A dual approach to some multiple exercise option problems 27th - PowerPoint PPT Presentation

A dual approach to some multiple exercise option problems 27th March 2009, Oxford-Princeton workshop Nikolay Aleksandrov D.Phil Mathematical Finance nikolay.aleksandrov@maths.ox.ac.uk Mathematical Institute Oxford University A dual approach to some multiple exercise option problems ⋄ 27th March 2009 ⋄ nikolay.aleksandrov@maths.ox.ac.uk – p. 1

Contents • Single stopping (American options) 1 Longstaff-Schwartz algorithm 2 Dual approach • Multiple stopping - Motivation and problem formulation. 1 Regression approach 2 Dual approach • Numerical example • Literature A dual approach to some multiple exercise option problems ⋄ 27th March 2009 ⋄ nikolay.aleksandrov@maths.ox.ac.uk – p. 2

American options American option - option, which exercise is allowed at any time prior to the expiration date. Problem formulation: � � h τ V ∗ t = sup E t B t , (1) B τ t ≤ τ ≤ T h t is the payoff from exersizing at time t . B t is the discount factor. h τ B τ is the payoff discounted to time zero. A dual approach to some multiple exercise option problems ⋄ 27th March 2009 ⋄ nikolay.aleksandrov@maths.ox.ac.uk – p. 3

Single stopping The optimal stopping formulation is equivalent to the dynamic programming equations V ∗ T ( X T ) = h T ( X T ) , (2) � B t � �� V ∗ V ∗ t ( X t ) = max h t ( X t ) , E t t +1 ( X t +1 ) (3) B t +1 The continuation value C ∗ t ( X t ) is defined by � B t � C ∗ V ∗ t ( X t ) = E t t +1 ( X t +1 ) t = 0 , 1 , ..., T − 1 . , (4) B t +1 X t are the state variables. A dual approach to some multiple exercise option problems ⋄ 27th March 2009 ⋄ nikolay.aleksandrov@maths.ox.ac.uk – p. 4

Longstaff-Schwartz algorithm • The difficulty in the general problem is in estimating the continuation value. • The Longstaff-Schwartz algorithm is a Monte Carlo method, which relies on least square regression of the continuation values from the simulated paths. • The fitted value from this regression then gives an estimate for the continuation value. • By estimating the continuation value an exercise rule is determined. • The stopping rule gives a lower bound for the option price. A dual approach to some multiple exercise option problems ⋄ 27th March 2009 ⋄ nikolay.aleksandrov@maths.ox.ac.uk – p. 5

Longstaff-Schwartz algorithm For all times t ∈ { 0 , 1 , 2 , ..., T } , at each point of the space set define an approximation to the continuation value by k ˆ X C t ( x ) = c t,i ψ i ( x ) . (5) i =1 Let ψ = ( ψ 1 , ψ 2 , ..., ψ k ) and ¯ c t = ( c t, 1 , c t, 2 , ..., c t,k ) . If n paths are simulated, an estimation for the regression coefficients would be n k C ( j ) c t,i ψ i ( X ( j ) ´ 2 , X X ` arg min − ) (6) t t c ∈ R k j =1 i =1 where B t C ( j ) V ( j ) = t +1 . (7) t B t +1 A dual approach to some multiple exercise option problems ⋄ 27th March 2009 ⋄ nikolay.aleksandrov@maths.ox.ac.uk – p. 6

Dual approach • The method relies on a dual represenaion of the value function. • The problem becomes equivalent to minimization of the dual representation over a set of martingales. • The optimal martingale (the one that achieves the infimum) is known. • The problem comes down to approximating the optimal martingale. A dual approach to some multiple exercise option problems ⋄ 27th March 2009 ⋄ nikolay.aleksandrov@maths.ox.ac.uk – p. 7

Dual approach V ∗ t /B t is a supermartingale » B t – V ∗ ≤ V ∗ E t t +1 ( X t +1 ) t ( X t ) , (8) B t +1 Here from the Doob-Meyer decomposition V ∗ = V ∗ 0 + M ∗ t − D ∗ t t , B t where M ∗ t is a martingale and D ∗ t is an increasing process, both vanishing at t = 0 . Theorem(Rogers; Haugh and Kogan) The value function V ∗ 0 at time zero is given by ( h t V ∗ 0 = inf E [ sup − M t )] , (9) B t M ∈ H 1 0 ≤ t ≤ T 0 where H 1 0 is the space of martingales M , for which sup 0 ≤ t ≤ T | M t | is integrable and such that M 0 = 0 . The infimum is attained by taking M = M ∗ . The optimal martingale here can be V ∗ V ∗ expressed by M ∗ t +1 − M ∗ t +1 t +1 t = B t +1 − E t [ B t +1 ] . A dual approach to some multiple exercise option problems ⋄ 27th March 2009 ⋄ nikolay.aleksandrov@maths.ox.ac.uk – p. 8

Multiple Stopping - Motivation The motivation for these optimal stopping problems comes from pricing swing contracts with the following features: 1 The swing option has maturity T days and can be exercised on days 1 , 2 , ..., T . 2 It can be exercised up to k t times on day t and the total number of exercise rights is m . 3 When exercising the option, its holder buys a certain number of units (usually 1MWh) of electricity for a prespecified fixed price K . A dual approach to some multiple exercise option problems ⋄ 27th March 2009 ⋄ nikolay.aleksandrov@maths.ox.ac.uk – p. 9

Multiple stopping We define an exercise policy π to be a set of stopping times { τ i } m i =1 with τ 1 ≤ τ 2 ≤ · · · ≤ τ m and # { j : τ j = s } ≤ k s . Then the value of the policy π at time t is given by m V π,m, k X = E t ( h τ i ( X τ i )) . t i =1 The value function is defined to be m V ∗ ,m, k V π,m, k X = sup = sup E t ( h ( X τ i )) . t t π π i =1 We denote the corresponding optimal policy π ∗ = { τ ∗ 1 , τ ∗ 2 , ..., τ ∗ m } . A dual approach to some multiple exercise option problems ⋄ 27th March 2009 ⋄ nikolay.aleksandrov@maths.ox.ac.uk – p. 10

Multiple stopping (Multiple exercise option price - Dynamic programming formulation) The price V ∗ ,m, k at t time t of an option with payoff function { h s , t ≤ s ≤ T } which could be exercised k s times per single exercise time s ∈ { t, . . . , T } with m exercise opportunities in total for m > k t is given by V ∗ ,m, k = k T h T , T V ∗ ,m, k = max { k t h t + E t [ V ∗ ,m − k t , k ] , ( k t − 1) h t + E t [ V ∗ ,m − ( k t − 1) , k ] , t t +1 t +1 ..., h t + E t [ V ∗ ,m − 1 , k ] , E t [ V ∗ ,m, k ] } . t +1 t +1 For m ≤ k t we have V ∗ ,m, k = mh T , T V ∗ ,m, k = max { mh t , ( m − 1) h t + E t [ V ∗ , 1 , k ] , ..., E t [ V ∗ ,m, k ] } . t t +1 t +1 A dual approach to some multiple exercise option problems ⋄ 27th March 2009 ⋄ nikolay.aleksandrov@maths.ox.ac.uk – p. 11

Multiple stopping (Multiple exercise option price - Optimal stopping problem formulation) The price V ∗ ,m, k t of an option, which could be exercised k t times per single exercise time t with m exercise opportunities in total is given by ] , ( k τ − 1) h τ + E τ [ V ∗ ,m − ( k τ − 1) , k V ∗ ,m, k max { k τ h τ + E τ [ V ∗ ,m − k τ , k ˆ = max ] , t ≤ τ ≤ T E t τ +1 τ +1 t ..., h τ + E τ [ V ∗ ,m − 1 , k ˜ ] } τ +1 (In the max bracket only those terms, which exist are taken) Marginal value The marginal value of one additional exercise opportunity is denoted by ∆ V ∗ ,m, k for m ≥ 1 : t ∆ V ∗ ,m, k = V ∗ ,m, k − V ∗ ,m − 1 , k . t t t The marginal value for m = 1 is just the option value for one exercise opportunity ∆ V ∗ , 1 , k = V ∗ , 1 , k . t t A dual approach to some multiple exercise option problems ⋄ 27th March 2009 ⋄ nikolay.aleksandrov@maths.ox.ac.uk – p. 12

Multiple stopping - lower Generalisation of Longstaff-Schwartz. Suppose that, working backwards in time and forward from one exercise opportunity, m − k t +1 +1 approximations ∆ ˆ t +1 , ∆ ˆ C m − 1 t +1 , ..., ∆ ˆ C m C to the m − th, m − 1 ,..., m − k t +1 + 1 t +1 marginal continuation value functions have been obtained. Then for path j define the approximate continuation value C m, ( j ) to be t 8 m − k t +1 , ( j ) k t +1 h t +1 ( X ( j ) t +1 ) + C , > t +1 > > > m − k t +1 +1 > if h t +1 ( X ( j ) ( X ( j ) t +1 ) ≥ ∆ ˆ > C t +1 ) > > t +1 > > > > > m − k t +1 +1 , ( j ) ( k t +1 − 1) h t +1 ( X ( j ) > t +1 ) + C , > > t +1 > > > m − k t +1 +1 m − k t +1 +2 > ( X ( j ) t +1 ) > h t +1 ( X ( j ) ( X ( j ) if ∆ ˆ t +1 ) ≥ ∆ ˆ < C C t +1 ) C m, ( j ) = t +1 t +1 t > > . > . > > . > > > > > > > C m, ( j ) > , > > t +1 > > > t +1 ( X ( j ) t +1 ) > h t +1 ( X ( j ) > if ∆ ˆ C m > t +1 ) : The non-optimal m − th marginal continuation values are also defined by ∆ C m, ( j ) = C m, ( j ) − C m − 1 , ( j ) . t t t A dual approach to some multiple exercise option problems ⋄ 27th March 2009 ⋄ nikolay.aleksandrov@maths.ox.ac.uk – p. 13

Multiple stopping - upper Theorem(Aleksandrov and Hambly; Bender) The marginal value ∆ V ∗ ,m, k is equal to 0 ∆ V ∗ ,m, k ˆ ˜ = inf inf E 0 u ∈ ( G 0 \{ τ m − 1 ,...,τ 1 } ) ( h u − M u ) max , 0 π M∈ H 0 where the infima are taken over all stopping policies π and over the set of integrable martingales H 0 . We define ¯ • N t ( τ m , ..., τ 1 ) to be the number of stopping time in the multiset τ m , ..., τ 1 that are less than or equal to t . • The optimal martingale is defined by m − 1 (∆ M ∗ ,m − l, k − ∆ M ∗ ,m − l, k X M ∗ t +1 − M ∗ t = ) 1 ¯ N ∗ t = l t +1 t l =0 • The optimal stopping policy π here is the optimal stopping policy for the problem with m − 1 exercise rights. • G 0 is the multiset of all possible stopping times. A dual approach to some multiple exercise option problems ⋄ 27th March 2009 ⋄ nikolay.aleksandrov@maths.ox.ac.uk – p. 14