Optimal investment to minimize the probability of drawdown

Transcription

1 Optial investent to iniize the probability of drawdown Bahan Angoshtari Joint work with E. Bayraktar and V. R. Young Financial/Actuarial Matheatics Seinar University of Michigan Noveber 2 nd, 2016

2 Drawdown dd t = ax W s W t =: M t W t s [0,t]

3 Drawdown dd t = ax W s W t =: M t W t s [0,t] Max. drawdown ax dd t t [0,T ] is a conservative easure for the potential loss of a portfolio

4 Drawdown dd t = ax W s W t =: M t W t s [0,t] Max. drawdown ax dd t t [0,T ] is a conservative easure for the potential loss of a portfolio Max. drawdown duration ax t [0,T ] { } t arg ax W s s [0,t] is a conservative easure of how long a loss lasts

5 Related Literature Drawdown constraint: Grossan and Zhou (1993), Cvitanic and Karatzas (1995),..., Cherny and Ob lój (2013), Kardaras et al. (2014),...

6 Related Literature Drawdown constraint: Grossan and Zhou (1993), Cvitanic and Karatzas (1995),..., Cherny and Ob lój (2013), Kardaras et al. (2014),... Goal-seeking probles: Typically to iniize probability of ruin or to axiize probability of reaching a bequest Dubins and Savage (1965, 1975), Pestien and Sudderth (1985), Karatzas (1997), Browne (1995, 1997, 1999a,b), Young (2004), Proislow and Young (2005), Bayraktar and Young (2007), Bäuerle and Bayraktar (2014), Bayraktar and Young (2016),... drift Maxiizing, axiizes the prob. of reaching b before reaching a < b volatility2

7 Related Literature Drawdown constraint: Grossan and Zhou (1993), Cvitanic and Karatzas (1995),..., Cherny and Ob lój (2013), Kardaras et al. (2014),... Goal-seeking probles: Typically to iniize probability of ruin or to axiize probability of reaching a bequest Dubins and Savage (1965, 1975), Pestien and Sudderth (1985), Karatzas (1997), Browne (1995, 1997, 1999a,b), Young (2004), Proislow and Young (2005), Bayraktar and Young (2007), Bäuerle and Bayraktar (2014), Bayraktar and Young (2016),... drift Maxiizing, axiizes the prob. of reaching b before reaching a < b volatility2 Miniizing probability of drawdown ( a fraction of the running axiu): Bäuerle and Bayraktar (2014): the optiizer is the sae as the ruin/bequest proble if the axiized ratio is independent of the state variable Chen et al. (2015), Angoshtari et al. (2016a), Angoshtari et al. (2016b)

8 Proble stateent A riskless with short-rate r > 0 and a stock ds t = µs tdt + σs tdb t

9 Proble stateent A riskless with short-rate r > 0 and a stock ds t = µs tdt + σs tdb t W t the value of the fund, and π t the aount invested in the stock

10 Proble stateent A riskless with short-rate r > 0 and a stock ds t = µs tdt + σs tdb t W t the value of the fund, and π t the aount invested in the stock The fund pays out at at deterinist rate c(w t) 0, e.g. c(w t) = c W t c(w t) c or

11 Proble stateent A riskless with short-rate r > 0 and a stock ds t = µs tdt + σs tdb t W t the value of the fund, and π t the aount invested in the stock The fund pays out at at deterinist rate c(w t) 0, e.g. c(w t) = c W t c(w t) c or The budget constraint: dw t = [r W t + (µ r)π t c(w t)] dt + σπ tdb t We assue (π t) to be progressively easurable and t 0 π2 sds < a.s.

12 Proble stateent A riskless with short-rate r > 0 and a stock ds t = µs tdt + σs tdb t W t the value of the fund, and π t the aount invested in the stock The fund pays out at at deterinist rate c(w t) 0, e.g. c(w t) = c W t c(w t) c or The budget constraint: dw t = [r W t + (µ r)π t c(w t)] dt + σπ tdb t We assue (π t) to be progressively easurable and t 0 π2 sds < a.s. The running axiu (or the } high-water-ark ): M t = ax {M 0, sup 0 s t W t

13 Proble stateent A riskless with short-rate r > 0 and a stock ds t = µs tdt + σs tdb t W t the value of the fund, and π t the aount invested in the stock The fund pays out at at deterinist rate c(w t) 0, e.g. c(w t) = c W t c(w t) c or The budget constraint: dw t = [r W t + (µ r)π t c(w t)] dt + σπ tdb t We assue (π t) to be progressively easurable and t 0 π2 sds < a.s. The running axiu (or the } high-water-ark ): M t = ax {M 0, sup 0 s t W t W 0 = w and M 0 = are given, where 0 < w

14 Proble stateent, cont w Given a constant 0 < α < 1, drawdown happens if W t α M t

15 Proble stateent, cont w Given a constant 0 < α < 1, drawdown happens if W t α M t Define τ α = inf{t 0 : W t α M t}

16 Proble stateent, cont w Given a constant 0 < α < 1, drawdown happens if W t α M t Define τ α = inf{t 0 : W t α M t} Miniu probability of (eternal) drawdown φ(w, ) = inf π P w, (τ α < ); 0 < w

17 Proble stateent, cont w Given a constant 0 < α < 1, drawdown happens if W t α M t Define τ α = inf{t 0 : W t α M t} Miniu probability of (eternal) drawdown φ(w, ) = inf π P w, (τ α < ); 0 < w

18 Proble stateent, cont w Given a constant 0 < α < 1, drawdown happens if W t α M t Define τ α = inf{t 0 : W t α M t} Miniu probability of (eternal) drawdown φ(w, ) = inf π P w, (τ α < ); 0 < w

19 Safe level w s w Assuption: c(w) is continuous, non-negative and non-decreasing There is a 0 < w s + such that { r w < c(w); w < ws r w > c(w); w > w s w s

20 Safe level w s w Assuption: c(w) is continuous, non-negative and non-decreasing There is a 0 < w s + such that { r w < c(w); w < ws r w > c(w); w > w s Exaple: c(w) c w s = c r c(w) = c w, c > r w s = + w s

21 Safe level w s w Assuption: c(w) is continuous, non-negative and non-decreasing There is a 0 < w s + such that { r w < c(w); w < ws r w > c(w); w > w s Exaple: c(w) c w s = c r c(w) = c w, c > r w s = + w s If W t w s, the consuption can be financed by investing risk-free For w s w, we have φ(w, ) = 0 and π 0

22 w s and α < w < w s w Drawdown is equivalent to hitting α = iniizing probability of ruin w s

23 w s and α < w < w s w Drawdown is equivalent to hitting α = iniizing probability of ruin Bäuerle and Bayraktar (2014)- The optiizer is obtain by axiizing r w + (µ r)π c(w) σ 2 π 2 which yields π (w) = 2( c(w) r w ) independent of and α µ r w s

24 w s and α < w < w s w Drawdown is equivalent to hitting α = iniizing probability of ruin Bäuerle and Bayraktar (2014)- The optiizer is obtain by axiizing r w + (µ r)π c(w) σ 2 π 2 which yields π (w) = 2( c(w) r w ) independent of and α µ r w s The optial wealth { } dw t = (c(w t) rw t) dt + 2σ µ r dbt and the in. prob. of ruin/drawdown is φ(w, ) = 1 g(w,) g(w, ) = w α exp ( y α ) δ du c(u) ru dy, δ := 1 2 where g(w s,) ) 2 ( µ r σ

25 < w s and α < w < w s w Drawdown ay happen at a level higher than α w s

26 < w s and α < w < w s w Drawdown ay happen at a level higher than α drift The axiu volatility 2 is (µ r) 2 4σ 2( c(w) r w ) not independent of w = Bäuerle and Bayraktar (2014) does not apply to the drawdown proble w s

27 < w s and α < w < w s w Drawdown ay happen at a level higher than α drift The axiu volatility 2 is (µ r) 2 4σ 2( c(w) r w ) not independent of w = Bäuerle and Bayraktar (2014) does not apply to the drawdown proble w s Let L π f(w, ) = [ r w + (µ r)π c(w) ] f w(w, ) σ2 π 2 f ww(w, ) and D = {(w, ) : 0 < α w in(, w s)}

28 Theore (Verification for the eternal proble) If h : D R is bounded and continuous and satisfies: (i) h(, ) C 2 is non-increasing and convex (ii) h(w, ) is continuously differentiable, except possibly at w s where it has right and left derivative (iii) h (, ) 0 if < w s (iv) h(α, ) = 1 (v) h(w s, ) = 0 if > w s w (vi) L π h 0 for all π Then, h(w, ) φ(w, ) on D w s

29 Theore (Verification for the eternal proble) If h : D R is bounded and continuous and satisfies: (i) h(, ) C 2 is non-increasing and convex (ii) h(w, ) is continuously differentiable, except possibly at w s where it has right and left derivative (iii) h (, ) 0 if < w s (iv) h(α, ) = 1 (v) h(w s, ) = 0 if > w s w (vi) L π h 0 for all π Then, h(w, ) φ(w, ) on D w s Proof: φ(w, ) = inf π E w, (1 {τα< }), apply Itô s forula to h(w π τ n, M π τ n )...

30 < w s : HJB equation For N w s and α w N sup L h N = ( r w c(w) ) h N w δ (hn w ) 2 π h N ww h N (α, ) = 1, h N (, ) = 0 h N (N, N) = 0 = 0; (BVP) Proposition The solution of (BVP) is h N (w, ) = 1 e N f(y)dy g(w, ) g(n, N) [ ] 1 where f() = α g(, ) δ. c(α) rα

31 < w s : The optial strategy h ws is the iniu probability of drawdown on α w w s Extra care if w s = +

32 < w s : The optial strategy h ws is the iniu probability of drawdown on α w w s Extra care if w s = + The optial strategy is π (w) = µ r h ws w = 2( c(w) r w ) σ 2 h ws ww µ r The sae as the one for probability of ruin!

33 < w s : The optial strategy h ws is the iniu probability of drawdown on α w w s Extra care if w s = + The optial strategy is π (w) = µ r σ 2 The sae as the one for probability of ruin! h ws w = 2( c(w) r w ) h ws ww µ r Note that for < w s, we have π () > 0 The optial strategy allows for the running ax to increase to w s

34 The optial strategy Theore The optial strategy is 0; W t w s π (W t) = ( ) 2 c(w) r w ; α < W µ r t < w s the iniu probability of drawdown is g(w, ) 1 g(w s, ), φ(w, ) = 1 e ws f(y)dy g(w,), g(w s,w s) w w s if α w ws, ws, if α w < ws,

35 The optial strategy Theore The optial strategy is 0; W t w s π (W t) = ( ) 2 c(w) r w ; α < W µ r t < w s the iniu probability of drawdown is g(w, ) 1 g(w s, ), φ(w, ) = 1 e ws f(y)dy g(w,), g(w s,w s) w w s if α w ws, ws, if α w < ws, The optial strategy is: independent of α and M t for w < w s, it is optial to let M t to increase up to w s

36 The lifetie drawdown proble The sae arket setting as before Introduce tie of death of the investor τ d Exp(λ)

37 The lifetie drawdown proble The sae arket setting as before Introduce tie of death of the investor τ d Exp(λ) Miniu probability of lifetie drawdown φ(w, ) = inf π P w, (τ α < τ d ); 0 < w

38 The lifetie drawdown proble The sae arket setting as before Introduce tie of death of the investor τ d Exp(λ) w Miniu probability of lifetie drawdown φ(w, ) = inf π P w, (τ α < τ d ); 0 < w Chen et al. (2015) considered the case of proportional consuption c(w) = κ w for κ > r = w s =

39 The lifetie drawdown proble The sae arket setting as before Introduce tie of death of the investor τ d Exp(λ) w Miniu probability of lifetie drawdown φ(w, ) = inf π P w, (τ α < τ d ); 0 < w Chen et al. (2015) considered the case of proportional consuption c(w) = κ w for κ > r = w s = They showed that it is not optial for M t to increase

40 The lifetie drawdown proble The sae arket setting as before Introduce tie of death of the investor τ d Exp(λ) w Miniu probability of lifetie drawdown φ(w, ) = inf π P w, (τ α < τ d ); 0 < w Chen et al. (2015) considered the case of proportional consuption c(w) = κ w for κ > r = w s = They showed that it is not optial for M t to increase We consider constant consuption rate: c(w) = c > 0

41 > c r w The safe level is c r : c < r w w > c r For W t c r, we have π t = 0 and ϕ(w, ) = 0 c r

42 > c r w The safe level is c r : c < r w w > c r For W t c r, we have π t = 0 and ϕ(w, ) = 0 For w < c r Drawdown is equivalent to hitting α = iniizing probability of lifetie ruin Young (2004)- πt = µ r 1 ( c ) σ 2 γ 1 r W t π independent of and α γ = 1 [ (r + λ + δ) + ] (r + λ + δ) 2r 2 4rλ > 1, δ = 1 2 c r ( µ r ) 2 σ

43 > c r w The safe level is c r : c < r w w > c r For W t c r, we have π t = 0 and ϕ(w, ) = 0 For w < c r Drawdown is equivalent to hitting α = iniizing probability of lifetie ruin Young (2004)- πt = µ r 1 ( c ) σ 2 γ 1 r W t π independent of and α γ = 1 [ (r + λ + δ) + ] (r + λ + δ) 2r 2 4rλ > 1, δ = 1 2 The optial wealth: dwt π = ( c W ) {( π 2δ r t the in. prob. of ruin/drawdown: φ(w, ) = γ 1 r ) ( c/r w c/r α dt + µ r σ ) γ c r ( µ r ) 2 σ } 1 γ 1 dbt

44 0 < c r : Main result w There exists a critical high-water-ark (0, c ) such that: r c r

45 0 < c r : Main result w There exists a critical high-water-ark (0, c ) such that: r (i) For (, c ), the optial strategy r lets M to increase above c r

46 0 < c r : Main result w There exists a critical high-water-ark (0, c ) such that: r (i) For (, c ), the optial strategy r lets M to increase above c r (ii) For (0, ], the optial strategy keeps M t =

47 Theore (Verification for the lifetie proble) If h : D R is bounded and continuous and satisfies: (i) h(, ) C 2 is non-increasing and convex (ii) h(w, ) is continuously differentiable, except possibly at finitely any points where it has right and left derivative (iii) h (, ) 0 if < w s (iv) h(α, ) = 1 (v) h(w s, ) = 0 if > w s w (vi) L π h λ h 0 for all π Then, h(w, ) φ(w, ) on D c r

48 Theore (Verification for the lifetie proble) If h : D R is bounded and continuous and satisfies: (i) h(, ) C 2 is non-increasing and convex (ii) h(w, ) is continuously differentiable, except possibly at finitely any points where it has right and left derivative (iii) h (, ) 0 if < w s (iv) h(α, ) = 1 (v) h(w s, ) = 0 if > w s w (vi) L π h λ h 0 for all π Then, h(w, ) φ(w, ) on D c r [ ] Proof: φ(w, ) = inf E w, 1 [ ] π {τα<t}λ e λ t dt = inf E w, e λ τα 0 π apply Itô s forula to e λ τn h(wτ π n, Mτ π n )...

49 For an arbitrary 0 (0, c/r), consider the BVP on 0 c/r and α w sup L h = (r w c)h w δ h2 w = λ h π h ww h(α, ) = 1, h (, ) = 0 li c/r h(, ) = 0 (BVP)

50 For an arbitrary 0 (0, c/r), consider the BVP on 0 c/r and α w sup L h = (r w c)h w δ h2 w = λ h π h ww h(α, ) = 1, h (, ) = 0 li c/r h(, ) = 0 Legendre transfor: φ(y, ) = in {h(w, ) + w y} w δy 2 φyy (r λ)y φ y λ φ + cy = 0 φ(ỹ α(), ) = 1 + α ỹ α(), φy(ỹ α(), ) = α φ y(ỹ (), ) =, φ(ỹ (), ) = 0 li φ (ỹ (), ) = c ( c ) c/r r ỹ r li φ y (ỹ (), ) = c c/r r (BVP) (FBP)

51 Ansatz: φ(y, ) = D1()y B 1 + D 2()y B 2 + c y r [ B 1 = 1 (r λ + δ) + ] (r λ + δ) 2δ 2 + 4λδ = γ > 1 γ 1 [ B 2 = 1 (r λ + δ) ] (r λ + δ) 2δ 2 + 4λδ < 0 Proposition (solution of FBP) Assue that z g 1(z)(c/r ) + g 0(z) () = h 2(z)(c/r ) 2 + h 1(z)(c/r ) + h 0(z) z(c/r) = 0 (ODE) has a solution on z : [ 0, c/r] [0, 1]. Here, g i and h i are known functions.

52 Proposition (solution of FBP, cont ) Then, the solution of (FBP) for (y, ) [ỹ (), ỹ α()] [ 0, c/r] is ) B1 φ(y, ) = c [ r y B ( 2 c ) ] ( 1 + B 1 B 2 r α B2 y ỹ α() B 1 B 2 ỹ α() [ B ( 1 c ) ] ( ) B2 + B 1 B 2 r α B1 1 y ỹ α(), B 1 B 2 ỹ α() where the free boundaries are 1 B ( 1B ) 2 z() B1 1 z() B 2 1 ỹ α() B 1 B 2 ( c ) = r and ỹ = z()ỹ α() ( c r α ) [ B 1(1 B 2) B 1 B 2 z() B ] B2(B1 1) z() B 2 1 B 1 B 2

53 z g 1(z)(c/r ) + g 0(z) () = h 2(z)(c/r ) 2 + h 1(z)(c/r ) + h 0(z) z(c/r) = 0 (ODE) A solution of (ODE) on [ 0, c/r] yields φ(w, ) for 0 c/r and α w

54 z g 1(z)(c/r ) + g 0(z) () = h 2(z)(c/r ) 2 + h 1(z)(c/r ) + h 0(z) z(c/r) = 0 (ODE) A solution of (ODE) on [ 0, c/r] yields φ(w, ) for 0 c/r and α w The inverse of z() satisfies an Able equation Any chance for a closed for solution? Functions g i and h i are too coplicated...

55 The auxiliary functions g 0(z) = (1 α) c r (zb2 z B1 ) [(B 2 1)z B 1 1 (B 1 1)z B 2 1] [ g 1(z) = (z B2 z B1 ) B 1 B 2 + α(b 2 1)z B 1 1 ] α(b 1 1)z B α(b 1 1)(B 2 1)(z B 1 1 z B 2 1 ) ( ) c 2 h 0(z) = (1 α) 2 (B 1 B 2)z B 1 +B 2 2[ (B 1 1)z B 2 1 (B 2 1)z B 1 1] r h 1(z) = (1 α) c r { [ (B 2 1)z B 1 1 (B 1 1)z B 2 1] [(B 1 1)z B 1 1 (B 2 1)z B 2 1 α(b 1 B 2)z B 1 +B 2 2] (B 1 B 2)z B 1 +B 2 2[ B 1 B 2 + α(b 2 1)z B 1 1 α(b 1 1)z B 2 1]} h 2(z) = [ (B 1 1)z B 1 1 (B 2 1)z B 2 1 α(b 1 B 2)z B 1 +B 2 2] [B 1 B 2 + α(b 2 1)z B 1 1 α(b 1 1)z B 2 1]

56 z Motivation & Lit. The Eternal Proble The Lifetie Proble Conclusion References z g 1(z)(c/r ) + g 0(z) G(, z) () = =: h 2(z)(c/r ) 2 + h 1(z)(c/r ) + h 0(z) H(, z) z(c/r) = 0 (ODE) ( ) 1 H, 0 for a known x() functions x() Sign of F (; z) = G(; z)=h(; z) 0.8 = 9(z) 1=x( b) z = 1 x() 0 0 b c r

57 z Motivation & Lit. The Eternal Proble The Lifetie Proble Conclusion References z g 1(z)(c/r ) + g 0(z) G(, z) () = =: h 2(z)(c/r ) 2 + h 1(z)(c/r ) + h 0(z) H(, z) z(c/r) = 0 (ODE) ( ) 1 H, 0 for a known x() functions x() G (ξ(z), z) 0 for a known functions ξ(z) = 9(z) Sign of F (; z) = G(; z)=h(; z) 1=x( b) z = 1 x() 0 0 b c r

58 z Motivation & Lit. The Eternal Proble The Lifetie Proble Conclusion References z g 1(z)(c/r ) + g 0(z) G(, z) () = =: h 2(z)(c/r ) 2 + h 1(z)(c/r ) + h 0(z) H(, z) z(c/r) = 0 (ODE) ( ) 1 H, 0 for a known x() functions x() G (ξ(z), z) 0 for a known functions ξ(z) = 9(z) Sign of F (; z) = G(; z)=h(; z) ( ) 1 (ξ(z), z) and, x() ( ) 1 intersect at, ˆ where x( ˆ) ˆ (0, c/r) 1=x( b) z = 1 x() 0 0 b c r

59 z Motivation & Lit. The Eternal Proble The Lifetie Proble Conclusion References z g 1(z)(c/r ) + g 0(z) G(, z) () = =: h 2(z)(c/r ) 2 + h 1(z)(c/r ) + h 0(z) H(, z) z(c/r) = 0 (ODE) ( ) 1 H, 0 for a known x() functions x() Integral Curves G (ξ(z), z) 0 for a known functions ξ(z) 0.8 ( ) 1 (ξ(z), z) and, x() ( ) 1 intersect at, ˆ where x( ˆ) ˆ (0, c/r) 1=x( b) b c r

60 z Motivation & Lit. The Eternal Proble The Lifetie Proble Conclusion References z g 1(z)(c/r ) + g 0(z) G(, z) () = =: h 2(z)(c/r ) 2 + h 1(z)(c/r ) + h 0(z) H(, z) z(c/r) = 0 (ODE) ( ) 1 H, 0 for a known x() functions x() The solution of BVP (5.22) G (ξ(z), z) 0 for a known functions ξ(z) ( ) 1 (ξ(z), z) and, x() ( ) 1 intersect at, ˆ where x( ˆ) ˆ (0, c/r) 0.8 1=x( $ ) 1=x( b) $ b c r

61 z Motivation & Lit. The Eternal Proble The Lifetie Proble Conclusion References Proposition (solution of ODE) Assue that there exist solutions z() and z() of (ODE) in D 0 such that z (respectively, z) satisfies the terinal condition z( 0) = z 0 for ( 0, z 0) D 0 (respectively, ( 0, z 0) + D 0) and extends on the left to D 0\ {(, 1/x( ) )}. Let (respectively, ) be the value of where z (respectively, z) intercepts D 0. Then, there exists a unique solution z() of (ODE) in D 0 satisfying the terinal condition z(c/r) = 0 and extending on the left to the boundary D 0 such that z( ) = 1/x( ) for soe (, ). In particular, z() is not defined on (0, ) =x( b) D 0 D 0 + D b c r

62 z Motivation & Lit. The Eternal Proble The Lifetie Proble Conclusion References Proposition (solution of ODE) Assue that there exist solutions z() and z() of (ODE) in D 0 such that z (respectively, z) satisfies the terinal condition z( 0) = z 0 for ( 0, z 0) D 0 (respectively, ( 0, z 0) + D 0) and extends on the left to D 0\ {(, 1/x( ) )}. Let (respectively, ) be the value of where z (respectively, z) intercepts D 0. Then, there exists a unique solution z() of (ODE) in D 0 satisfying the terinal condition z(c/r) = 0 and extending on the left to the boundary D 0 such that z( ) = 1/x( ) for soe (, ). In particular, z() is not defined on (0, ) =x( b) b c r

63 z Motivation & Lit. The Eternal Proble The Lifetie Proble Conclusion References Proposition (solution of ODE) Assue that there exist solutions z() and z() of (ODE) in D 0 such that z (respectively, z) satisfies the terinal condition z( 0) = z 0 for ( 0, z 0) D 0 (respectively, ( 0, z 0) + D 0) and extends on the left to D 0\ {(, 1/x( ) )}. Let (respectively, ) be the value of where z (respectively, z) intercepts D 0. Then, there exists a unique solution z() of (ODE) in D 0 satisfying the terinal condition z(c/r) = 0 and extending on the left to the boundary D 0 such that z( ) = 1/x( ) for soe (, ). In particular, z() is not defined on (0, ) =x( $ ) 1=x( b) $ b c r

64 Proposition (Miniu probability of DD for c r ) Assue that z() is the solution of (ODE) on [, c/r] Then, for α w and c/r φ(w, ) = B1 1 [ ( c ) ] ( B 2 + B 1 B 2 r α y(w) (1 B 2) ỹ α() ỹ α() + 1 B2 B 1 B 2 [ B 1 ( c r α ) (B 1 1) ỹ α() ) B1 ] ( y(w) ỹ α() where y(w) [ỹ (), ỹ α()] uniquely solves [ c r w = B 1 B ( 2 c ) ] ( y(w) B 1 B 2 ỹ + α() r α (1 B 2) ỹ α() [ B 2 B ( 1 c ) ] ( y(w) B 1 B 2 ỹ α() r α (B 1 1) ỹ α() ) B1 1 ) B2 ) B2 1

65 : $ (; ) Motivation & Lit. The Eternal Proble The Lifetie Proble Conclusion References For α w and c/r π (w, ) = µ r [ B 1(B 1 1) B ( 2 c ) σ 2 B 1 B 2 ỹ + α() r α + µ r [ B 2(1 B 2) σ 2 B 1 B 2 ] ( (1 B 2) B ( 1 c ) ỹ α() r α (B 1 1) y ỹ α() ] ( y ỹ α() ) B1 1 ) B2 1 w $ c r c r

66 z Motivation & Lit. The Eternal Proble The Lifetie Proble Conclusion References 0 < < : The restricted proble No solution for (ODE) on 0 < < =x( $ ) 1=x( b) $ b c r

67 0 < < : The restricted proble No solution for (ODE) on 0 < < w However, it sees that π (, ) = 0 for 0 < c r

68 0 < < : The restricted proble No solution for (ODE) on 0 < < w However, it sees that π (, ) = 0 for 0 < Consider the restricted proble where M is not allowed to increase sup L h = (r w c)h w δ h2 w = λ h h ww π h(α, ) = 1, h w(w, ) li w h ww(w, ) = 0 c r

69 0 < < : The restricted proble No solution for (ODE) on 0 < < w However, it sees that π (, ) = 0 for 0 < Consider the restricted proble where M is not allowed to increase sup L h = (r w c)h w δ h2 w = λ h h ww π h(α, ) = 1, h w(w, ) li w h ww(w, ) = 0 c r The dual FBP corresponds to an optial controller-stopper proble This is where we got the ansatz...

70 0 < < : The restricted proble No solution for (ODE) on 0 < < w However, it sees that π (, ) = 0 for 0 < Consider the restricted proble where M is not allowed to increase sup L h = (r w c)h w δ h2 w = λ h h ww π h(α, ) = 1, h w(w, ) li w h ww(w, ) = 0 c r The dual FBP corresponds to an optial controller-stopper proble This is where we got the ansatz... The dual proble reduces to an ODE = its solution is 1 x()!

71 The solution for 0 < c r The equations for φ(w, ) and π (w, ) in the restricted proble are the sae as the one for [, c/r], expect that we replace z() with 1 x()!

72 The solution for 0 < c r The equations for φ(w, ) and π (w, ) in the restricted proble are the sae as the one for [, c/r], expect that we replace z() with Let us glue the solutions of the two probles 1 x()!

73 z Motivation & Lit. The Eternal Proble The Lifetie Proble Conclusion References The solution for 0 < c r The equations for φ(w, ) and π (w, ) in the restricted proble are the sae as the one for [, c/r], expect that we replace z() with Let us glue the solutions of the two probles 1 x()! { 1/x(), 0 η() = z(), c/r =x( $ ) 1=x( b) $ b c r

74 The solution for 0 < c r The equations for φ(w, ) and π (w, ) in the restricted proble are the sae as the one for [, c/r], expect that we replace z() with Let us glue the solutions of the two probles { 1/x(), 0 η() = z(), c/r The free boundary : 1 B ( 1B ) 2 η() B1 1 η() B 2 1 y α() B 1 B 2 ( c ) = r ( c r α ) [ B 1(1 B 2) B 1 B 2 η() B x()! ] B2(B1 1) η() B 2 1 B 1 B 2

75 Theore (Miniu probability of DD for 0 < c r ) Assue that z() is the solution of (ODE) on [, c/r] and define η and y α. Then, for α w and 0 < c/r φ(w, ) = B1 1 [ ( c ) ] ( B 2 + B 1 B 2 r α y(w) (1 B 2) y α() y α() ] ( y(w) + 1 B2 B 1 B 2 [ B 1 ( c r α ) (B 1 1) y α() y α() where y(w) [y (), y α()] uniquely solves [ c r w = B 1 B ( 2 c ) ] ( y(w) B 1 B 2 y + α() r α (1 B 2) y α() [ B 2 B ( 1 c ) ] ( y(w) B 1 B 2 y α() r α (B 1 1) y α() ) B1 ) B1 1 ) B2 ) B2 1

76 For α w and 0 < c/r π (w, ) = µ r [ B 1(B 1 1) B ( 2 c ) σ 2 B 1 B 2 y + α() r α + µ r [ B 2(1 B 2) σ 2 B 1 B 2 ] ( (1 B 2) B ( 1 c ) y α() r α (B 1 1) y y α() ] ( y y α() ) B1 1 ) B2 1

77 : $ (; ) Motivation & Lit. The Eternal Proble The Lifetie Proble Conclusion References For α w and 0 < c/r π (w, ) = µ r [ B 1(B 1 1) B ( 2 c ) σ 2 B 1 B 2 y + α() r α + µ r [ B 2(1 B 2) σ 2 B 1 B 2 ] ( (1 B 2) B ( 1 c ) y α() r α (B 1 1) y y α() ] ( y y α() ) B1 1 ) B2 1 w $ c r c r

78 : $ (w; ) Motivation & Lit. The Eternal Proble The Lifetie Proble Conclusion References For α w and 0 < c/r π (w, ) = µ r [ B 1(B 1 1) B ( 2 c ) σ 2 B 1 B 2 y + α() r α + µ r [ B 2(1 B 2) σ 2 B 1 B 2 ] ( (1 B 2) B ( 1 c ) y α() r α (B 1 1) y y α() ] ( y y α() ) B1 1 ) B w w c r c r $ 0 c r

79 ?(w; ) Motivation & Lit. The Eternal Proble The Lifetie Proble Conclusion References w 20 c r c r $ 0

80 ?(w; ) Motivation & Lit. The Eternal Proble The Lifetie Proble Conclusion References w c 10 5 c 20 $ r r

81 Conclusion We deterined the optial strategy to iniize the probability of drawdown in two scenarios For the infinite tie horizon proble, the optial strategy is the sae as the one for infinite tie ruin proble For the lifetie proble with constant consuption, there is a trade-off in allowing the high-water-ark to increase 0 < : increasing high-water-ark level akes DD ore probable (0 < ) < < c/r: letting wealth increase helps fund the consuption, thus reducing the probability of DD Will this behavior exist for other types of consuption where w s <? Adding trade-off between risk and return?

82 Thank you for your attention! Angoshtari, B., E. Bayraktar, and V. R. Young: Miniizing the probability of lifetie drawdown under constant consuption. Insurance: Matheatics and Econoics (2016a). Optial investent to iniize the probability of drawdown. Stochastics, pp (2016b).

83 Bäuerle, N. and E. Bayraktar: A note on applications of stochastic ordering to control probles in insurance and finance. Stochastics An International Journal of Probability and Stochastic Processes, volue 86, no. 2: pp (2014). Bayraktar, E. and V. R. Young: Correspondence between lifetie iniu wealth and utility of consuption. Finance and Stochastics, volue 11, no. 2: pp (2007). Optially investing to reach a bequest goal. Insurance: Matheatics and Econoics, volue 70: pp (2016). Browne, S.: Optial investent policies for a fir with a rando risk process: exponential utility and iniizing the probability of ruin. Matheatics of operations research, volue 20, no. 4: pp (1995). Survival and growth with a liability: Optial portfolio strategies in continuous tie. Matheatics of Operations Research, volue 22, no. 2: pp (1997). Beating a oving target: Optial portfolio strategies for outperforing a stochastic benchark. Finance and Stochastics, volue 3, no. 3: pp (1999a).

84 Reaching goals by a deadline: Digital options and continuous-tie active portfolio anageent. Advances in Applied Probability, volue 31, no. 2: pp (1999b). Chen, X., D. Landriault, B. Li, and D. Li: On iniizing drawdown risks of lifetie investents. Insurance: Matheatics and Econoics, volue 65: pp (2015). Cherny, V. and J. Ob lój: Portfolio optiisation under non-linear drawdown constraints in a seiartingale financial odel. Finance and Stochastics, volue 17, no. 4: pp (2013). Cvitanic, J. and I. Karatzas: On portfolio optiization under drawdown constraints. IMA Volues in Matheatics and its Applications, volue 65: pp (1995). Dubins, L. E. and L. J. Savage: How to gable if you ust: Inequalities for stochastic processes edition McGraw-Hill, New York edition Dover, New York (1965, 1975). Grossan, S. J. and Z. Zhou: Optial investent strategies for controlling drawdowns. Matheatical finance, volue 3, no. 3: pp (1993).

85 Karatzas, I.: Adaptive control of a diffusion to a goal and a parabolic onge-apere-type equation. Asian Journal of Matheatics, volue 1: pp (1997). Kardaras, C., J. Ob lój, and E. Platen: The nuéraire property and long-ter growth optiality for drawdown-constrained investents. Matheatical Finance (2014). Pestien, V. C. and W. D. Sudderth: Continuous-tie red and black: How to control a diffusion to a goal. Matheatics of Operations Research, volue 10, no. 4: pp (1985). Proislow, D. S. and V. R. Young: Miniizing the probability of ruin when clais follow brownian otion with drift. North Aerican Actuarial Journal, volue 9, no. 3: pp (2005). Young, V. R.: Optial investent strategy to iniize the probability of lifetie ruin. North Aerican Actuarial Journal, volue 8, no. 4: pp (2004).