A Numerical Approach to Price Path Dependent Asian Options Tatiana - PowerPoint PPT Presentation

Motivation Splitting method Finite difference approximations Numerical experiments and results Summary A Numerical Approach to Price Path Dependent Asian Options Tatiana Chernogorova 1 Lubin Vulkov 2 1 Faculty of Mathematics and Informatics University of Sofia, Bulgaria, 2 Faculty of Natural Sciences and Education University of Rousse, Bulgaria 10th International Conference on "Large-Scale Scientific Computations" LSSC’15 Tatiana Chernogorova, Lubin Vulkov A Numerical Approach to Price Path Dependent Asian Options

Motivation Splitting method Finite difference approximations Numerical experiments and results Summary Outline Motivation 1 Options. Asian options Mathematical model of the problem to determine the price of Asian option Splitting method 2 Parabolic subproblem (PSP) Hyperbolic subproblem (HSP) Finite difference approximations 3 First difference approximation of the PSP Second difference approximation for PSP Difference approximation for HSP Numerical experiments and results 4 Summary 5 Tatiana Chernogorova, Lubin Vulkov A Numerical Approach to Price Path Dependent Asian Options

Motivation Splitting method Options. Asian options Finite difference approximations Mathematical model of the problem to determine the price of Asian option Numerical experiments and results Summary Option. Call and Put options Option An option is a contract between the writer and the holder of the option about trading the stock at a prespecified fixed price K (exercise price) within a specified period (from the date of signing the contract to the maturity date T ). Depending on what an option concern: Call and Put options The call option gives the holder the right (but not the obligation) to buy the stock for the price K by the date (or at the date) of the maturity. The put option gives the holder the right (but not the obligation) to sell the stock for the price K by the date (or at the date) of the maturity. Tatiana Chernogorova, Lubin Vulkov A Numerical Approach to Price Path Dependent Asian Options

Motivation Splitting method Options. Asian options Finite difference approximations Mathematical model of the problem to determine the price of Asian option Numerical experiments and results Summary Option. Call and Put options Option An option is a contract between the writer and the holder of the option about trading the stock at a prespecified fixed price K (exercise price) within a specified period (from the date of signing the contract to the maturity date T ). Depending on what an option concern: Call and Put options The call option gives the holder the right (but not the obligation) to buy the stock for the price K by the date (or at the date) of the maturity. The put option gives the holder the right (but not the obligation) to sell the stock for the price K by the date (or at the date) of the maturity. Tatiana Chernogorova, Lubin Vulkov A Numerical Approach to Price Path Dependent Asian Options

Motivation Splitting method Options. Asian options Finite difference approximations Mathematical model of the problem to determine the price of Asian option Numerical experiments and results Summary European and American style of option; Asian option Depending on when an option may be exercised European option exercise is only at the date of the maturity. American style of option can be exercised at any time up to and including the date of the maturity. The payoff depends on the underlying asset price in the moment of its exercise. Asian option An Asian option can be of European or American style. An Asian option is an option whose payoff depends on the average of an underlying asset price over some time period, for t example A = A ( t ) = 1 � S ( θ ) d θ , where S ( θ ) is the price of the t 0 underlying stock. Tatiana Chernogorova, Lubin Vulkov A Numerical Approach to Price Path Dependent Asian Options

Motivation Splitting method Options. Asian options Finite difference approximations Mathematical model of the problem to determine the price of Asian option Numerical experiments and results Summary European and American style of option; Asian option Depending on when an option may be exercised European option exercise is only at the date of the maturity. American style of option can be exercised at any time up to and including the date of the maturity. The payoff depends on the underlying asset price in the moment of its exercise. Asian option An Asian option can be of European or American style. An Asian option is an option whose payoff depends on the average of an underlying asset price over some time period, for t example A = A ( t ) = 1 � S ( θ ) d θ , where S ( θ ) is the price of the t 0 underlying stock. Tatiana Chernogorova, Lubin Vulkov A Numerical Approach to Price Path Dependent Asian Options

Motivation Splitting method Options. Asian options Finite difference approximations Mathematical model of the problem to determine the price of Asian option Numerical experiments and results Summary Mathematical model Asian call option of European style P . Wilmott et al.,Mathematical Models and Computation, (1993): S γ ∂ 2 V ∂ V ∂τ = 1 S ∂ V S ∂ V 1 ¯ S 2 + r ¯ S − ¯ 2 σ 2 x − rV , ∂ ¯ ∂ ¯ ∂ ¯ (¯ S , ¯ x , τ ) ∈ ( 0 , ∞ ) × ( 0 , ∞ ) × ( 0 , T ] , ¯ V is the Asian option prise; S is the underlying stock price; τ = T − t , is the time to maturity T ( t is the time); σ 1 is the volatility; r is the interest rate; t ¯ � x = ¯ ¯ x ( t ) = S ( θ ) d θ , γ is the order of degeneracy, 0 < γ ≤ 2; 0 (¯ S , ¯ x , τ ) ∈ ( 0 , S max ) × ( 0 , x max ) × ( 0 , T ] . Tatiana Chernogorova, Lubin Vulkov A Numerical Approach to Price Path Dependent Asian Options

Motivation Splitting method Options. Asian options Finite difference approximations Mathematical model of the problem to determine the price of Asian option Numerical experiments and results Summary Mathematical model Initial and boundary conditions V (¯ x ) − K , 0 } ≡ V 0 (¯ S , ¯ x , 0 ) = max { X (¯ S , ¯ x ) , x , τ ) = e − r τ max { X (¯ V ( 0 , ¯ x ) − K , 0 } ≡ V 1 (¯ x , τ ) , � x ) − K ) + S max � e − r τ ( X (¯ 1 − e − r τ � � V ( S max , ¯ x , τ ) = max , 0 rT ≡ V 2 (¯ x , τ ) , ¯ S V (¯ ≡ V 3 (¯ 1 − e − r τ � � S , 0 , τ ) = S , τ ) , rT X (¯ x ) = ( x max − ¯ x ) / T . Tatiana Chernogorova, Lubin Vulkov A Numerical Approach to Price Path Dependent Asian Options

Motivation Splitting method Options. Asian options Finite difference approximations Mathematical model of the problem to determine the price of Asian option Numerical experiments and results Summary Previous Work FDM and FEM, constructed for ultra-parabolic equations without degeneration : Vabishchevich, P . N.: The numerical simulation of unsteady convective-diffusion processes in a countercurrent. Zh. Vychisl. Mat. Mat. Fiz. 35 (1), 46–52 (1995) Akrivis, G., Grouzlix, M., Thomee, V.: Numerical methods for ultra-parabolic equations. CALCOLO 31, 179–190 (1996) Ashyralyev, A., Yilmaz, S.: Modified Crank-Nicholson difference schemes for ultra-parabolic equations. Comp. and Math. Appls. 64, 2756–2764 (2012) Tatiana Chernogorova, Lubin Vulkov A Numerical Approach to Price Path Dependent Asian Options

Motivation Splitting method Options. Asian options Finite difference approximations Mathematical model of the problem to determine the price of Asian option Numerical experiments and results Summary Previous Work A number of techniques to price Asian options have been proposed: Monte-Carlo method (Y.-K.Kwok, R.Seydel); analytical methods (I.Sengypta, M.Fu, D.Madan, T.Wang); modified binomial tree approach (P .Wilmott, J.Dewyne, S. Howison); finite difference schemes (Z.Cen, A.Le, A.Xu, J.Hugger, T.Chernogorova, L.Vulkov); PDE approach (G.Meyer, L.Chan, S.-P .Zhu, W.Bao, C.-L.Chen, J.Zhang ), etc. Tatiana Chernogorova, Lubin Vulkov A Numerical Approach to Price Path Dependent Asian Options

Motivation Splitting method Options. Asian options Finite difference approximations Mathematical model of the problem to determine the price of Asian option Numerical experiments and results Summary Previous Work A number of techniques to price Asian options have been proposed: Monte-Carlo method (Y.-K.Kwok, R.Seydel); analytical methods (I.Sengypta, M.Fu, D.Madan, T.Wang); modified binomial tree approach (P .Wilmott, J.Dewyne, S. Howison); finite difference schemes (Z.Cen, A.Le, A.Xu, J.Hugger, T.Chernogorova, L.Vulkov); PDE approach (G.Meyer, L.Chan, S.-P .Zhu, W.Bao, C.-L.Chen, J.Zhang ), etc. Tatiana Chernogorova, Lubin Vulkov A Numerical Approach to Price Path Dependent Asian Options

Motivation Splitting method Options. Asian options Finite difference approximations Mathematical model of the problem to determine the price of Asian option Numerical experiments and results Summary Previous Work A number of techniques to price Asian options have been proposed: Monte-Carlo method (Y.-K.Kwok, R.Seydel); analytical methods (I.Sengypta, M.Fu, D.Madan, T.Wang); modified binomial tree approach (P .Wilmott, J.Dewyne, S. Howison); finite difference schemes (Z.Cen, A.Le, A.Xu, J.Hugger, T.Chernogorova, L.Vulkov); PDE approach (G.Meyer, L.Chan, S.-P .Zhu, W.Bao, C.-L.Chen, J.Zhang ), etc. Tatiana Chernogorova, Lubin Vulkov A Numerical Approach to Price Path Dependent Asian Options