Jean-Paul Murara, Västeras, 26-April Mälardalen University, Sweden. Pricing EO under 2-dim. B S PDE by. using the Crank-Nicolson Method

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1 Prcng EO under Mälardalen Unversty, Sweden Västeras, 26-Aprl / 15

2 Outlne / 15

3 Optons - contracts that gve to the holder the rght but not the oblgaton to buy/sell an asset sometmes n the future at a predetermned prce (exercse/strke prce: K). European Optons - When the executon tme of the contract s fxed (at maturty: T). Payoff functons: Call opton: h(s T ) = (S T K ) + = max(s T K,0) Put opton: h(s T ) = (K S T ) + = max(k S T,0) 3 / 15

4 B S (1973) consdered n ther model a CONSTANT VOLATILITY. For S t that follows: ds t = rs t dt + σs t dw t (1) C BS = S t N(d 1 ) Ke r(t t) N(d 2 ) (2) P BS = S t N( d 1 ) + Ke r(t t) N( d 2 ) (3) Ths s far from realty. Hence STOCHASTIC VOLATILITY models. ds t = µs t dt + σ t S t dw t (4) dσ t = a(σ t,t)dt + b(σ t,t)dw j t (5) 4 / 15

5 Chrstoffersen et al. (2009) proved that a 2-factor SV model for optons prcng captures very well the behavour of fnancal markets. ds = µsdt + v 1 SdZ 1 + v 2 SdZ 2, (6) dv 1 = κ 1 (θ 1 v 1 )dt + σ 1 v1 dz 3, (7) dv 2 = κ 2 (θ 2 v 2 )dt + σ 2 v2 dz 4, (8) µ - the nst. return per unt tme of the underlyng asset, θ 1 and θ 2 - the long-run means of v 1 and v 2, κ 1 and κ 2 are the speeds of mean-reverson; σ 1 and σ 2 - the nst. vol. of v 1 and v 2 per unt tme resp.; Z are correlated WP wth a correlaton s.t. E Q (dz 1 dz 3 ) = ρ 13 dt, E Q (dz 2 dz 4 ) = ρ 24 dt and all other correlatons are zeros. 5 / 15

6 Let V(S(t),t) be the value of the dervatve at tme t, S(t) the prce of the underlyng at tme t, r the zero rsk nterest rate and σ the volatlty of the underlyng. Wth V(S(t), t) where S(t) satsfes then ds t = µs t dt + σs t dw t (9) dv = (µv S + V t + σ ) 2 2 V SS dt + σv S dw t (10) after calculatons we end up wth the Black-Scholes-Merton PDE: V t + σ 2 2 S2 V SS + rsv S = rv (11) 6 / 15

7 1D Crank Ncolson Fn. Dff. In one dmenson, the CNM for heat equaton (: V t = av tt ) comes to V n+1 V n t = a[ (V n 1 n V + V n+1 1 ) + (V n 2( x) 2 n s the tme step and s the poston. = (12) +1 2V n + V n 1 )] Wth r = a t 2( x) 2, ths (12) smplfes to: rv n+1 n (1+2rV ) rv n+1 1 = rv n +1 +(1 2rV n )+rv n 1 (13) whch can be solved for V n+1. 7 / 15

8 Fgure: CNM n 1D Heat equaton 8 / 15

9 Crank Ncholson Usng CNM we obtan the followng transformatons C t = Cj+1 C j t (14) 2 C x = 1 ( ) 2 2( x) 2 (C j Cj+1 + C j+1 1 ) + (Cj +1 2Cj + C j 1 ) C x = 1 2 ( j+1 (C +1 Cj+1 1 ) + (Cj +1 Cj 1 ) ) 2( x) 2( x) (15) (16) 9 / 15

10 2D Crank Ncolson Fn. Dff. In two dmensons, the CNM for the heat equaton comes to: = V n+1 V n t = (17) [ a (V n+1 +1,j + V n+1 1,j + V n+1,j+1 + V n+1 n+1,j 1 4V ) 2( x) 2 + a[ (V n +1,j + V n 1,j + V n,j+1 + V n,j 1 4V n ) ] 2( x) 2 ] Eq. (17) after some transformatons can be wrtten as (1 µ 2 δ 2 x µ 2 δ 2 y )V n+1,j = (1 µ 2 δ 2 x µ 2 δ 2 y )V n,j (18) 10 / 15

11 The Model Wth V(S(t),t) we have V t σ 2 S 2 V ss + rsv s = rv (19) An opton wth two assets V(s 1,s 2,t) satsfes the B S PDE: V t σ 2 1 S2 1 V s 1 s 1 + rs 1 V s σ 2 2 S2 2 V s 2 s 2 + (20) +rs 2 V s2 + ρs 1 S 2 σ 1 σ 2 V s1 s 2 = rv wth payoff V T (S 1,S 2 ) = V(S 1,S 2,T ) at maturty:.e. V T (s 1,s 2 ) = max[max(s 1,s 2 ) K,0] (21) 11 / 15

12 The Model: Boundary condtons The B.C. are: V(S 1,0,t) = s 1 N(d 1 ) Ke rt N(d 2 ) (22) V(0,S 2,t) = s 2 N(d 1 ) Ke rt N(d 2 ) (23) V(S 1,s 2,t) = S 1 Ke rt (24) V(s 1,S 2,t) = S 2 Ke rt (25) the frst two are obtaned by usng Black-Scholes formula (1973). 12 / 15

13 System U t = Uj+1 U j t U s = U U = U v 2 j+1 U j s j+1 (26) (27) U j v 2 (28) 2 U s = 1 ( ) 2 2( s) 2 (C j Cj+1 + C j+1 1 ) + (Cj +1 2Cj + C j 1 ) (29) Let s = x and v 2 = y wth x = x 0 + x and y = y 0 + j y. The obaned system must be solved. 13 / 15

14 We have 3 varables: tme, volatlty and prce. Startng by boundary condtons. Current Work Arrange equatons nto 2 tme steps. Obtan matrces U and solve the obtaned systems gettng U,j,k s. Compare wth other results (Conze) 14 / 15

15 Comments + Hnts +??? Thanks for your attenton! and HBD to Anatoly! 15 / 15