation horizon ∆, confidence level α and interest rate r . In other words, increasing the risk limit alone is not sufficient to guarantee theeffectiveness of a dynamic risk measure. The risk manager must impose a short evaluation horizon window(small ∆) and emphasise on the extreme tail of the loss distribution (small α ) to ensure the necessary andsufficient condition of dynamic risk measure effectiveness | µ − rσ | < M i is satisfied. But given a dynamicrisk constraint is effective, the risk limit R will play a role in controlling the implied delta notional limitas per the expressions of k i and k i in Lemma 1.The main driver behind the effectiveness of a dynamic risk measure is that the risk constraint impliesa hard bound on the delta notional to be taken by the trader. Indeed, there is no economic differencebetween imposing a delta limit and a more complicated risk measure such as VaR or ES, as the followingcorollary shows. Corollary 1.
An effective dynamic risk constraint i ∈ { VaR , ES } is equivalent to imposing a delta notionallimit on the underlying risky asset. i.e. if a dynamic constraint i is effective, then there exists a boundedset D ⊆ R such that V i ( t, x ) = V d ( t, x ) := sup Π ∈A ( D ) E ( t,x ) [ U ( X T )] . Proof.
This follows immediately from Lemma 1. (cid:3)
The next proposition gives a theoretical characterisation of the value function.
Proposition 3.
Suppose the model parameters are such that | µ − rσ | < M i . Then the value function ofthe optimisation problem (7) under dynamic risk constraint i is the unique viscosity solution to the HJBequation − V t − H i ( x, V x , V xx ) = 0 , t < T, (12) subject to terminal condition V ( T, x ) = U ( x ) and linear growth condition V ( t, x ) ≤ c (1 + | x | ) for some c > . Here H i is the Hamiltonian defined as H i ( x, p, M ) := sup π ∈ K i (cid:26) [( µ − r ) π + rx ] p + σ M π (cid:27) . (13) HE IMPORTANCE OF DYNAMIC RISK CONSTRAINTS FOR LIMITED LIABILITY OPERATORS 13
Proof.
Provided that | µ − rσ | < M i , the set K i is bounded and hence by (11) we can deduce that V ( t, x ) ≤ α + β x for some constant α > β >
0. On the other hand, the utility function U is a negativeconvex increasing function on x <
0. Hence there exists α > β > U ( x ) > − α − β x − =: G ( x ) for all x . Then since ˜Π t = 0 for all t is an admissible strategy in A ( K i ), we have V i ( t, x ) = sup Π ∈A ( K i ) E ( t,x ) (cid:2) U ( X Π T ) (cid:3) ≥ E ( t,x ) h G ( X ˜Π T ) i = − α − β e r ( T − t ) x − for all ( t, x ). Thus we conclude V i ( t, x ) ≤ c (1 + | x | ) for some c >
0, i.e. the value function has at mosta linear growth. Finally, since K i is bounded the Hamiltonian in (13) is always finite. Standard theoryof stochastic control suggests that the value function V i is a viscosity solution to the HJB equation (12).Moreover, the solution is indeed unique in the class of viscosity solutions with linear growth due to strongcomparison principle. See Theorem 4.4.5 of Pham (2009). (cid:3) Proposition 3 provides a characterisation of the value function in terms of viscosity solution, whichserves as a useful basis for implementation of numerical methods to solve the HJB equation. In general, itis difficult to make further analytical progress to extract meaningful economic intuitions from the solutionstructure. Nonetheless, in Section 4 we will show that further characterisation of the optimal portfoliostrategy is indeed possible under a special case of µ = r .For now, we numerically solve the portfolio optimisation problem for the more general case of µ = r .Two specifications of utility function are considered: the Kahneman and Tversky (1979) piecewise powerform of U ( x ) = x β , x ≥ − k | x | β , x < < β , β < k >
0, and the piecewise exponential form of U ( x ) = φ (1 − e − γ x ) , x ≥ φ ( e γ x − , x < φ , φ , γ , γ > , T ] × R . As an approximation, we only solve for thenumerical solutions on a bounded domain [0 , T ) × [ − x min , x max ] for some large x min > x max > V ( t, x ) = U ( x ) is imposed along [0 , T ) × {− x min , x max } and then wefocus on the solution behaviours on a narrow range away from the boundary points. We observe thatthe numerical results are not sensitive to the choice of x min and x max provided that their values are HE IMPORTANCE OF DYNAMIC RISK CONSTRAINTS FOR LIMITED LIABILITY OPERATORS 14 sufficiently large. Second, we focus on a parameter choice of r = 0 to ensure that the “positive coefficientcondition” of the finite difference scheme (Condition 4.1 of Forsyth and Labahn (2007)) is satisfied whenthe step size along the x -axis is sufficiently small. But the more general case of non-zero interest rate canbe recovered by change of numeraire. −3 −2 −1 0 1 2 3x−3−2−1012 Value function V(t, x) t = 0t = 0.5t = 0.75U(x) (a)
S-shaped power utility function. −3 −2 −1 0 1 2 3x1.01.52.02.53.03.54.04.5
Optimal investment in risky asset Π * (t, x) t = 0t = 0.5t = 0.75 (b) S-shaped power utility function. −3 −2 −1 0 1 2 3x−1.5−1.0−0.50.00.5
Value function V(t, x) t = 0t = 0.5t = 0.75U(x) (c)
S-shaped exponential utility function. −3 −2 −1 0 1 2 3x3.003.253.503.754.004.254.504.75
Optimal investment in risky asset Π * (t, x) t = 0t = 0.5t = 0.75 (d) S-shaped exponential utility function.
Figure 1.
Value function and optimal investment level at different selected time pointsunder S-shaped power and exponential utility function. Parameters used are µ = 0 . σ = 0 . r = 0, T = 1, α = 0 . R = 1, ∆ = 30 / β = β = 0 . k = 2, γ = γ = 0 . φ = 1 and φ = 2.Figure 1 shows the value functions and the corresponding optimal investment levels at several differenttime points. In general, the agents will adopt the largest possible risk exposure when the portfoliovalue is negative due to risk-seeking over losses induced by the convex segment of the utility function.Investment level is the lowest when the portfolio value is at a small positive level. It is perhaps not toosurprising because local risk-aversion is typically the highest for small positive wealth level. Meanwhile,the investment behaviours for larger positive wealth depend on the precise utility function of the agents. Inthe piecewise power (i.e. constant relative risk aversion alike) specification, investment level increases with HE IMPORTANCE OF DYNAMIC RISK CONSTRAINTS FOR LIMITED LIABILITY OPERATORS 15 wealth until it hits the delta limit implied by the dynamic risk constraint. For the piecewise exponential(i.e. constant absolute risk aversion alike) specification, the investment level will flat out at a constantlevel as wealth increases. −3 −2 −1 0 1 2 3x4.04.55.05.56.0
Optimal investment in risky asset Π * (0, x) α = 0.01α = 0.025α = 0.05 (a) S-shaped power utility function. −3 −2 −1 0 1 2 3x4.254.504.755.005.255.505.756.006.25
Optimal investment in risky asset Π * (0, x) α = 0.01α = 0.025α = 0.05 (b) S-shaped exponential utility function.
Figure 2.
Optimal investment level at t = 0 under S-shaped power and exponentialutility function for different values of Expected Shortfall confidence level α . Based pa-rameters used are µ = 0 . σ = 0 . r = 0, T = 1, α = 0 . R = 1, ∆ = 30 / β = β = 0 . k = 2, γ = γ = 0 . φ = 1 and φ = 2.Figure 2 shows how the optimal investment level changes with the Expected Shortfall significance level.The results are intuitive: tighter the risk limit, more conservative the portfolio strategy.We can measure in monetary terms the impact of a dynamic risk constraint on both the tail-risk-seekingtrader and a risk averse manager who derives utility from the terminal value of the portfolio managed bythe trader. Under a given set of model parameters, the maximal expected utility of the trader V ( t, x )and the optimal trading strategy Π ∗ can be computed numerically. The certainty equivalent (CE) of thetrader (with capital x at time t ) is defined as the value C such that V ( t, x ) = U ( C ). Economically, it isthe fixed amount of wealth to be endowed by the trader to make him indifferent between this endowmentand the opportunity to trade under a dynamic risk constraint. Likewise, the CE of the manager is definedas the value of C solving E ( t,x ) [ U m ( X Π ∗ T )] = U m ( C ) where U m ( · ) is the concave utility function of themanager.As an example, consider a tail-risk-seeking trader with a unit of initial capital x = 1 and his utilityfunction has a piecewise power form. The risk averse manager has a utility function of U m ( x ) = − e − ηx and he imposes a dynamic ES constraint to risk-control the trader. Figure 3 shows the time-zero CE ofboth the trader and the manager as a function of the risk limit R .When R is very close to zero, the CE of the trader and the risk manager are both around unity which isthe initial trading capital. This is not surprising because under a very tight risk limit the trader essentially HE IMPORTANCE OF DYNAMIC RISK CONSTRAINTS FOR LIMITED LIABILITY OPERATORS 16
Certainty Equivalent (CE)
Trader's CEManager's CE
Figure 3.
The time-zero certainty equivalent of the trader (with piecewise power utilityfunction) and the risk averse manager (with exponential utility function) against the risklimit of a dynamic ES constraint. Parameters used are µ = 0 . σ = 0 . r = 0, T = 1, α = 0 .
01, ∆ = 30 / k = 2, α = α = 0 . η = 1 and x = 1.cannot purchase any risky asset. The portfolio under an admissible strategy is then almost riskless andthe CE simply becomes the initial capital available (multiplied by the interest rate factor).As R increases, the CE of the trader gradually increases because a larger value of R means the traderbecomes less risk-constrained and therefore must be better off economically. On the other hand, the CEof the manager first increases slightly but then drops significantly. The CE of the manager improves atthe beginning because a small but non-zero risk limit encourages the trader to invest conservatively inthe risky asset which in turn creates value for the risk averse manager. However, when the risk limit isfurther relaxed, the trader takes more and more risk which starts becoming detrimental to the risk aversemanager. Once R goes above around 130% of the initial capital, the CE of the manager goes below unitymeaning that the trading activity now causes value destruction from the perspective of the manager.Indeed, when R becomes arbitrarily large, Proposition 1 implies that the trader’s CE will go to positiveinfinity while Theorem 5.4 of Armstrong and Brigo (2019) suggests the CE of the risk averse manager willbecome negative infinity. Figure 3 highlights the conflict of interests between a tail-risk-seeking traderand a risk averse manager, and a slack risk management policy could easily result in drastic economiclosses faced by the bank. 4. A special case of zero excess return
Proposition 3 provides a theoretical characterisation of the value function. However, it does not tellus much about the behaviours of the optimal portfolio strategy. In this section, we focus on a setup with µ = r = 0 (the assumption of r = 0 is imposed for convenience only. The slightly more general case of HE IMPORTANCE OF DYNAMIC RISK CONSTRAINTS FOR LIMITED LIABILITY OPERATORS 17 µ = r can be handled by a change of numeraire technique.) The key idea is that in this special case wecan exploit an equivalence between the risk-constrained portfolio optimisation problem and an optimalstopping problem. We show that the optimal trading strategy can be characterised in terms of a stoppingtime. As we will see soon, the state-space [0 , T ] × R of the problem can be split into two regions: a tradingregion where the maximum possible amount is invested in the risky asset (Π ∗ = k for some constant k )and a no-trade region where the agent opts to hold a pure cash position (Π ∗ = 0).As a preliminary discussion, investment motive vanishes in the case of µ = r = 0. Then whether thetrader would participate in a fair gamble is purely driven by his risk appetite. Due to the S-shaped utilityfunction, the trader is risk seeking over the domain of losses whereas he is risk averse over the domain ofgains. Simple economic intuitions suggest that the trader prefers to gambling when the portfolio value islow, and prefers to taking all the risk off when the portfolio value is high. We therefore postulate thatthe optimal portfolio strategy has a “bang-bang” feature where the agent invests the maximum possibleamount in the risky asset when the portfolio value is low. Once the portfolio value becomes sufficientlyhigh, the trader’s risk aversion dominates and he will immediately liquidate the entire holding in the riskyasset. The postulated strategy can be stated in terms of a stopping time: the portfolio value evolves asa Brownian motion with maximum volatility (under the most aggressive admissible strategy) and stopswhen the agent decides to sell his entire risky asset holding and the portfolio value will remain unchangedthereafter. This inspires us to consider a simple optimal stopping problem introduced in the followingsubsection.4.1. An optimal stopping problem.
We introduce below an optimal stopping problem and verify someproperties of its solution structure. Towards the end of this subsection, we will show that this optimalstopping problem and the risk-constrained portfolio optimisation problem (7) are indeed equivalent. Beforeproceeding, we need to impose some slightly stronger assumptions on the utility function U throughoutthis section. Assumption 2.
The utility function U : R → R is a continuous, strictly increasing and strictly concave(resp. convex) C function on x > (resp. x < ) with U (0) = 0 and lim x → + ∞ U ′ ( x ) = 0 . Proposition 4.
Suppose X = ( X t ) t ≥ has the dynamics of dX t = νdB t where ν > is a constant.Define an optimal stopping problem W ( t, x ) := sup τ ∈T t,T E ( t,x ) [ U ( X τ )] (14) where T t,T is the set of F t -stopping times valued in [ t, T ] . The value function of problem (14) is the uniqueviscosity solution to the HJB variational inequality min (cid:16) − W t − ν W xx , W − U (cid:17) = 0 , t < T ; W ( T, x ) = U ( x ) , t = T. (15) HE IMPORTANCE OF DYNAMIC RISK CONSTRAINTS FOR LIMITED LIABILITY OPERATORS 18
Define the continuation set C and the stopping set S as C := { ( t, x ) ∈ [0 , T ) × R : W ( t, x ) > U ( x ) } , S := { ( t, x ) ∈ [0 , T ] × R : W ( t, x ) = U ( x ) } . (16) The optimal stopping time is given by τ ∗ = inf { u ≥ t : ( u, X u ) ∈ S} .Proof. The relationship between the solution of an optimal stopping problem and the viscosity solutionof the corresponding HJB variational inequality as well as the characterisation of the optimal stoppingrule are standard - see for example Øksendal and Reikvam (1998). Note that the techniques used in theproofs of Proposition 2 and 3 can be adopted here to show that the value function W ( t, x ) has at most alinear growth in x , which in turn confirms the uniqueness of the viscosity solution. (cid:3) The below important result characterises the optimal stopping region in a more economically intuitivemanner. In particular, the optimal stopping rule is a simple time-varying threshold strategy where theagent stops the process when its value is sufficiently high.
Proposition 5.
There exists a continuous and decreasing function b : [0 , T ) → (0 , ∞ ) with lim t ↑ T b ( t ) = 0 such that the stopping set in (16) admits a representation of S = { ( t, x ) ∈ [0 , T ) × R : x ≥ b ( t ) } ∪ {{ T } × R } . (17) Proof.
See the appendix. (cid:3)
Finally, we verify the equivalence of the portfolio optimisation problem (7) and the optimal stoppingproblem (14) under µ = r = 0. Proposition 6.
Suppose µ = r = 0 . For i ∈ { VaR , ES } , let V i be the value function of the portfoliooptimisation problem (7) under K = K i . Then V i ( t, x ) = W ( t, x ; k i ) (18) where k VaR = − R Φ − ( α ) √ ∆ > , k ES = Rαφ (Φ − ( α )) √ ∆ > , (19) and W ( t, x ; ν ) is the value function of the optimal stopping problem (14) with diffusion constant ν . More-over, an optimal portfolio strategy is Π ∗ t = k i σ ( X t
Proof.
When µ = r = 0, Lemma 1 implies that the set K i simplifies to K i = [ − k i σ , k i σ ] where the k i ’s aredefined in (19). Moreover, the Hamiltonian in (12) becomes H i ( p, M ) = H i ( M ) = sup π ∈ [ − kiσ , kiσ ] σ M π = ( k i ) M, M ≥ , M < . To verify (18), it is sufficient to show that W ( t, x ; ν ), the solution to (14), is also a solution to (12)under the choice of ν = k i . For ( t, x ) ∈ C , we have W xx ≥ W t + H ( W xx ) = W t + ( k i ) W = W t + ν W = 0. For ( t, x ) ∈ S , we have W ( t, x ) = U ( x ) and W t + ( k i ) W xx = W t + ν W xx ≤ W t = 0 and W xx ≤
0. Then W t + H ( W xx ) = 0 + 0 = 0. Hence W solves (12).The candidate strategy Π ∗ defined by (20) is clearly in the admissible set A ( K i ). To verify its optimality,one can compute E ( t,x ) [ U ( X Π ∗ T )] and show that it attains the same value as W ( t, x ). But it is clear since X Π T = X t + Z Tt σ Π ∗ s dB s = X t + Z Tt k i ( X s s ) dB s = X t + ν Z τ ∗ t dB s = X t + ν ( B τ ∗ − B t )where the portfolio process coincides (in distribution) with the optimally stopped process in problem(14). (cid:3) Figure 4 gives a stylised plot of the optimal portfolio strategy. t (Time) X t (Portfolio value) b ( t ) T ∗ t = k i σ No-trade region Π ∗ t = 0 Figure 4.
A graphical illustration of the optimal portfolio strategy under the specialcase µ = r = 0. When the portfolio value is low, the agent invests the maximum possibleamount in the risky asset by taking Π ∗ t = k i /σ , whereas when the portfolio value is highthe agent takes all the risk off and sets Π ∗ t = 0. The critical boundary between risk-onand risk-off is given by a non-negative continuous, and decreasing function b ( t ). HE IMPORTANCE OF DYNAMIC RISK CONSTRAINTS FOR LIMITED LIABILITY OPERATORS 20
Comparative statics.
Some comparative statics can be established to shed light on the policyimplications of the dynamic risk constraint. We begin by offering a useful lemma.
Lemma 2.
Consider problem (14) and let b ( t ; ν ) be the optimal stopping boundary defined in (17) undera fixed diffusion constant ν . Then b ( t ; ν ) is increasing in ν .Proof. Let W ( j ) be the value function and b j ( t ) be the corresponding optimal stopping boundary associ-ated with problem (14) under parameter ν j . Similarly, define G j f := − f t − ν j f xx . Fix ν > ν and define ¯ W := W (2) − W (1) .On x ≥ b ( t ), we have ¯ W ( t, x ) = W (2) ( t, x ) − W (1) ( t, x ) = W (2) ( t, x ) − U ( x ) ≥
0. On x < b ( t ), wehave G W (1) = 0 and G W (2) ≥ ≤ G W (2) − G W (1) = − W (2) t + W (1) t − ν W (1) xx + ν W (1) xx = − ∂∂t ( W (2) − W (1) ) − ν ∂ ∂x ( W (2) − W (1) ) − ν − ν W (1) xx = G ¯ W − ν − ν W (1) xx and therefore G ¯ W ≥ ν − ν W (1) xx ≥ x < b ( t ) since W (1) xx ≥ W ( t, x ) ≥ x ≥ b ( t ) and ¯ W ( T, x ) = U ( x ) − U ( x ) = 0, it follows from maximum principle that W (2) − W (1) = ¯ W ≥ x < b ( t ). Hence W (2) ( t, x ) ≥ W (1) ( t, x ) > U ( x ) on x < b ( t ) from whichwe can conclude b ( t ) ≥ b ( t ), i.e. b ( t ; ν ) is increasing in ν . (cid:3) Proposition 7.
In the special case of µ = r = 0 , denote by b ( t ; θ ) the trading boundary associated withthe optimal strategy of the VaR/ES-constrained problem (7) introduced in Proposition 6 under a particularmodel parameter θ . For t being fixed, we have the following:(1) b ( t ; T ) is increasing in the trading horizon T ;(2) b ( t ; R ) is increasing in the risk limit level R ;(3) b ( t ; α ) is increasing in the significance level of the VaR/ES measure α ;(4) b ( t ; ∆) is decreasing in the risk evaluation window ∆ ;(5) b ( t ; σ ) does not depend on σ .Proof. From Proposition 6, the trading boundary of the optimal strategy is given by b ( t ; k i ) which can becharacterised by the optimal stopping boundary of problem (14) with parameter ν = k i . HE IMPORTANCE OF DYNAMIC RISK CONSTRAINTS FOR LIMITED LIABILITY OPERATORS 21
Property 1 can simply be inferred from the fact that b ( t ) is decreasing in t , and Property 2 to 5immediately follow from Lemma 2 by observing that both k VaR = − R Φ − ( α ) √ ∆ and k ES = Rαφ (Φ − ( α )) √ ∆ are increasing in R and α (for α < . σ . (cid:3) Recall the optimal strategy is in form of Π ∗ t = k i σ ( X t
In the context of portfolio optimisation under a complete market, it is typically not important todistinguish a “delta-one” trader (who is constrained to trade only in the underlying stock and a risk-freeaccount) and a derivatives trader (who can purchase any payoff structure contingent on the underlyingstock price). It is because market completeness implies that perfect replication of any arbitrary claim isfeasible and hence derivatives securities are redundant. This insight is exploited heavily to facilitate themartingale duality method where a dynamic portfolio selection problem is converted into a static problemof optimal payoff design.Our main results in Section 3 and 4 apply to a delta-one trader, in which case the expected shortfallof the portfolio is determined by the delta of the portfolio and this is a key ingredient in our calculation.However, the results will change drastically if the trader has access to the derivatives market. Atrader with limited liability who is allowed to purchase arbitrary derivative securities at the Black-Scholesprice will be able to achieve arbitrarily high expected utilities under any expected shortfall constraint bypursuing a martingale type strategy. The essential idea is to use Theorem 4.1 of Armstrong and Brigo(2019) to find a derivative which comfortably meets the expected shortfall constraint and provides thedesired utility. If at some future point the market moves so that the expected shortfall constraint hits thelimit, then the trader may apply Theorem 4.1 Armstrong and Brigo (2019) to find a new derivative whichstill yields the desired expected utility and which ensures that the constraint again comfortably met. Itis possible to construct a strategy so that with probability 1, the trader will only need to rebalance theirportfolio in this way a finite number of times. We give a proof of this in Appendix B.Why is a dynamic risk constraint effective against a delta-one trader but not a derivatives trader? It isbecause the replication of large quantities of out-of-money digital options will involve trading a massive
HE IMPORTANCE OF DYNAMIC RISK CONSTRAINTS FOR LIMITED LIABILITY OPERATORS 22 notional of the underling stock in the bad state of the world, which the delta-one trader understands ex-ante will not be feasible under a given dynamic risk constraint. In contrast, the feasibility of a derivativeposition only depends on the current statistical profile of the payoff. The derivatives trader can thereforeexploit the blindspot of a risk measure to ensure the massive tail-risk is not detected. Finally, thepossibility to roll-over a derivative position allows the risk constraint to be satisfied throughout the entiretrading horizon.One might ask what alternative types of risk limits would be effective against such a trader. Expectedutility constraints give one possible answer. For example, one can choose a concave increasing function U m of the form U m ( x ) = − ( − x ) γ x ≤ for γ ∈ (1 , ∞ ) and require that at each time E [ U m ( X ( t ) t +∆ )] ≥ R where X ( t ) s is the time- s value of the derivatives portfolio held by the trader at time t and R ∈ ( −∞ , T needed to achieve such a utility constraint. This would implies that thetrading strategy must achieve a minimum expected u M at time T and one may then apply Theorem 5.3of Armstrong and Brigo (2019). 6. Concluding remarks
While VaR and ES are widely adopted by practitioners, the impact of such risk constraints on traders’behaviours are not necessarily well understood. This paper addresses the negative result of Armstrong and Brigo(2019) that a static VaR/ES measure does not work at all on a tail-risk-seeking trader. Our key resulthighlights that dynamic monitoring of the trading positions is crucial. Continuous re-evaluation of port-folio exposure demands traders to respect a delta notional limit at all time. This alone is sufficient todiscourage excessive risk taking during market distress which is naturally attractive to a tail-risk-seekingtrader.However, the dangerous combination of tail-risk-seeking preference and derivatives trading can posechallenges to risk management. The possibility to rebalance a derivative position allows the trader topursue a martingale strategy where the trading losses and risk limit breaches can be indefinitely deferred.As the possible alternatives to statistical measures like VaR or ES, utility-based risk measures or otherscenario-based assessments such as stress testing might be the superior tools for risk managing derivativestraders. It will be of both theoretical and practical interests to further explore the desirable features ofan effective risk control mechanism which performs well beyond delta-one trading.
HE IMPORTANCE OF DYNAMIC RISK CONSTRAINTS FOR LIMITED LIABILITY OPERATORS 23
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Appendix A. Proofs
We first provide some prior properties of the value function (14) in the following lemma.
Lemma 3.
The value function (14) has the following properties:(1) W ( t, x ) is continuous in t and x .(2) W ( t, x ) is decreasing in t and is increasing in x ;(3) W ( t, x ) ∈ C , ( C ) with W xx ( t, x ) ≥ for all ( t, x ) ∈ C where C is defined in (16) .Proof. Property 1 is due to the standard comparison principle of viscosity solution. Property 2 can beeasily inferred from the structure of the optimal stopping problem. Here we will prove Property 3.
HE IMPORTANCE OF DYNAMIC RISK CONSTRAINTS FOR LIMITED LIABILITY OPERATORS 25
Fix a bounded open domain O in C and consider a boundary value problem G f := − f t − ν f xx = 0 , ( t, x ) ∈ O f = W, ( t, x ) ∈ ∂ O (21)Since the operator G is linear, standard PDE theory suggests that there exists a unique smooth solution f ∈ C , to (21) on O . But this f also solves (15) on O . By uniqueness of the viscosity solution, wededuce W = f on O such that W ∈ C , ( O ). Finally, C is an open set and thus by the arbitrariness of O the smoothness property of W can be extended to the entire C .Thanks to the C , ( O ) property, (15) can be interpreted in the classical sense such that on C we have ν W xx = − W t ≥ W is decreasing in t . (cid:3) Proof of Proposition 5.
We first prove a preliminary result that [0 , T ) × ( −∞ , ⊆ C , i.e. it is alwayssuboptimal to stop on the negative regime before the terminal time. Suppose on contrary there exists( t ′ , x ′ ) with 0 ≤ t ′ < T and x ′ < t ′ , x ′ ) ∈ S . Then W ( t ′ , x ′ ) = U ( x ′ ). Now consider analternative stopping rule τ ǫ := inf { s ≥ t ′ : X s / ∈ ( x ′ − ǫ, x ′ + ǫ ) } ∧ T for some 0 < ǫ < − x ′ . Let p ǫ := P ( τ ǫ < T ). Then W ( t ′ , x ′ ) ≥ E ( t ′ ,x ′ ) [ U ( X τ ǫ )] = P ( τ ǫ < T, X τ ǫ = x ′ − ǫ ) U ( x ′ − ǫ ) + P ( τ ǫ < T, X τ ǫ = x ′ + ǫ ) U ( x ′ + ǫ )+ P ( τ ǫ = T ) E [ U ( X T ) | τ ǫ = T ]= 12 U ( x ′ − ǫ ) + 12 U ( x ′ + ǫ ) + (1 − p ǫ ) (cid:26) E [ U ( X T ) | τ ǫ = T ] − U ( x ′ − ǫ ) − U ( x ′ + ǫ ) (cid:27) > U ( x ′ ) + (1 − p ǫ ) (cid:26) U ( x ′ − ǫ ) − U ( x ′ − ǫ ) − U ( x ′ + ǫ ) (cid:27) . using the strict convexity of U on x < X . It is not hardto observe that p ǫ ↑ ǫ ↓
0. We immediately obtain the required contradiction W ( t ′ , x ′ ) > U ( x ′ ) = W ( t ′ , x ′ ).The rest of the proof goes as follows:(i) Existence and non-negativity of b :We first show that W ( t, x ) − U ( x ) is decreasing in x over x ≥ t ∈ [0 , T ). Fix an arbitrary β > F ( t, x ) := W ( t, x + β ). By the linear structure of the underlying Brownianmotion, it can be easily seen that F ( t, x ) = sup τ ∈T t,T E [ U ( X τ + β )] and hence F is the (unique)viscosity solution to min {G F, F − U ( x + β ) } = 0 , t < T ; F ( t, x ) = U ( x + β ) , t = T, (22) HE IMPORTANCE OF DYNAMIC RISK CONSTRAINTS FOR LIMITED LIABILITY OPERATORS 26 where G f := − f t − ν f xx = 0. Let G ( t, x ) := W ( t, x ) + U ( x + β ) − U ( x ). Then whenever G W = 0,we have G G = G W + G U ( x + β ) − G U ( x ) = ν U ′′ ( x ) − U ′′ ( x + β )) ≥ U is concave, and whenever W ( t, x ) = U ( x ) we have G ( t, x ) = U ( x + β ). Moreover, G ( T, x ) = U ( x + β ). Hence G is a supersolution to (22). By maximum principle (in a viscosity sense), wededuce G ≥ F leading to W ( t, x ) − U ( x ) ≥ W ( x + β ) − U ( x + β ) , (23)i.e. W ( t, x ) − U ( x ) is decreasing.Now we show that that for each fixed t ∈ [0 , T ) there always exists x such that W ( t, x ) = U ( x ). Such x , if exists, must be strictly positive since it is suboptimal to stop in the negativeregime. Then together with the fact that W ( t, x ) − U ( x ) is decreasing in x on x ≥
0, we concludethere exists a unique b ( t ) ∈ (0 , ∞ ) such that W ( t, x ) = U ( x ) ⇐⇒ x ≥ b ( t ). This will be sufficientto justify the existence of a positive boundary function b which characterises the stopping set (17).To complete the proof, suppose on contrary that W ( t, x ) > U ( x ) for all x . Then { t } × R ∈ C onwhich W is a C increasing convex function in x . W ( t, x ) − U ( x ) being decreasing in x now implies W x ( t, x ) ≤ U ′ ( x ) for all x . In turn lim s →∞ W x ( t, s ) ≤ lim s →∞ U ′ ( s ) = 0 and hence W ( t, x ) mustbe a constant independent of x . But with W ( t, x ) > U ( x ) this must imply W ( t, x ) = sup s U ( s )for all ( t, x ) ∈ [0 , T ) × R . This can easily shown to be false based on the same ideas used in theproof for Proposition 2.(ii) Monotonicity of b : Consider ( t , x ) ∈ S such that W ( t , x ) = U ( x ). Then for any t > t ,we have 0 ≤ W ( t , x ) − U ( x ) ≤ W ( t , x ) − U ( x ) = 0 since W ( t, x ) is decreasing in t . Hence W ( t , x ) = U ( x ) and ( t , x ) ∈ S as well such that b ( t ) must be decreasing.(iii) Continuity of b : We begin by showing that b is right-continuous. Fix t < T and consider adecreasing sequence ( t n ) n ≥ with t n ↓ t . Then for each n we have ( t n , b ( t n )) ∈ S . Since the set S is closed, we have ( t, b ( t +)) ∈ S as well such that b ( t +) ≥ b ( t ). But b ( t +) ≤ b ( t ) as b is decreasing.We hence conclude b ( t ) = b ( t +).Now we show that b ( t ) is left-continuous. Suppose on contrary that there exists t < T suchthat b ( t − ) > b ( t ). Define ξ := b ( t − )+ b ( t )2 such that 0 ≤ b ( t ) < ξ < b ( t − ). Choose s ∈ (0 , t ) andthen 0 ≤ b ( t ) < b ( t − ) ≤ b ( s ). By definition of b ( · ) and the smooth-pasting property, we have W ( s, b ( s )) = U ( b ( s )) and W x ( s, b ( s )) = U ′ ( b ( s )). Then W ( s, ξ ) − U ( ξ ) = Z ξb ( s ) ( W x ( s, y ) − U ′ ( y )) dy = Z ξb ( s ) Z yb ( s ) ( W xx ( s, z ) − U ′′ ( z )) dzdy Smooth-pasting must hold at b ( s ) because b ( s ) > U ′ ( x ) exists for any x >
0. See Peskir and Shiryaev (2006).
HE IMPORTANCE OF DYNAMIC RISK CONSTRAINTS FOR LIMITED LIABILITY OPERATORS 27 = Z b ( s ) ξ Z b ( s ) y ( W xx ( s, z ) − U ′′ ( z )) dzdy ≥ Z b ( s ) ξ Z b ( s ) y Cdzdy = C ξ − b ( s )) for some constant C > s where we have used Lemma 3 that W xx ≥ C and U is a strictly concave C function on the positive domain. Since W ( s, x ) is continuous in s , ifwe let s ↑ t we deduce W ( t, ξ ) − U ( ξ ) ≥ C ξ − b ( t − )) > . But b ( t ) < ξ and hence ( t, ξ ) ∈ S which implies W ( t, ξ ) = U ( ξ ). We arrive at the requiredcontradiction.(iv) Limiting behaviour of b : Suppose b ( T ) := lim t ↑ T b ( t ) >
0. Then let ξ := b ( T )2 > W ( s, ξ ) − U ( ξ ) > C ( ξ − b ( s )) for someconstant C > s < T . Contradiction can be obtained again by letting s ↑ T on recalling theterminal condition that W ( T, x ) = U ( x ) for all x . (cid:3) Appendix B. Ineffectiveness of dynamic Expected Shortfall constraint on a derivativetrader
We will assume that U ( x ) = 0 for all negative x , so we are specializing to the case of an investor withlimited liability. We will consider a Black–Scholes market with µ > r .Consider a derivatives trader who has a given budget and who wishes to purchase a sequence ofEuropean options all with maturity T to maximize their expected utility. We suppose that this traderis subject only to a self-financing constraint and that at all times t ∈ [0 , T ) they must meet an expectedshortfall constraint at confidence level α with time interval T − t . We will show that such a trader canachieve an expected utility greater than or equal to u for any u < sup U .Let f : (0 , ∞ ) → R be the payoff function of a European derivative with maturity T . We will write P ( f, S, t ), U ( f, S, t ) and ES α ( f, S, t ) for the price, expected U utility and expected shortfall of this de-rivative at time t given that S t = S (the time horizon for the expected shortfall calculation being theremaining time to maturity, T − t ).Given constants k, h, ℓ ∈ R with ℓ < < h and we define a digital payoff function f k,h,ℓ by f k,h,ℓ ( S T ) = h S T ≥ e k ℓ otherwise . We will write D for the set of all such digital options. HE IMPORTANCE OF DYNAMIC RISK CONSTRAINTS FOR LIMITED LIABILITY OPERATORS 28
The proof of Theorem 4.1 in Armstrong and Brigo (2019) shows that we may find f k,h,ℓ ∈ D satisfying P ( f k,h,ℓ , S, t ) ≤ − α ( f k,h,ℓ , S, t ) ≤ − U ( f k,h,ℓ , S, t ) ≥ u (24)for any u < sup U . Moreover, we may take k to be arbitrarily small and one may also require | ℓ | > h . Weprove an extension of this result in the lemma below. Lemma 4.
Given any constants λ , λ with λ > and λ ∈ (0 , µ − r ) , there exists f k,h,ℓ ∈ D such that P ( f k,h,ℓ , Se λ ( T − t ) , t ) ≥ P ( f k,h,ℓ , S, t ) = − α ( f k,h,ℓ , Se − λ ( T − t ) , t ) ≤ − U ( f k,h,ℓ , S, t ) ≥ u (25) for any u < sup U . The k can be arbitrarily small.Proof. Calculating the price, expected shortfall and expected utility explicitly one finds that P ( f k,h,ℓ , S, t ) = e − r ( T − t ) (cid:18) ℓ Φ (cid:18) k − ln S − ν ( T − t ) σ √ T − t (cid:19) + h (cid:18) − Φ (cid:18) k − ln S − ν ( T − t ) σ √ T − t (cid:19)(cid:19)(cid:19) ES α ( f k,h,ℓ , S, t ) = − α (cid:18) ℓ Φ (cid:18) k − ln S − ˜ ν ( T − t ) σ √ T − t (cid:19) + h (cid:18) α − Φ (cid:18) k − ln S − ˜ ν ( T − t ) σ √ T − t (cid:19)(cid:19)(cid:19) U ( f k,h,ℓ , S, t ) = U ( h ) (cid:18) − Φ (cid:18) k − ln S − ˜ ν ( T − t ) σ √ T − t (cid:19)(cid:19) (26)for k < ln S + ˜ ν ( T − t ) + σ √ T − t Φ − ( α ), where ν := r − σ and ˜ ν := µ − σ . Fix h >
0. The budgetconstraint P ( f k,h,ℓ , S, t ) = − ℓ in terms of k and h via ℓ = h − h + e r ( T − t ) Φ (cid:16) k − ln S − ν ( T − t ) σ √ T − t (cid:17) . Then P ( f k,h,ℓ , Se λ ( T − t ) , t ) and ES α ( f k,h,ℓ , Se − λ ( T − t ) , t ) can be written as P ( f k,h,ℓ , Se λ ( T − t ) , t ) = e − r ( T − t ) h − ( h + e r ( T − t ) ) Φ (cid:16) k − ln S − λ ( T − t ) − ν ( T − t ) σ √ T − t (cid:17) Φ (cid:16) k − ln S − ν ( T − t ) σ √ T − t (cid:17) , ES α ( f k,h,ℓ , Se − λ ( T − t ) , t ) = − h + h + e r ( T − t ) α Φ (cid:16) k − ln S + λ ( T − t ) − ˜ ν ( T − t ) σ √ T − t (cid:17) Φ (cid:16) k − ln S − ν ( T − t ) σ √ T − t (cid:17) . For as long as λ > λ ∈ (0 , µ − r ), we havelim k →−∞ P ( f k,h,ℓ , Se λ ( T − t ) , t ) = e − r ( T − t ) h, lim k →−∞ ES α ( f k,h,ℓ , Se − λ ( T − t ) , t ) = − h, lim k →−∞ U ( f k,h,ℓ , S, t ) = U ( h ) . The result immediately follows since h is arbitrary. (cid:3) HE IMPORTANCE OF DYNAMIC RISK CONSTRAINTS FOR LIMITED LIABILITY OPERATORS 29
Let us write d ν ( S, k, ξ ) := σ − ( ξ − ( k − ln S ) − νξ ) . This will have a positive partial derivative with respect to ξ whenever k < ln S − νξ . Hence if choose k such that k < ln S − | ν | T = ln S − | µ − σ | T then d ν ( S, k, ξ ) will be an increasing function of ξ in the range ξ ∈ (0 , √ T ] . Since ES α ( f k,h,ℓ , S, t ) = − α h ℓ Φ( d ν ( S, k, √ T − t )) + h ( α − Φ( d ν ( S, k, √ T − t ))) i we see that for sufficiently small k , when | ℓ | > | h | , ES α ( f k,j,ℓ , S, t ) will be decreasing in t for t ∈ [0 , T )(consider S and f k,h,ℓ as fixed). Similarly, we can deduce P ( f k,h,ℓ , S, t ) is increasing in t for t ∈ [0 , T ) forsufficiently small k . It now follows from Lemma 4 that given S t = S ∈ R ≥ and t ∈ [0 , T ) we may find f k,h,ℓ ∈ D satisfying P ( f k,h,ℓ , Se λ ( T − t ) , s ) ≥ P ( f k,h,ℓ , S, t ) ≤ − α ( f k,h,ℓ , Se − λ ( T − t ) , s ) ≤ − U ( f k,h,ℓ , S, t ) ≥ u. (27)for all s ∈ [ t, T ), λ > λ ∈ (0 , µ − r ) and u < sup U .Suppose that at a given time t the stock price is S t . By purchasing an arbitrarily large quantity of theoption with payoff f k,h,ℓ satisfying (27) we can meet any cost or expected shortfall constraint at time t . Bychoosing k sufficiently small we may ensure that the probability this option has a negative payoff is as smallas we like. Thus we may find an option that meets our current budget, meets a given expected shortfalland ensures that the expected utility for the trader is greater than or equal to u < sup U . Furthermore,when the stock price rises to S t e λ ( T − t ) the option position can be liquidated for an arbitrarily highpositive value. On the other hand, this option will continue to meet the expected shortfall constraintuntil the stock price falls to S t e − λ ( T − t ) . When this occurs, the trader can opt to rebalance the optionposition. Let π be the probability that a rebalancing occurs, which is the probability that the stock pricelevel visits S t e − λ ( T − t ) before S t e λ ( T − t ) in the time interval [ t, T ]. By the scaling properties of geometricBrownian motion, one can show that if µ ≥ σ / π is bounded above by the constant λ / ( λ + λ ). We define a sequence of stopping times ( t i ) i ∈ N inductively as follows. We define t = 0 and construct f ∈ D such that equation (27) holds. Let t be the smaller of T and the first time t ′ > t satisfying S t ′ = S t e − λ ( T − t ) or S t ′ = S t e λ ( T − t ) . If S t = S t e λ ( T − t ) , the option is liquidated at a positive For the case of µ < σ /
2, an upper bound can be derived by using the fact that the probability of a drifting Brownianmotion X t := ηt + σB t hitting level − a before b over an infinite horizon is − exp( − ησ b )exp( ησ a ) − exp( − ησ b ) . HE IMPORTANCE OF DYNAMIC RISK CONSTRAINTS FOR LIMITED LIABILITY OPERATORS 30 value. Then the trader deposits the proceed in the riskfree account and no further action is taken untilthe maturity time t = T (equivalent to purchasing a claim with constant payoff equal to the future valueof the trader’s current wealth). Else if S t = S t e − λ ( T − t ) , the trader roll into a new position f ∈ D satisfying equation (27) using the current market value of f at t = t as the new initial budget.Once t i has been defined and the position is not yet liquidated, we choose f i ∈ D such that equation(27) holds again for S = S t i and t = t i . We define t i +1 to be the smaller of T and the first time t ′ > t i satisfying S t ′ = S t i e − λ ( T − t i ) or S t ′ = S t i e λ ( T − t i ) . At each time t i < T where the optionis not liquidated, the trader may rebalance by purchasing α > f i toguarantee that u i := U ( αf i , S t , t ) ≥ u , to meet the budget constraint and to ensure that expected shortfallconstraint will hold until the stopping time t i +1 . If the option is liquidated at t = t i , the resulting proceedis arbitrarily large by construction of f i − and hence depositing this amount of wealth in the riskfreeaccount can deliver any (riskfree) utility level u at maturity.The probability that the trader needs to rebalance the portfolio n or more times is π n . So withprobability 1, the trader will only need to rebalance finitely often. The expected utility conditioned onthe stock price at time t will be an increasing function of S t . Hence the expected utility conditioned onrebalancing the portfolio exactly n times will be greater than or equal to u n , since rebalancing only occursif the stock price drops. Hence the overall expected utility of the strategy will be greater than or equalto u . Our assumption of limited liability ensures that our trader is unconcerned by events of probability0, so it is acceptable to assign an expected utility to this strategy even though it is a martingale strategy.This shows that a derivatives trader subject only to an expected shortfall constraint with the expectedshortfall always calculated at time T is able to achieve any expected utility less than sup U . We nowconsider instead a trader who invests on a time interval [0 , T ] subject to to expected shortfall constraintswith a fixed time interval ∆ ≤ T used to calculate the expected shortfall. This trader may choose anyacceptable strategy until T − ∆. Over the time interval [ T − ∆ , T ) they may pursue an essentially identicalstrategy to the one given above except using derivatives whose payoff at time t + ∆ is given by a function f k,h,ℓ ( S T ) for some f k,h,ℓ ∈ D . Hence such a trader can also achieve a utility greater than or equal to u . Appendix C. Extension to multiple risky assets
We discuss how several key results in this paper will change when there are multiple risky asset.Suppose there are n risky assets in the Black-Scholes economy and the price process vector S t :=( S (1) t , S (1) t , . . . , S ( n ) t ) ′ has the dynamics dS t = Diag ( S t ) µdt + σdB t where Diag ( S t ) := Diag ( S (1) t , S (2) t , . . . , S ( n ) t ) is a n × n diagonal matrix, µ is a n × σ is a n × m constant matrix and B is a m -dimensional Brownian motion. The portfolio strategy Π is now HE IMPORTANCE OF DYNAMIC RISK CONSTRAINTS FOR LIMITED LIABILITY OPERATORS 31 valued in R n and the portfolio value process X has dynamics of dX t = [ rX t + Π ′ t ( µ − re )] dt + Π ′ t σdB t (28)with e being a vector of unity. Following a similar derivation as in Section 2, if the risk manager assumesthe risky assets holding Π is held fixed over the risk evaluation horizon then the projected portfolio losson [ t, t + ∆] is a normal random variable L t with E [ L t ] = − e r ∆ − r Π ′ t ( µ − re ) , Var( L t ) = e r ∆ − r Π ′ t ΣΠ t where Σ := σσ ′ is the variance-covariance matrix of the risky assets return.It is now straightforward to write down the new admissible sets under the VaR and ES constraint as K VaR := ( π ∈ R n : − e r ∆ − r π ′ ( µ − re ) − Φ − ( α ) r e r ∆ − r π ′ Σ π ≤ R ) (29)and K ES := ( π ∈ R n : − e r ∆ − r π ′ ( µ − re ) + φ (Φ − ( α )) α r e r ∆ − r π ′ Σ π ≤ R ) (30)respectively. The portfolio optimisation problem is to solve V ( t, x ) := sup Π ∈A ( K ) E ( t,x ) [ U ( X T )] (31)where A ( K ) := { Π : R T | Π t | dt < ∞ P − a.s, Π( t, ω ) ∈ K L ⊗ P -a.e. ( t, ω ) } subject to the dynamics(28). A dynamic VaR and ES constraint can be incorporated by the choice of K = K VaR and K ES .Now we show that Proposition 2 can be extended to the multi-asset setup. Proposition 8.
For the optimisation problem (31) , if the set K is bounded then for every t < T thereexists x such that V ( t, x ) < sup s U ( s ) Proof.
Based on the same ideas in the proof of Proposition 2, for any S-shaped U there exists some m > C > U ( x ) ≤ C + mx + =: ¯ U ( x ). For as long as K is bounded, there exists b > K ⊆ { ( π , π , . . . , π n ) ′ ∈ R n : − b ≤ π i ≤ b ∀ i = 1 , , . . . , n } =: ¯ K. Let ¯ µ := b P ni =1 | µ i − r | . Given X t = x , for any Π ∈ A ( ¯ K ) we have X Π T = xe r ( T − t ) + Z Tt e r ( T − s ) Π ′ t ( µ − re ) ds + Z Tt e r ( T − s ) Π ′ s σdB s ≤ xe r ( T − t ) + ¯ µ Z Tt e r ( T − s ) ds + Z Tt e r ( T − s ) Π ′ s σdB s =: ¯ X Π T almost surely. Hence V ( t, x ) ≤ sup Π t ∈A ( ¯ K ) E ( t,x ) [ ¯ U ( X Π T )] ≤ sup Π ∈A ( ¯ K ) E ( t,x ) [ ¯ U ( ¯ X Π T )] =: ¯ V ( t, x ) (32) HE IMPORTANCE OF DYNAMIC RISK CONSTRAINTS FOR LIMITED LIABILITY OPERATORS 32 where ¯ X has dynamics of d ¯ X t = ( r ¯ X t + ¯ µ ) dt + Π ′ t σdB t , ¯ X t = x. Since ¯ U is convex, based on the same augment in the proof of Proposition 2 we expect the optimal controlfor problem (32) is a constant process given byΠ ∗ t = sup π : π i ∈{− K,K } π ′ Σ π for all t . i.e. one should make the volatility of the process to be as large as possible. Moreover, theoptimally controlled process ¯ X Π ∗ is simply a drifting Brownian motion. Hence we can show that thecorresponding value function ¯ V has the same form as in (11) and in turn the same result follows imme-diately. (cid:3) In the single-asset case, whether the set K i is bounded is solely determined by the Sharpe ratio of theasset relative to the risk management parameter M i as shown in Lemma 1. We illustrate that a similarsufficient condition holds in the case with two risky assets. Lemma 5.
Consider a two-asset economy where the mean return vector is µ = ( µ , µ ) ′ and the variance-covariance matrix is Σ = ( σ , σ σ ρ ; σ σ ρ, σ ) . Recall the definitions of M i for i ∈ { VaR , ES } as per (6) . If vuut (cid:16) µ − rσ (cid:17) + (cid:16) µ − rσ (cid:17) − (cid:16) µ − rσ (cid:17) (cid:16) µ − rσ (cid:17) ρ − ρ < M i , (33) then the set K i defined by (29) or (30) is bounded.Proof. We prove the result for the case of ES constraint i = ES. The result under VaR constraint can beobtained similarly.If ( π , π ) ′ ∈ K ES then − e r ∆ − r [( µ − r ) π + ( µ − r ) π ] + φ (Φ − ( α )) α r ( e r ∆ − r ( σ π + σ π + 2 σ σ ρπ π ) ≤ R ⇐⇒ φ (Φ − ( α )) α r ( e r ∆ − r ( σ π + σ π + 2 σ σ ρπ π ) ≤ R + e r ∆ − r [( µ − r ) π + ( µ − r ) π ]= ⇒ " φ (Φ − ( α )) α r ( e r ∆ − r ( σ π + σ π + 2 σ σ ρπ π ) ≤ (cid:20) R + e r ∆ − r [( µ − r ) π + ( µ − r ) π ] (cid:21) . On rearranging, the last inequality becomes f ( π , π ) := Aπ + Bπ π + Cπ + Dπ + Eπ ≤ R where A := (cid:18) φ (Φ − ( α )) α (cid:19) (cid:18) e r ∆ − r (cid:19) σ − (cid:18) e r ∆ − r (cid:19) ( µ − r ) , HE IMPORTANCE OF DYNAMIC RISK CONSTRAINTS FOR LIMITED LIABILITY OPERATORS 33 B := 2 (cid:18) φ (Φ − ( α )) α (cid:19) (cid:18) e r ∆ − r (cid:19) σ σ ρ − (cid:18) e r ∆ − r (cid:19) ( µ − r )( µ − r ) ,C := (cid:18) φ (Φ − ( α )) α (cid:19) (cid:18) e r ∆ − r (cid:19) σ − (cid:18) e r ∆ − r (cid:19) ( µ − r ) ,D := − R (cid:18) e r ∆ − r (cid:19) ( µ − r ) ,E := − R (cid:18) e r ∆ − r (cid:19) ( µ − r ) . The set ˜ K := { ( π , π ) ′ ∈ R : f ( π , π ) ≤ R } is bounded if and only if the conic section f ( π , π ) = R is an ellipse, or equivalently B − AC <
0. This condition can be explicitly written in terms of the modelparameters as in (33). The result immediately follows on noticing that K ES is a subset of ˜ K . (cid:3) Corollary 2.
A dynamic risk constraint i ∈ { VaR , ES } is effective in a two-asset economy if condition (33) holds.Proof. This immediately follows from Lemma 5 and Proposition 8. (cid:3)
Comparing the results in Lemma 5 to the single-asset case in Lemma 1, we can see that a dynamic riskconstraint will translate into a bound on the trading strategy provided that the same risk managementparameter M i is sufficiently strict relative to the quality of the assets. But the criteria of assets qualitywill now take the correlation ρ into account to reflect the benefits of diversification. John Armstrong: King’s College London
Email address : [email protected] Damiano Brigo: Imperial College London
Email address : [email protected] Alex S.L. Tse: University of Exeter
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