Long Run Investment: Opportunities and Frictions Paolo Guasoni Boston University and Dublin City University July 2-6, 2012 CoLab Mathematics Summer School
Outline• Long Run and Stochastic Investment Opportunities• High-water Marks and Hedge Fund Fees• Transaction Costs• Price Impact• Open Problems
Long Run and Stochastic Investment Opportunities One Long Run and Stochastic Investment Opportunities Based on Portfolios and Risk Premia for the Long Run with Scott Robertson
Long Run and Stochastic Investment Opportunities Independent Returns? • Higher yields tend to be followed by higher long-term returns. • Should not happen if returns independent!
Long Run and Stochastic Investment Opportunities Utility Maximization • Basic portfolio choice problem: maximize utility from terminal wealth: π max E[U(XT )] π • Easy for logarithmic utility U(x) = log x. Myopic portfolio π = Σ−1 µ optimal. Numeraire argument. • Portfolio does not depend on horizon (even random!), and on the dynamics of the the state variable, but only its current value. • But logarithmic utility leads to counterfactual predictions. And implies that unhedgeable risk premia η are all zero. • Power utility U(x) = x 1−γ /(1 − γ) is more ﬂexible. Portfolio no longer myopic. Risk premia η nonzero, and depend on γ. • Power utility far less tractable. Joint dependence on horizon and state variable dynamics. • Explicit solutions few and cumbersome. • Goal: keep dependence from state variable dynamics, lose from horizon. • Tool: assume long horizon.
Long Run and Stochastic Investment Opportunities Asset Prices and State Variables t • Safe asset St0 = exp 0 r (Ys )ds , d risky assets, k state variables. dSti =r (Yt )dt + dRti 1≤i ≤d Sti n dRti =µi (Yt )dt + σij (Yt )dZtj 1≤i ≤d j=1 k dYti =bi (Yt )dt + aij (Yt )dWtj 1≤i ≤k j=1 d Zi, Wj t =ρij (Yt )dt 1 ≤ i ≤ d, 1 ≤ j ≤ k • Z , W Brownian Motions. • Σ(y ) = (σσ )(y ), Υ(y ) = (σρa )(y ), A(y ) = (aa )(y ).Assumptionr ∈ C γ (E, R), b ∈ C 1,γ (E, Rk ), µ ∈ C 1,γ (E, Rn ), A ∈ C 2,γ (E, Rk ×k ),Σ ∈ C 2,γ (E, Rn×n ) and Υ ∈ C 2,γ (E, Rn×k ). The symmetric matrices A and Σ ¯are strictly positive deﬁnite for all y ∈ E. Set Σ = Σ−1
Long Run and Stochastic Investment Opportunities State Variables • Investment opportunities: safe rate r , excess returns µ, volatilities σ, and correlations ρ. • State variables: anything on which investment opportunities depend. • Example with predictable returns: dRt =Yt dt + σdZt dYt = − λYt dt + dWt • State variable is expected return. Oscillates around zero. • Example with stochastic volatility: dRt =νYt dt + Yt dZt dYt =κ(θ − Yt )dt + a Yt dWt • State variable is squared volatility. Oscillates around positive value. • State variables are generally stationary processes.
Long Run and Stochastic Investment Opportunities (In)Completeness • Υ Σ−1 Υ: covariance of hedgeable state shocks: Measures degree of market completeness. • A = Υ Σ−1 Υ: complete market. State variables perfectly hedgeable, hence replicable. • Υ = 0: fully incomplete market. State shocks orthogonal to returns. • Otherwise state variable partially hedgeable. • One state: Υ Σ−1 Υ/a2 = ρ ρ. Equivalent to R 2 of regression of state shocks on returns.
Long Run and Stochastic Investment Opportunities Well PosednessAssumptionThere exists unique solution P (r ,y ) ) r ∈Rn ,y ∈E to martingale problem: n+k n+k 1 ˜ ∂2 ˜ ∂ ˜ Σ Υ ˜ µ L= Ai,j (x) + bi (x) A= b= 2 ∂xi ∂xj ∂xi Υ A b i,j=1 i=1 • Ω = C([0, ∞), Rn+k ) with uniform convergence on compacts. • B Borel σ-algebra, (Bt )t≥0 natural ﬁltration.Deﬁnition(P x )x∈Rn ×E on (Ω, B) solves martingale problem if, for all x ∈ Rn × E: • P x (X0 = x) = 1 • P x (Xt ∈ Rn × E, ∀t ≥ 0) = 1 t • f (Xt ) − f (X0 ) − 0 (Lf )(Xu )du is P x -martingale for all f ∈ C0 (Rn × E) 2
Long Run and Stochastic Investment Opportunities Trading and PayoffsDeﬁnitionTrading strategy: process (πti )1≤i≤d , adapted to Ft = Bt+ , the right-continuous t≥0envelope of the ﬁltration generated by (R, Y ), and R-integrable. • Investor trades without frictions. Wealth dynamics: dXtπ = r (Yt )dt + πt dRt Xtπ • In particular, Xtπ ≥ 0 a.s. for all t. Admissibility implied by R-integrability.
Long Run and Stochastic Investment Opportunities Equivalent Safe Rate • Maximizing power utility E (XT )1−γ /(1 − γ) equivalent to maximizing the π 1 certainty equivalent E (XT )1−γ π 1−γ . • Observation: in most models of interest, wealth grows exponentially with the horizon. And so does the certainty equivalent. • Example: with r , µ, Σ constant, the certainty equivalent is exactly 1 exp (r + 2γ µ Σ−1 µ)T . Only total Sharpe ratio matters. • Intuition: an investor with a long horizon should try to maximize the rate at which the certainty equivalent grows: 1 1 β = max lim inf log E (XT )1−γ π 1−γ π T →∞ T • Imagine a “dream” market, without risky assets, but only a safe rate ρ. • If β < ρ, an investor with long enough horizon prefers the dream. • If β > ρ, he prefers to wake up. • At β = ρ, his dream comes true.
Long Run and Stochastic Investment Opportunities Equivalent Annuity • Exponential utility U(x) = −e−αx leads to a similar, but distinct idea. • Suppose the safe rate is zero. • Then optimal wealth typically grows linearly with the horizon, and so does the certainty equivalent. • Then it makes sense to consider the equivalent annuity: 1 π β = max lim inf − log E e−αXT π T →∞ αT • The dream market now does not offer a higher safe rate, but instead a stream of ﬁxed payments, at rate ρ. The safe rate remains zero. • The investor is indifferent between dream and reality for β = ρ. • For positive safe rate, use deﬁnition with discounted quantities. • Undiscounted equivalent annuity always inﬁnite with positive safe rate.
Long Run and Stochastic Investment Opportunities Solution Strategy • Duality Bound. • Stationary HJB equation and ﬁnite-horizon bounds. • Criteria for long-run optimality.
Long Run and Stochastic Investment Opportunities Stochastic Discount FactorsDeﬁnitionStochastic discount factor: strictly positive adapted M = (Mt )t≥0 , such that: y EP Mt Sti Fs = Ms Ss i for all 0 ≤ s ≤ t, 0 ≤ i ≤ dMartingale measure: probability Q, such that Q|Ft and P y |Ft equivalent for allt ∈ [0, ∞), and discounted prices S i /S 0 Q-martingales for 1 ≤ i ≤ d. • Martingale measures and stochastic discount factors related by: dQ t = exp 0 r (Ys )ds Mt dP y Ft • Local martingale property: all stochastic discount factors satisfy t · · Mtη = exp − 0 rdt E − 0 (µ Σ−1 + η Υ Σ−1 )σdZ + 0 η adW t for some adapted, Rk -valued process η. • η represents the vector of unhedgeable risk premia. • Intuitively, the Sharpe ratios of shocks orthogonal to dR.
Long Run and Stochastic Investment Opportunities Duality Bound π η • For any payoff X = and any discount factor M = MT , E[XM] ≤ x. XT Because XM is a local martingale. • Duality bound for power utility: γ 1 1−γ E X 1−γ 1−γ ≤ xE M 1−1/γ • Proof: exercise with Hölder’s inequality. • Duality bound for exponential utility: 1 x 1 M M − log E e−αX ≤ + E log α E[M] α E[M] E[M] • Proof: Jensen inequality under risk-neutral densities. • Both bounds true for any X and for any M. Pass to sup over X and inf over M. • Note how α disappears from the right-hand side. • Both bounds in terms of certainty equivalents. • As T → ∞, bounds for equivalent safe rate and annuity follow.
Long Run and Stochastic Investment Opportunities Long Run OptimalityDeﬁnition (Power Utility)An admissible portfolio π is long run optimal if it solves: 1 1 max lim inf log E (XT )1−γ 1−γ π π T →∞ TThe risk premia η are long run optimal if they solve: γ 1 η 1−γ min lim sup log E (MT )1−1/γ η T →∞ TPair (π, η) long run optimal if both conditions hold, and limits coincide. • Easier to show that (π, η) long run optimal together. • Each η is an upper bound for all π and vice versa.Deﬁnition (Exponential Utility)Portfolio π and risk premia η long run optimal if they solve: 1 π η η max lim inf − log E e−αXT 1 min lim sup T E MT log MT π T →∞ T η T →∞
Long Run and Stochastic Investment Opportunities HJB Equation • V (t, x, y ) depends on time t, wealth x, and state variable y . • Itô’s formula: 1 dV (t, Xt , Yt ) = Vt dt + Vx dXt + Vy dYt + (Vxx d X t + Vxy d X , Y t + Vyy d Y t ) 2 • Vector notation. Vy , Vxy k -vectors. Vyy k × k matrix. • Wealth dynamics: dXt = (r + πt µt )Xt dt + Xt πt σt dZt • Drift reduces to: 1 x2 Vt + xVx r + Vy b + tr(Vyy A) + xπ (µVx + ΥVxy ) + Vxx π Σπ 2 2 • Maximizing over π, the optimal value is: Vx −1 Vxy −1 π=− Σ µ− Σ Υ xVxx xVxx • Second term is new. Interpretation?
Long Run and Stochastic Investment Opportunities Intertemporal Hedging V V • π = − xVx Σ−1 µ − Σ−1 Υ xVxy xx xx • First term: optimal portfolio if state variable frozen at current value. • Myopic solution, because state variable will change. • Second term hedges shifts in state variables. • If risk premia covary with Y , investors may want to use a portfolio which covaries with Y to control its changes. • But to reduce or increase such changes? Depends on preferences. • When does second term vanish? • Certainly if Υ = 0. Then no portfolio covaries with Y . Even if you want to hedge, you cannot do it. • Also if Vy = 0, like for constant r , µ and σ. But then state variable is irrelevant. • Any other cases?
Long Run and Stochastic Investment Opportunities HJB Equation • Maximize over π, recalling max(π b + 2 π Aπ = − 1 b A−1 b). 1 2 HJB equation becomes: 1 1 Σ−1 Vt + xVx r + Vy b + tr(Vyy A) − (µVx + ΥVxy ) (µVx + ΥVxy ) = 0 2 2 Vxx • Nonlinear PDE in k + 2 dimensions. A nightmare even for k = 1. • Need to reduce dimension. • Power utility eliminates wealth x by homogeneity.
Long Run and Stochastic Investment Opportunities Homogeneity • For power utility, V (t, x, y ) = x 1−γ 1−γ v (t, y ). x 1−γ Vt = vt Vx = x −γ v Vxx = −γx −γ−1 v 1−γ x 1−γ x 1−γ Vxy = x −γ vy Vy = vy Vyy = vyy 1−γ 1−γ • Optimal portfolio becomes: 1 −1 1 vy π= Σ µ + Σ−1 Υ γ γ v • Plugging in, HJB equation becomes: 1 1−γ vt + (1 − γ) r + µ Σ−1 µ v + b + Υ Σ−1 µ vy 2γ γ 1 1 − γ vy Υ Σ−1 Υvy + tr(vyy A) + =0 2 γ 2v • Nonlinear PDE in k + 1 variables. Still hard to deal with.
Long Run and Stochastic Investment Opportunities Long Run Asymptotics • For a long horizon, use the guess v (t, y ) = e(1−γ)(β(T −t)+w(y )) . • It will never satisfy the boundary condition. But will be close enough. • Here β is the equivalent safe, to be found. • We traded a function v (t, y ) for a function w(y ), plus a scalar β. • The HJB equation becomes: 1 1−γ −β + r + µ Σ−1 µ + b + Υ Σ−1 µ wy 2γ γ 1 1−γ 1−γ + tr(wyy A) + wy A− Υ Σ−1 Υ wy = 0 2 2 γ • And the optimal portfolio: 1 −1 1 π= Σ µ+ 1− γ Σ−1 Υwy γ • Stationary portfolio. Depends on state variable, not horizon. • HJB equation involved gradient wy and Hessian wyy , but not w. • With one state, ﬁrst-order ODE. • Optimality? Accuracy? Boundary conditions?
Long Run and Stochastic Investment Opportunities Example • Stochastic volatility model: dRt =νYt dt + Yt dZt dYt =κ(θ − Yt )dt + ε Yt dWt • Substitute values in stationary HJB equation: ν2 1−γ −β + r + y + κ(θ − y ) + ρενy wy 2γ γ ε2 ε2 y 2 1−γ 2 + wyy + (1 − γ) wy 1 − ρ =0 2 2 γ • Try a linear guess w = λy . Set constant and linear terms to zero. • System of equations in β and λ: −β + r + κθλ =0 2 1−γ 2 1−γ ν2 λ2 (1 − γ) 1− ρ +λ νρ − κ + =0 2 γ γ 2γ • Second equation quadratic, but only larger solution acceptable. Need to pick largest possible β.
Long Run and Stochastic Investment Opportunities Example (continued) • Optimal portfolio is constant, but not the usual constant. 1 π= (ν + βρε) γ • Hedging component depends on various model parameters. • Hedging is zero if ρ = 0 or ε = 0. • ρ = 0: hedging impossible. Returns do not covary with state variable. • ε = 0: hedging unnecessary. State variable deterministic. • Hedging zero also if β = 0, which implies logarithmic utility. • Logarithmic investor does not hedge, even if possible. • Lives every day as if it were the last one. • Equivalent safe rate: θν 2 1 νρ β= 1− 1− ε + O(ε2 ) 2γ γ κ • Correction term changes sign as γ crosses 1.
Long Run and Stochastic Investment Opportunities Martingale Measure • Many martingale measures. With incomplete market, local martingale condition does not identify a single measure. • For any arbitrary k -valued ηt , the process: · · Mt = E − (µ Σ−1 + η Υ Σ−1 )σdZ + η adW 0 0 t is a local martingale such that MR is also a local martingale. π • Recall that MT = yU (XT ). • If local martingale M is a martingale, it deﬁnes a stochastic discount factor. 1 −1 • π= γΣ (µ + Υ(1 − γ)wy ) yields: T T µ− 1 π Σπ)dt−γ U (XT ) = (XT )−γ =x −γ e−γ π π 0 (π 2 0 π σdZt T +(1−γ)wy Υ )Σ−1 σdZt + T =e− 0 (µ 0 (... )dt • Mathing the two expressions, we guess η = (1 − γ)wy .
Long Run and Stochastic Investment Opportunities Risk Neutral Dynamics • To ﬁnd dynamics of R and Y under Q, recall Girsanov Theorem. • If Mt has previous representation, dynamics under Q is: ˜ dRt =σd Zt ˜ dYt =(b − Υ Σ−1 µ + (A − Υ Σ−1 Υ)η)dt + ad Wt • Since η = (1 − γ)wy , it follows that: ˜ dRt =σd Zt ˜ dYt = b − Υ Σ−1 µ + (A − Υ Σ−1 Υ)(1 − γ)wy dt + ad Wt • Formula for (long-run) risk neutral measure for a given risk aversion. • For γ = 1 (log utility) boils down to minimal martingale measure. • Need to ﬁnd w to obtain explicit solution. • And need to check that above martingale problem has global solution.
Long Run and Stochastic Investment Opportunities Exponential Utility • Instead of homogeneity, recall that wealth factors out of value function. • Long-run guess: V (x, y , t) = e−αx+αβt+w(y ) . β is now equivalent annuity. • Set r = 0, otherwise safe rate wipes out all other effects. • The HJB equation becomes: 1 1 1 −β + µ Σ−1 µ + b − Υ Σ−1 µ wy + tr(wyy A)− wy A − Υ Σ−1 Υ wy = 0 2 2 2 • And the optimal portfolio: 1 −1 Σ µ − Σ−1 Υwy xπ = α • Rule of thumb to obtain exponential HJB equation: ˜ write power HJB equation in terms of w = γw, then send γ ↑ ∞. Then remove the˜ • Exponential utility like power utility with “∞” relative risk aversion. • Risk-neutral dynamics is minimal entropy martingale measure: ˜ dYt = b − Υ Σ−1 µ − (A − Υ Σ−1 Υ)wy dt + ad Wt
Long Run and Stochastic Investment Opportunities HJB EquationAssumptionw ∈ C 2 (E, R) and β ∈ R solve the ergodic HJB equation: 1 ¯ 1−γ 1 ¯ r+ µ Σµ + w A − (1 −)Υ ΣΥ w+ 2γ 2 γ 1 ¯ 1 w b − (1 − )Υ Σµ + tr AD 2 w = β γ 2 • Solution must be guessed one way or another. • PDE becomes ODE for a single state variable • PDE becomes linear for logarithmic utility (γ = 1). • Must ﬁnd both w and β
Long Run and Stochastic Investment Opportunities Myopic ProbabilityAssumption ˆThere exists unique solution (P r ,y )r ∈Rn ,y ∈Rk to to martingale problem n+k n+k ˆ 1 ˜ ∂2 ˆ ∂ L= Ai,j (x) + bi (x) 2 ∂xi ∂xj ∂xi i,j=1 i=1 1 γ (µ + (1 − γ)Υ w) ˆ b= b − (1 − 1 ¯ 1 ¯ Σµ + A − (1 − γ )Υ ΣΥ (1 − γ) w γ )Υ ˆ • Under P, the diffusion has dynamics: 1 ˆ dRt = γ (µ + (1 − γ)Υ w) dt + σd Zt 1 ¯ 1 ¯ ˆ dYt = b − (1 − γ )Υ Σµ + (A − (1 − γ )Υ ΣΥ)(1 − γ) w dt + ad Wt ˆ • Same optimal portfolio as a logarthmic investor living under P.
Long Run and Stochastic Investment Opportunities Finite horizon boundsTheoremUnder previous assumptions: 1¯ π = Σ (µ + (1 − γ)Υ w) , η = (1 − γ) w γsatisfy the equalities: 1 1 y y 1−γ EP (XT )1−γ π 1−γ = eβT +w(y ) EP e−(1−γ)w(YT ) ˆ γ γ γ−1 1−γ 1−γ 1−γ y y η EP (MT ) γ = eβT +w(y ) EP e− ˆ γ w(YT ) • Bounds are almost the same. Differ in Lp norm • Long run optimality if expectations grow less than exponentially.
Long Run and Stochastic Investment Opportunities Path to Long Run solution • Find candidate pair w, β that solves HJB equation. • Different β lead to to different solutions w. • Must ﬁnd w corresponding to the lowest β that has a solution. • Using w, check that myopic probabiity is well deﬁned. ˆ Y does not explode under dynamics of P. • Then ﬁnite horizon bounds hold. • To obtain long run optimality, show that: 1 y 1−γ lim sup log EP e− γ w(YT ) = 0 ˆ T →∞ T 1 y lim sup log EP e−(1−γ)w(YT ) = 0 ˆ T →∞ T
Long Run and Stochastic Investment Opportunities Proof of wealth bound (1) ˆ dP dP |Ft , which equals to E(M), where: • Z = ρW + ρB. Deﬁne Dt = ¯ t Mt = ¯ ¯ −qΥ Σµ + A − qΥ ΣΥ v (a )−1 dWt 0 t − ¯ ¯ q Σµ + ΣΥ v σ ρdBt ¯ 0 • For the portfolio bound, it sufﬁces to show that: p (XT ) = ep(βT +w(y )−w(YT )) DT π π 1 which is the same as log XT − p log DT = βT + w(y ) − w(YT ). • The ﬁrst term on the left-hand side is: T T π 1 log XT = r +π µ− π Σπ dt + π σdZt 0 2 0
Long Run and Stochastic Investment Opportunities Proof of wealth bound (2) • Set π = 1 ¯ π 1−p Σ (µ + pΥ w). log XT becomes: T 1−2p ¯ p2 ¯ 1 p 2 ¯ 0 r+ 2(1−p) µ Σµ − (1−p)2 µ ΣΥ w − 2 (1−p)2 w Υ ΣΥ w dt 1 T ¯ 1 T ¯ ¯ + 1−p 0 (µ + pΥ w) ΣσρdWt − 1−p 0 (µ + pΥ w) Σσ ρdBt • Similarly, log DT /p becomes: T p ¯ p ¯ p p(2−p) ¯ 0 − 2(1−p)2 µ Σµ − (1−p)2 µ ΣΥ w − 2 w A+ (1−p)2 Υ ΣΥ w dt+ T 1 ¯ 1 T ¯ ¯ 0 w a+ 1−p (µ + pΥ w) Σσρ dWt + 1−p 0 (µ + pΥ w) Σσ ρdBt π • Subtracting yields for log XT − log DT /p T 1 ¯ p ¯ p p ¯ 0 r+ 2(1−p) µ Σµ + 1−p µ ΣΥ w + 2 w A+ 1−p Υ ΣΥ w dt T − 0 w adWt
Long Run and Stochastic Investment Opportunities Proof of wealth bound (3) • Now, Itô’s formula allows to substitute: T T T 1 − w adWt = w(y ) − w(YT ) + w bdt + tr(AD 2 w)dt 0 0 2 0 • The resulting dt term matches the one in the HJB equation. π • log XT − log DT /p equals to βT + w(y ) − w(YT ).
Long Run and Stochastic Investment Opportunities Proof of martingale bound (1) • For the discount factor bound, it sufﬁces to show that: 1 η 1 1 p−1 log MT − p log DT = 1−p (βT + w(y ) − w(YT )) 1 η • The term p−1 log MT equals to: 1 T 1 ¯ p2 ¯ 1−p 0 r + 2 µ Σµ + 2 w A − Υ ΣΥ w dt+ 1 T ¯ 1 T ¯ ¯ p−1 0 p w a − (µ + pΥ w) Σσρ dWt + 1−p 0 (µ + pΥ w) Σσ ρdBt 1 1 η 1 • Subtracting p log DT yields for p−1 log MT − p log DT : 1 T 1 ¯ p ¯ p p ¯ 1−p 0 r+ 2(1−p) µ Σµ + 1−p µ ΣΥ w + 2 w A+ 1−p Υ ΣΥ w dt 1 T − 1−p 0 w adWt
Long Run and Stochastic Investment Opportunities Proof of martingale bound (2) T • Replacing again 0 w adWt with Itô’s formula yields: 1 T 1 ¯ p ¯ 1−p 0 (r + 2(1−p) µ Σµ + ( 1−p µ ΣΥ + b ) w+ 1 p p ¯ 2 tr(AD 2 w) + 2 w A+ 1−p Υ ΣΥ w)dt 1 + 1−p (w(y ) − w(YT )) • And the integral equals 1 1−p βT by the HJB equation.
Long Run and Stochastic Investment Opportunities Exponential UtilityTheoremIf r = 0 and w solves equation: 1 ¯ 1 ¯ ¯ 1 µ Σµ − w A − Υ ΣΥ w+ w b − Υ Σµ + tr AD 2 w = β 2 2 2and myopic dynamics is well posed: ˆ dRt =σd Zt ¯ ¯ ˆ dYt = b − Υ Σµ − (A − Υ ΣΥ) w dt + ad WtThen for the portfolio and risk premia (π, η) given by: 1 xπ = Σ−1 µ − Σ−1 Υ w η=− w αﬁnite-horizon bounds hold as: 1 y π 1 y − log EP e−α(XT −x) =βT + log EP ew(y )−w(YT ) ˆ α α 1 y η 1 y EP [M log M η ] =βT + EP [w(y ) − w(YT )] 2 α ˆ
Long Run and Stochastic Investment Opportunities Long-Run OptimalityTheoremIf, in addition to the assumptions for ﬁnite-horizon bounds, ˆ • the random variables (Yt )t≥0 are P y -tight in E for each y ∈ E; • supy ∈E (1 − γ)F (y ) < +∞, where F ∈ C(E, R) is deﬁned as: µ Σ−1 µ (1−γ)2 r −β+ 2γ − 2γ w Υ Σ−1 Υ w e−(1−γ)w γ>1 F = µ Σ−1 µ (1−γ)2 − 1−γ w r −β+ 2γ − 2 w A − Υ Σ−1 Υ w e γ γ<1Then long-run optimality holds. • Straightforward to check, once w is known. • Tightness checked with some moment condition. • Does not require transition kernel for Y under any probability.
Long Run and Stochastic Investment Opportunities Proof of Long-Run Optimality (1) • By the duality bound: 1 1 y η 1 1−p y 0 ≤ lim inf log EP (MT )q log EP [(XT )p ] − π T →∞ p T T 1 y η 1−p 1 y ≤ lim sup log EP (MT )q − lim inf log EP [(XT )p ] π T →∞ pT T →∞ pT 1−p y 1 1 y = lim sup log EP e− 1−p v (YT ) − lim inf ˆ log EP e−v (YT ) ˆ T →∞ pT T →∞ pT • For p < 0 enough to show lower bound 1 y 1 lim inf log EP exp − 1−p v (YT ) ˆ ≥0 T →∞ T and upper bound: 1 y lim supT →∞ T log EP [exp (−v (YT ))] ≤ 0 ˆ • Lower bound follows from tightness.
Long Run and Stochastic Investment Opportunities Proof of Long-Run Optimality (2) • For upper bound, set: 1 Lf = f b − qΥ Σ−1 µ + A − qΥ Σ−1 Υ v + 2 tr AD 2 f • Then, for α ∈ R, the HJB equation implies that L (eαv ) equals to: 1 αeαv v b − qΥ Σ−1 µ + A − qΥ Σ−1 Υ v + 2 tr AD 2 v + 1 α v A v 2 = αeαv 1 2 v (1 + α)A − qΥ Σ−1 Υ v + pβ − pr + q µ Σ−1 µ 2 • Set α = −1, to obtain that: q q L e−v = e−v v Υ Σ−1 Υ v − λ + pr − µ Σ−1 µ 2 2 • The boundedness hypothesis on F allows to conclude that: y EP e−v (YT ) ≤ e−v (y ) + (K ∨ 0) T ˆ whence 1 y log EP e−v (YT ) ≤ 0 ˆ lim sup T →∞ T • 0 < p < 1 similar. Reverse inequalities for upper and lower bounds. 1 Use α = − 1−p for upper bound.
High-water Marks and Hedge Fund Fees Two High-water Marks and Hedge Fund Fees Based on The Incentives of Hedge Fund Fees and High-Water Marks with Jan Obłoj
High-water Marks and Hedge Fund Fees Two and Twenty • Hedge Funds Managers receive two types of fees. • Regular fees, like Mutual Funds. • Unlike Mutual Funds, Performance Fees. • Regular fees: a fraction ϕ of assets under management. 2% typical. • Performance fees: a fraction α of trading proﬁts. 20% typical. • High-Water Marks: Performance fees paid after losses recovered.
High-water Marks and Hedge Fund Fees High-Water Marks Time Gross Net High-Water Mark Fees 0 100 100 100 0 1 110 108 108 2 2 100 100 108 2 3 118 116 116 4 • Fund assets grow from 100 to 110. The manager is paid 2, leaving 108 to the fund. • Fund drops from 108 to 100. No fees paid, nor past fees reimbursed. • Fund recovers from 100 to 118. Fees paid only on increase from 108 to 118. Manager receives 2.
High-water Marks and Hedge Fund Fees High-Water Marks2.52.01.51.00.5 20 40 60 80 100
High-water Marks and Hedge Fund Fees Risk Shifting? • Manager shares investors’ proﬁts, not losses. Does manager take more risk to increase proﬁts? • Option Pricing Intuition: Manager has a call option on the fund value. Option value increases with volatility. More risk is better. • Static, Complete Market Fallacy: Manager has multiple call options. • High-Water Mark: future strikes depend on past actions. • Option unhedgeable: cannot short (your!) hedge fund.
High-water Marks and Hedge Fund Fees Questions • Portfolio: Effect of fees and risk-aversion? • Welfare: Effect on investors and managers? • High-Water Mark Contracts: consistent with any investor’s objective?
High-water Marks and Hedge Fund Fees Answers • Goetzmann, Ingersoll and Ross (2003): Risk-neutral value of management contract (future fees). Exogenous portfolio and fund ﬂows. • High-Water Mark contract worth 10% to 20% of fund. • Panageas and Westerﬁeld (2009): Exogenous risky and risk-free asset. Optimal portfolio for a risk-neutral manager. Fees cannot be invested in fund. • Constant risky/risk-free ratio optimal. Merton proportion does not depend on fee size. Same solution for manager with Hindy-Huang utility.
High-water Marks and Hedge Fund Fees This Model • Manager with Power Utility and Long Horizon. Exogenous risky and risk-free asset. Fees cannot be invested in fund. • Optimal Portfolio: 1 µ π= γ ∗ σ2 γ ∗ =(1 − α)γ + α γ =Manager’s Risk Aversion α =Performance Fee (e.g. 20%) • Manager behaves as if owned fund, but were more myopic (γ ∗ weighted average of γ and 1). • Performance fees α matter. Regular fees ϕ don’t.
High-water Marks and Hedge Fund Fees Three Problems, One Solution • Power utility, long horizon. No regular fees. 1 Manager maximizes utility of performance fees. Risk Aversion γ. 2 Investor maximizes utility of wealth. Pays no fees. Risk Aversion γ ∗ = (1 − α)γ + α. 3 Investor maximizes utility of wealth. Pays no fees. Risk Aversion γ. Maximum Drawdown 1 − α. • Same optimal portfolio: 1 µ π= γ ∗ σ2
High-water Marks and Hedge Fund Fees Price Dynamics dSt = (r + µ)dt + σdWt (Risky Asset) St dSt α ∗ dXt = (r − ϕ)Xt dt + Xt πt St − rdt − 1−α dXt (Fund) α dFt = rFt dt + ϕXt dt + dX ∗ (Fees) 1−α t Xt∗ = max Xs (High-Water Mark) 0≤s≤t • One safe and one risky asset. • Gain split into α for the manager and 1 − α for the fund. • Performance fee is α/(1 − α) of fund increase.
High-water Marks and Hedge Fund Fees Dynamics Well Posed? • Problem: fund value implicit. Find solution Xt for dSt α dXt = Xt πt − ϕXt dt − dX ∗ St 1−α t • Yes. Pathwise construction.Proposition ∗The unique solution is Xt = eRt −αRt , where: t t σ2 2 Rt = µπs − π − ϕ ds + σ πs dWs 0 2 s 0is the cumulative log return.
High-water Marks and Hedge Fund Fees Fund Value ExplicitLemmaLet Y be a continuous process, and α > 0.Then Yt + 1−α Yt∗ = Rt if and only if Yt = Rt − αRt∗ . αProof.Follows from: α α 1 Rt∗ = sup Ys + sup Yu = Yt∗ + Yt∗ = Y∗ s≤t 1 − α u≤s 1−α 1−α t • Apply Lemma to cumulative log return.
High-water Marks and Hedge Fund Fees Long Horizon • The manager chooses the portfolio π which maximizes expected power utility from fees at a long horizon. • Maximizes the long-run objective: 1 p max lim log E[FT ] = β π T →∞ pT • Dumas and Luciano (1991), Grossman and Vila (1992), Grossman and Zhou (1993), Cvitanic and Karatzas (1995). Risk-Sensitive Control: Bielecki and Pliska (1999) and many others. 2 1 µ • β=r+ γ 2σ 2 for Merton problem with risk-aversion γ = 1 − p.
High-water Marks and Hedge Fund Fees Solving It • Set r = 0 and ϕ = 0 to simplify notation. • Cumulative fees are a fraction of the increase in the fund: α Ft = (X ∗ − X0 ) ∗ 1−α t • Thus, the manager’s objective is equivalent to: 1 ∗ max lim log E[(XT )p ] π T →∞ pT • Finite-horizon value function: 1 V (x, z, t) = sup E[XT p |Xt = x, Xt∗ = z] ∗ π p 1 dV (Xt , Xt∗ , t) = Vt dt + Vx dXt + Vxx d X t + Vz dXt∗ 2 2 = Vt dt + Vz − α 1−α Vx dXt∗ + Vx Xt (πt µ − ϕ)dt + Vxx σ πt2 Xt2 dt 2
High-water Marks and Hedge Fund Fees Dynamic Programming • Hamilton-Jacobi-Bellman equation: 2 Vt + supπ xVx (πµ − ϕ) + Vxx σ π 2 x 2 ) x <z 2 α Vz = 1−α Vx x =z p V = z /p x =0 V = z p /p t =T • Maximize in π, and use homogeneity V (x, z, t) = z p /pV (x/z, 1, t) = z p /pu(x/z, 1, t). 2 ut − ϕxux − µ22 ux = 0 x ∈ (0, 1) 2σ uxx ux (1, t) = p(1 − α)u(1, t) t ∈ (0, T ) u(x, T ) = 1 x ∈ (0, 1) u(0, t) = 1 t ∈ (0, T )
High-water Marks and Hedge Fund Fees Long-Run Heuristics • Long-run limit. Guess a solution of the form u(t, x) = ce−pβt w(x), forgetting the terminal condition: 2 µ2 wx −pβw − ϕxwx − 2σ2 wxx = 0 for x < 1 wx (1) = p(1 − α)w(1) • This equation is time-homogeneous, but β is unknown. • Any β with a solution w is an upper bound on the rate λ. • Candidate long-run value function: the solution w with the lowest β. 1−α µ2 • w(x) = x p(1−α) , for β = (1−α)γ+α 2σ 2 − ϕ(1 − α).
High-water Marks and Hedge Fund Fees VeriﬁcationTheorem µ2 1 1If ϕ − r < 2σ 2 min γ∗ , γ∗2 , then for any portfolio π: 1 π p 1 µ2 lim log E (FT ) ≤ max (1 − α) + r − ϕ ,r T ↑∞ pT γ ∗ 2σ 2 2 α 1 µUnder the nondegeneracy condition ϕ + 1−α r < γ∗ 2σ 2 , the unique optimal µsolution is π = γ1∗ σ2 . ˆ • Martingale argument. No HJB equation needed. • Show upper bound for any portfolio π (delicate). • Check equality for guessed solution (easy).
High-water Marks and Hedge Fund Fees Upper Bound (1) • Take p > 0 (p < 0 symmetric). • For any portfolio π: T σ2 2 T ˜ RT = − π dt 0 2 t + 0 σπt d Wt ˜ • Wt = Wt + µ/σt risk-neutral Brownian Motion • Explicit representation: ∗ ∗ µ ˜ µ2 E[(XT )p ] = E[ep(1−α)RT ] = EQ ep(1−α)RT e σ WT − 2σ2 T π • For δ > 1, Hölder’s inequality: δ−1 δ µ ˜ µ2 δ µ2 σ WT − 2σ 2 µ ˜ 1 T ∗ ∗ σ WT − 2σ 2 T p(1−α)RT δp(1−α)RT δ δ−1 EQ e e ≤ EQ e EQ e 1 µ2 • Second term exponential normal moment. Just e δ−1 2σ2 T .
High-water Marks and Hedge Fund Fees Upper Bound (2) ∗ • Estimate EQ eδp(1−α)RT . • Mt = eRt strictly positive continuous local martingale. Converges to zero as t ↑ ∞. • Fact: inverse of lifetime supremum (M∞ )−1 uniform on [0, 1]. ∗ • Thus, for δp(1 − α) < 1: ∗ ∗ 1 EQ eδp(1−α)RT ≤ EQ eδp(1−α)R∞ = 1 − δp(1 − α) • In summary, for 1 < δ < 1 p(1−α) : 1 π p 1 µ2 lim log E (FT ) ≤ T ↑∞ pT p(δ − 1) 2σ 2 • Thesis follows as δ → 1 p(1−α) .
High-water Marks and Hedge Fund Fees High-Water Marks and Drawdowns • Imagine fund’s assets Xt and manager’s fees Ft in the same account Ct = Xt + Ft . dSt dCt = (Ct − Ft )πt St • Fees Ft proportional to high-water mark Xt∗ : α Ft = (X ∗ − X0 ) ∗ 1−α t • Account increase dCt∗ as fund increase plus fees increase: t t α 1 Ct∗ − C0 = ∗ ∗ (dXs + dFs ) = ∗ + 1 dXs = (X ∗ − X0 ) 0 0 1−α 1−α t • Obvious bound Ct ≥ Ft yields: Ct ≥ α(Ct∗ − X0 ) • X0 negligible as t ↑ ∞. Approximate drawdown constraint. Ct ≥ αCt∗
Transaction Costs Three Transaction Costs Based on Transaction Costs, Trading Volume, and the Liquidity Premium with Stefan Gerhold, Johannes Muhle-Karbe, and Walter Schachermayer
Transaction Costs Model • Safe rate r . • Ask (buying) price of risky asset: dSt = (r + µ)dt + σdWt St • Bid price (1 − ε)St . ε is the spread. • Investor with power utility U(x) = x 1−γ /(1 − γ). • Maximize equivalent safe rate: 1 1 1−γ 1−γ max lim log E XT π T →∞ T
Transaction Costs Wealth Dynamics • Number of shares must have a.s. locally ﬁnite variation. • Otherwise inﬁnite costs in ﬁnite time. • Strategy: predictable process (ϕ0 , ϕ) of ﬁnite variation. • ϕ0 units of safe asset. ϕt shares of risky asset at time t. t • ϕt = ϕ↑ − ϕ↓ . Shares bought ϕ↑ minus shares sold ϕ↓ . t t t t • Self-ﬁnancing condition: St St dϕ0 = − t dϕ↑ + (1 − ε) 0 dϕ↓ t St0 St • Xt0 = ϕ0 St0 , Xt = ϕt St safe and risky wealth, at ask price St . t dXt0 =rXt0 dt − St dϕ↑ + (1 − ε)St dϕ↓ , t t dXt =(µ + r )Xt dt + σXt dWt + St dϕ↑ − St dϕ↓ t
Transaction Costs Control Argument • V (t, x, y ) value function. Depends on time, and on asset positions. • By Itô’s formula: 1 dV (t, Xt0 , Xt ) = Vt dt + Vx dXt0 + Vy dXt + Vyy d X , X t 2 σ2 2 = Vt + rXt0 Vx + (µ + r )Xt Vy + X Vyy dt 2 t + St (Vy − Vx )dϕ↑ + St ((1 − ε)Vx − Vy )dϕ↓ + σXt dWt t t • V (t, Xt0 , Xt ) supermartingale for any ϕ. • ϕ↑ , ϕ↓ increasing, hence Vy − Vx ≤ 0 and (1 − ε)Vx − Vy ≤ 0 Vx 1 1≤ ≤ Vy 1−ε
Transaction Costs No Trade Region Vx 1 • When 1 ≤ Vy ≤ 1−ε does not bind, drift is zero: σ2 2 Vx 1 Vt + rXt0 Vx + (µ + r )Xt Vy + X Vyy = 0 if 1 < < . 2 t Vy 1−ε • This is the no-trade region. • Long-run guess: V (t, Xt0 , Xt ) = (Xt0 )1−γ v (Xt /Xt0 )e−(1−γ)(β+r )t (1−γ)v (z) 1 • Set z = y /x. For 1 + z < v (z) < 1−ε + z, HJB equation is 2 σ 2 z v (z) + µzv (z) − (1 − γ)βv (z) = 0 2 • Linear second order ODE. But β unknown.
Transaction Costs Smooth Pasting (1−γ)v (z) 1 • Suppose 1 + z < v (z) < 1−ε + z same as l ≤ z ≤ u. • For l < u to be found. Free boundary problem: σ2 2 z v (z) + µzv (z) − (1 − γ)βv (z) = 0 if l < z < u, 2 (1 + l)v (l) − (1 − γ)v (l) = 0, (1/(1 − ε) + u)v (u) − (1 − γ)v (u) = 0. • Conditions not enough to ﬁnd solution. Matched for any l, u. • Smooth pasting conditions. • Differentiate boundary conditions with respect to l and u: (1 + l)v (l) + γv (l) = 0, (1/(1 − ε) + u)v (u) + γv (u) = 0.
Transaction Costs Solution Procedure • Unknown: trading boundaries l, u and rate β. • Strategy: ﬁnd l, u in terms of β. • Free boundary problem becomes ﬁxed boundary problem. • Find unique β that solves this problem.
Transaction Costs Trading Boundaries • Plug smooth-pasting into boundary, and result into ODE. Obtain: 2 2 l l − σ (1 − γ)γ (1+l)2 v + µ(1 − γ) 1+l v − (1 − γ)βv = 0. 2 • Setting π− = l/(1 + l), and factoring out (1 − γ)v : γσ 2 2 − π + µπ− − β = 0. 2 − • π− risky weight on buy boundary, using ask price. • Same argument for u. Other solution to quadratic equation is: u(1−ε) π+ = 1+u(1−ε) , • π+ risky weight on sell boundary, using bid price.
Transaction Costs Gap • Optimal policy: buy when “ask" weight falls below π− , sell when “bid" weight rises above π+ . Do nothing in between. • π− and π+ solve same quadratic equation. Related to β via µ µ2 − 2βγσ 2 π± = ± . γσ 2 γσ 2 • Set β = (µ2 − λ2 )/2γσ 2 . β = µ2 /2γσ 2 without transaction costs. • Investor indifferent between trading with transaction costs asset with volatility σ and excess return µ, and... • ...trading hypothetical frictionless asset, with excess return µ2 − λ2 and same volatility σ. • µ− µ2 − λ2 is liquidity premium. µ±λ • With this notation, buy and sell boundaries are π± = γσ 2 .
Transaction Costs Symmetric Trading Boundaries • Trading boundaries symmetric around frictionless weight µ/γσ 2 . • Each boundary corresponds to classical solution, in which expected return is increased or decreased by the gap λ. • With l(λ), u(λ) identiﬁed by π± in terms of λ, it remains to ﬁnd λ. • This is where the trouble is.
Transaction Costs First Order ODE • Use substitution: log(z/l(λ)) w(y )dy l(λ)ey v (l(λ)ey ) v (z) = e(1−γ) , i.e. w(y ) = (1−γ)v (l(λ)ey ) • Then linear second order ODE becomes ﬁrst order Riccati ODE 2µ µ−λ µ+λ w (x) + (1 − γ)w(x)2 + σ2 − 1 w(x) − γ γσ 2 γσ 2 =0 µ−λ w(0) = γσ 2 µ+λ w(log(u(λ)/l(λ))) = γσ 2 2 u(λ) 1 π+ (1−π− ) 1 (µ+λ)(µ−λ−γσ ) where l(λ) = 1−ε π− (1−π+ ) = 1−ε (µ−λ)(µ+λ−γσ 2 ) . • For each λ, initial value problem has solution w(λ, ·). µ+λ • λ identiﬁed by second boundary w(λ, log(u(λ)/l(λ))) = γσ 2 .
Transaction Costs Shadow Market • Find shadow price to make argument rigorous. ˜ ˜ • Hypothetical price S of frictionless risky asset, such that trading in S withut transaction costs is equivalent to trading in S with transaction costs. For optimal policy. • For all other policies, shadow market is better. • Use frictionless theory to show that candidate optimal policy is optimal in shadow market. • Then it is optimal also in transaction costs market.
Transaction Costs Shadow Price as Marginal Rate of Substitution ˜ • Imagine trading is now frictionless, but price is St . • Intuition: the value function must not increase by trading δ ∈ R shares. • In value function, risky position valued at ask price St . Thus ˜ V (t, Xt0 − δ St , Xt + δSt ) ≤ V (t, Xt0 , Xt ) ˜ • Expanding for small δ, −δVx St + δVy St ≤ 0 ˜ • Must hold for δ positive and negative. Guess for St : ˜ Vy St = St Vx log(Xt /lXt0 ) • Plugging guess V (t, Xt0 , Xt ) = e−(1−γ)(r +β)t (Xt0 )1−γ e(1−γ) 0 w(y )dy , ˜ w(Yt ) St = St leYt (1 − w(Yt )) • Shadow portfolio weight is precisely w: w(Yt ) ˜ ϕt St 1−w(Yt ) πt = ˜ = = w(Yt ), ˜ ϕ0 St0 + ϕt St 1 w(Yt ) + 1−w(Yt ) t
Transaction Costs Shadow Value Function ˜ ˜ • In terms of shadow wealth Xt = ϕ0 St0 + ϕt St , the value function becomes: t 1−γ Xt0 Yt V (t, Xt , Yt ) = V (t, Xt0 , Xt ) = e−(1−γ)(r +β)t Xt1−γ ˜ ˜ ˜ e(1−γ) 0 w(y )dy . ˜ Xt ˜ ˜ • Note that Xt0 /Xt = 1 − w(Yt ) by deﬁnition of St , and rewrite as: 1−γ Xt0 Yt w(y )dy Yt e(1−γ) 0 = exp (1 − γ) log(1 − w(Yt )) + 0 w(y )dy ˜ Xt Yt w (y ) = (1 − w(0))γ−1 exp (1 − γ) 0 w(y ) − 1−w(y ) dy . ˜ • Set w = w − w 1−w to obtain Yt V (t, Xt , Yt ) = e−(1−γ)(r +β)t Xt1−γ e(1−γ) ˜ ˜ ˜ 0 ˜ w(y )dy (1 − w(0))γ−1 . • We are now ready for veriﬁcation.
Transaction Costs Construction of Shadow Price • So far, a string of guesses. Now construction. • Deﬁne Yt as reﬂected diffusion on [0, log(u/l)]: dYt = (µ − σ 2 /2)dt + σdWt + dLt − dUt , Y0 ∈ [0, log(u/l)],Lemma ˜ w(Y )The dynamics of S = S leY (1−w(Y )) is given by ˜ ˜ d S(Yt )/S(Yt ) = (˜(Yt ) + r ) dt + σ (Yt )dWt , µ ˜where µ(·) and σ (·) are deﬁned as ˜ ˜ σ 2 w (y ) w (y ) σw (y )µ(y ) =˜ − (1 − γ)w(y ) , σ (y ) = ˜ . w(y )(1 − w(y )) 1 − w(y ) w(y )(1 − w(y )) ˜And the process S takes values within the bid-ask spread [(1 − ε)S, S].
Transaction Costs Finite-horizon BoundTheorem ˜The shadow payoff XT of the portfolio π = w(Yt ) and the shadow discount ˜ · µ ˜ y ˜factor MT = E(− 0 σ dWt )T satisfy (with q (y ) = ˜ ˜ w(z)dz): ˜ 1−γ =e(1−γ)βT E e(1−γ)(q (Y0 )−q (YT )) , E XT ˆ ˜ ˜ 1 γ γ 1− γ 1 ˜ ˜ E MT =e(1−γ)βT E e( γ −1)(q (Y0 )−q (YT )) ˆ . ˆ ˆwhere E[·] is the expectation with respect to the myopic probability P: ˆ T T 2 dP µ ˜ 1 µ ˜ = exp − + σ π dWt − ˜˜ − + σπ ˜˜ dt . dP 0 σ ˜ 2 0 σ ˜
Transaction Costs First Bound (1) ˜ ˜ ˜ ˜ • µ, σ , π , w functions of Yt . Argument omitted for brevity. ˜ • For ﬁrst bound, write shadow wealth X as: ˜ 1−γ = exp (1 − γ) T σ2 2 ˜ T XT 0 µπ − ˜˜ 2 π ˜ dt + (1 − γ) 0 σ π dWt . ˜˜ • Hence: ˆ T 2 ˜ 1−γ = d P exp σ2 2 ˜ 1 µ ˜ XT dP 0 (1 − γ) µπ − ˜˜ 2 π ˜ + 2 − σ + σπ ˜ ˜˜ dt T µ ˜ × exp 0 (1 − γ)˜ π − − σ + σ π σ˜ ˜ ˜˜ dWt . 1 µ ˜ • Plug π = ˜ γ σ2 ˜ + (1 − γ) σ w . Second integrand is −(1 − γ)σ w. σ ˜ ˜ ˜ 1µ˜2 2 • First integrand is 2 σ2 ˜ + γ σ π 2 − γ µπ , which equals to ˜ 2 ˜ ˜˜ 2 2 1−γ µ ˜ (1 − γ)2 σ w 2 + 2 ˜ 2γ σ ˜ ˜ + σ(1 − γ)w .
Transaction Costs First Bound (2) 1−γ ˜ • In summary, XT equals to: ˆ T 2 2 dP 1 µ ˜ dP exp (1 − γ) 0 (1 − γ) σ w 2 + 2 ˜ 2γ σ ˜ ˜ + σ(1 − γ)w dt T × exp −(1 − γ) 0 ˜ σ wdWt . ˜ ˜ • By Itô’s formula, and boundary conditions w(0) = w(log(u/l)) = 0, T 1 T ˜ ˜ q (YT ) − q (Y0 ) = 0 ˜ w(Yt )dYt + 2 0 ˜ w (Yt )d Y , Y t ˜ ˜ + w(0)LT − w(u/l)UT T σ2 2 σ T = 0 µ− 2 ˜ w+ 2 ˜ w dt + 0 ˜ σ wdWt . T ˜ 1−γ equals to: • Use identity to replace 0 ˜ σ wdWt , and XT ˆ dP T dP exp (1 − γ) 0 ˜ ˜ (β) dt) × exp (−(1 − γ)(q (YT ) − q (Y0 ))) . 2 σ2 ˜ 2 σ2 1 µ ˜ as 2 w + (1 − γ) σ w 2 + µ − 2 ˜ 2 ˜ w+ 2γ σ ˜ ˜ + σ(1 − γ)w = β. • First bound follows.