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In both cases, from V(0,t) = V(Smax,t) = 0, Vt > 0 we have the boundary conditions U (0, 4) = U™ (Sax, 4) = 0, Vk > Regardless of the relationship between o? and r, the matrix Apw is an M-matrix [7, p. 16] and then Apy > 0. From (3.2) fhe property (2.9) is guaranteed according to the financial meaning of V(S,t), ie., V(S,t) > 0. Diam: comp, Dew on tema: of San FE PK FT 4 Al ae 2 N nem merereseewmerry thee ween. mb YT (OC #\ RLWC #£\ fn Rn heterivn nd Lene than

Figure 1 In both cases, from V(0,t) = V(Smax,t) = 0, Vt > 0 we have the boundary conditions U (0, 4) = U™ (Sax, 4) = 0, Vk > Regardless of the relationship between o? and r, the matrix Apw is an M-matrix [7, p. 16] and then Apy > 0. From (3.2) fhe property (2.9) is guaranteed according to the financial meaning of V(S,t), ie., V(S,t) > 0. Diam: comp, Dew on tema: of San FE PK FT 4 Al ae 2 N nem merereseewmerry thee ween. mb YT (OC #\ RLWC #£\ fn Rn heterivn nd Lene than

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Fig. 1. Truncated call option value just before the first monitoring date t; = T. Parameters: L= 0, K = 40, U=60, r=0.05, o = 0.1 (the upper graphic) and o = 0.001 (the lower graphic), T = 1. The numerical solutions are obtained by the fully-implicit scheme (FD) with AS = 0.2, At = 0.001, Snax = 16C (the upper graphic) and Sia, = 80 (the lower graphic), according to (5.2), with R = 4 and R = 2 respectively. Indeed, the requested condition of M-matrix, in order to be avoided spurious oscillations, leads to a low order scheme, < he convection term 2 is discretized through a forward difference. Such method introduces numerical diffusion when (3.2). olved. By standard analysis of consistency such a term amounts to rs af . Then in (3.2) the coefficient } o?S? is replaced b les? + S48), so that under the condition o* < r the numerical aitiston becomes particularly significant. The equivalenc tween the two methods allows us to interpret the numerical solution Vy(S;,T) obtained by means of the Post-Widd« nversion formula. Under the condition o? <r the discrepancy between exact V(S;,T) and approximate one Vj(Sj,T) - jue to the numerical diffusion introduced by the forward difference of the convection term &. Fig. 1, where two distinct va ies 0; = 0.1 and o2 = 0.001 are chosen (in the latter case the numerical diffusion becomes euriiolerdse significant), confirm he statement in the previous sentence. In Fig. 1, T = 1 and At = 0.001 are assumed, so that the numerical solution obtaine y finite difference scheme is equivalent to the one obtained by the mixed method with N = 1000. Fig. 2, where geomett nd parameters coincide with Fig. 1, compares the finite difference solution and the mixed method one for increasing ! \ccording to results in literature, the Post-Widder inversion formula has an extremely slow convergence rate. In both Figs. nd 2 the spatial domain [0, 00) has been replaced by [0, Smax|. The criterion choice of Sima, will be discussed later. Because « he slow convergence of the sequence Vy(S;, T) it is natural to seek extrapolation methods to speed up convergence. The lat er will be discussed later too.
Fig. 1. Truncated call option value just before the first monitoring date t; = T. Parameters: L= 0, K = 40, U=60, r=0.05, o = 0.1 (the upper graphic) and o = 0.001 (the lower graphic), T = 1. The numerical solutions are obtained by the fully-implicit scheme (FD) with AS = 0.2, At = 0.001, Snax = 16C (the upper graphic) and Sia, = 80 (the lower graphic), according to (5.2), with R = 4 and R = 2 respectively. Indeed, the requested condition of M-matrix, in order to be avoided spurious oscillations, leads to a low order scheme, < he convection term 2 is discretized through a forward difference. Such method introduces numerical diffusion when (3.2). olved. By standard analysis of consistency such a term amounts to rs af . Then in (3.2) the coefficient } o?S? is replaced b les? + S48), so that under the condition o* < r the numerical aitiston becomes particularly significant. The equivalenc tween the two methods allows us to interpret the numerical solution Vy(S;,T) obtained by means of the Post-Widd« nversion formula. Under the condition o? <r the discrepancy between exact V(S;,T) and approximate one Vj(Sj,T) - jue to the numerical diffusion introduced by the forward difference of the convection term &. Fig. 1, where two distinct va ies 0; = 0.1 and o2 = 0.001 are chosen (in the latter case the numerical diffusion becomes euriiolerdse significant), confirm he statement in the previous sentence. In Fig. 1, T = 1 and At = 0.001 are assumed, so that the numerical solution obtaine y finite difference scheme is equivalent to the one obtained by the mixed method with N = 1000. Fig. 2, where geomett nd parameters coincide with Fig. 1, compares the finite difference solution and the mixed method one for increasing ! \ccording to results in literature, the Post-Widder inversion formula has an extremely slow convergence rate. In both Figs. nd 2 the spatial domain [0, 00) has been replaced by [0, Smax|. The criterion choice of Sima, will be discussed later. Because « he slow convergence of the sequence Vy(S;, T) it is natural to seek extrapolation methods to speed up convergence. The lat er will be discussed later too.
Fig. 2. Truncated call option value just before the first monitoring date t; = T. Parameters: L=0, K = 40, U=60, r=0.05, o=0.001, T=1. Th Numerical solutions are obtained by the fully-implicit finite difference scheme (FD), with AS = 0.2, At = 0.001, Sinax = 80 (the upper graphic) and mixe method (PW) with N = 10,20, 30, AS = 0.2, Sax = 80 (the lower graphic), according to (5.2), with R = 2.
Fig. 2. Truncated call option value just before the first monitoring date t; = T. Parameters: L=0, K = 40, U=60, r=0.05, o=0.001, T=1. Th Numerical solutions are obtained by the fully-implicit finite difference scheme (FD), with AS = 0.2, At = 0.001, Sinax = 80 (the upper graphic) and mixe method (PW) with N = 10,20, 30, AS = 0.2, Sax = 80 (the lower graphic), according to (5.2), with R = 2.
Fig. 3. Truncated call option value just before the first monitoring date t; = T. The numerical solutions are obtained using the mixed method witt AS=0.05,N=100,N=1000 (the upper’ graphic) and N=1000,N=10000 (the lower graphic), respectively. Parameters L=90, K= 100, U=110, r=0.05, ¢ = 0.001, T=1, Smnax = 200, according to (5.2), with R = 2. 4. In literature spurious oscillations, arising from discontinuity in initial/boundary conditions are usually associated with finite difference or finite element methods. It is surprising how the mixed method, which combines ae Laplace Trans- form on the real axis with a particular inversion technique and finite difference method (where the term “— is discretized by a centered difference), exhibits an analogous phenomenon (see Fig. 3). A posteriori we can say that avieiing spurious oscillations is a consequence of the equivalence of the adopted methods. Now we illustrate and prove this statement. In other terms, we ask what happens if an unsuitable finite difference scheme is adopted. More specifically, under the condition o? <r, what solution does the mixed method, with centered difference for discretizing the convection term 2%, arise? For reasons that will be evident in the sequel, the condition o? < 4 is assumed. Through mixed method, it the convection term % is discretized by a centered difference, we have the sac tebe Apw given by (3.3).
Fig. 3. Truncated call option value just before the first monitoring date t; = T. The numerical solutions are obtained using the mixed method witt AS=0.05,N=100,N=1000 (the upper’ graphic) and N=1000,N=10000 (the lower graphic), respectively. Parameters L=90, K= 100, U=110, r=0.05, ¢ = 0.001, T=1, Smnax = 200, according to (5.2), with R = 2. 4. In literature spurious oscillations, arising from discontinuity in initial/boundary conditions are usually associated with finite difference or finite element methods. It is surprising how the mixed method, which combines ae Laplace Trans- form on the real axis with a particular inversion technique and finite difference method (where the term “— is discretized by a centered difference), exhibits an analogous phenomenon (see Fig. 3). A posteriori we can say that avieiing spurious oscillations is a consequence of the equivalence of the adopted methods. Now we illustrate and prove this statement. In other terms, we ask what happens if an unsuitable finite difference scheme is adopted. More specifically, under the condition o? <r, what solution does the mixed method, with centered difference for discretizing the convection term 2%, arise? For reasons that will be evident in the sequel, the condition o? < 4 is assumed. Through mixed method, it the convection term % is discretized by a centered difference, we have the sac tebe Apw given by (3.3).
The sign pattern of the entries of Apw is as follows [7, formula (4.33)]
The sign pattern of the entries of Apw is as follows [7, formula (4.33)]
Fig. 4. Real/complex eigenvalues j4,(Apw) for different values of r = aMo?, with « = 0,1,1.2, and o = 0.01, M=50, 4 = 100 fixed. terms like Nw, N* uN, ..., which amount to Np" cos(N@), N* p% cos(N0),..., No% sin(N0), N?p% sin(NO), ...AS N > oo, from (4.3 it follows p — 1, from which spurious oscillations increasing with N. Similar analysis if 1, (Apw) is multiple and real. Eigen values pattern of Apy for different r values is illustrated in Fig. 4.
Fig. 4. Real/complex eigenvalues j4,(Apw) for different values of r = aMo?, with « = 0,1,1.2, and o = 0.01, M=50, 4 = 100 fixed. terms like Nw, N* uN, ..., which amount to Np" cos(N@), N* p% cos(N0),..., No% sin(N0), N?p% sin(NO), ...AS N > oo, from (4.3 it follows p — 1, from which spurious oscillations increasing with N. Similar analysis if 1, (Apw) is multiple and real. Eigen values pattern of Apy for different r values is illustrated in Fig. 4.
Fig. 5. Truncated call option. Geometry and parameters as upper graphic of Fig. 1. Numerical solution Vy(S,T) and V2n(S,T) from Post-Widder formula, with N = 1000. Errors |V(S,T) — Vy(S,T)| and |V(S,T) — V(S,T)}. condition o* > 2r, the far field boundary condition reads Siqx = K exp(vV2To? In 100), involving strike price K, maturity T and volatility o. Numerical experiments in literature show that the estimates at least incorporate the usual rule of thumb (e.g., “three or four times the strike price”). So that in the numerical experiments above, Smnax is assumed according to the widely used criterion.
Fig. 5. Truncated call option. Geometry and parameters as upper graphic of Fig. 1. Numerical solution Vy(S,T) and V2n(S,T) from Post-Widder formula, with N = 1000. Errors |V(S,T) — Vy(S,T)| and |V(S,T) — V(S,T)}. condition o* > 2r, the far field boundary condition reads Siqx = K exp(vV2To? In 100), involving strike price K, maturity T and volatility o. Numerical experiments in literature show that the estimates at least incorporate the usual rule of thumb (e.g., “three or four times the strike price”). So that in the numerical experiments above, Smnax is assumed according to the widely used criterion.
Related topics:
Behavioral FinanceDerivatives MarketsBlack Scholes ModelLaplace transformationOption pricingFixed Income Markets
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