Bounds for Asian basket options Griselda Deelstra (ULB) Ibrahima - PowerPoint PPT Presentation

Bounds for Asian basket options Griselda Deelstra (ULB) Ibrahima Diallo (ULB) Mich` ele Vanmaele (Ghent University) Mid-Term Conference on Advanced Mathematical Methods for Finance Vienna, September, 17th-22nd, 2007

Agenda 1. Introduction: problem description and motivation 2. Bounds based on conditioning and/or comonotonicity reasoning 1 , 2 3. Lower bound and upper bound UBRS in the Non-Comonotonic Case 4. Generalization of the upper bound of Thompson 3 5. Generalization of an upper bound of Lord 4 6. Numerical results 7. Conclusions 1 Vanmaele M., Deelstra G., Liinev J., Dhaene J. and Goovaerts M. J. (2006). “Bounds for the price of discrete sampled arithmetic Asian options”, Journal of Computational and Applied Mathematics, 185(1), 51-90. 2 Deelstra G., Liinev J. and Vanmaele M. (2004). “Pricing of arithmetic basket options by conditioning”, Insur- ance: Mathematics & Economics, 34, 55-57. 3 Thompson G.W.P. (1999a). “Fast narrow bounds on the value of Asian options”, Working paper, University of Cambridge. 4 Lord R. (2006). “Partially exact and bounded approximations for arithmetic Asian options”, Journal of Com- putational Finance, Vol 10(2).

1. Introduction: problem description and motivation • Bounds for European-style discrete arithmetic Asian basket options in a Black & Scholes framework • Consider a basket with n assets whose prices S i ( t ) , i = 1 , . . . , n , are described, under the risk neutral measure Q and with r some risk-neutral interest rate, by dS i ( t ) = rS i ( t ) dt + σ i S i ( t ) dW i ( t ) , • Assume that the different asset prices are instantaneously correlated in a constant way i.e. corr ( dW i , dW j ) = ρ ij dt. • Given the above dynamics, the i -th asset price at time t equals S i ( t ) = S i (0) e ( r − 1 2 σ 2 i ) t + σ i W i ( t )

• Price of a discrete arithmetic Asian basket call option with a fixed strike K and maturity T on m averaging dates at current time t = 0 is determined by     n m − 1 � � ABC ( n, m, K, T ) = e − rT E Q b j S l ( T − j ) − K a l     j =0 l =1 + with a l and b j positive coefficients, which both sum up to 1 . • For T ≤ m − 1 , the Asian basket call option is said to be in progress and for T > m − 1 , we call it forward starting • Suitable for hedging as their payoff depend on an average of asset prices at different times and of different assets, and takes the correlations between the assets in the basket into account • No closed-form solutions available in the Black & Scholes setting • Dahl and Benth (2001a,b) 5 value such options by quasi-Monte Carlo techniques and singular value decomposition • In the setting of Asian options, Thompson (1999a) used a first order approximation of the arithmetic sum and derived an upper bound that sharpens those of Rogers and Shi (1995) 6 5 Dahl L.O. and Benth F.E. (2001a). “Valuing Asian basket options with quasi-Monte Carlo techniques and singular value decomposition”, Pure Mathematics, 2 February, 1-21. Dahl L.O. and Benth F.E. (2001b). “Fast evaluation of the Asian basket option by singular value decomposition”, Pure Mathematics, 8 March, 1-14. 6 Rogers L.C.G. and Shi Z. (1995). “The value of an Asian option”, Journal of Applied Probability, 32, 1077-1088.

• Lord (2006) revised Thompson’s method and proposed a shift lognormal approximation to the sums and he included a supplementary parameter which is estimated by an optimization algorithm • Deelstra et al. (2004) and Vanmaele et al. (2006) used techniques based on comonotonic risks for deriving upper and lower bounds for stop-loss premiums of sums of dependent random variables, as explained in Kaas et al. (2000) 7 and Dhaene et al. (2002a) 8 , combined with conditioning techniques and splitting up like in Curran (1994) 9 , Rogers and Shi (1995) and Nielsen and Sandmann (2003) 10 • A random vector ( X c 1 , . . . , X c k ) is comonotonic if each two possible outcomes ( x 1 , . . . , x k ) and ( y 1 , . . . , y k ) of ( X c 1 , . . . , X c k ) are ordered componentwise. • Consider S = � k i =1 X i with X = ( X 1 , . . . , X k ) with known marginal distributions but unknown dependence structure. Then the sum S is bounded below and above in convex order ( � cx ) by sums of comonotonic variables: S ℓ � cx S � cx S u � cx S c , 7 Kaas R., Dhaene J. and Goovaerts M.J. (2000). “Upper and lower bounds for sums of random variables”, Insurance: Mathematics & Economics, 27, 151-168. 8 Dhaene J., Denuit M., Goovaerts M.J., Kaas R. and Vyncke D. (2002a). “The concept of comonotonicity in actuarial science and finance: theory”, Insurance: Mathematics & Economics, 31(1), 3-33. 9 Curran M. (1994). “Valuing Asian and portfolio by conditioning on the geometric mean price”, Management science, 40, 1705-1711. 10 Nielsen J.A. and Sandmann K. (2003). “Pricing Bounds on Asian options”, Journal of Financial and Quanti- tative Analysis, 38(2), 449-473.

which implies by definition of convex order that S ℓ − d ( S u − d ) + ( S c − d ) + �� � � � � � � � � ≤ E ( S − d ) + ≤ E ≤ E E + for all d in R + , while E � S ℓ � = E [ S ] = E [ S u ] = E [ S c ] .

2. Bounds based on conditioning and/or comonotonicity reasoning � m − 1 • Write the double sum S = � n j =0 b j S l ( T − j ) as: l =1 a l mn mn � � α i e Y i X i = S = i =1 i =1 with ( r − 1 2 σ 2 m ⌉ )( T − ( i − 1) mod m ) ⌈ i α i = a ⌈ i m ⌉ b ( i − 1) mod m S ⌈ i m ⌉ (0) e and m ⌉ ( T − ( i − 1) mod m ) ∽ N (0 , σ 2 Y i = σ 2 m ⌉ ( T − ( i − 1) mod m )) Y i = σ ⌈ i m ⌉ W ⌈ i ⌈ i for all i = 1 , . . . , mn, where ⌈ x ⌉ is the smallest integer greater than or equal to x and y mod m = y − ⌊ y/m ⌋ m ; ⌊ y ⌋ denotes the greatest integer less than or equal to y.

Comonotonic Upper Bound • Define S c = F − 1 X 1 ( U ) + F − 1 X 2 ( U ) + · · · + F − 1 X nm ( U ) with U a Uniform (0 , 1) random variable and consider ( S c − K ) + ABC ( n, m, K, T ) ≤ e − rT E Q � � • Comonotonic upper bound given by n m − 1 � � � � a l b j S l (0) e − rj Φ � T − j − Φ − 1 ( F S c ( K )) ABC ( n, m, K, T ) ≤ σ l j =0 l =1 − e − rT K (1 − F S c ( K )) where the cdf of the comonotonic sum F S c ( K ) solves n m − 1 � � ( r − 1 � � � T − j Φ − 1 ( F S c ( K )) 2 σ 2 a l b j S l (0) exp l )( T − j ) + σ l = K j =0 l =1 where Φ is the cumulative distribution function of a normally distributed random variable.

Interpretation of comonotonic upper bound The payoff of the Asian basket call option satisfies     m − 1 m − 1 n n � � � � b j S l ( T − j ) − K ≤ b j S l ( T − j ) − K l a l a l     l =1 j =0 l =1 j =0 + + m − 1 n � � ≤ a l b j ( S l ( T − j ) − K lj ) + l =1 j =0 as well as � n   n m − 1 m − 1 � � � � � a l b j S l ( T − j ) − K ≤ b j a l S l ( T − j ) − K j   j =0 j =0 l =1 l =1 + + n m − 1 � � ≤ a l b j ( S l ( T − j ) − K lj ) + , j =0 l =1 with m − 1 m − 1 n n � � � � a l K l = b j K j = a l b j K lj = K. l =1 j =0 l =1 j =0

By a no-arbitrage argument we find that the time zero price of such Asian basket option should satisfy the following two relations: m − 1 n n � � � a l b j e − rj C l ( K lj , T − j ) ABC ( n, m, K, T ) ≤ a l AC l ( m, K l , T ) ≤ l =1 l =1 j =0 m − 1 m − 1 n � � � b j e − rj BC ( n, K j , T − j ) ≤ a l b j e − rj C l ( K lj , T − j ) . ABC ( n, m, K, T ) ≤ j =0 l =1 j =0 Superreplication by a static portfolio of vanilla call options C l on the underlying assets S l in the basket and with different maturities and strikes. Also an average of Asian options AC l or a combination of basket options BC with different maturity dates form a superreplicating strategy. Simon et al. (2000), Albrecher et al. (2005) Hobson et al. (2005) for a basket option in a model-free framework Chen et al. (2007) for a class of exotic options in a model-free framework.

Comonotonic Lower Bound • A lower bound, in the sense of convex order, for S = � mn i =1 X i is S ℓ = E [ S | Λ] where Λ is a normally distributed random variable such that ( W l ( T − j ) , Λ) are bivariate normally distributed for all l and j , which we assume in the sequel. • Define r l,j by r l,j = Cov ( W l ( T − j ) , Λ) √ T − j σ Λ • Assume that all r l,j are positive: then the comonotonic lower bound is determined by (and the case of all r l,j negative can be treated in a similar way): n m − 1 � � � � a l b j S l (0) e − rj Φ � T − j − Φ − 1 ( F S ℓ ( K )) ABC ( n, m, K, T ) ≥ r l,j σ l j =0 l =1 − e − rT K (1 − F S ℓ ( K )) where F S ℓ ( K ) , solves m − 1 n � � T − j + ( r − 1 � � r l,j σ l Φ − 1 ( F S ℓ ( K )) � 2 r 2 l,j σ 2 l )( T − j ) K = a l b j S l (0) exp l =1 j =0