A dual approach to some multiple exercise option problems

Transcription

1 A dual approach o some muliple exercise opion problems 27h March 2009, Oxford-Princeon workshop Nikolay Aleksandrov D.Phil Mahemaical Finance nikolay.aleksandrov@mahs.ox.ac.uk Mahemaical Insiue Oxford Universiy A dual approach o some muliple exercise opion problems 27h March 2009 nikolay.aleksandrov@mahs.ox.ac.uk p. 1

2 Conens Single sopping (American opions) 1 Longsaff-Schwarz algorihm 2 Dual approach Muliple sopping - Moivaion and problem formulaion. 1 Regression approach 2 Dual approach Numerical example Lieraure A dual approach o some muliple exercise opion problems 27h March 2009 nikolay.aleksandrov@mahs.ox.ac.uk p. 2

3 American opions American opion - opion, which exercise is allowed a any ime prior o he expiraion dae. Problem formulaion: V = sup τ T E [ B h τ B τ h is he payoff from exersizing a ime. B is he discoun facor. h τ B τ is he payoff discouned o ime zero. ], (1) A dual approach o some muliple exercise opion problems 27h March 2009 nikolay.aleksandrov@mahs.ox.ac.uk p. 3

4 Single sopping The opimal sopping formulaion is equivalen o he dynamic programming equaions V T (X T ) = h T (X T ), (2) { [ ]} V B (X ) = max h (X ),E V B (X ) (3) The coninuaion value C (X ) is defined by [ ] C B (X ) = E V B (X ), = 0, 1,...,T 1. (4) X are he sae variables. A dual approach o some muliple exercise opion problems 27h March 2009 nikolay.aleksandrov@mahs.ox.ac.uk p. 4

5 Longsaff-Schwarz algorihm The difficuly in he general problem is in esimaing he coninuaion value. The Longsaff-Schwarz algorihm is a Mone Carlo mehod, which relies on leas square regression of he coninuaion values from he simulaed pahs. The fied value from his regression hen gives an esimae for he coninuaion value. By esimaing he coninuaion value an exercise rule is deermined. The sopping rule gives a lower bound for he opion price. A dual approach o some muliple exercise opion problems 27h March 2009 nikolay.aleksandrov@mahs.ox.ac.uk p. 5

6 Longsaff-Schwarz algorihm For all imes {0, 1,2,..., T }, a each poin of he space se define an approximaion o he coninuaion value by Ĉ (x) = kx c,i ψ i (x). (5) i=1 Le ψ = (ψ 1, ψ 2,..., ψ k ) and c = (c,1, c,2,..., c,k ). If n pahs are simulaed, an esimaion for he regression coefficiens would be arg min c R k nx j=1 `C(j) kx i=1 c,i ψ i (X (j) ) 2, (6) where C (j) = B B V (j). (7) A dual approach o some muliple exercise opion problems 27h March 2009 nikolay.aleksandrov@mahs.ox.ac.uk p. 6

7 Dual approach The mehod relies on a dual represenaion of he value funcion. The problem becomes equivalen o minimizaion of he dual represenaion over a se of maringales. The opimal maringale (he one ha achieves he infimum) is known. The problem comes down o approximaing he opimal maringale. A dual approach o some muliple exercise opion problems 27h March 2009 nikolay.aleksandrov@mahs.ox.ac.uk p. 7

8 Dual approach V /B is a supermaringale» B E V(X ) B Here from he Doob-Meyer decomposiion V B = V 0 + M D, V (X ), (8) where M is a maringale and D is an increasing process, boh vanishing a = 0. Theorem(Rogers; Haugh and Kogan) The value funcion V0 a ime zero is given by V 0 = inf M H 1 0 E[ sup ( h M )], (9) 0 T B where H0 1 is he space of maringales M, for which sup 0 T M is inegrable and such ha M 0 = 0. The infimum is aained by aking M = M. The opimal maringale here can be expressed by M M = V E B [ V ]. B A dual approach o some muliple exercise opion problems 27h March 2009 nikolay.aleksandrov@mahs.ox.ac.uk p. 8

9 Muliple Sopping - Moivaion The moivaion for hese opimal sopping problems comes from pricing swing conracs wih he following feaures: 1 The swing opion has mauriy T days and can be exercised on days 1, 2,..., T. 2 I can be exercised up o k imes on day and he oal number of exercise righs is m. 3 When exercising he opion, is holder buys a cerain number of unis (usually 1MWh) of elecriciy for a prespecified fixed price K. A dual approach o some muliple exercise opion problems 27h March 2009 nikolay.aleksandrov@mahs.ox.ac.uk p. 9

10 Muliple sopping We define an exercise policy π o be a se of sopping imes {τ i } m i=1 wih τ 1 τ 2 τ m and #{j : τ j = s} k s. Then he value of he policy π a ime is given by The value funcion is defined o be V π,m,k = E ( mx h τi (X τi )). i=1 V,m,k = sup π V π,m,k = sup π E ( mx h(x τi )). i=1 We denoe he corresponding opimal policy π = {τ 1, τ 2,..., τ m }. A dual approach o some muliple exercise opion problems 27h March 2009 nikolay.aleksandrov@mahs.ox.ac.uk p. 10

11 Muliple sopping (Muliple exercise opion price - Dynamic programming formulaion) The price V,m,k a ime of an opion wih payoff funcion {h s, s T } which could be exercised k s imes per single exercise ime s {,..., T } wih m exercise opporuniies in oal for m > k is given by V,m,k T =k T h T, V,m,k For m k we have = max{k h + E [V,m k,k ], (k 1)h + E [V,m (k 1),k ],..., h + E [V,m 1,k ], E [V,m,k ]}. V,m,k T =mh T, V,m,k =max{mh,(m 1)h + E [V,1,k ],..., E [V,m,k ]}. A dual approach o some muliple exercise opion problems 27h March 2009 nikolay.aleksandrov@mahs.ox.ac.uk p. 11

12 Muliple sopping (Muliple exercise opion price - Opimal sopping problem formulaion) The price V,m,k of an opion, which could be exercised k imes per single exercise ime wih m exercise opporuniies in oal is given by V,m,k = max E ˆ max{kτ h τ + E τ [V,m k τ,k τ T τ+1 ],(k τ 1)h τ + E τ [V,m (k τ 1),k τ+1 ],..., h τ + E τ [V,m 1,k τ+1 ]} (In he max bracke only hose erms, which exis are aken) Marginal value The marginal value of one addiional exercise opporuniy is denoed by V,m,k for m 1: V,m,k = V,m,k V,m 1,k. The marginal value for m = 1 is jus he opion value for one exercise opporuniy V,1,k = V,1,k. A dual approach o some muliple exercise opion problems 27h March 2009 nikolay.aleksandrov@mahs.ox.ac.uk p. 12

13 Muliple sopping - lower Generalisaion of Longsaff-Schwarz. Suppose ha, working backwards in ime and forward from one exercise opporuniy, approximaions Ĉm, Ĉm 1,..., Ĉm k +1 o he m h, m 1,...,m k + 1 marginal coninuaion value funcions have been obained. Then for pah j define he approximae coninuaion value C m,(j) C m,(j) = 8 >< o be k h (X (j) ) + Cm k,(j), if h (X (j) ) Ĉm k +1 (X (j) ) (k 1)h (X (j) ) + Cm k +1,(j), if Ĉm k +1 (X (j) ) > h (X (j) ) Ĉm k +2 (X (j) ). >: C m,(j), if Ĉm (X(j) ) > h (X (j) ) The non-opimal m h marginal coninuaion values are also defined by = C m,(j) C m,(j) C m 1,(j). A dual approach o some muliple exercise opion problems 27h March 2009 nikolay.aleksandrov@mahs.ox.ac.uk p. 13

14 Muliple sopping - upper Theorem(Aleksandrov and Hambly; Bender) The marginal value V,m,k 0 is equal o V,m,k 0 = inf π inf E 0ˆ M H 0 max (h u M u ), u (G 0 \{τ m 1,...,τ 1 }) where he infima are aken over all sopping policies π and over he se of inegrable maringales H 0. We define N (τ m,..., τ 1 ) o be he number of sopping ime in he mulise τ m,..., τ 1 ha are less han or equal o. The opimal maringale is defined by m 1 M M = X l=0 ( M,m l,k )1 N =l M,m l,k The opimal sopping policy π here is he opimal sopping policy for he problem wih m 1 exercise righs. G0 is he mulise of all possible sopping imes. A dual approach o some muliple exercise opion problems 27h March 2009 nikolay.aleksandrov@mahs.ox.ac.uk p. 14

15 Muliple sopping - upper Propery 1. V,m+1,k V,m,k,. Propery 2. V,m,k = max E ˆmin(hτ, E τ [ V,m k τ,k τ T τ+1 ]) Dτ,m 1,k + D,m 1,k. Propery 3. V,m,k 0 = inf 0 τ T inf ` E 0ˆ max min(h, E [ V,m k,k M H 0 0 τ ])1 <τ + max(min(h, E [ V,m k,k ]), E [ V,m 1,k ])1 =τ M, where he infima are aken over all sopping imes τ and over he se of inegrable maringales H 0. The infimum is aained for he maringale M,m,k and sopping ime τ defined as τ = min{ : D,m 1,k > 0}. A dual approach o some muliple exercise opion problems 27h March 2009 nikolay.aleksandrov@mahs.ox.ac.uk p. 15

16 Numerical Example We consider a elecriciy swing opion wih a lifeime T = The opion can be exercised once on a weekend and wice on a weekday. The underlying process (elecriciy spo price) is he exponenial of a discree mean revering process log S = (1 α) log S + σw. (10) Wih parameers S 0 = 1, σ = 0.5 and α = 0.9. The payoff is aken o be he spo price iself. The basis funcions used for approximaing he marginal coninuaion values are Ψ = {1, log S}. (11) We use pre-simulaion pah o deermine he sopping sraegy and pahs for he lower bound, given he sopping sraegy. For he upper bound we use 1000 pahs and 50 inner pahs for he maringale approximaion. A dual approach o some muliple exercise opion problems 27h March 2009 nikolay.aleksandrov@mahs.ox.ac.uk p. 16

17 Numerical Resuls Exercise 1 ex. righ lower upper sandard dev relaive upper bound possibiliies per ime upper bound difference marginal value A dual approach o some muliple exercise opion problems 27h March 2009 nikolay.aleksandrov@mahs.ox.ac.uk p. 17

18 Numerical Resuls one exercise per ime lower bound upper bound n single exercise opions 80 Opion Value Number of Exercise Righs A dual approach o some muliple exercise opion problems 27h March 2009 nikolay.aleksandrov@mahs.ox.ac.uk p. 18

19 Advanages & Disadvanages Advanages: The approach is model independen. Provides upper and lower bounds for he value funcion. Mone Carlo is linear in he dimensionaliy. Disadvanages: Basis funcions are in some cases hard o choose. A dual approach o some muliple exercise opion problems 27h March 2009 nikolay.aleksandrov@mahs.ox.ac.uk p. 19

20 Lieraure Aleksandrov, N. and B.Hambly(2008): A dual approach o muliple exercise opion problems. Working Paper. Bender, C.(2008): Dual pricing of muli-exercise opions under volume consrains. Working Paper. Haugh, M.B. and L.Kogan(2001): Pricing American Opions: A dualiy approach. Technical repor, Operaion Research Cener, MIT and The Wharon School, Universiy of Pennsylvania. Longsaff, F.A. and E.S.Schwarz.(2002): Valuing american opions by simulaion: A leas-square approach. The Review of Financial Sudies 5, A dual approach o some muliple exercise opion problems 27h March 2009 nikolay.aleksandrov@mahs.ox.ac.uk p. 20

21 Lieraure Meinshausen N. and Hambly B.M. (2004) Mone Carlo Mehods for he Valuaion of Muliple-Exercise Opions. Mah. Finance 14, Rogers, L.C. (2002): Mone Carlo valuaion of american opions. Mahemaical Finance 12, Tsisiklis, J.N. and B. van Roy(2001): Regression mehods for pricing complex American-syle opions. IEEE Transacions on Neural Neworks 12, A dual approach o some muliple exercise opion problems 27h March 2009 nikolay.aleksandrov@mahs.ox.ac.uk p. 21

22 Quesions Thank you for your aenion. Quesions? A dual approach o some muliple exercise opion problems 27h March 2009 nikolay.aleksandrov@mahs.ox.ac.uk p. 22