Optimal investment to minimize the probability of drawdownOptimal investment to minimize the probability of drawdown Bahman Angoshtari Joint work with E. Bayraktar and V. R. Young

Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References

Optimal investment to minimize the

probability of drawdown

Bahman Angoshtari

Joint work with E. Bayraktar and V. R. Young

Financial/Actuarial Mathematics Seminar

University of Michigan

November 2nd, 2016


Drawdown

ddt = maxs∈[0,t]

Ws −Wt =: Mt −Wt

Max. drawdown

maxt∈[0,T ]

ddt

is a conservative measure for thepotential loss of a portfolio

Max. drawdown duration

maxt∈[0,T ]

{t− arg max

s∈[0,t]Ws

}is a conservative measure of howlong a loss lasts


Drawdown

ddt = maxs∈[0,t]


Max. drawdown

maxt∈[0,T ]

ddt



maxt∈[0,T ]

{t− arg max

s∈[0,t]Ws



Drawdown

ddt = maxs∈[0,t]


Max. drawdown

maxt∈[0,T ]

ddt



maxt∈[0,T ]

{t− arg max

s∈[0,t]Ws



Related Literature

Drawdown constraint: Grossman and Zhou (1993), Cvitanic and Karatzas(1995), ... , Cherny and Ob loj (2013), Kardaras et al. (2014), ...

Goal-seeking problems: Typically to minimize probability of ruin or tomaximize probability of reaching a bequest

Dubins and Savage (1965, 1975), Pestien and Sudderth (1985), Karatzas (1997),Browne (1995, 1997, 1999a,b), Young (2004), Promislow and Young (2005),Bayraktar and Young (2007), Bauerle and Bayraktar (2014), Bayraktar andYoung (2016), ...

Maximizingdrift

volatility2, maximizes the prob. of reaching b before reaching a < b

Minimizing probability of drawdown (≥ a fraction of the running maximum):

Bauerle and Bayraktar (2014): the optimizer is the same as the ruin/bequestproblem if the maximized ratio is independent of the state variable

Chen et al. (2015), Angoshtari et al. (2016a), Angoshtari et al. (2016b)


Related Literature




Maximizingdrift






Related Literature




Maximizingdrift






Problem statement

A riskless with short-rate r > 0 and a stock dSt = µStdt+ σStdBt

Wt the value of the fund, and πt the amount invested in the stock

The fund pays out at at determinist rate c(Wt) ≥ 0, e.g. c(Wt) ≡ c orc(Wt) = c Wt

The budget constraint: dWt = [rWt + (µ− r)πt − c(Wt)] dt+ σπtdBt

We assume (πt) to be progressively measurable and∫ t0π2sds <∞ a.s.

The running maximum (or the “high-water-mark”):

Mt = max{M0, sup0≤s≤tWt

}W0 = w and M0 = m are given, where 0 < w ≤ m


Problem statement










Problem statement










Problem statement










Problem statement








}

W0 = w and M0 = m are given, where 0 < w ≤ m


Problem statement










Problem statement, cont’

Given a constant 0 < α < 1, drawdownhappens if Wt ≤ α Mt

Define τα = inf{t ≥ 0 : Wt ≤ α Mt}

Minimum probability of (eternal)drawdown

φ(w,m) = infπ

Pw,m(τα <∞); 0 < w ≤ m

m

w






φ(w,m) = infπ

Pw,m(τα <∞); 0 < w ≤ m

m

w






φ(w,m) = infπ

Pw,m(τα <∞); 0 < w ≤ m

m

w






φ(w,m) = infπ

Pw,m(τα <∞); 0 < w ≤ m

m

w






φ(w,m) = infπ

Pw,m(τα <∞); 0 < w ≤ m

m

w


Safe level ws

Assumption: c(w) is continuous,non-negative and non-decreasing

There is a 0 < ws ≤ +∞ such that{r w < c(w); ∀w < ws

r w > c(w); ∀w > ws

Example: c(w) ≡ c ⇒ ws =c

r

c(w) = c w, c > r ⇒ ws = +∞

If Wt ≥ ws, the consumption can be financed by investing risk-free

For ws ≤ w ≤ m, we have φ(w,m) = 0 and π∗ ≡ 0

m

w

ws


Safe level ws



r w > c(w); ∀w > ws


r

c(w) = c w, c > r ⇒ ws = +∞



m

w

ws


Safe level ws



r w > c(w); ∀w > ws


r

c(w) = c w, c > r ⇒ ws = +∞



m

w

ws


m ≥ ws and α m < w < ws

Drawdown is equivalent to hitting αm=⇒ minimizing probability of ruin

Bauerle and Bayraktar (2014)-The optimizer is obtain by maximizing

r w + (µ− r)π − c(w)

σ2π2

which yields π∗(w) =2(c(w)− r w

)µ− r

independent of m and α

The optimal wealth

dWt = (c(Wt)− rWt){dt+ 2σ

µ−r dBt}

and the min. prob. of ruin/drawdown is φ(w,m) = 1− g(w,m)g(ws,m)

where

g(w,m) =

∫ w

αm

exp

(−∫ y

αm

δ du

c(u)− ru

)dy, δ :=

1

2

(µ− rσ

)2

m

w

ws





r w + (µ− r)π − c(w)

σ2π2


)µ− r


The optimal wealth


µ−r dBt}


where

g(w,m) =

∫ w

αm

exp

(−∫ y

αm

δ du

c(u)− ru

)dy, δ :=

1

2

(µ− rσ

)2

m

w

ws





r w + (µ− r)π − c(w)

σ2π2


)µ− r


The optimal wealth


µ−r dBt}


where

g(w,m) =

∫ w

αm

exp

(−∫ y

αm

δ du

c(u)− ru

)dy, δ :=

1

2

(µ− rσ

)2

m

w

ws


m < ws and α m < w < ws

Drawdown may happen at a levelhigher than α m

The maximumdrift

volatility2 is(µ− r)2

4σ2(c(w)− r w

)not independent of w =⇒Bauerle and Bayraktar (2014) does not applyto the drawdown problem

Let Lπ f(w,m) =[r w + (µ− r)π − c(w)

]fw(w,m) + 1

2σ2π2fww(w,m)

and D = {(w,m) : 0 < α m ≤ w ≤ min(m,ws)}

m

w

ws




The maximumdrift


4σ2(c(w)− r w



]fw(w,m) + 1

2σ2π2fww(w,m)


m

w

ws




The maximumdrift


4σ2(c(w)− r w



]fw(w,m) + 1

2σ2π2fww(w,m)


m

w

ws


Theorem (Verification for the eternal problem)

If h : D→ R is bounded and continuous and satisfies:

(i) h(·,m) ∈ C2 is non-increasing and convex

(ii) h(w, ·) is continuously differentiable,except possibly at wswhere it has right and left derivative

(iii) hm(m,m) ≥ 0 if m < ws

(iv) h(α m,m) = 1

(v) h(ws,m) = 0 if m > ws

(vi) Lπ h ≥ 0 for all π

Then, h(w,m) ≤ φ(w,m) on Dm

w

ws


Theorem (Verification for the eternal problem)



(ii) h(w, ·) is continuously differentiable,except possibly at wswhere it has right and left derivative


(iv) h(α m,m) = 1


(vi) Lπ h ≥ 0 for all π

Then, h(w,m) ≤ φ(w,m) on D

Proof: φ(w,m) = infπ

Ew,m(1{τα<∞}), apply Ito’s formula to h(Wπτn ,M

πτn) ...

m

w

ws


m < ws: HJB equation

For N ≤ ws and α m ≤ w ≤ m ≤ NsupπL hN =

(r w − c(w)

)hNw − δ

(hNw )2

hNww= 0;

hN (α m,m) = 1, hNm(m,m) = 0

hN (N,N) = 0

(BVP)

Proposition

The solution of (BVP) is hN (w,m) = 1− e−∫Nm f(y)dy g(w,m)

g(N,N)

where f(m) = α

[1

g(m,m)− δ

c(αm)− rαm

].


m < ws: The optimal strategy

hws is the minimum probability of drawdown on α m ≤ w ≤ m ≤ wsExtra care if ws = +∞

The optimal strategy is π∗(w) = −µ− rσ2

hwswhwsww

=2(c(w)− r w

)µ− r

The same as the one for probability of ruin!

Note that for m < ws, we have π∗(m) > 0The optimal strategy allows for the running max to increase to ws





hwswhwsww

=2(c(w)− r w

)µ− r







hwswhwsww

=2(c(w)− r w

)µ− r




The optimal strategy

TheoremThe optimal strategy is

π∗(Wt) =

0; Wt ≥ ws2(c(w)−r w

)µ−r ; α m < Wt < ws

the minimum probability of drawdown is

φ(w,m) =

1− g(w,m)

g(ws,m), if αm ≤ w ≤ ws,m ≥ ws,

1− e−∫wsm f(y)dy g(w,m)

g(ws,ws), if αm ≤ w ≤ m < ws,

The optimal strategy is:

independent of α and Mt

for w < ws, it is optimal to let Mt to increase up to ws

m

w

ws

m


The optimal strategy

TheoremThe optimal strategy is

π∗(Wt) =

0; Wt ≥ ws2(c(w)−r w

)µ−r ; α m < Wt < ws

the minimum probability of drawdown is

φ(w,m) =

1− g(w,m)

g(ws,m), if αm ≤ w ≤ ws,m ≥ ws,

1− e−∫wsm f(y)dy g(w,m)

g(ws,ws), if αm ≤ w ≤ m < ws,

The optimal strategy is:

independent of α and Mt

for w < ws, it is optimal to let Mt to increase up to ws

m

w

ws

m


The lifetime drawdown problem

The same market setting as before

Introduce time of death of the investor τd ∼ Exp(λ)

Minimum probability of lifetime drawdown

φ(w,m) = infπ

Pw,m(τα < τd); 0 < w ≤ m

Chen et al. (2015) considered the case of

proportional consumption

c(w) = κ w for κ > r =⇒ ws =∞

They showed that it is not optimal for Mt to increase

We consider constant consumption rate: c(w) = c > 0






φ(w,m) = infπ

Pw,m(τα < τd); 0 < w ≤ m



c(w) = κ w for κ > r =⇒ ws =∞








φ(w,m) = infπ

Pw,m(τα < τd); 0 < w ≤ m



c(w) = κ w for κ > r =⇒ ws =∞



m

w






φ(w,m) = infπ

Pw,m(τα < τd); 0 < w ≤ m



c(w) = κ w for κ > r =⇒ ws =∞



m

w






φ(w,m) = infπ

Pw,m(τα < τd); 0 < w ≤ m



c(w) = κ w for κ > r =⇒ ws =∞



m

w


m > cr

The safe level is cr: c < r w ⇔ w > c

r

For Wt ≥ cr, we have π∗t = 0 and ϕ(w,m) = 0

For w < cr≤ m

Drawdown is equivalent to hitting αm=⇒ minimizing probability of lifetime ruin

Young (2004)- π∗t =µ− rσ2

1

γ − 1

( cr−Wπ

t

)independent of m and α

γ =1

2r

[(r + λ+ δ) +

√(r + λ+ δ)2 − 4rλ

]> 1, δ =

1

2

(µ− rσ

)2The optimal wealth: dWπ

t =(cr−Wπ

t

){(2δγ−1− r)dt+ µ−r

σ1

γ−1dBt

}the min. prob. of ruin/drawdown: φ(w,m) =

(c/r−wc/r−α m

)γ

m

w

cr


m > cr


r


For w < cr≤ m



1

γ − 1

( cr−Wπ

t


γ =1

2r

[(r + λ+ δ) +

√(r + λ+ δ)2 − 4rλ

]> 1, δ =

1

2

(µ− rσ

)2

The optimal wealth: dWπt =

(cr−Wπ

t

){(2δγ−1− r)dt+ µ−r

σ1

γ−1dBt


(c/r−wc/r−α m

)γ

m

w

cr


m > cr


r


For w < cr≤ m



1

γ − 1

( cr−Wπ

t


γ =1

2r

[(r + λ+ δ) +

√(r + λ+ δ)2 − 4rλ

]> 1, δ =

1

2

(µ− rσ

)2The optimal wealth: dWπ

t =(cr−Wπ

t

){(2δγ−1− r)dt+ µ−r

σ1

γ−1dBt


(c/r−wc/r−α m

)γ

m

w

cr


0 < m ≤ cr : Main result

There exists a critical high-water-mark

m∗ ∈ (0,c

r) such that:

(i) For m ∈ (m∗, cr), the optimal strategy

lets M to increase above m

(ii) For m ∈ (0,m∗], the optimal strategy

keeps Mt = m

m

w

crm∗




m∗ ∈ (0,c

r) such that:




keeps Mt = m

m

w

crm∗




m∗ ∈ (0,c

r) such that:




keeps Mt = m

m

w

crm∗


Theorem (Verification for the lifetime problem)



(ii) h(w, ·) is continuously differentiable,except possibly at finitely many pointswhere it has right and left derivative


(iv) h(α m,m) = 1


(vi) Lπ h− λ h ≥ 0 for all π


m

w

cr


Theorem (Verification for the lifetime problem)



(ii) h(w, ·) is continuously differentiable,except possibly at finitely many pointswhere it has right and left derivative


(iv) h(α m,m) = 1


(vi) Lπ h− λ h ≥ 0 for all π


Proof: φ(w,m) = infπ

Ew,m

[∫ ∞0

1{τα<t}λ e−λ tdt

]= inf

πEw,m

[e−λ τα

]apply Ito’s formula to e−λ τnh(Wπ

τn ,Mπτn) ...

m

w

cr


For an arbitrary m0 ∈ (0, c/r),consider the BVP on m0 ≤ m ≤ c/r and α m ≤ w ≤ m

supπL h = (r w − c)hw − δ

h2w

hww= λ h

h(α m,m) = 1, hm(m,m) = 0

limm→c/r−

h(m,m) = 0

(BVP)

Legendre transform: φ(y,m) = minw{h(w,m) + w y}

δy2φyy − (r − λ)yφy − λφ+ cy = 0

φ(yαm(m),m) = 1 + αm yαm(m), φy(yαm(m),m) = αm

φy(ym(m),m) = m, φm(ym(m),m) = 0

limm→c/r−

φ (ym(m),m) =c

rym( cr

)lim

m→c/r−φy (ym(m),m) =

c

r

(FBP)


For an arbitrary m0 ∈ (0, c/r),consider the BVP on m0 ≤ m ≤ c/r and α m ≤ w ≤ m

supπL h = (r w − c)hw − δ

h2w

hww= λ h

h(α m,m) = 1, hm(m,m) = 0

limm→c/r−

h(m,m) = 0

(BVP)

Legendre transform: φ(y,m) = minw{h(w,m) + w y}

δy2φyy − (r − λ)yφy − λφ+ cy = 0

φ(yαm(m),m) = 1 + αm yαm(m), φy(yαm(m),m) = αm

φy(ym(m),m) = m, φm(ym(m),m) = 0

limm→c/r−

φ (ym(m),m) =c

rym( cr

)lim

m→c/r−φy (ym(m),m) =

c

r

(FBP)


Ansatz: φ(y,m) = D1(m)yB1 +D2(m)yB2 + cry

B1 = 12δ

[(r − λ+ δ) +

√(r − λ+ δ)2 + 4λδ

]= γ

γ−1> 1

B2 = 12δ

[(r − λ+ δ)−

√(r − λ+ δ)2 + 4λδ

]< 0

Proposition (solution of FBP)

Assume that z′(m) =g1(z)(c/r −m) + g0(z)

h2(z)(c/r −m)2 + h1(z)(c/r −m) + h0(z)

z(c/r) = 0

(ODE)

has a solution on z : [m0, c/r]→ [0, 1]. Here, gi and hi are known functions.


Proposition (solution of FBP, cont’)

Then, the solution of (FBP) for (y,m) ∈ [ym(m), yαm(m)]× [m0, c/r] is

φ(y,m) =c

ry −

[B2

B1 −B2+( cr− αm

) 1−B2

B1 −B2yαm(m)

](y

yαm(m)

)B1

+

[B1

B1 −B2−( cr− αm

) B1 − 1

B1 −B2yαm(m)

](y

yαm(m)

)B2

,

where the free boundaries are

1

yαm(m)

B1B2

B1 −B2

(z(m)B1−1 − z(m)B2−1

)=( cr−m

)−( cr− αm

)[B1(1−B2)

B1 −B2z(m)B1−1 +

B2(B1 − 1)

B1 −B2z(m)B2−1

]and ym = z(m)yαm(m)


z′(m) =g1(z)(c/r −m) + g0(z)

h2(z)(c/r −m)2 + h1(z)(c/r −m) + h0(z)

z(c/r) = 0

(ODE)

A solution of (ODE) on [m0, c/r] yields φ(w,m) for m0 ≤ m ≤ c/r andαm ≤ w ≤ m

The “inverse” of z(m) satisfies an Able equation

Any chance for a closed form solution?

Functions gi and hi are too complicated...


z′(m) =g1(z)(c/r −m) + g0(z)

h2(z)(c/r −m)2 + h1(z)(c/r −m) + h0(z)

z(c/r) = 0

(ODE)

A solution of (ODE) on [m0, c/r] yields φ(w,m) for m0 ≤ m ≤ c/r andαm ≤ w ≤ m

The “inverse” of z(m) satisfies an Able equation

Any chance for a closed form solution?

Functions gi and hi are too complicated...


The auxiliary functions

g0(z) = (1 − α)c

r(zB2 − z

B1 )[(B2 − 1)z

B1−1 − (B1 − 1)zB2−1

]g1(z) = (z

B2 − zB1 )[B1 − B2 + α(B2 − 1)z

B1−1

− α(B1 − 1)zB2−1

+ α(B1 − 1)(B2 − 1)(zB1−1 − z

B2−1)]

h0(z) = (1 − α)2

(c

r

)2

(B1 − B2)zB1+B2−2

[(B1 − 1)z

B2−1 − (B2 − 1)zB1−1

]h1(z) = (1 − α)

c

r

{[(B2 − 1)z

B1−1 − (B1 − 1)zB2−1

]×[

(B1 − 1)zB1−1 − (B2 − 1)z

B2−1 − α(B1 − B2)zB1+B2−2

]− (B1 − B2)z

B1+B2−2[B1 − B2 + α(B2 − 1)z

B1−1 − α(B1 − 1)zB2−1

]}h2(z) =

[(B1 − 1)z

B1−1 − (B2 − 1)zB2−1 − α(B1 − B2)z

B1+B2−2]×[

B1 − B2 + α(B2 − 1)zB1−1 − α(B1 − 1)z

B2−1]


z′(m) =g1(z)(c/r −m) + g0(z)

h2(z)(c/r −m)2 + h1(z)(c/r −m) + h0(z)=:

G(m, z)

H(m, z)

z(c/r) = 0

(ODE)

0 bm 5 10 15 20 c

rm

0

0.2

0.4

1=x(bm)

0.8

1

1.2

z

Sign of F (m; z) = G(m; z)=H(m; z)

m = 9(z)

z = 1x(m)

H(

1x(m)

,m)≡ 0 for a known

functions x(m)

G (ξ(z), z) ≡ 0 for a knownfunctions ξ(z)

(ξ(z), z) and(

1x(m)

,m)

intersect at(

1x(m)

, m)

where

m ∈ (0, c/r)


z′(m) =g1(z)(c/r −m) + g0(z)

h2(z)(c/r −m)2 + h1(z)(c/r −m) + h0(z)=:

G(m, z)

H(m, z)

z(c/r) = 0

(ODE)

0 bm 5 10 15 20 c

rm

0

0.2

0.4

1=x(bm)

0.8

1

1.2

z


m = 9(z)

z = 1x(m)

H(

1x(m)


functions x(m)


(ξ(z), z) and(

1x(m)

,m)

intersect at(

1x(m)

, m)

where

m ∈ (0, c/r)


z′(m) =g1(z)(c/r −m) + g0(z)

h2(z)(c/r −m)2 + h1(z)(c/r −m) + h0(z)=:

G(m, z)

H(m, z)

z(c/r) = 0

(ODE)

0 bm 5 10 15 20 c

rm

0

0.2

0.4

1=x(bm)

0.8

1

1.2

z


m = 9(z)

z = 1x(m)

H(

1x(m)


functions x(m)


(ξ(z), z) and(

1x(m)

,m)

intersect at(

1x(m)

, m)

where

m ∈ (0, c/r)


z′(m) =g1(z)(c/r −m) + g0(z)

h2(z)(c/r −m)2 + h1(z)(c/r −m) + h0(z)=:

G(m, z)

H(m, z)

z(c/r) = 0

(ODE)

0 bm 5 10 15 20 c

rm

0

0.2

0.4

1=x(bm)

0.8

1

1.2

z

Integral Curves

H(

1x(m)


functions x(m)


(ξ(z), z) and(

1x(m)

,m)

intersect at(

1x(m)

, m)

where

m ∈ (0, c/r)


z′(m) =g1(z)(c/r −m) + g0(z)

h2(z)(c/r −m)2 + h1(z)(c/r −m) + h0(z)=:

G(m, z)

H(m, z)

z(c/r) = 0

(ODE)

0 m$ bm 5 10 15 20 c

rm

0

0.2

0.4

1=x(bm)

1=x(m$)

0.8

1

1.2

z

The solution of BVP (5.22)

H(

1x(m)


functions x(m)


(ξ(z), z) and(

1x(m)

,m)

intersect at(

1x(m)

, m)

where

m ∈ (0, c/r)


Proposition (solution of ODE)

Assume that there exist solutions z(m) and z(m) of (ODE) in D0 suchthat z (respectively, z) satisfies the terminal condition z(m0) = z0 for(m0, z0) ∈ ∂D0 (respectively, (m0, z0) ∈ ∂+D0) and extends on the left to∂−D0\

{(m, 1/x(m)

)}. Let m∗ (respectively, m∗) be the value of m where z

(respectively, z) intercepts ∂−D0.

Then, there exists a unique solution z(m) of (ODE) in D0 satisfying theterminal condition z(c/r) = 0 and extending on the left to the boundary∂−D0 such that z(m∗) = 1/x(m∗) for some m∗ ∈ (m∗,m∗). In particular,z(m) is not defined on (0,m∗).

0 bm 5 10 15 20 c

rm

0

0.2

0.4

1=x(bm)

0.8

1

1.2

z

D0 and its boundaries

@!D0

@D0

@D0

@+D0

D0




{(m, 1/x(m)




0 bm 5 10 15 20 c

rm

0

0.2

0.4

1=x(bm)

0.8

1

1.2

z

Integral Curves




{(m, 1/x(m)




0 m$ bm 5 10 15 20 c

rm

0

0.2

0.4

1=x(bm)

1=x(m$)

0.8

1

1.2

z



Proposition (Minimum probability of DD for m∗ ≤ m ≤ cr)

Assume that z(m) is the solution of (ODE) on [m∗, c/r]

Then, for αm ≤ w ≤ m and m∗ ≤ m ≤ c/r

φ(w,m) =B1 − 1

B1 −B2

[B2 +

( cr− αm

)(1−B2) yαm(m)

]( y(w)

yαm(m)

)B1

+1−B2

B1 −B2

[B1 −

( cr− αm

)(B1 − 1) yαm(m)

]( y(w)

yαm(m)

)B2

where y(w) ∈ [ym(m), yαm(m)] uniquely solves

c

r− w =

B1

B1 −B2

[B2

yαm(m)+( cr− αm

)(1−B2)

](y(w)

yαm(m)

)B1−1

− B2

B1 −B2

[B1

yαm(m)−( cr− αm

)(B1 − 1)

](y(w)

yαm(m)

)B2−1


For αm ≤ w ≤ m and m∗ ≤ m ≤ c/r

π∗(w,m) =µ− rσ2

B1(B1 − 1)

B1 −B2

[B2

yαm(m)+( cr− αm

)(1−B2)

](y

yαm(m)

)B1−1

+µ− rσ2

B2(1−B2)

B1 −B2

[B1


)(B1 − 1)

](y

yαm(m)

)B2−1

0 m$ 5 10 15 20 c

rm

0

0.5

1

?(m

;m)

0 m$ 5 10 15 20 c

rm

0

10

20

?m(m

;m)

#10!3

0 m$ 5 10 15 20 c

rm

-0.1

-0.05

0

?w(m

;m)

0 m$ 5 10 15 20 c

rm

0

5000

10000

?w

w(m!0;

m)

0 m$ 5 10 15 20 c

rm

0

0.05

0.1

?w

w(m

;m)

0 m$ 5 10 15 20 c

rm

0

5

10

:$(m

;m)

m

w

crm∗


0 < m < m∗: The restricted problem

No solution for (ODE) on 0 < m < m∗

However, it seems that

π∗(m,m) = 0 for 0 < m ≤ m∗

Consider the restricted problem where

M is not allowed to increasesupπL h = (r w − c)hw − δ

h2w

hww= λ h

h(α m,m) = 1, limw→m−

hw(w,m)

hww(w,m)= 0

The dual FBP corresponds to an optimal controller-stopper problem

This is where we got the ansatz ...

The dual problem reduces to an ODE =⇒ its solution is1

x(m)!

0 m$ bm 5 10 15 20 c

rm

0

0.2

0.4

1=x(bm)

1=x(m$)

0.8

1

1.2

z






π∗(m,m) = 0 for 0 < m ≤ m∗



h2w

hww= λ h


hw(w,m)

hww(w,m)= 0




x(m)!

m

w

crm∗





π∗(m,m) = 0 for 0 < m ≤ m∗



h2w

hww= λ h


hw(w,m)

hww(w,m)= 0




x(m)!

m

w

crm∗





π∗(m,m) = 0 for 0 < m ≤ m∗



h2w

hww= λ h


hw(w,m)

hww(w,m)= 0




x(m)!

m

w

crm∗





π∗(m,m) = 0 for 0 < m ≤ m∗



h2w

hww= λ h


hw(w,m)

hww(w,m)= 0




x(m)!

m

w

crm∗


The solution for 0 < m ≤ cr

The equations for “φ(w,m)” and π∗(w,m) in the restricted problem are thesame as the one for m ∈ [m∗, c/r], expect that we replace z(m) with 1

x(m)!

Let us glue the solutions of the two problems

η(m) =

{1/x(m), 0 ≤ m ≤ m∗

z(m), m∗ ≤ m ≤ c/r

The “free boundary”:

1

yαm(m)

B1B2

B1 −B2

(η(m)B1−1 − η(m)B2−1

)=( cr−m

)−( cr− αm

)[B1(1−B2)

B1 −B2η(m)B1−1 +

B2(B1 − 1)

B1 −B2η(m)B2−1

]




x(m)!


η(m) =

{1/x(m), 0 ≤ m ≤ m∗

z(m), m∗ ≤ m ≤ c/r


1

yαm(m)

B1B2

B1 −B2

(η(m)B1−1 − η(m)B2−1

)=( cr−m

)−( cr− αm

)[B1(1−B2)

B1 −B2η(m)B1−1 +

B2(B1 − 1)

B1 −B2η(m)B2−1

]




x(m)!


η(m) =

{1/x(m), 0 ≤ m ≤ m∗

z(m), m∗ ≤ m ≤ c/r


1

yαm(m)

B1B2

B1 −B2

(η(m)B1−1 − η(m)B2−1

)=( cr−m

)−( cr− αm

)[B1(1−B2)

B1 −B2η(m)B1−1 +

B2(B1 − 1)

B1 −B2η(m)B2−1

]

0 m$ bm 5 10 15 20 c

rm

0

0.2

0.4

1=x(bm)

1=x(m$)

0.8

1

1.2

z





x(m)!


η(m) =

{1/x(m), 0 ≤ m ≤ m∗

z(m), m∗ ≤ m ≤ c/r


1

yαm(m)

B1B2

B1 −B2

(η(m)B1−1 − η(m)B2−1

)=( cr−m

)−( cr− αm

)[B1(1−B2)

B1 −B2η(m)B1−1 +

B2(B1 − 1)

B1 −B2η(m)B2−1

]


Theorem (Minimum probability of DD for 0 < m ≤ cr)

Assume that z(m) is the solution of (ODE) on [m∗, c/r]

and define η and yαm. Then, for αm ≤ w ≤ m and 0 < m ≤ c/r

φ(w,m) =B1 − 1

B1 −B2

[B2 +

( cr− αm

)(1−B2) yαm(m)

]( y(w)

yαm(m)

)B1

+1−B2

B1 −B2

[B1 −

( cr− αm

)(B1 − 1) yαm(m)

]( y(w)

yαm(m)

)B2

where y(w) ∈ [ym(m), yαm(m)] uniquely solves

c

r− w =

B1

B1 −B2

[B2

yαm(m)+( cr− αm

)(1−B2)

](y(w)

yαm(m)

)B1−1

− B2

B1 −B2

[B1


)(B1 − 1)

](y(w)

yαm(m)

)B2−1


For αm ≤ w ≤ m and 0 < m ≤ c/r


B1(B1 − 1)

B1 −B2

[B2

yαm(m)+( cr− αm

)(1−B2)

](y

yαm(m)

)B1−1

+µ− rσ2

B2(1−B2)

B1 −B2

[B1


)(B1 − 1)

](y

yαm(m)

)B2−1




B1(B1 − 1)

B1 −B2

[B2

yαm(m)+( cr− αm

)(1−B2)

](y

yαm(m)

)B1−1

+µ− rσ2

B2(1−B2)

B1 −B2

[B1


)(B1 − 1)

](y

yαm(m)

)B2−1

0 m$ 5 10 15 20 c

rm

0

0.5

1

?(m

;m)

0 m$ 5 10 15 20 c

rm

0

10

20

?m(m

;m)

#10!3

0 m$ 5 10 15 20 c

rm

-0.1

-0.05

0

?w(m

;m)

0 m$ 5 10 15 20 c

rm

0

5000

10000

?w

w(m!0;

m)

0 m$ 5 10 15 20 c

rm

0

0.05

0.1

?w

w(m

;m)

0 m$ 5 10 15 20 c

rm

0

5

10

:$(m

;m)

m

w

crm∗




B1(B1 − 1)

B1 −B2

[B2

yαm(m)+( cr− αm

)(1−B2)

](y

yαm(m)

)B1−1

+µ− rσ2

B2(1−B2)

B1 −B2

[B1


)(B1 − 1)

](y

yαm(m)

)B2−1

0

0

5

5

10

:$(w

;m)

10

15

w

15

20

m

c

r0m$5101520c

r

m

w

crm∗


0

0.2

0

0.4

0.6

5

0.8

?(w

;m)

1

10

w

15

20

m

c

r0m$5101520c

r


0

0

0.2

0.4

5

0.6

?(w

;m)

0.8

10

1

w

15

20 0

m

510m$c

r20c

r


Conclusion

We determined the optimal strategy to minimize the probability ofdrawdown in two scenarios

For the infinite time horizon problem, the optimal strategy is thesame as the one for infinite time ruin problem

For the lifetime problem with constant consumption, there is atrade-off in allowing the high-water-mark to increase

0 < m ≤ m∗: increasing high-water-mark level makes DD moreprobable (0 < m ≤ m∗)m∗ < m < c/r: letting wealth increase helps fund theconsumption, thus reducing the probability of DD

Will this behavior exist for other types of consumption wherews <∞?

Adding trade-off between risk and return?


Thank you for your attention!

Angoshtari, B., E. Bayraktar, and V. R. Young: Minimizing the probabilityof lifetime drawdown under constant consumption. Insurance:Mathematics and Economics (2016a).

——— Optimal investment to minimize the probability of drawdown.Stochastics, pp. 1–13 (2016b).


Bauerle, N. and E. Bayraktar: A note on applications of stochastic orderingto control problems in insurance and finance. Stochastics AnInternational Journal of Probability and Stochastic Processes, volume 86,no. 2: pp. 330–340 (2014).

Bayraktar, E. and V. R. Young: Correspondence between lifetime minimumwealth and utility of consumption. Finance and Stochastics, volume 11,no. 2: pp. 213–236 (2007).

——— Optimally investing to reach a bequest goal. Insurance:Mathematics and Economics, volume 70: pp. 1–10 (2016).

Browne, S.: Optimal investment policies for a firm with a random riskprocess: exponential utility and minimizing the probability of ruin.Mathematics of operations research, volume 20, no. 4: pp. 937–958(1995).

——— Survival and growth with a liability: Optimal portfolio strategies incontinuous time. Mathematics of Operations Research, volume 22, no. 2:pp. 468–493 (1997).

——— Beating a moving target: Optimal portfolio strategies foroutperforming a stochastic benchmark. Finance and Stochastics,volume 3, no. 3: pp. 275–294 (1999a).


——— Reaching goals by a deadline: Digital options and continuous-timeactive portfolio management. Advances in Applied Probability,volume 31, no. 2: pp. 551–577 (1999b).

Chen, X., D. Landriault, B. Li, and D. Li: On minimizing drawdown risksof lifetime investments. Insurance: Mathematics and Economics,volume 65: pp. 46–54 (2015).

Cherny, V. and J. Ob loj: Portfolio optimisation under non-linear drawdownconstraints in a semimartingale financial model. Finance and Stochastics,volume 17, no. 4: pp. 771–800 (2013).

Cvitanic, J. and I. Karatzas: On portfolio optimization under” drawdown”constraints. IMA Volumes in Mathematics and its Applications,volume 65: pp. 35–35 (1995).

Dubins, L. E. and L. J. Savage: How to gamble if you must: Inequalities forstochastic processes. 1965 edition McGraw-Hill, New York. 1976 editionDover, New York (1965, 1975).

Grossman, S. J. and Z. Zhou: Optimal investment strategies for controllingdrawdowns. Mathematical finance, volume 3, no. 3: pp. 241–276 (1993).


Karatzas, I.: Adaptive control of a diffusion to a goal and a parabolicmonge-ampere-type equation. Asian Journal of Mathematics, volume 1:pp. 295–313 (1997).

Kardaras, C., J. Ob loj, and E. Platen: The numeraire property andlong-term growth optimality for drawdown-constrained investments.Mathematical Finance (2014).

Pestien, V. C. and W. D. Sudderth: Continuous-time red and black: Howto control a diffusion to a goal. Mathematics of Operations Research,volume 10, no. 4: pp. 599–611 (1985).

Promislow, D. S. and V. R. Young: Minimizing the probability of ruinwhen claims follow brownian motion with drift. North AmericanActuarial Journal, volume 9, no. 3: pp. 110–128 (2005).

Young, V. R.: Optimal investment strategy to minimize the probability oflifetime ruin. North American Actuarial Journal, volume 8, no. 4: pp.106–126 (2004).

Documents

Optimal investment to minimize the probability of drawdownOptimal investment to minimize the probability of drawdown Bahman Angoshtari Joint work with E. Bayraktar and V. R. Young