Partial Differential Equations - Modelling and ... - ResearchGate

More documents

Recommendations

Info

An Operator Splitting Method for Pricing American Options 281 paper demonstrates that the operator splitting method is suitable for pricing American options with different models and that the computation of a sufficiently accurate price for most purposes requires only a few milliseconds on a contemporary PC. Outline of the paper is the following. We begin by describing the three models and the resulting P(I)DEs for European options. After this we formulate linear complementarity problems for the value of American options. Next we sketch finite difference discretizations for the partial differential operators. Then the operator splitting method is described and after this methods for solving the resulting systems of linear equations are discussed. The paper ends with numerical examples with all of the considered models and conclusions. 2 Models 2.1 Black–Scholes Model By assuming that the value of the underlying asset denoted by x follows a geometric Brownian motion with a drift, the Black–Scholes PDE [BS73] v t = A BS v = − 1 2 (σx)2 v xx − rxv x + rv (1) can be derived for the value of an option denoted by v, whereσ is the volatility of the value of the asset and r is the risk free interest rate. In practice, the market prices of options do not satisfy (1). One possible way to make the model to match the markets is to use a volatility function σ which depends on the value of the underlying asset and time; see [AP05, Dup94], for example. In this case, the volatility function has to be calibrated with the market data. 2.2 Jump-Diffusion Models When there is a high market stress like the crash of 1987 the value of assets can move faster than a geometric Brownian motion would predict. Partly due to this, models which allow also jumps for the value of asset have become more common; see [CT04] and references therein. Already in 1976 Merton considered such a model in [Mer76]. With independent and identically distributed jumps a partial integro-differential equation (PIDE) v t = A JD v = − 1 ∫ 2 (σx)2 v xx −(r−µζ)xv x +(r+µ)v−µ v(t, xy)f(y) dy (2) R + can be derived for the value of an option, where µ is the rate of jumps, the function f defines the distributions of jumps, and ζ is the mean jump amplitude.
282 S. Ikonen and J. Toivanen Merton used a Gaussian distribution for jumps in [Mer76]. Kou considered in [Kou02] a log-double-exponential distribution for jumps which leads a more flexible and tractable model. In this case, the density is { qα 2 y α2−1 , y < 1, f(y) = pα 1 y −α1−1 (3) , y ≥ 1, where p, q, α 1 > 1, and α 2 are positive constants such that p + q =1.The mean jump amplitude is ζ = pα1 α + qα2 1−1 α 2+1 − 1. We will employ this model in the numerical experiments. Also in this case one possible way to calibrate the model is to let the volatility σ be a function of time and asset value like in [AA00]. 2.3 Stochastic Volatility Models In practice, the volatility of the value of an asset is not a constant over time. Several models have been also developed for the behavior of the volatility. Among several stochastic volatility models probably the one developed by Heston in [Hes93] is the most popular. It assumes the volatility to be a meanreverting process. Under the assumption that the market price of risk is zero Heston’s model leads to the two-dimensional PDE v t = A SV v = − 1 2 yx2 v xx − ργyxv xy − 1 2 γ2 yv yy − rxv x − α(β − y)v y + rv, (4) where y is the variance, that is, the square of the volatility, β is the mean level of the variance, α is the rate of reversion on the mean level, and γ is the volatility of the variance. The correlation between the price of the underlying asset and its variance is ρ. 3 Linear Complementarity Problems The value of an option at the expiry time T is given by v(T,x)=g(x), (5) where the payoff function g depends on the type of the option. For example, for a put option with a strike price K it is g(x) =max{K − x, 0}. (6) The value v of an American option satisfies a linear complementarity problem (LCP) { (v t − Av) ≥ 0, v ≥ g, (7) (v t − Av)(v − g) =0,
Page 1 and 2:
Partial Differential Equations
Page 3 and 4:
Partial Differential Equations Mode
Page 5 and 6:
Dedicated to Olivier Pironneau
Page 7 and 8:
VIII Preface computers has been at
Page 9 and 10:
Contents List of Contributors .....
Page 11 and 12:
List of Contributors Yves Achdou UF
Page 13 and 14:
List of Contributors XV Claude Le B
Page 15 and 16:
Discontinuous Galerkin Methods Vive
Page 17 and 18:
Discontinuous Galerkin Methods 5 2
Page 19 and 20:
Discontinuous Galerkin Methods 7 g
Page 21 and 22:
Discontinuous Galerkin Methods 9 (
Page 23 and 24:
Page 25 and 26:
Page 27 and 28:
Discontinuous Galerkin Methods 15 W
Page 29 and 30:
Let a h and b h denote the bilinear
Page 31 and 32:
Discontinuous Galerkin Methods 19 t
Page 33 and 34:
Discontinuous Galerkin Methods 21 l
Page 35 and 36:
Table 1. Primal DG for transport Di
Page 37 and 38:
Discontinuous Galerkin Methods 25 [
Page 39 and 40:
Mixed Finite Element Methods on Pol
Page 41 and 42:
Mixed FE Methods on Polyhedral Mesh
Page 43 and 44:
Page 45 and 46:
Page 47 and 48:
4 Hybridization and Condensation Mi
Page 49 and 50:
Page 51 and 52:
is symmetric and positive definite,
Page 53 and 54:
with some coefficient α ∈ R wher
Page 55 and 56:
44 E.J. Dean and R. Glowinski so fa
Page 57 and 58:
46 E.J. Dean and R. Glowinski 2 A L
Page 59 and 60:
48 E.J. Dean and R. Glowinski S:T=
Page 61 and 62:
50 E.J. Dean and R. Glowinski minim
Page 63 and 64:
52 E.J. Dean and R. Glowinski 6 On
Page 65 and 66:
54 E.J. Dean and R. Glowinski Fig.
Page 67 and 68:
56 E.J. Dean and R. Glowinski and
Page 69 and 70:
58 E.J. Dean and R. Glowinski 7 Num
Page 71 and 72:
60 E.J. Dean and R. Glowinski Fig.
Page 73 and 74:
62 E.J. Dean and R. Glowinski Assum
Page 75 and 76:
Higher Order Time Stepping for Seco
Page 77 and 78:
u n+1 h Optimal Higher Order Time D
Page 79 and 80:
Optimal Higher Order Time Discretiz
Page 81 and 82:
Page 83 and 84:
Page 85 and 86:
Page 87 and 88:
Page 89 and 90:
Page 91 and 92:
Page 93 and 94:
Page 95 and 96:
Page 97 and 98:
Page 99 and 100:
Page 101 and 102:
Page 103 and 104:
96 I. Sazonov et al. To provide a p
Page 105 and 106:
98 I. Sazonov et al. In the first s
Page 107 and 108:
100 I. Sazonov et al. Fig. 1. An ex
Page 109 and 110:
102 I. Sazonov et al. H z 1 exact F
Page 111 and 112:
104 I. Sazonov et al. (a) (b) Fig.
Page 113 and 114:
106 I. Sazonov et al. Scattering Wi
Page 115 and 116:
108 I. Sazonov et al. 6.4 Scatterin
Page 117 and 118:
110 I. Sazonov et al. (a) (b) Fig.
Page 119 and 120:
112 I. Sazonov et al. [MHP96] K. Mo
Page 121 and 122:
114 R. Sanders and A.M. Tesdall I R
Page 123 and 124:
116 R. Sanders and A.M. Tesdall imp
Page 125 and 126:
118 R. Sanders and A.M. Tesdall alo
Page 127 and 128:
120 R. Sanders and A.M. Tesdall (a)
Page 129 and 130:
122 R. Sanders and A.M. Tesdall D C
Page 131 and 132:
124 R. Sanders and A.M. Tesdall 8.6
Page 133 and 134:
126 R. Sanders and A.M. Tesdall 0.3
Page 135 and 136:
128 R. Sanders and A.M. Tesdall [TR
Page 137 and 138:
132 S. Lapin et al. Ω R γ Ω 2 Γ
Page 139 and 140:
134 S. Lapin et al. ∫ ∂ 2 ∫
Page 141 and 142:
136 S. Lapin et al. Ω R γ Fig. 3.
Page 143 and 144:
138 S. Lapin et al. 4 Energy Inequa
Page 145 and 146:
140 S. Lapin et al. 5 Numerical Exp
Page 147 and 148:
142 S. Lapin et al. Fig. 6. Contour
Page 149 and 150:
144 S. Lapin et al. Fig. 9. Obstacl
Page 151 and 152:
Domain Decomposition and Electronic
Page 153 and 154:
Domain Decomposition Approach for C
Page 155 and 156:
Page 157 and 158:
Page 159 and 160:
Page 161 and 162:
Page 163 and 164:
Page 165 and 166:
Page 167 and 168:
Page 169 and 170:
Numerical Analysis of a Finite Elem
Page 171 and 172:
Page 173 and 174:
Page 175 and 176:
Page 177 and 178:
Page 179 and 180:
Page 181 and 182:
so that |u| 2 1,ω h ≤ 2 Numerica
Page 183 and 184:
Page 185 and 186:
Page 187 and 188:
Page 189 and 190:
188 A. Bonito et al. of the model a
Page 191 and 192:
190 A. Bonito et al. Fig. 1. The sp
Page 193 and 194:
192 A. Bonito et al. 3 1 16 4 1 1 1
Page 195 and 196:
194 A. Bonito et al. ∫ v n+1 h
Page 197 and 198:
196 A. Bonito et al. −pn +2µD(v)
Page 199 and 200:
198 A. Bonito et al. each of its pa
Page 201 and 202:
200 A. Bonito et al. The normal vec
Page 203 and 204:
202 A. Bonito et al. with initial c
Page 205 and 206:
204 A. Bonito et al. Fig. 8. Jet bu
Page 207 and 208:
206 A. Bonito et al. References [AM
Page 209 and 210:
208 A. Bonito et al. [Set96] J. A.
Page 211 and 212:
210 J. Hao et al. due to shear flow
Page 213 and 214:
212 J. Hao et al. The backward reac
Page 215 and 216:
214 J. Hao et al. where u and p den
Page 217 and 218:
216 J. Hao et al. * * * * * * * * *
Page 219 and 220:
218 J. Hao et al. and solve for V n
Page 221 and 222:
220 J. Hao et al. Table 2. The calc
Page 223 and 224:
222 J. Hao et al. References [ASS80
Page 225 and 226: Computing the Eigenvalues of the La
Page 227 and 228: Eigenvalues of the Laplace-Beltrami
Page 233 and 234: A Fixed Domain Approach in Shape Op
Page 235 and 236: Shape Optimization Problems with Ne
Page 243 and 244: Reduced-Order Modelling of Dispersi
Page 255 and 256: Calibration of Lévy Processes with
Page 263 and 264: We have proved Calibration of Lévy
Page 271 and 272: Note that p ∗ satisfies Calibrati
Page 275: 280 S. Ikonen and J. Toivanen the p
Page 279 and 280: 284 S. Ikonen and J. Toivanen For H
Page 281 and 282: 286 S. Ikonen and J. Toivanen and {
Page 283 and 284: 288 S. Ikonen and J. Toivanen 1.6 1
Page 285 and 286: 290 S. Ikonen and J. Toivanen 8 Con
Page 287: 292 S. Ikonen and J. Toivanen [Mer7
show all

Partial Differential Equations - Modelling and ... - ResearchGate

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?