Numerical Methods for Hamilton Jacobi Bellman Equations in Finance Lecture 1 Introduction

Transcription

1 Lecture 1 Numercal Methods for Hamlton Jacob Bellman Equatons n Fnance Lecture 1 Introducton Peter Forsyth Unversty of Waterloo Cherton School of Computer Scence January / 31

2 Lecture 1 Overvew of Course Lecture 1: Examples of HJB Equatons n Fnance Lecture 2: Vscosty Solutons and Basc Propertes of a Numercal Scheme Monotoncty, Stablty and Consstency Lecture 3: Examples Passport optons, Penson Investment Lecture 4: Guaranteed Mnmum Wthdrawal Benefts Lecture 5: Gas Storage 2 / 31

3 Lecture 1 Outlne For Lecture 1 Overvew Uncertan Volatlty Stock Borrowng Fees, Unequal Borrowng Lendng Rates Contnuous Tme Mean Varance Asset Allocaton GMWB General Form Vscosty Solutons Summary 3 / 31

4 Lecture 1 Overvew Many problems n fnance can be vewed as some form of optmal stochastc control (Survey artcle, see (Pham, 2005)). These problems often nvolve modellng realstc market effects, and/or complex decson makng. Contnuous tme mean-varance portfolo optmzaton Optmal trade executon, hedgng wth lqudty effects (.e. prce mpact) and transacton costs Investments n endowments and penson plans Optmal operaton of gas storage facltes Varable annuty products wth market guarantees (GMWB, GMDB) These problems gve rse to non-lnear Hamlton Jacob Bellman (HJB) Partal Dfferental Equatons (PDEs), or HJB Partal Integro Dfferental Equatons (PIDEs). 4 / 31

5 Lecture 1 Uncertan Volatlty Uncertan Volatlty model proposed n (Avellenada et al (1995), Lyons (1995)). Suppose that an asset prce S follows the rsk neutral process ds = rs dt + σs dz r = nterest rate σ = volatlty dz = ncrement of a Wener process where volatlty s stochastc, but we only know that σ [σ mn, σ max ] What s the far prce to charge for a contngent clam? 5 / 31

6 Lecture 1 Uncertan Volatlty cont d Let V (S, τ = T t) be the value of a contngent clam, then Short Poston: Long Poston: { Q 2 S 2 V τ = sup Q ˆQ 2 { Q 2 S 2 V τ = nf Q ˆQ 2 ˆQ = [σ mn, σ max ] } V SS + SV S rv } V SS + SV S rv [σ mn, σ max ] s the admssble control set Q s the control for ths HJB equaton 6 / 31

7 Lecture 1 Worst Case Hedge Suppose we are short the opton Consder a strategy where we value the opton (solve the HJB equaton) We delta hedge, usng delta values from our HJB soluton No matter what happens, as long as σ mn σ σ max, we wll always end up wth a non-negatve balance n the hedgng portfolo So the HJB equaton soluton s an upper bound to the cost of constructng a self-fnancng hedge Ths s the least upper bound,.e. there exsts at least one asset path, volatlty scenaro where our hedgng portfolo s worth exactly zero at expry 7 / 31

8 Lecture 1 Bd-Ask Prces If buyers sellers prced optons based on worst case hedgng (from ther own perspectves) Long/short prces would correspond to bd-ask prces Smlar HJB PDE for other cases Uncertan dvdends Hedgng wth transacton costs Dfferent nterest rates for lendng/borrowng 10 Long European Butterfly Opton Value 10 Short European Butterfly Opton Value Opton Value 5 4 Opton Value Asset Prce Asset Prce 8 / 31

9 Lecture 1 Stock Borrowng Fees, Unequal Borrowng Lendng Rates Assume that rsky asset S follows GBM (under the rsk neutral measure) ds = rs dt + σs dz Contngent clams are hedged usng delta hedgng, usual Black Scholes assumptons wth the exceptons Hedger can borrow at rate r b. Hedger can receve r l < r b for cash on depost. Stock borrowng fee r f charged for shortng stock. Effectvely, the hedger receves nterest at rate r l r f, on the proceeds of a short sale. 9 / 31

10 Lecture 1 HJB Equaton: Short Poston on Contngent Clam Let the no-arbtrage value of a contngent clam be V = V (S, τ = T t). V τ = { σ 2 S 2 sup Q ˆQ 2 V SS + q 3 q 1 (SV S V ) } +(1 q 3 )[(r l r f )SV S q 2 V ] Q = (q 1, q 2, q 3 ) ˆQ = ({r l, r b }, {r l, r b }, {0, 1}) (1) For Long poston on clam, replace sup by nf n (1). 10 / 31

11 Lecture 1 Contnuous Tme Mean Varance Asset Allocaton Suppose an nvestor saves for retrement by contrbutng to a penson account at a rate π per year. She can dvde her wealth W n the penson account nto A fracton p nvested n a rsky asset X whch follows dx = (r + ξσ)x dt + σx dz ξ = the market prce of rsk A fracton (1 p) n a rskless asset B whch follows db dt = rb The process followed by W = B + X s dw = (r + pξσ)w dt + π dt + pσw dz. 11 / 31

12 Lecture 1 Optmal Strategy Defne Let p(w, t) = dynamc fracton nvested n the rsky asset W T = termnal wealth Ep( ) t=0 T ] = Expected gan under strategy p( ) Var t=0 p( ) [W T ] = Varance under strategy p( ) So that ( ) 2 Var t=0 p( ) [W T ] = Ep( ) t=0 [(W T ) 2 ] Ep( ) t=0 [W T ] 12 / 31

13 Lecture 1 Mnmum Varance The objectve s to determne the strategy p( ) such that mn Varp( ) t=0 [W T ] = Ep( ) t=0 [(W T ) 2 ] d 2 { E t=0 subject to p( ) [W T ] = d p( ) Z Z = set of admssble controls Gven an expected return d = E t=0 p( ) [W t], strategy p( ) produces the smallest possble varance. Varyng the parameter d traces out a curve n the expected value - standard devaton plane. 13 / 31

14 Lecture 1 Fgure: Typcal Effcent Fronters Allow bankruptcy E[W(t=T)] at t= No bankruptcy 8 6 Bounded control std[w(t=t)] at t = 0 14 / 31

15 Lecture 1 Effcent Fronter Each pont on the fronter s optmal n the sense that no other strategy gves smaller rsk for gven expected gan. Any ratonal nvestor wll choose ponts on ths curve. Dfferent nvestors wll choose dfferent ponts dependng on her rsk preferences. E[W(t=T)] at t= Allow bankruptcy No bankruptcy Bounded control std[w(t=t)] at t = 0 15 / 31

16 Lecture 1 Elmnate Constrant Orgnal problem s convex optmzaton, use Lagrange multpler γ to elmnate constrant. max γ mn E p( ) t=0 p( ) Z [ ] (W T ) 2 d 2 γ(ep( ) t=0 [W T ] d). (2) Suppose somehow we know the γ whch solves (2), for fxed d. Then the optmal strategy p ( ) whch solves (2) s gven by (for fxed γ) Note d has dsappeared from (3). mn E p( ) t=0 [(W T γ p( ) Z 2 )2 ]. (3) 16 / 31

17 Lecture 1 Constructon of Effcent Fronter We can alternatvely regard γ as a parameter, and determne the optmal strategy p ( ) whch solves mn E p( ) t=0 [(W T γ p( ) Z 2 )2 ]. Once p ( ) s known, we can easly determne E t=0 p ( ) [W T ], E t=0 p ( ) [(W T ) 2 ]. Varyng γ traces out the effcent fronter. Let V (W, τ) = E t=t τ [(W T γ/2) 2 ], then p ( ) s determned from soluton of HJB equaton { V τ = sup [(pµ + (1 p)r)w + π]vw + (pσ) 2 W 2 } V WW p z V (W, τ = 0) = (W γ 2 )2 17 / 31

18 Lecture 1 Guaranteed Mnmum Wthdrawal Beneft (GMWB) Popular varable annuty product sold n Canada/US. Desgned for holders of defned contrbuton penson plans. Investor hands over lump sum to an nsurance company, nsurance company nvests sum n rsky assets. The nvestor can wthdraw up to a contract amount each year, up to total amount of lump sum, regardless of actual amount n nvestment account. At end of contract, nvestor gets amount remanng n the nvestment account (net of wthdrawals). The nvestor can partcpate n market gans, but stll has a guaranteed cash flow, n the case of market losses. Ths nsulates pensoners from losses n the early years of retrement. 18 / 31

19 Lecture 1 Some more detals Denote the amount n the rsky nvestment by W. The nvestor also has a vrtual guarantee account A. The nvestor can choose to wthdraw up to the specfed contract rate G r wthout penalty. Usually, a penalty (κ > 0) s charged for any wthdrawal above rate G r (nstantaneous wthdrawal allowed). 19 / 31

20 Lecture 1 The rsk neutral process followed by W s then (ncludng wthdrawals da). dw = (r α)wdt + σwdz + da, f W > 0 dw = 0, f W = 0 α = fee pad for guarantee ; A = guarantee account Let the value of the guarantee be V = V (W, A, τ = T t). Defne LV = 1 2 σ2 W 2 V WW + (r α)wv W rv. 20 / 31

21 Lecture 1 Impulse Control (HJB Varatonal Inequalty) Optmal strategy: wthdraw at fnte rate ( G r ) or nstantaneously wthdraw a fnte amount (nfnte rate). { ) mn V τ LV max (ˆγ ˆγVW ˆγV A, ˆγ [0,G r ] [ ] } V sup V (max(w γ, 0), A γ, τ) + (1 κ)γ c = 0 γ (0,A] c s a small fxed cost whch ensures that the Impulse Control problem s well-posed. 21 / 31

22 Lecture 1 Fgure: Optmal wthdrawal strategy: GMWB 22 / 31

23 Lecture 1 GMWB: Sngular Control Formulaton Ths problem can also be posed as a sngular control [ ] mn V τ LV G max(fv, 0), κ FV = 0 LV = 1 2 σ2 W 2 V WW + (r α)wv W rv FV = 1 V W V A Ths sngular control formulaton can be solved numercally usng a penalty method (Da, Kwok, Zong: Mathematcal Fnance (2008)). The penalty method can vewed as a partcular type of control, hence the methods descrbed n these lectures can be used. 23 / 31

24 Lecture 1 HJB Equatons: A General Form Many of the HJB equatons we have seen can be wrtten as L Q V a(x, τ, Q)V xx + b(x, τ, Q)V x c(x, τ, Q)V Q = control c(x, τ, Q) 0 { } V τ = sup Q ˆQ L Q V + d(x, τ, Q) ˆQ = set of admssble controls. Standard approach: determne optmal Q(t) by dfferentatng {L Q V + d(q)} w.r.t. Q and settng to zero. From a computatonal perspectve: ths s a bad dea. We wll see why later! 24 / 31

25 Lecture 1 Vscosty Soluton In general, the HJB equaton may not have smooth solutons. What does t mean to solve a dfferental equaton when the soluton s not dfferentable? We can wrte our general HJB equaton n the form { } g(v xx, V x, V τ, V, x, τ) = V τ sup Q ˆQ L Q V + d(x, τ, Q) = 0 Suppose we have a C 2,1 test functon φ such that φ V, and φ touches V at a sngle pont (x 0, τ 0 ). For smplcty, assume that V s contnuous. Otherwse, φ should touch the upper sem-contnuous envelope of V. 25 / 31

26 Lecture 1 Subsoluton Fgure: If, for any pont (x 0, τ 0 ), for any test functon φ V, where φ touches V at the sngle pont (x 0, τ 0 ), g(φ xx, φ x, φ τ, φ, x 0, τ 0 ) 0, then V s a vscosty subsoluton g <= Test Functon Vscosty Subsoluton / 31

27 Lecture 1 Supersoluton Fgure: If, for any pont (x 0, τ 0 ), for any test functon φ V, where φ touches V at the sngle pont (x 0, τ 0 ), g(φ xx, φ x, φ τ, φ, x 0, τ 0 ) 0, then V s a vscosty supersoluton Vscosty Supersoluton g >= Test Functon / 31

28 Lecture 1 Vscosty Soluton Any soluton whch s both a subsoluton and a supersoluton s a vscosty soluton Note that we never evaluate g(v xx, V x, V τ,...) but only g(φ xx, φ x, φ τ,...), no problems wth non-dfferentable V. Numercal ssues: We want to ensure that our numercal scheme converges to the vscosty soluton For examples of cases where seemngly reasonable dscretzatons converge to non-vscosty solutons, see Pooley, Forsyth, Vetzal, IMA J. Num. Anal. (2003) Suffcent condtons known whch ensure that a numercal scheme converges to the vscosty soluton (Barles, Sougands (1991)) 28 / 31

29 Lecture 1 Techncal Pont I Remark There may also be some ponts where a smooth C 2,1 test functon cannot touch the soluton from ether above or below. As a pathologcal example, consder the functon f (x) = { x x 0, x x < 0. (4) Ths functon cannot be touched at the orgn from below (or above) by any smooth functon wth bounded dervatves. Note that the defnton of a vscosty soluton only specfes what happens when the test functon touches the vscosty soluton at a sngle pont (from ether above or below). The defnton s slent about cases where ths cannot happen. 29 / 31

30 Lecture 1 Techncal Pont II From now on, we make the followng Assumpton Assumpton (Strong Comparson) We assume that any HJB PDE we wll examne, along wth approprate boundary condtons, satsfes the strong comparson property, whch then mples that there exsts a unque, contnuous vscosty soluton to the HJB equaton. Ths has been proven for many cases, but not precsely all the problems we wll look at. 30 / 31

31 Lecture 1 Summary Many problems n fnance can be formulated n terms of optmal stochastc control nonlnear HJB equaton/varatonal nequalty. Solutons are n general non-dfferentable vscosty soluton. There are precse rules to follow whch wll guarantee convergence of a numercal scheme to the vscosty soluton It s a bad dea (numercally) to analytcally determne the optmal control (.e. by dfferentatng and settng equal to zero), and then to dscretze the PDE A better approach: frst dscretze and then optmze the dscrete equatons. 31 / 31

32 Lecture 2 Numercal Methods for Hamlton Jacob Bellman Equatons n Fnance Lecture 2 Suffcent Condtons for Convergence to the Vscosty Soluton Peter Forsyth Unversty of Waterloo Cherton School of Computer Scence January / 36

33 Lecture 2 Outlne For Lecture 2 A Cautonary Tale Defntons Dscretzaton Suffcent Condtons: Convergence Matrx Form Arbtrage Inequalty A Cautonary Tale: Contnued Summary 2 / 36

34 Lecture 2 A cautonary tale Recall the uncertan volatlty model { q 2 S 2 } V τ = max q ˆQ 2 V SS + SV S rv ˆQ = [σ mn, σ max ] Suppose we solve ths equaton usng a standard fnte dfference method wth Crank-Ncolson tmesteppng. Parameter Value S nt 100 σ max.25 σ mn.15 r.10 T.25 Payoff Butterfly Poston Short Table: Data used for the uncertan volatlty model. 3 / 36

35 Lecture 2 Convergence Study Solve ths problem on a sequence of grds/tmesteps At each refnement level we add new fne grd nodes and double tmesteps Nodes Tmesteps Opton value Table: Value of the uncertan volatlty clam at S = 100, Crank-Ncolson tmesteppng. Appears to converge to about $ / 36

36 Lecture 2 Crank-Ncolson Soluton: Base Intal Grd 10 Short Poston European Butterfly Opton Value 9 8 Opton Value Payoff Asset Prce Fgure: Opton value, uncertan volatlty model, Crank-Ncolson tmesteppng, base ntal grd. 5 / 36

37 Lecture 2 Add Nodes to Base Grd Convergence was a bt slow, let s add more nodes near the strke on the base grd, and repeat the convergence study. Nodes Tmesteps Opton value Table: Convergence data, uncertan volatlty, Crank-Ncolson tmesteppng, S = 100, dfferent ntal grd. Appears to converge to about $0.85, compare wth $2.05 before! 6 / 36

38 Lecture 2 Crank-Ncolson Soluton: Intal grd wth more nodes near the strke 10 Short Poston European Butterfly Opton Value 9 8 Opton Value Payoff Asset Prce Fgure: Opton value, uncertan volatlty model, Crank-Ncolson tmesteppng, more nodes near the strke. 7 / 36

39 Lecture 2 What Happened? For a lnear PDE Crank-Ncolson s uncondtonally stable, and converges to the exact soluton, regardless of the ntal grd/tmesteps For ths nonlnear HJB equaton It appears that we can converge to dfferent solutons dependng on our startng grd! We need to be careful to converge to the correct,.e. vscosty soluton 8 / 36

40 Lecture 2 Defntons Defne a soluton doman Ω = Ω n Ω bnd. Let Ω n = Interor ponts Ω bnd = Boundary ponts x = (x, τ) ; V = soluton D 2 V = V xx ; DV = (V x, V τ ) We can wrte the general form for the HJB equaton n a way that ncludes boundary condtons g(d 2 V (x), DV (x), V (x), x) = g n (D 2 V (x), DV (x), V (x), x) x Ω n g(d 2 V (x), DV (x), V (x), x) = g bnd (D 2 V (x), DV (x), V (x), x) x Ω bnd 9 / 36

41 Lecture 2 Defntons cont d where { } g n (D 2 V, DV, V, x) = V τ sup Q ˆQ L Q V + d(x, τ, Q) = 0 g bnd (D 2 V, DV, V, x) = specfed boundary condton Suppose we defne a grd {x 0, x 1,..., x,...} and a set of tmesteps {τ 0, τ 1,..., τ n,...}. Let the dscretzaton parameter h be gven by (C 1, C 2 = const.) max τ n ) n = C 1 h max(x +1 x ) = C 2 h 10 / 36

42 Lecture 2 Dscretzaton: General Form Let V n be the approxmate value of the soluton,.e. V n V (x, τ n ) = V (x n ). Then we can wrte a general dscretzaton of the HJB equaton g(d 2 V, DV, V, x) at node x n+1 = (x, τ n+1 ) G n+1 (h, V n+1, V n+1 +1, V n+1 1, V n, V n = G n+1 (h, V n+1, { Vm p } p n+1 ) or m +1, V 1, n...) = 0 { V p} m p n+1 refers to the dscrete soluton values at nodes or m neghbourng (n space and tme) node (x, τ n+1 ). 11 / 36

43 Lecture 2 Example: Lnear Heat Equaton Dscretze V t = V xx V (x = 0, t) = V (x = 1, t) = 1 usng a mesh x = x, = 0,.., max, t n = n t, V (x, t n ) V n. V n+1 t V n = V n+1 1 2V n+1 + V n+1 +1 ( x) 2 t = h ; x = h G n+1 =0 = V n+1 =0 1 G n+1 = V n+1 V n h G n+1 = max = V n+1 = max 1 ( V n+1 n+1 1 2V + V n+1 +1 h 2 ) ; 0, max 12 / 36

44 Lecture 2 Suffcent Condtons: Convergence We use the results from (Barles, Sougandns, 1991). Theorem (Convergence) Any numercal scheme whch s consstent, l stable, and monotone, converges to the vscosty soluton. Stablty Usual requrement (dscrete soluton bounded n l as mesh, tmestep 0) Can prove usng maxmum analyss Usually, only fully mplct tmesteppng s uncondtonally l stable (but not always, e.g. jump dffusons). 13 / 36

45 Lecture 2 Monotoncty Defnton (Monotoncty) The dscretzaton s monotone f G n+1 (h, V n+1 G n+1 (h, V n+1, { Xm p }, { Vm p } p n+1 ) or m p n+1 ) G n+1 or m (h, V n+1, { Ym p } X p m Y p m, m, p p n+1 ) or m 14 / 36

46 Lecture 2 Monotoncty cont d A dscretzaton whch s a postve coeffcent method s monotone (usually) For 1-d problems, mult-dmensonal problems wth dffuson only n one drecton, forward/backward/central dfferencng generate a postve coeffcent scheme Often results n low order schemes (frst order) Has nce fnancal nterpretaton: enforces dscrete no-arbtrage nequaltes..e. f payoff from contngent clam A s greater than clam B (on the same underlyng) then the value of Clam A must always be greater than Clam B at all earler tmes. More later 15 / 36

47 Lecture 2 Monotoncty: Heat equaton Recall that = V n+1 V n h G n+1 ( V n+1 n+1 1 2V + V n+1 +1 h 2 ) ; 0, max We have to show that G n+1 (V n+1 +1, V n+1 1, V n ). Frst step, consder s a nonncreasng functon of G n+1 (h, V n+1, V n+1 n ɛ, V 1, V n ) G n+1 (h, V n+1, V n+1 +1, V n+1 1, V n ( V n+1 n+1 1 2V + V n+1 +1 = + ɛ ) ( V n+1 n+1 1 2V + V n+1 ) +1 + h 2 = ɛ h 2 < 0 ; ɛ > 0 h 2 ) Smlar result f we perturb V n, V n / 36

48 Lecture 2 Consstency Consstency n the vscosty sense defned a techncal way. I nclude ths for completeness. Defnton (Sem-contnuous envelopes) If C s a closed set, and f (x) s a functon defned on C, then the upper sem-contnuous envelope f (x) and the lower sem-contnuous envelope f (x), x C, s defned by f (x) = lm sup f (y) ; y x y C f (x) = lm nf y x y C f (y) Remark Note that C s closed here, whch means that we consder the neghbours of x and x tself. 17 / 36

49 Lecture 2 Consstency The scheme G n+1 (h, V n+1, { Xm p } p n+1 m ) s consstent f, for all ˆx = (ˆx, ˆτ) n the computatonal doman, and for all smooth, bounded test functons φ(x), we have lm sup h 0 x n+1 ˆx ξ 0 ( G n+1 lm nf G n+1 h 0 x n+1 ˆx ξ 0 h, φ(x n+1 ) + ξ, { φ(x p m) + ξ } p n+1 or m g ( D 2 φ(ˆx), Dφ(ˆx), φ(ˆx), ˆx ), ( h, φ(x n+1 ) + ξ, { φ(x p m) + ξ } p n+1 or m g ( D 2 φ(ˆx), Dφ(ˆx), φ(ˆx), ˆx ), ) ) (1) 18 / 36

50 Lecture 2 Why do we ever need ths complex defnton of consstency? At ponts near the boundary, the vscosty defnton of consstency s more relaxed than the classcal defnton of consstency. Sometmes, a scheme (.e. a sem-lagrangan method) may never be consstent (n the classcal sense) near the boundary. But the scheme s consstent n the vscosty sense! Consstency n the vscosty sense can be very useful n such stuatons. In fact, f you do anythng reasonable n a complex stuaton, you usually fnd that everythng s fne n the vscosty sense. The whole dea of vscosty solutons allows you to do what you were gong to do anyway, except now you know t s OK. 19 / 36

51 Lecture 2 Dscretzaton Recall that we can wrte our HJB equaton as { } g(v xx, V x, V τ, V, x, τ) = V τ sup Q ˆQ L Q V + d(x, τ, Q) = 0 (2) where L Q V a(x, τ, Q)V xx + b(x, τ, Q)V x c(x, τ, Q)V Dscretze (2) Fully mplct tmesteppng Forward/backward/central dfferencng 20 / 36

52 Lecture 2 Fully Implct Tmesteppng Defne a grd [x 0, x 1,.., x max ], and let V n = [V0 n, V 1 n,..., V n max ]. Let L Q h be the dscrete form of the operator LQ, so that (L Q h V n+1 ) = α n+1 (Q)V n βn+1 (Q)V n+1 +1 (α n+1 (Q) + β n+1 (Q) + c n+1 (Q))V n+1. The dscrete form of the HJB equaton s then V n+1 τ V n { } = sup Q n+1 ˆQ (L Qn+1 h V n+1 ) + d n / 36

53 Lecture 2 General Form We can wrte the dscretzed PDE n our general form G n+1 (h, V n+1, V n+1 +1, V n+1 1, V n ) = V n+1 + nf Q ˆQ V n { τ (α n+1 (Q) + β n+1 (Q) + c n+1 (Q))V n+1 α n+1 (Q)V n+1 1 βn+1 (Q)V n+1 +1 d n+1 } (Q) = 0 (3) 22 / 36

54 Lecture 2 Assume c n+1 0, and that the followng condton holds (usng central/forward/backward dfferencng) Condton (Postve Coeffcent) α n+1 (Q) 0, β n+1 (Q) 0, c n+1 (Q) 0 ; Q ˆQ. It s now easy to show that ths dscretzaton s monotone. We have to show that G n+1 ( ) s a nonncreasng functon of the neghbour nodes Vm, p (m or p n + 1). (4) 23 / 36

55 Lecture 2 Dscretzed equatons Recall that the dscretzed equatons are G n+1 (h, V n+1, V n+1 +1, V n+1 = V n+1 τ V n + nf Q ˆQ 1, V n ) { (α n+1 (Q) + β n+1 (Q) + c n+1 (Q))V n+1 } α n+1 (Q)V n+1 1 βn+1 (Q)V n+1 +1 d n+1 (Q) 24 / 36

56 Lecture 2 Monotoncty Consder the case where we perturb V n+1 +1 show by ɛ > 0, we need to G n+1 (h, V n+1, V n+1 n ɛ, V 1, V n ) G n+1 (h, V n+1, V n+1 +1, V n+1 1, V n ) 0 Gven two functons S(z), T (z), z D, then a useful fact s that ( ) nf S(x) nf T (y) sup S(z) T (z) x D y D z D 25 / 36

57 Lecture 2 G n+1 (h, V n+1, V n+1 +1 = nf Q ˆQ + ɛ, V n+1 1, V n ) G n+1 (h, V n+1, V n+1 +1, V n+1 1, V n ) { (α n+1 (Q) + β n+1 (Q) + c n+1 (Q))V n+1 α n+1 (Q)V n+1 1 nf Q ˆQ βn+1 (Q)V n+1 +1 βn+1 (Q)ɛ d n+1 { (α n+1 (Q ) + β n+1 (Q ) + c n+1 (Q ))V n+1 } α n+1 (Q )V n+1 1 βn+1 (Q )V n+1 +1 d n+1 (Q ) { } { } sup Q ˆQ β n+1 (Q)ɛ = ɛ nf Q ˆQ β n+1 (Q) 0 whch follows from β n+1 (Q) 0 and we have used (nf S nf T ) sup(s T ). } (Q) 26 / 36

58 Lecture 2 Smlarly, we can show that a postve perturbaton of any of {V n+1 1, V n } causes a decrease n the value of G n+1 ( ). We can now state the followng result. Proposton (Monotoncty) If the postve coeffcent condton (4) s satsfed, then (3) s a monotone dscretzaton, as defned n Defnton 1. Note: In order to ensure a postve coeffcent dscretzaton our choce of central/forward/backward dfferencng wll depend, n general, on the control Q. more about ths later 27 / 36

59 Lecture 2 What about Consstency? How do we show ths? Accordng to the defnton of lm sup, there exst sequences h k, k, n k, ξ k, such that h k 0, ξ k 0, x n k+1 k ˆx as k and lm sup k 0 = lm sup ( G n k+1 k k>k 0 h 0 x n+1 ˆx ξ 0 G n+1 ( h k, φ(x n k+1 k ) + ξ k, { ) φ(x p m) + ξ k }p n k +1 or m k h, φ(x n+1 ) + ξ, { φ(x p m) + ξ } ) p n+1 or m 28 / 36

60 Lecture 2 At nteror ponts, t s usually easy to show (usng Taylor seres) that (recall that φ s nfntely dfferentable) ( G n k+1 k h k, φ(x n k+1 k ) + ξ k, { ) φ(x p m) + ξ k }p n k +1 or m ( k ) = g n D 2 φ(x n k+1 k ), Dφ(x n k+1 k ), φ(x n k+1 k ), x n k+1 k + O(h k ) + O(ξ k ) Snce usually g n ( ) s contnuous, then at nteror ponts g = g = g n, and we are done. Note: Usually, the complex defnton of consstency bols down to a classc defnton of consstency, whch s easly proved usng Taylor seres. 29 / 36

61 Lecture 2 Matrx Form of the Dscrete Equatons Let the value of the optmal control at node, tmestep n be Q n, and defne Q n = [Q n 0, Q n,.., Q n max ] Defne the matrx A(Q n ), so that [A(Q n )V n ] = (L Qn h V n+1 ) and the vector D(Q n+1 ) n+1 = [d0 n, d 1 n,..., d n max ] so that fully mplct tmesteppng can be wrtten as { } V n+1 = V n + τ sup Q n+1 ˆQ A n+1 (Q n+1 )V n+1 + D n+1 (Q n+1 ) 30 / 36

62 Lecture 2 Matrx Form of the Dscrete Equatons II If ˆQ s compact, and the objectve functon s contnuous, then we can wrte the dscretzed equatons as [I τa n+1 (Q n+1 )]V n+1 = V n + τd n+1 (Q n+1 ) { [A where Q n+1 arg max n+1 (Q n+1 )V n+1 + D n+1 (Q n+1 ) ] } Q n+1 ˆQ From the postve coeffcent condton, [I τa n+1 (Q n+1 )] s an M matrx, hence [I τa n+1 (Q n+1 )] / 36

63 Lecture 2 Nonlnear Algebrac Equatons { } V n+1 = V n + τ sup Q n+1 ˆQ A n+1 (Q n+1 )V n+1 + D n+1 (Q n+1 ) (5) Note that ths s a set of nonlnear algebrac equatons But, from M matrx property, the soluton exsts and s unque Arbtrage Inequalty Suppose we have two dfferent ntal vectors W n, U n, so that W n+1 = W n + τ sup Q n+1 ˆQ { } A n+1 (Q n+1 )W n+1 + D n+1 (Q n+1 ) (6) { } U n+1 = U n + τ sup Q n+1 ˆQ A n+1 (Q n+1 )U n+1 + D n+1 (Q n+1 ) (7) 32 / 36

64 Lecture 2 Arbtrage Inequalty Proposton (Dscrete Comparson) If a postve coeffcent dscretzaton s used to dscretze the HJB PDE, then f W n, U n are dscrete solutons to (6)-(7), wth W n U n, then W n+1 U n+1. Proof. Manpulate (6)-(7), use the M matrx property of [I τa n+1 (Q n+1 )]. Remark (Monotoncty) We can see here that monotoncty arbtrage nequalty,.e. nequalty of payoffs translates nto nequalty of value at all earler tmes. Note that ths holds regardless of tmestep or mesh sze. 33 / 36

65 Lecture 2 Back to Our Uncertan Volatlty Equaton Recall that, wth C-N tmesteppng, usng dfferent ntal grds, we appeared to converge to two dfferent values Grd1 $2.05 Grd2 $0.85 What went wrong here? C-N tmesteppng does not satsfy two of the condtons we need C-N s not monotone (for tmesteps > explct tmestep sze) C-N s not l stable. However, fully mplct tmesteppng, postve coeffcent dscretzaton s Uncondtonally monotone, l stable Consstent 34 / 36

66 Lecture 2 Fully Implct Tmesteppng 10 Short Poston European Butterfly Opton Value Nodes Tme Opton steps value Opton Value Payoff Asset Prce We now know for sure that $2.30 s the correct soluton! 35 / 36

67 Lecture 2 Summary: Lecture II Convergence to the vscosty soluton ensured f dscretzaton s Consstent, l stable and monotone. Consstency: most of the tme, ths s just classcal consstency, appled to nfntely dfferentable test functons Consstency n the vscosty sense s very forgvng, when t comes to ponts near the boundares l Stablty: standard maxmum analyss can be used to prove ths Monotoncty: most nterestng condton: preserves dscrete arbtrage nequaltes Postve coeffcent dscretzaton guarantees ths property Seemngly reasonable dscretzatons can converge to wrong values f these condtons are not satsfed! 36 / 36

68 Lecture 3 Numercal Methods for Hamlton Jacob Bellman Equatons n Fnance Lecture 3 Examples: Passport Optons, Asset Allocaton n Penson Plans Peter Forsyth Unversty of Waterloo Cherton School of Computer Scence January / 33

69 Lecture 3 Outlne For Lecture 3 Example: Passport optons Example: Penson Plan Dscretzaton Polcy Iteraton Central Dfferencng as Much as Possble Numercal Results Dscretzed Control Summary 2 / 33

70 Lecture 3 Example I: Passport Optons Opton on a tradng account Holder s allowed to go long/short underlyng asset S at any tme durng [0, T ] Assume S follows Geometrc Brownan Moton Holder can hold q shares at any tme, where q < C Let W be the accumulated gan n the portfolo At maturty, holder s poston s max(w, 0) 3 / 33

71 Lecture 3 Passport Optons Opton Value V (S, W, t) = Su(x = W /S, t) u τ = γu + sup [ ((r γ r c )q (r γ r t )x)u x q 1 ] + σ2 2 (x q)2 u xx, x = W /S ; τ = T t ; r = rsk-free rate, γ = dvdend rate ; r c = cost of carry, r t = nterest rate on tradng account 4 / 33

72 Lecture 3 Asset Allocaton: Penson Plan Carns et al, JEDC, 2006 Stochastc Lfestylng: Optmal Dynamc Asset Allocaton for Defned Contrbuton Penson Plans Holder of defned contrbuton penson plan has annual salary Y (t) (stochastc), contrbutes π Y per year to penson plan Invests a fracton p of accumulated wealth W n rsky asset S, (1 p) n rskfree asset Desres to maxmze expected utlty of wealth/ncome rato at retrement T Power Law Utlty = U(x, T ) = x γ γ ; γ < 0 x = W Y ; Wealth/Income rato 5 / 33

73 Lecture 3 Penson Plan II Bond process: db = rb dt B = Bond prce ; r = rsk free rate Underlyng rsky asset process: ds = (r + ξ 1 σ 1 )S dt + σ 1 S dz 1, S = rsky asset ; ξ 1 = market prce of rsk σ 1 = volatlty; dz 1 = ncrement Wener process Wealth process: dw = (r + pξ 1 σ 1 )W dt + pσ 1 W dz 1 + πydt. W = accumulated wealth; πy = contrbuton rate, p = fracton nvested n S; (1 p) = fracton n bond 6 / 33

74 Lecture 3 Stochastc Salary Assume that yearly salary s also stochastc: dy = (r + µ Y )Y dt + σ Y0 Y dz 0 + σ Y1 Y dz 1, Y = yearly salary; r + µ Y = salary drft rate, σ Y1 = random salary component correlated wth market. σ Y0 = random salary component uncorrelated wth market Z 0, Z 1 are uncorrelated,.e. dz 0 dz 1 = 0. 7 / 33

75 Lecture 3 Maxmze expected utlty V (x, τ) Studes show: we are happy f our wealth W at retrement s large compared to our pre-retrement yearly salary Y. Let x = W /Y, and maxmze expected termnal utlty by determnng V (x, τ) such that V (x, τ) = sup E[U(x(T )) x(t τ) = x] p ˆP τ = T t p(t) = fracton of W n rsky asset ˆP = set of admssble allocaton strateges U(x) = U(W /Y ) = utlty T = retrement date 8 / 33

76 Lecture 3 HJB Equaton: Penson plan x = Wealth Yearly ncome Standard dynamc programmng arguments: V τ = sup {µ px V x + 12 } (σpx )2 V xx p ˆP ; x [0, ), where V (x, τ = 0) = γ 1 x γ ; γ < 0, µ p X = π + x( µ Y + pσ 1 (ξ 1 σ Y1 ) + σy σy 2 1 ), (σ p X )2 = x 2 (σy (pσ 1 σ Y1 ) 2 ). 9 / 33

77 Lecture 3 No-bankruptcy Note that the salary process s always non-negatve. In order to ensure that W 0, whch mples that x 0 V τ = πv x ; x 0 As we shall see, ths means that (px) 0 as x 0. The amount nvested n the rsky asset tends to zero as the (wealth-ncome rato 0). But, the rato of wealth nvested n S becomes nfnte. 10 / 33

78 Lecture 3 General Form (Passport and Penson Allocaton) Both of these PDEs can be be wrtten n the general form L Q V a(x, τ, Q)V xx + b(x, τ, Q)V x c(x, τ, Q)V Q = control { } V τ = sup Q ˆQ L Q V + d(x, τ, Q) ˆQ = set of admssble controls. Textbook approach: determne optmal Q(t) by dfferentatng {L Q V + d(q)} w.r.t. Q and settng to zero. Typcal method used to get analytc solutons. 11 / 33

79 Lecture 3 Standard Approach Apply ths dea (dfferentatng w.r.t. p and settng to zero) to Penson Plan example V τ = (π + δx)v x + σ2 Y 0 x 2 V xx (ξ 1 σ Y 1 ) 2 ( ) V 2 x 2 2 V xx δ = µ Y + σy σ Y 1ξ 1 (1) Now, we have taken a nce HJB equaton, and made a mess of t! It wll be very dffcult to construct a monotone, consstent, stable method for (1). Moral of story: Frst dscretze Then, optmze Fnd max/mn of dscrete equatons not analytc approxmatons to dscrete equatons. 12 / 33

80 Lecture 3 Dscretzaton Fully mplct dscretzaton: Let (L Q h V n ) denote the dscrete form of the dfferental operator L Q V at node (x, τ n ). Use central, forward, backward dfferencng to obtan (L Q h V n+1 ) = α(q) n+1 V n β(q)n+1 V n+1 +1 (α(q) n+1 + β(q) n+1 + c(q) n+1 )V n+1 The dscrete form of the HJB equaton s then V n+1 τ V n { } = sup Q n+1 ˆQ (L Qn+1 h V n+1 ) + d n / 33

81 Lecture 3 Central, Forward, Backward Recall our postve coeffcent condton (whch ensures monotoncty) α(q) n+1 0 ; β(q) n+1 0 ; c(q) n+1 0 (2) In order to ensure ths condton, we have to be careful how we dscretze the term n the PDE. b(x, τ, Q)V x A smple-mnded dea s to use ether forward or backward dfferencng dependng on the sgn of b(x, τ, Q). However, ths s only frst order correct, and s not always necessary. We would lke to use central dfferencng as much as possble, and yet stll have a postve coeffcent scheme. 14 / 33

82 Lecture 3 Convergence Lemma (Stablty) The fully mplct, postve coeffcent scheme s uncondtonally stable. Proof. Follows from a straghtforward maxmum analyss. Lemma (Consstency) The fully mplct postve coeffcent scheme s consstent. Proof. Taylor seres appled to smooth test functons. Lemma (Monotoncty) The fully mplct postve coeffcent scheme s monotone. Proof. Follows from lecture / 33

83 Lecture 3 Nonlnear dscretzed equatons Recall the matrx form of the dscretzaton of L Q : (L Q h V n ) = [A n (Q n )V n ] so that the dscretzed equatons are [ I τa n+1 (Q n+1 ) n+1] V n+1 = V n + τd n+1 (Q n+1 ) n+1 where { [A Q n+1 = arg max n+1 (Q n+1 )V n+1 + D n+1 (Q n+1 ) ] } Q n+1 ˆQ 16 / 33

84 Lecture 3 Polcy Iteraton We use the followng teratve scheme to solve the nonlnear equatons. Let (V n+1 ) 0 = V n Let ˆV k = (V n+1 ) k For k = 0, 1, 2,... untl convergence EndFor Solve [ I τa n+1 (Q k ) ] ˆV k+1 = V n + τd n+1 (Q k ) { } = arg max F (Q k ) Q k F (Q k ) = Q k ˆQ [ ] A n+1 (Q k ) ˆV k + D n+1 (Q k ) 17 / 33

85 Lecture 3 Central Dfferencng as Much as Possble Gven a control Q n at node, the followng algorthm s used to determne the dfferencng method dff = central IF α < 0 or β < 0 THEN dff = backward IF α < 0 or β < 0 THEN dff = forward ENDIF ENDIF 18 / 33

86 Lecture 3 Local Objectve Functon, Passport Opton, Central Dfferencng as much as possble Forward F(Q) Central Change Pont, nonsmooth Backward Central Q Local objectve functon s a dscontnuous functon of the control. 19 / 33

87 Lecture 3 Local Objectve Functon, Passport Opton, Forward/Backward only Forward F(Q) Change pont, nonsmooth Backward Q Local objectve functon s a contnuous functon of the control. 20 / 33

88 Lecture 3 Polcy Iteraton In vew of the fact that the local objectve functon can be a dscontnuous functon of the control At ponts of dscontnuty, we want the lmtng value of the objectve functon whch produces the supremum At ponts of dscontnuty, we can specfy the supremum by specfyng the control and the dfferencng method whch gves ths lmtng value (abuse of notaton: arg sup( )) A subtle pont whch may cause non-convergence f gnored [ I τa n+1 (Q )] k ˆV k+1 = V n + τd n+1 (Q k ) { [ Q k arg sup A n+1 (Q k ) ˆV k + D n+1 (Q )] k Q k ˆQ } (3) 21 / 33

89 Lecture 3 Convergence of Polcy Iteraton Manpulaton of algorthm (3) gves [ I τa n+1 (Q k ) ] ( ˆV k+1 ˆV k ) [ (A = τ n+1 (Q k ) ˆV k + D n+1 (Q k ) ) ( A n+1 (Q k 1 ) ˆV k + D n+1 (Q k 1 ) )] Postve coeffcent dscretzaton LHS s an M matrx RHS s non-negatve (Why?) Easy to show that ˆV k+1 s bounded ndependent of k Iterates form a bounded non-decreasng sequence If ˆV k+1 = ˆV k, resdual s zero 22 / 33

90 Lecture 3 Theorem (Convergence of Polcy Iteraton) If a postve coeffcent dscretzaton s used, then the Polcy teraton algorthm converges to the unque soluton of the nonlnear dscretzed equatons for any ntal terate ˆV 0. Note: We do not requre that the local objectve functon be a contnuous functon of the control. Usng central dfferencng as much as possble should be more accurate. But we have no guarantee on how many teratons requred to solve the nonlnear equatons. We have no guarantee on the order of convergence as the mesh s refned Forward/backward dfferencng only, can be at most frst order Central weghtng as much as possble may be second order Is central dfferencng as much as possble useful n practce? 23 / 33

91 Lecture 3 Penson Allocaton Example Convergence study: Table: Data from Carns et al (2006) µ y 0.0 ξ σ σ Y σ Y π 0.1 T 20 years γ -5 Sequence of tests, on each grd refnement, add new node between each coarse grd node Dvde tmestep by four (not requred for stablty, but fully mplct method s only frst order) If we are gettng beneft from central weghtng as much as possble, then rato of changes 4 24 / 33

92 Lecture 3 Computatonal Issues Orgnal doman x [0, ). Computatonal doman [0, 80]. Drchlet boundary condton at x = x max. Utlty functon undefned at x = 0. V (x, τ = 0) = γ 1 (max(x, ε)) γ ε = 10 3 ; γ < 0 Relatve convergence tolerance for Polcy Iteraton Modfyng these parameters made no change n the soluton to 7 dgts. 25 / 33

93 Lecture 3 Convergence Study: Penson Allocaton Example Nodes Tmesteps Nonlnear CPU Tme Utlty Rato teratons (Sec) Central Dfferencng as much as possble, x = Forward/backward dfferencng only, x = Smlar results for Passport Optons 26 / 33

94 Lecture 3 Penson Allocaton: Optmal Equty fracton p Optmal Equty Proporton Years to retrement T = 20 fracton of salary per year nvested 0.10 p as wealth/ncome rato Wealth-ncome Rato, X(0) = x 27 / 33

95 Lecture 3 Penson Allocaton: Optmal Equty Amount (px) 0.3 Optmal Equty Amount p, px 0, when x 0 Recall x = wealth n unts of yearly ncome Wealth-ncome Rato, X(0) = x 28 / 33

96 Lecture 3 How do we solve the local optmzaton problem? Recall that we need to determne (at each node) { [ } Q k arg sup A n+1 (Q k ) ˆV k + D n+1 (Q )] k Q k ˆQ For Passport optons, Penson Allocaton problem, we can form an analytc expresson for the local objectve functon, as a functon of the control and dfferencng method Determne the subntervals of control values where central, forward, backward gve a postve coeffcent dscretzaton (central has prorty) On each subnterval, objectve functon s smooth, use standard methods to fnd maxmum. Global maxmum found by comparng the maxma on each subnterval But, ths approach may not always be possble, f the objectve functon s complex 29 / 33

97 Lecture 3 Dscretzaton of the Control Suppose that q ˆQ, where ˆQ = [q mn, q max ], q mn, q max bounded. Dscretze control,.e. replace ˆQ by ˆQ ˆQ = [q 0, q 1, q 2,..., q k ] Let max (q +1 q ) = h, then we have the followng Proposton If the coeffcents of the HJB equaton are contnuous, bounded functons of the control, then the dscretzed control problem { } V τ = sup Q ˆQ L Q V + d(s, τ, Q) (4) s consstent wth the orgnal control problem ( ˆQ = [q mn, q max ]), as h / 33

98 Lecture 3 Dscretzed Control II So, f the local objectve functon s too complex to analytcally determne the maxmum, we can Dscretze the control wth parameter h Solve the local optmzaton problem by lnear search Let h 0 as the mesh, tmesteps 0 Ths wll converge to the vscosty soluton Advantage: Trval mplementaton, can be appled to PDE wth complex coeffcents. Dsadvantage: Increase of computatonal complexty, effectvely we ncrease the dmensonalty of the problem by one. 31 / 33

99 Lecture 3 Convergence Study: Dscretzed Control, Penson Allocaton x-nodes p-nodes Tmesteps CPU Utlty Rato (Sec) x = x = Note: we replace admssble set p [0, ] by p [0, 200]. On fnest grd, soluton same to 5 dgts compared to contnuous control. 32 / 33

100 Lecture 3 Lecture 3: Summary Dscretze frst, then maxmze (NOT the other way around) Maxmze the dscrete equatons drectly ensures monotoncty, constrants on control easly handled Postve Coeffcent Dscretzaton Guarantees convergence to vscosty soluton Guarantees convergence of polcy teraton Central weghtng as much as possble Not usually done Hgher convergence rate at lttle cost HJB PDEs wth complex coeffcents Dscretze control, solve optmzaton problem by lnear search Guaranteed to converge, easy to mplement But more expensve 33 / 33

101 Lecture 4 Numercal Methods for Hamlton Jacob Bellman Equatons n Fnance Lecture 4 Guaranteed Mnmum Wthdrawal Benefts (GMWB) Peter Forsyth Unversty of Waterloo Cherton School of Computer Scence January / 41

102 Lecture 4 Outlne For Lecture 4 Motvaton Impulse Control Formulaton Dscrete Wthdrawal Model Convergence to Impulse Control Problem A Unfed Method Numercal Example Summary 2 / 41

103 Lecture 4 Motvaton Varable annuty products: sold by nsurance companes to retal nvestors. These products are guarantees on nvestments n penson plans. From a paper we wrote n 2002 (segregated funds are a Canadan verson of Varable Annutes) If one adopts the no-arbtrage perspectve...n many cases these contracts appear to be sgnfcantly underprced, n the sense that the current deferred fees beng charged are nsuffcent to establsh a dynamc hedge for provdng the guarantee. Ths s partcularly true for cases where the underlyng asset has relatvely hgh volatlty. Ths fndng mght rase concerns at nsttutons wrtng such contracts. Wndclff, Forsyth, LeRoux, Vetzal, North Amercan Actuaral J., 6 (2002) / 41

104 Lecture 4 What Happened? As descrbed n a Globe and Mal artcle (Report on Busness, December 2, 2008, Manulfe, n red, rases new equty, ), one of the large Canadan nsurance companes, Manulfe, posted a large mark-to-market wrtedown to account for losses assocated wth these segregated fund guarantees. From the Globe and Mal Streetwse Blog, November 7, 2008 Concerns that the market selloff wll translate nto massve future losses at Canada s largest nsurer sent Manulfe shares reelng last month. Those concerns were a result of Manulfe s strategy of not fully hedgng products such as annutes and segregated funds, whch promse nvestors ncome no matter what markets do. 4 / 41

105 Lecture 4 Why dd ths happen? These products contan embedded optons whch allow the nvestors many opportuntes to optmze the value of the guarantee. Prcng of these products requres soluton of an optmal stochastc control problem (an HJB PDE). Ths was beyond the techncal abltes of most nsurance companes Insurance companes used smplstc models whch underestmated the rsk nvolved. These models showed that there was no need to hedge these products, (Quote from actuary: ) Over any ten year perod, the market never goes down. Insurance company executves were able to boast of large (apparent) profts, whch then trggered rch bonus payments to traders and executves. 5 / 41

106 Lecture 4 Retrement Rsk Zone Consder an nvestor wth a retrement account, whch s nvested n the stock market Over the long run (before retrement), t does not matter f the market frst drops by 10% per year over several years and then goes up by 20% per year for several years; or the market frst goes up by 20% per year and then drops by 10% per year (.9)(.9)...(1.2)(1.2)... = (1.2)(1.2)...(.9)(.9)... 6 / 41

107 Lecture 4 The Retrement Rsk Zone II Ths s not the case once the nvestor retres, and begns to make wthdrawals from the retrement account The outcomes wll be very dfferent n the cases: n the frst few years after retrement, the market has losses, and the account s further depleted by wthdrawals, followed by some years of good market returns; compared to a few years of good market returns, after retrement (ncludng wthdrawals), followed by some years of losses Losses n the early years of retrement can be devastatng n the long run! Early bad returns can cause complete depleton of the account. 7 / 41

108 Lecture 4 A Typcal GMWB Example Investor pays $100 to an nsurance company, whch s nvested n a rsky asset. Denote amount n rsky asset sub-account by W = 100. The nvestor also has a vrtual guarantee account A = 100. Suppose that the contract runs for 10 years, and the guaranteed wthdrawal rate s $10 per year. 8 / 41

109 Lecture 4 A Typcal GMWB Example II At the end of each year, the nvestor can choose to wthdraw up to $10 from the account. If $γ [0, 10] s wthdrawn, then W new = max(w old γ, 0) ; Actual nvestment A new = A old γ ; Vrtual account Ths contnues for 10 years. At the end of 10 years, the nvestor can wthdraw anythng left,.e. max(w new, A new ). Note: the nvestor can contnue to wthdraw cash as long as A > 0, even f W = 0 (recall that W s nvested n a rsky asset). 9 / 41

110 Lecture 4 Example: Order of Random Returns Good Returns at Start Tme Return (%) Balance ($) Wthdrawal ($) Total Wthdrawal Amount ($) Ten year balance f no wthdrawal ($) / 41

111 Lecture 4 Same Random Returns: Dfferent Order No GMWB: poor returns at start Tme Return (%) Balance ($) Wthdrawal ($) Total Wthdrawal Amount ($) Ten year balance f no wthdrawal ($) / 41

112 Lecture 4 Unlucky Order of Returns: Wth GMWB GMWB Protecton Tme Return (%) Balance ($) Wthdrawal ($) Total Wthdrawal Amount ($) 100 Ten year balance f no wthdrawal ($) / 41

113 Lecture 4 Why s ths useful? The nvestor can partcpate n market gans, but stll has a guaranteed cash flow, n the case of market losses. Ths nsulates pensoners from losses n the early years of retrement. Ths protecton s pad for by deductng a yearly fee α from the amount n the rsky account W each year. The smple form of GMWB descrbed has many varants n practce: Guaranteed Lfetme Wthdrawal Beneft (GLWB), ratchet ncrease of vrtual account A f no wthdrawals, etc. We wll keep thngs smple here, and look at the basc GMWB. Most varable annutes sold n North Amerca have some type of market guarantee. 13 / 41

114 Lecture 4 Some More Detals The nvestor can choose to wthdraw up to the specfed contract rate G r wthout penalty. Usually, a penalty (κ > 0) s charged for wthdrawals above G r. Let ˆγ be the rate of wthdrawal selected by the holder. Then, the rate of actual cash receved by the holder of the GMWB s ˆf (ˆγ) = { ˆγ f 0 ˆγ Gr, ˆγ κ(ˆγ G r ) f ˆγ > G r. 14 / 41

115 Lecture 4 Stochastc Process Let S denote the value of the rsky asset, we assume that the rsk neutral process followed by S s ds = rsdt + σsdz r = rsk free rate; dz = φ dt ; φ N (0, 1) σ = volatlty The rsk neutral process followed by W s then (ncludng wthdrawals da). dw = (r α)wdt + σwdz + da, f W > 0 dw = 0, f W = 0 α = fee pad for guarantee ; A = guarantee account 15 / 41

116 Lecture 4 No-arbtrage Value Let V (W, A, τ) (τ = T t, T s contract expry) be the no-arbtrage value of the GMWB contract (.e. the cost of hedgng). At contract expry (τ = 0) we have (payoff = wthdrawal) V (W, A, τ = 0) = max(w, A(1 κ)) It turns out that t s optmal to wthdraw at a rate ˆγ ˆγ [0, G r ], or ˆγ = (nstantaneously wthdraw a fnte amount) 16 / 41

117 Lecture 4 Impulse Control Let LV = 1 2 σ2 W 2 V WW + (r α)wv W rv. Snce we have the opton of wthdrawng at a fnte rate at each pont n (W, A, τ), Ito s Lemma and no-arbtrage arguments gve ) V τ LV max (ˆγ ˆγVW ˆγV A 0 ˆγ [0,G r ] Note that ˆγ s a fnte wthdrawal rate. Wthdrawals only allowed f A > / 41

118 Lecture 4 Impulse Control II We also have the opton of wthdrawng a fnte amount nstantaneously (wthdrawng at an nfnte rate) at each pont n (W, A, τ) [ ] V (W, A, τ) sup V (max(w γ, 0), A γ, τ) + (1 κ)γ c 0. γ (0,A] where γ s a fnte wthdrawal amount. c > 0 s a fxed cost (whch can be very small). Ths s requred to make the Impulse Control problem well-posed. Note that ths equaton specfes that any amount n the remanng guarantee account can be wthdrawn nstantaneously (.e. γ (0, A]) wth a penalty. 18 / 41

119 Lecture 4 HJB Varatonal Inequalty Snce t must be optmal to ether wthdraw at a fnte rate or wthdraw a fnte amount at each pont, then ths can all be wrtten compactly as a Hamlton Jacob Bellman Varatonal Inequalty { ) mn V τ LV max (ˆγ ˆγVW ˆγV A, ˆγ [0,G r ] [ ] } V sup V (max(w γ, 0), A γ, τ) + (1 κ)γ c = 0 γ (0,A] 19 / 41

120 Lecture 4 Prevous Work Mlevsky, Salsbury, (2006, Insurance: Mathematcs and Economcs), pose GMWB prcng problem as a sngular control. Da, Kwok, Zong, (Mathematcal Fnance, 2008), solve sngular control formulaton usng a penalty method. Zakamoulne (Mathematcal Methods Operatons Research, 2005) argues that n general one can pose sngular control problems as mpulse control wth neglgble dfference (nfntesmal fxed cost) Clams that mpulse control s more general Chen, Forsyth (Numersche Mathematk, 2008), solve mpulse control formulaton (ths lecture) 20 / 41

121 Lecture 4 Alternatve Approach: Dscrete Wthdrawal Tmes Rather than attempt to solve the HJB Impulse Control problem drectly, let s replace ths problem by a dscrete wthdrawal problem Assume that the holder can only wthdraw at dscrete wthdrawal tmes τ 1,..., τ N, wth τ +1 τ = t w Use dynamc programmng dea, work backwards from t = T (τ = 0), so that V (W, A, 0) = max(w, A(1 κ)) Durng the nterval from τ = 0 to τ = τ 1 (the frst wthdrawal tme gong backwards) we solve V τ LV = 0 ; LV = 1 2 σ2 W 2 V WW + (r α)wv W rv. 21 / 41

122 Lecture 4 Optmum Strategy: Dscrete Wthdrawals At τ 1, we assume that the holder wthdraws the optmum amount γ V (W, A, τ 1 + [ ( ) = ) ] V max(w γ, 0), A γ, τ1 + f (γ), max γ [0,A] where now the cash flow term s { γ f 0 γ G, f (γ) = γ κ(γ G) c f γ > G. G = G r t w 22 / 41

123 Lecture 4 Dscrete Wthdrawals Then, from τ + 1 to τ 2, we solve V τ LV = 0 ; No A dependence n LV Then, we determne the optmum wthdrawal at τ 2 +, and so on, back down to τ = T (t = 0) today. Ths would appear to be a reasonable approxmaton to realty. In fact, most real contracts allow only dscrete wthdrawals. 23 / 41

124 Lecture 4 Dscrete Wthdrawal: A Numercal Scheme Defne nodes n the W drecton [W 0, W 1,..., W max ], and n the A drecton [A 0, A 1,..., A jmax ]. Let V n,j V (W, A j, τ n ). [V n ],j = V n,j. Let (L h V ) n,j be a dscrete form of the operator LV. Away from wthdrawal tmes, we solve V n+1,j V n,j τ = (L h V ) n+1,j 24 / 41

125 Lecture 4 A Numercal Scheme II At wthdrawal tme τ n, we then solve the local optmzaton problem at each node V n+ [,j = max I,j (γ,j)v n n + f ( γ n )] γ,j n [0,A,j, j ] where I s an nterpolaton operator I,j (γ)v n = V n (max(w γ, 0), A j γ) + nterpolaton error We use a lnear nterpolant of V n,j to determne the optmum wthdrawal at each node γ n,j. 25 / 41

126 Lecture 4 Numercal Scheme III Away from wthdrawal tmes, we solve a decoupled set of 1-d PDEs. (V j+1 ) τ = L (V j+1 ) A j+1 At wthdrawal tmes, we solve a set of decoupled optmzaton problems. A (V j ) τ = L (V j ) (V j-1 ) τ = L (V j-1 ) A j A j-1 Vast majorty of CPU tme spent solvng the local optmzaton problem at each node: W V n+ [,j = max I,j (γ,j)v n n + f ( γ n )] γ,j n [0,A,j, j ] Ths s embarrassngly parallel, but requres access to global data. 26 / 41

127 Lecture 4 Obvous Queston If we let τ w 0, does ths dscrete wthdrawal approxmaton converge to the soluton of the Impulse Control HJB equaton? If we allow dscrete wthdrawals every tmestep, then our numercal method s V n+1 [,j max I,j (γ,j)v n n + f ( γ n (,j)] τ Lh V ) n+1 = 0. γ,j n [0,A j ],j where the cash flow term f ( γ n,j) s { γ f 0 γ G, f (γ) = γ κ(γ G) c f γ > G. G = G r τ and I s a lnear nterpolaton operator 27 / 41

128 Lecture 4 Does t Converge? We want to show that ths scheme converges as τ, A, W 0 to the vscosty soluton of { ) mn V τ LV max (ˆγ ˆγVW ˆγV A, ˆγ [0,G r ] [ ] } V sup V (max(w γ, 0), A γ, τ) + (1 κ)γ c = 0 γ (0,A] Ths seems ntutvely obvous, but there are some subtle ponts. 28 / 41

129 Lecture 4 Monotoncty, Stablty and Consstency Lemma (Monotoncty and Stablty) Provded (L h V n+1 ) s dscretzed usng a postve coeffcent method and lnear nterpolaton s used when solvng the local optmzaton problem at each node, then the scheme s uncondtonally l stable and monotone. Proof. Straghtforward Lemma (Consstency) Provded the dscrete operator (L h V n+1 ) s consstent n the classcal sense, and lnear nterpolaton s used to solve the local optmzaton problem at each node, then the numercal scheme s consstent as defned n (Barles, Sougands (1991)). Proof. Not so straghtforward (lm nf, lm sup form needed for boundary condtons) 29 / 41

130 Lecture 4 Convergence Theorem (Strong Comparson Result) The GMWB Impulse Control problem satsfes the Strong Comparson Result,.e. there s a unque, contnuous vscosty soluton to the Impulse Control Problem. (Seydel, 2008) Theorem (Convergence to the Vscosty Soluton) The dscrete wthdrawal numercal method, wth wthdrawal nterval t w 0 converges to the unque vscosty soluton of the Impulse Control problem. Proof. Ths scheme s consstent, stable, and monotone, hence converges to the vscosty soluton (Barles, Sougands (1991)). 30 / 41

131 Lecture 4 One scheme for all problems So, we now have a sngle scheme whch Can be used to prce GMWB contracts wth fnte wthdrawal ntervals (the usual case n real contracts,.e. wthdrawals only allowed once or twce a year) We can also prce GMWB contracts n the lmt as the wthdrawal nterval 0 Convergence to the Impulse Control problem guaranteed No need for dfferent method for these two cases! Scheme s smple and ntutve to mplement mght be actually used by practtoners. 31 / 41

132 Lecture 4 Local Optmzaton Recall that at each node, at each tmestep, we have to solve [ V n+1,j max V n γ,j n [0,A j ] î,ĵ + f ( γ,j n ) ] τ ( L h V ) n+1 = 0.,j where V n s a lnear nterpolaton of î,ĵ V n( max(w γ,j n, 0), A j γ,j) n. Obvous method: use one-d optmzaton method at each node But, these methods are not guaranteed to get the global max of the objectve functon. We have seen some problems where only a local max was found, convergence to the wrong soluton of the PDE. 32 / 41

133 Lecture 4 Local Optmzaton In order to maxmze [ max V n γ,j n [0,A j ] î,ĵ + f ( γ,j n ) ] τ ( L h V ) n+1,j We dscretze the control values γ n,j n [0, A j ], and fnd the maxmum value by evaluatng the objectve functon at all the dscrete control values. Provded the dscretzaton step n [0, A j ], 0 as τ 0, then ths s a consstent (hence convergent) method. 33 / 41

134 Lecture 4 Examples Recall that the nvestor pays no extra up-front fee for the guarantee (only the ntal premum w 0 ). The nsurance company deducts an annual fee α from the balance n the sub-account W. Problem: let V (α, W, A, τ) be the value of the GMWB contract, for gven yearly guarantee fee α. Assume that the nvestor pays an ntal premum w 0 at t = 0 (τ = T ). Fnd the no-arbtrage fee α such that V (α, w 0, w 0, T ) = w 0 (we do ths by a Newton teraton). 34 / 41

135 Data Lecture 4 Parameter Value Expry tme T 10.0 years Interest rate r.05 Maxmum wthdrawal rate G r 10/year Wthdrawal penalty κ.10 Volatlty σ.30 Intal Lump-sum premum w Intal guarantee account balance 100 Intal sub-account value 100 Contnuous Wthdrawal Yes 35 / 41

136 Lecture 4 The No-arbtrage Fee (t = 0, A = 100) 36 / 41

137 Lecture 4 Optmal Wthdrawal Strategy t = 0, far fee charged for w 0 = 100. Indetermnate regon: appears to converge to optmal wthdrawal rate ˆγ = 0? 37 / 41

138 Lecture 4 Indetermnate Regon? In the contnuous wthdrawal regon, we have V τ = LV + max [ˆγ(1 VW V A ) ] (1) ˆγ [0,G r ] In the ndetermnate regon, we observe that (mesh, tmestep 0) [ ] 1 V W (W, A j, τ) V A (W, A j, τ) 0 If (1 V W V A ) 0, then any wthdrawal rate n [0, G r ] s optmal. Control may not be unque (but value s unque).,j 38 / 41

139 Lecture 4 No-arbtrage Fee σ =.15 α =.007 (70 bps) σ =.20 α =.014 (140 bps) σ =.30 α =.031 (310 bps) Current volatlty of S&P.25 Typcal fees charged: α =.005 (50 bps) too low for current market condtons. Insurance companes seem to be chargng fees based on marketng consderatons, not hedgng costs. Fee should be even hgher f other (typcal) contract optons consdered 39 / 41

140 Lecture 4 Other Issues Can easly use the same method f we assume underlyng process s a jump dffuson (Chen, Forsyth, SIAM J. Scentfc Computng (2007)). Effect of dscrete wthdrawals, volatlty, non-optmal wthdrawals, etc. (Chen, Vetzal, Forsyth, Insurance: Mathematcs and Economcs (2008)). A penalty method for sngular control formulaton of a GMWB (Da et al, Mathematcal Fnance (2008)), (Huang, Forsyth, Workng paper (2009)). Impulse control for a Guaranteed Mnmum Death Beneft (Belanger, Forsyth, Labahn, Appled Mathematcal Fnance (forthcomng)). 40 / 41

141 Lecture 4 Summary We have developed a sngle scheme whch can be used to prce GMWB contracts wth fnte wthdrawal ntervals, and the lmtng case of nfntesmal wthdrawal ntervals In the case of nfntesmal wthdrawal ntervals, we have proven convergence to the vscosty soluton of the Impulse Control problem For an nfntesmal fxed cost, solutons agree wth a sngular control formulaton Insurance companes seem to be chargng fees whch are too low to cover hedgng costs. Another subprme problem? 41 / 41

142 Lecture 5 Numercal Methods for Hamlton Jacob Bellman Equatons n Fnance Lecture 5 Gas Storage Peter Forsyth Unversty of Waterloo Cherton School of Computer Scence January / 34

143 Lecture 5 Outlne For Lecture 5 Motvaton HJB Equaton Sem-Lagrangan Dscretzaton Convergence Numercal Examples Summary: Gas Storage Summary: Lectures Open Problems 2 / 34

144 Lecture 5 Introducton Natural gas prces show seasonalty effects (hgher n wnter, lower n summer). Natural gas storage facltes are constructed to provde a cushon of avalable gas Gas s released durng perods of hgh demand Gas s stored n seasons of low demand Underground caverns are used for long term storage Objectve: determne the no-arbtrage value of leasng a storage faclty for a fxed term A by-product of ths valuaton s the optmal operatng strategy (.e. when to nject, produce or do nothng) Ths njecton strategy s the control n ths case 3 / 34

145 Lecture 5 An Underground Storage Faclty 4 / 34