Actual source code: blackscholes.c

```  1: /*
2:     American Put Options Pricing using the Black-Scholes Equation

4:    Background (European Options):
5:      The standard European option is a contract where the holder has the right
6:      to either buy (call option) or sell (put option) an underlying asset at
7:      a designated future time and price.

9:      The classic Black-Scholes model begins with an assumption that the
10:      price of the underlying asset behaves as a lognormal random walk.
11:      Using this assumption and a no-arbitrage argument, the following
12:      linear parabolic partial differential equation for the value of the
13:      option results:

15:        dV/dt + 0.5(sigma**2)(S**alpha)(d2V/dS2) + (r - D)S(dV/dS) - rV = 0.

17:      Here, sigma is the volatility of the underling asset, alpha is a
18:      measure of elasticity (typically two), D measures the dividend payments
19:      on the underling asset, and r is the interest rate.

21:      To completely specify the problem, we need to impose some boundary
22:      conditions.  These are as follows:

24:        V(S, T) = max(E - S, 0)
25:        V(0, t) = E for all 0 <= t <= T
26:        V(s, t) = 0 for all 0 <= t <= T and s->infinity

28:      where T is the exercise time and E the strike price (price paid
29:      for the contract).

31:      An explicit formula for the value of an European option can be
32:      found.  See the references for examples.

34:    Background (American Options):
35:      The American option is similar to its European counterpart.  The
36:      difference is that the holder of the American option can exercise
37:      their right to buy or sell the asset at any time prior to the
38:      expiration.  This additional ability introduce a free boundary into
39:      the Black-Scholes equation which can be modeled as a linear
40:      complementarity problem.

42:        0 <= -(dV/dt + 0.5(sigma**2)(S**alpha)(d2V/dS2) + (r - D)S(dV/dS) - rV)
43:          complements
44:        V(S,T) >= max(E-S,0)

46:      where the variables are the same as before and we have the same boundary
47:      conditions.

49:      There is not explicit formula for calculating the value of an American
50:      option.  Therefore, we discretize the above problem and solve the
51:      resulting linear complementarity problem.

53:      We will use backward differences for the time variables and central
54:      differences for the space variables.  Crank-Nicholson averaging will
55:      also be used in the discretization.  The algorithm used by the code
56:      solves for V(S,t) for a fixed t and then uses this value in the
57:      calculation of V(S,t - dt).  The method stops when V(S,0) has been
58:      found.

60:    References:
61: + * - Huang and Pang, "Options Pricing and Linear Complementarity,"
62:        Journal of Computational Finance, volume 2, number 3, 1998.
63: - * - Wilmott, "Derivatives: The Theory and Practice of Financial Engineering,"
64:        John Wiley and Sons, New York, 1998.
65: */

67: /*
68:   Include "petsctao.h" so we can use TAO solvers.
69:   Include "petscdmda.h" so that we can use distributed meshes (DMs) for managing
70:   the parallel mesh.
71: */

73: #include <petscdmda.h>
74: #include <petsctao.h>

76: static char help[] = "This example demonstrates use of the TAO package to\n\
77: solve a linear complementarity problem for pricing American put options.\n\
78: The code uses backward differences in time and central differences in\n\
79: space.  The command line options are:\n\
80:   -rate <r>, where <r> = interest rate\n\
81:   -sigma <s>, where <s> = volatility of the underlying\n\
82:   -alpha <a>, where <a> = elasticity of the underlying\n\
83:   -delta <d>, where <d> = dividend rate\n\
84:   -strike <e>, where <e> = strike price\n\
85:   -expiry <t>, where <t> = the expiration date\n\
86:   -mt <tg>, where <tg> = number of grid points in time\n\
87:   -ms <sg>, where <sg> = number of grid points in space\n\
88:   -es <se>, where <se> = ending point of the space discretization\n\n";

90: /*
91:   User-defined application context - contains data needed by the
92:   application-provided call-back routines, FormFunction(), and FormJacobian().
93: */

95: typedef struct {
96:   PetscReal *Vt1; /* Value of the option at time T + dt */
97:   PetscReal *c;   /* Constant -- (r - D)S */
98:   PetscReal *d;   /* Constant -- -0.5(sigma**2)(S**alpha) */

100:   PetscReal rate;                /* Interest rate */
101:   PetscReal sigma, alpha, delta; /* Underlying asset properties */
102:   PetscReal strike, expiry;      /* Option contract properties */

104:   PetscReal es;     /* Finite value used for maximum asset value */
105:   PetscReal ds, dt; /* Discretization properties */
106:   PetscInt  ms, mt; /* Number of elements */

108:   DM dm;
109: } AppCtx;

111: /* -------- User-defined Routines --------- */

113: PetscErrorCode FormConstraints(Tao, Vec, Vec, void *);
114: PetscErrorCode FormJacobian(Tao, Vec, Mat, Mat, void *);
115: PetscErrorCode ComputeVariableBounds(Tao, Vec, Vec, void *);

117: int main(int argc, char **argv)
118: {
119:   Vec        x;    /* solution vector */
120:   Vec        c;    /* Constraints function vector */
121:   Mat        J;    /* jacobian matrix */
122:   PetscBool  flg;  /* A return variable when checking for user options */
123:   Tao        tao;  /* Tao solver context */
124:   AppCtx     user; /* user-defined work context */
125:   PetscInt   i, j;
126:   PetscInt   xs, xm, gxs, gxm;
127:   PetscReal  sval = 0;
128:   PetscReal *x_array;
129:   Vec        localX;

131:   /* Initialize PETSc, TAO */
132:   PetscFunctionBeginUser;
133:   PetscCall(PetscInitialize(&argc, &argv, (char *)0, help));

135:   /*
136:      Initialize the user-defined application context with reasonable
137:      values for the American option to price
138:   */
139:   user.rate   = 0.04;
140:   user.sigma  = 0.40;
141:   user.alpha  = 2.00;
142:   user.delta  = 0.01;
143:   user.strike = 10.0;
144:   user.expiry = 1.0;
145:   user.mt     = 10;
146:   user.ms     = 150;
147:   user.es     = 100.0;

149:   /* Read in alternative values for the American option to price */
150:   PetscCall(PetscOptionsGetReal(NULL, NULL, "-alpha", &user.alpha, &flg));
151:   PetscCall(PetscOptionsGetReal(NULL, NULL, "-delta", &user.delta, &flg));
152:   PetscCall(PetscOptionsGetReal(NULL, NULL, "-es", &user.es, &flg));
153:   PetscCall(PetscOptionsGetReal(NULL, NULL, "-expiry", &user.expiry, &flg));
154:   PetscCall(PetscOptionsGetInt(NULL, NULL, "-ms", &user.ms, &flg));
155:   PetscCall(PetscOptionsGetInt(NULL, NULL, "-mt", &user.mt, &flg));
156:   PetscCall(PetscOptionsGetReal(NULL, NULL, "-rate", &user.rate, &flg));
157:   PetscCall(PetscOptionsGetReal(NULL, NULL, "-sigma", &user.sigma, &flg));
158:   PetscCall(PetscOptionsGetReal(NULL, NULL, "-strike", &user.strike, &flg));

160:   /* Check that the options set are allowable (needs to be done) */

162:   user.ms++;
163:   PetscCall(DMDACreate1d(PETSC_COMM_WORLD, DM_BOUNDARY_NONE, user.ms, 1, 1, NULL, &user.dm));
164:   PetscCall(DMSetFromOptions(user.dm));
165:   PetscCall(DMSetUp(user.dm));
166:   /* Create appropriate vectors and matrices */

168:   PetscCall(DMDAGetCorners(user.dm, &xs, NULL, NULL, &xm, NULL, NULL));
169:   PetscCall(DMDAGetGhostCorners(user.dm, &gxs, NULL, NULL, &gxm, NULL, NULL));

171:   PetscCall(DMCreateGlobalVector(user.dm, &x));
172:   /*
173:      Finish filling in the user-defined context with the values for
174:      dS, dt, and allocating space for the constants
175:   */
176:   user.ds = user.es / (user.ms - 1);
177:   user.dt = user.expiry / user.mt;

179:   PetscCall(PetscMalloc1(gxm, &user.Vt1));
180:   PetscCall(PetscMalloc1(gxm, &user.c));
181:   PetscCall(PetscMalloc1(gxm, &user.d));

183:   /*
184:      Calculate the values for the constant.  Vt1 begins with the ending
185:      boundary condition.
186:   */
187:   for (i = 0; i < gxm; i++) {
188:     sval        = (gxs + i) * user.ds;
189:     user.Vt1[i] = PetscMax(user.strike - sval, 0);
190:     user.c[i]   = (user.delta - user.rate) * sval;
191:     user.d[i]   = -0.5 * user.sigma * user.sigma * PetscPowReal(sval, user.alpha);
192:   }
193:   if (gxs + gxm == user.ms) user.Vt1[gxm - 1] = 0;
194:   PetscCall(VecDuplicate(x, &c));

196:   /*
197:      Allocate the matrix used by TAO for the Jacobian.  Each row of
198:      the Jacobian matrix will have at most three elements.
199:   */
200:   PetscCall(DMCreateMatrix(user.dm, &J));

202:   /* The TAO code begins here */

204:   /* Create TAO solver and set desired solution method  */
205:   PetscCall(TaoCreate(PETSC_COMM_WORLD, &tao));
206:   PetscCall(TaoSetType(tao, TAOSSILS));

208:   /* Set routines for constraints function and Jacobian evaluation */
209:   PetscCall(TaoSetConstraintsRoutine(tao, c, FormConstraints, (void *)&user));
210:   PetscCall(TaoSetJacobianRoutine(tao, J, J, FormJacobian, (void *)&user));

212:   /* Set the variable bounds */
213:   PetscCall(TaoSetVariableBoundsRoutine(tao, ComputeVariableBounds, (void *)&user));

215:   /* Set initial solution guess */
216:   PetscCall(VecGetArray(x, &x_array));
217:   for (i = 0; i < xm; i++) x_array[i] = user.Vt1[i - gxs + xs];
218:   PetscCall(VecRestoreArray(x, &x_array));
219:   /* Set data structure */
220:   PetscCall(TaoSetSolution(tao, x));

222:   /* Set routines for function and Jacobian evaluation */
223:   PetscCall(TaoSetFromOptions(tao));

225:   /* Iteratively solve the linear complementarity problems  */
226:   for (i = 1; i < user.mt; i++) {
227:     /* Solve the current version */
228:     PetscCall(TaoSolve(tao));

230:     /* Update Vt1 with the solution */
231:     PetscCall(DMGetLocalVector(user.dm, &localX));
232:     PetscCall(DMGlobalToLocalBegin(user.dm, x, INSERT_VALUES, localX));
233:     PetscCall(DMGlobalToLocalEnd(user.dm, x, INSERT_VALUES, localX));
234:     PetscCall(VecGetArray(localX, &x_array));
235:     for (j = 0; j < gxm; j++) user.Vt1[j] = x_array[j];
236:     PetscCall(VecRestoreArray(x, &x_array));
237:     PetscCall(DMRestoreLocalVector(user.dm, &localX));
238:   }

240:   /* Free TAO data structures */
241:   PetscCall(TaoDestroy(&tao));

243:   /* Free PETSc data structures */
244:   PetscCall(VecDestroy(&x));
245:   PetscCall(VecDestroy(&c));
246:   PetscCall(MatDestroy(&J));
247:   PetscCall(DMDestroy(&user.dm));
248:   /* Free user-defined workspace */
249:   PetscCall(PetscFree(user.Vt1));
250:   PetscCall(PetscFree(user.c));
251:   PetscCall(PetscFree(user.d));

253:   PetscCall(PetscFinalize());
254:   return 0;
255: }

257: /* -------------------------------------------------------------------- */
258: PetscErrorCode ComputeVariableBounds(Tao tao, Vec xl, Vec xu, void *ctx)
259: {
260:   AppCtx   *user = (AppCtx *)ctx;
261:   PetscInt  i;
262:   PetscInt  xs, xm;
263:   PetscInt  ms   = user->ms;
264:   PetscReal sval = 0.0, *xl_array, ub = PETSC_INFINITY;

266:   PetscFunctionBeginUser;
267:   /* Set the variable bounds */
268:   PetscCall(VecSet(xu, ub));
269:   PetscCall(DMDAGetCorners(user->dm, &xs, NULL, NULL, &xm, NULL, NULL));

271:   PetscCall(VecGetArray(xl, &xl_array));
272:   for (i = 0; i < xm; i++) {
273:     sval        = (xs + i) * user->ds;
274:     xl_array[i] = PetscMax(user->strike - sval, 0);
275:   }
276:   PetscCall(VecRestoreArray(xl, &xl_array));

278:   if (xs == 0) {
279:     PetscCall(VecGetArray(xu, &xl_array));
280:     xl_array[0] = PetscMax(user->strike, 0);
281:     PetscCall(VecRestoreArray(xu, &xl_array));
282:   }
283:   if (xs + xm == ms) {
284:     PetscCall(VecGetArray(xu, &xl_array));
285:     xl_array[xm - 1] = 0;
286:     PetscCall(VecRestoreArray(xu, &xl_array));
287:   }
288:   PetscFunctionReturn(PETSC_SUCCESS);
289: }
290: /* -------------------------------------------------------------------- */

292: /*
293:     FormFunction - Evaluates gradient of f.

295:     Input Parameters:
296: .   tao  - the Tao context
297: .   X    - input vector
298: .   ptr  - optional user-defined context, as set by TaoAppSetConstraintRoutine()

300:     Output Parameters:
301: .   F - vector containing the newly evaluated gradient
302: */
303: PetscErrorCode FormConstraints(Tao tao, Vec X, Vec F, void *ptr)
304: {
305:   AppCtx    *user = (AppCtx *)ptr;
306:   PetscReal *x, *f;
307:   PetscReal *Vt1 = user->Vt1, *c = user->c, *d = user->d;
308:   PetscReal  rate = user->rate;
309:   PetscReal  dt = user->dt, ds = user->ds;
310:   PetscInt   ms = user->ms;
311:   PetscInt   i, xs, xm, gxs, gxm;
312:   Vec        localX, localF;
313:   PetscReal  zero = 0.0;

315:   PetscFunctionBeginUser;
316:   PetscCall(DMGetLocalVector(user->dm, &localX));
317:   PetscCall(DMGetLocalVector(user->dm, &localF));
318:   PetscCall(DMGlobalToLocalBegin(user->dm, X, INSERT_VALUES, localX));
319:   PetscCall(DMGlobalToLocalEnd(user->dm, X, INSERT_VALUES, localX));
320:   PetscCall(DMDAGetCorners(user->dm, &xs, NULL, NULL, &xm, NULL, NULL));
321:   PetscCall(DMDAGetGhostCorners(user->dm, &gxs, NULL, NULL, &gxm, NULL, NULL));
322:   PetscCall(VecSet(F, zero));
323:   /*
324:      The problem size is smaller than the discretization because of the
325:      two fixed elements (V(0,T) = E and V(Send,T) = 0.
326:   */

328:   /* Get pointers to the vector data */
329:   PetscCall(VecGetArray(localX, &x));
330:   PetscCall(VecGetArray(localF, &f));

332:   /* Left Boundary */
333:   if (gxs == 0) {
334:     f[0] = x[0] - user->strike;
335:   } else {
336:     f[0] = 0;
337:   }

339:   /* All points in the interior */
340:   /*  for (i=gxs+1;i<gxm-1;i++) { */
341:   for (i = 1; i < gxm - 1; i++) {
342:     f[i] = (1.0 / dt + rate) * x[i] - Vt1[i] / dt + (c[i] / (4 * ds)) * (x[i + 1] - x[i - 1] + Vt1[i + 1] - Vt1[i - 1]) + (d[i] / (2 * ds * ds)) * (x[i + 1] - 2 * x[i] + x[i - 1] + Vt1[i + 1] - 2 * Vt1[i] + Vt1[i - 1]);
343:   }

345:   /* Right boundary */
346:   if (gxs + gxm == ms) {
347:     f[xm - 1] = x[gxm - 1];
348:   } else {
349:     f[xm - 1] = 0;
350:   }

352:   /* Restore vectors */
353:   PetscCall(VecRestoreArray(localX, &x));
354:   PetscCall(VecRestoreArray(localF, &f));
355:   PetscCall(DMLocalToGlobalBegin(user->dm, localF, INSERT_VALUES, F));
356:   PetscCall(DMLocalToGlobalEnd(user->dm, localF, INSERT_VALUES, F));
357:   PetscCall(DMRestoreLocalVector(user->dm, &localX));
358:   PetscCall(DMRestoreLocalVector(user->dm, &localF));
359:   PetscCall(PetscLogFlops(24.0 * (gxm - 2)));
360:   /*
361:   info=VecView(F,PETSC_VIEWER_STDOUT_WORLD);
362:   */
363:   PetscFunctionReturn(PETSC_SUCCESS);
364: }

366: /* ------------------------------------------------------------------- */
367: /*
368:    FormJacobian - Evaluates Jacobian matrix.

370:    Input Parameters:
371: .  tao  - the Tao context
372: .  X    - input vector
373: .  ptr  - optional user-defined context, as set by TaoSetJacobian()

375:    Output Parameters:
376: .  J    - Jacobian matrix
377: */
378: PetscErrorCode FormJacobian(Tao tao, Vec X, Mat J, Mat tJPre, void *ptr)
379: {
380:   AppCtx    *user = (AppCtx *)ptr;
381:   PetscReal *c = user->c, *d = user->d;
382:   PetscReal  rate = user->rate;
383:   PetscReal  dt = user->dt, ds = user->ds;
384:   PetscInt   ms = user->ms;
385:   PetscReal  val[3];
386:   PetscInt   col[3];
387:   PetscInt   i;
388:   PetscInt   gxs, gxm;
389:   PetscBool  assembled;

391:   PetscFunctionBeginUser;
392:   /* Set various matrix options */
393:   PetscCall(MatSetOption(J, MAT_IGNORE_OFF_PROC_ENTRIES, PETSC_TRUE));
394:   PetscCall(MatAssembled(J, &assembled));
395:   if (assembled) PetscCall(MatZeroEntries(J));

397:   PetscCall(DMDAGetGhostCorners(user->dm, &gxs, NULL, NULL, &gxm, NULL, NULL));

399:   if (gxs == 0) {
400:     i      = 0;
401:     col[0] = 0;
402:     val[0] = 1.0;
403:     PetscCall(MatSetValues(J, 1, &i, 1, col, val, INSERT_VALUES));
404:   }
405:   for (i = 1; i < gxm - 1; i++) {
406:     col[0] = gxs + i - 1;
407:     col[1] = gxs + i;
408:     col[2] = gxs + i + 1;
409:     val[0] = -c[i] / (4 * ds) + d[i] / (2 * ds * ds);
410:     val[1] = 1.0 / dt + rate - d[i] / (ds * ds);
411:     val[2] = c[i] / (4 * ds) + d[i] / (2 * ds * ds);
412:     PetscCall(MatSetValues(J, 1, &col[1], 3, col, val, INSERT_VALUES));
413:   }
414:   if (gxs + gxm == ms) {
415:     i      = ms - 1;
416:     col[0] = i;
417:     val[0] = 1.0;
418:     PetscCall(MatSetValues(J, 1, &i, 1, col, val, INSERT_VALUES));
419:   }

421:   /* Assemble the Jacobian matrix */
422:   PetscCall(MatAssemblyBegin(J, MAT_FINAL_ASSEMBLY));
423:   PetscCall(MatAssemblyEnd(J, MAT_FINAL_ASSEMBLY));
424:   PetscCall(PetscLogFlops(18.0 * (gxm) + 5));
425:   PetscFunctionReturn(PETSC_SUCCESS);
426: }

428: /*TEST

430:    build:
431:       requires: !complex

433:    test:
434:       args: -tao_monitor -tao_type ssils -tao_gttol 1.e-5
435:       requires: !single

437:    test:
438:       suffix: 2
439:       args: -tao_monitor -tao_type ssfls -tao_max_it 10 -tao_gttol 1.e-5
440:       requires: !single

442:    test:
443:       suffix: 3
444:       args: -tao_monitor -tao_type asils -tao_subset_type subvec -tao_gttol 1.e-5
445:       requires: !single

447:    test:
448:       suffix: 4
449:       args: -tao_monitor -tao_type asils -tao_subset_type mask -tao_gttol 1.e-5
450:       requires: !single

452:    test:
453:       suffix: 5
454:       args: -tao_monitor -tao_type asils -tao_subset_type matrixfree -pc_type jacobi -tao_max_it 6 -tao_gttol 1.e-5
455:       requires: !single

457:    test:
458:       suffix: 6
459:       args: -tao_monitor -tao_type asfls -tao_subset_type subvec -tao_max_it 10 -tao_gttol 1.e-5
460:       requires: !single

462:    test:
463:       suffix: 7
464:       args: -tao_monitor -tao_type asfls -tao_subset_type mask -tao_max_it 10 -tao_gttol 1.e-5
465:       requires: !single

467: TEST*/
```