THE TIME-DISCRETE METHOD OF LINES FOR OPTIONS AND BONDS APDEApproach % " 24 BSV ViSfVs^i + pbi<rbs 1S 2V Sls a + + (T 9I)5'IVS 1 + (r q2)s2vs 3 rv Vt = 0 V(Si, %, 0) = max{0, Kmin(aiSi, 02%)} for «1,03 > 0 i 14 / // / ' 14 31 Gunter H. Meyer Georgia Institute of Technology, USA World Scientific NEW JERSEY. LONDON S1NGAP0RE. BEIJING SHANGHAI HONG KONG TAIPEI CHENNAI
Contents Preface Acknowledgment v xv 1. Comments on the Pricing Equations in Finance 1 1.1 Solutions and their properties 2 Example 1.1 Positivity of option prices and the Black Scholes formulas 5 Example 1.2 The early exercise boundary for piain American puts and calls 8 Example 1.3 Exercise boundaries for options with jump diffusion 10 Example 1.4 The early exercise premium for an American put 12 Example 1.5 The early exercise premium for an American call 14 Example 1.6 Strike price convexity 15 Example 1.7 Put-call parity 16 Example 1.8 Put-call symmetry for a CEV and Heston model 19 Example 1.9 Equations with an uncertain parameter... 23 1.2 Boundary conditions for the pricing equations 27 1.2.1 The Fichera function for degenerate equations.. 29 Example 1.11 Boundary conditions for the heat equation. 33 Example 1.12 Boundary condition for the CEV Black Scholes equation at S = 0 34 xi
xii TAe T:me-ßwcre(e Met/iod o/.linea /or Optiona and Bonda ;4 f AppmacA Example 1.13 Boundary conditions for a discount bond at Example 1.14 Boundary conditions for the Black Scholes equation on two assets 37 Example 1.15 Boundary conditions for the Black Scholes equation with stochastic volatility v at S = 0 and v 0 38 Example 1.16 Boundary conditions for an Asian option.. 40 1.2.2 The boundary condition at "infinity" 42 Example 1.17 CEV puts and calls 42 Example 1.18 Puts and calls with stochastic volatility.. 44 Example 1.19 The European max option 45 Example 1.20 An Asian average price call 47 1.2.3 The Venttsel boundary conditions on "far but finite" boundaries 48 Example 1.21 A defaultable bond 50 Example 1.22 The Black Scholes equation with stochastic volatility 52 1.2.4 Free boundaries 54 2. The Method of Lines (MOL) for the Diffusion Equation 57 2.1 The method of lines with continuous time (the vertical MOL) 58 2.2 The method of lines with continuous x (the horizontal MOL) 61 Appendix 2.2 Stability of the time discrete three-level scheine for the heat equation 63 2.3 The method of lines with continuous x for multidimensional problems 64 Appendix 2.3 Convergence of the line Gauss Seidel iteration for a model problem 69 2.4 Free boundaries and the MOL in two dimensions 71 3. The Riccati Transformation Method for Linear Two Point Boundary Value Problems 75 3.1 The Riccati transformation on a fixed interval 76 3.2 The Riccati transformation for a free boundary problem. 79 3.3 The numerical Solution of the sweep equations 81
Contents xiii Example 3.1 A real option for interest rate sensitive Investments 87 Appendix 3.3 Connection between the Riccati transformation, Gaussian elimination and the Brennan- Schwartz method 88 4. European Options 93 Example 4.1 A piain European call 96 Example 4.2 A binary cash or nothing European call... 101 Example 4.3 A binary call with low volatility 107 Example 4.4 The Black Scholes Barenblatt equation for a CEV process 111 5. American Puts and Calls 117 Example 5.1 An American put 117 Example 5.2 An American put with sub-optimal early exercise 123 Example 5.3 A put on an asset with a fixed dividend... 125 Example 5.4 An American lookback call 129 Example 5.5 An American strangle for power options... 135 Example 5.6 Jump diffusion with uncertain volatility... 141 6. Bonds and Options for One-Factor Interest Rate Models 153 Example 6.1 The Ho Lee model 158 Example 6.2 A one-factor CEV model 161 Example 6.3 An implied volatility for a call on a discount bond 165 Example 6.4 An American put on a discount bond... 171 7. Two-Dimensional Diffusion Problems in Finance 181 7.1 Front tracking in Cartesian coordinates 185 Example 7.1 An American call on an asset with stochastic volatility. 185 Example 7.2 A European put on a combination of two assets 190 Example 7.3 A perpetual American put - MOL with overrelaxation 197 Example 7.4 An American call, its deltas and a vega... 199
xiv The Time-Discrete Method of Lines for Options and Bonds A PDE Approach Example 7.5 American spread and exchange options... 207 Example 7.6 An American call option on the maximum of two assets 212 7.2 American calls and puts in polar coordinates 220 Example 7.7 The basket call in polar coordinates 221 Example 7.8 A call on the minimum of two assets 223 Example 7.9 A put on the minimum of two assets 229 Example 7.10 A perpetual put on the minimum of two assets with uncertain correlation 237 Example 7.11 Implied correlation for a put on the sum of two assets 239 7.3 A three-dimensional problem 245 Example 7.12 An American call with Heston volatility and a stochastic interest rate 246 Bibliography 261 fndez 265 About the Author 269