Actual source code: blackscholes.c

petsc-3.9.0 2018-04-07
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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 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 excercise
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[] =
77: "This example demonstrates use of the TAO package to\n\
78: solve a linear complementarity problem for pricing American put options.\n\
79: The code uses backward differences in time and central differences in\n\
80: space.  The command line options are:\n\
81:   -rate <r>, where <r> = interest rate\n\
82:   -sigma <s>, where <s> = volatility of the underlying\n\
83:   -alpha <a>, where <a> = elasticity of the underlying\n\
84:   -delta <d>, where <d> = dividend rate\n\
85:   -strike <e>, where <e> = strike price\n\
86:   -expiry <t>, where <t> = the expiration date\n\
87:   -mt <tg>, where <tg> = number of grid points in time\n\
88:   -ms <sg>, where <sg> = number of grid points in space\n\
89:   -es <se>, where <se> = ending point of the space discretization\n\n";

91: /*T
92:    Concepts: TAO^Solving a complementarity problem
93:    Routines: TaoCreate(); TaoDestroy();
94:    Routines: TaoSetJacobianRoutine(); TaoAppSetConstraintRoutine();
95:    Routines: TaoSetFromOptions();
96:    Routines: TaoSolveApplication();
97:    Routines: TaoSetVariableBoundsRoutine(); TaoSetInitialSolutionVec();
98:    Processors: 1
99: T*/

104: /*
105:   User-defined application context - contains data needed by the
106:   application-provided call-back routines, FormFunction(), and FormJacobian().
107: */

109: typedef struct {
110:   PetscReal *Vt1;                /* Value of the option at time T + dt */
111:   PetscReal *c;                  /* Constant -- (r - D)S */
112:   PetscReal *d;                  /* Constant -- -0.5(sigma**2)(S**alpha) */

114:   PetscReal rate;                /* Interest rate */
115:   PetscReal sigma, alpha, delta; /* Underlying asset properties */
116:   PetscReal strike, expiry;      /* Option contract properties */

118:   PetscReal es;                  /* Finite value used for maximum asset value */
119:   PetscReal ds, dt;              /* Discretization properties */
120:   PetscInt  ms, mt;               /* Number of elements */

122:   DM        dm;
123: } AppCtx;

125: /* -------- User-defined Routines --------- */

127: PetscErrorCode FormConstraints(Tao, Vec, Vec, void *);
128: PetscErrorCode FormJacobian(Tao, Vec, Mat, Mat, void *);
129: PetscErrorCode ComputeVariableBounds(Tao, Vec, Vec, void*);

131: int main(int argc, char **argv)
132: {
134:   Vec            x;             /* solution vector */
135:   Vec            c;             /* Constraints function vector */
136:   Mat            J;                  /* jacobian matrix */
137:   PetscBool      flg;         /* A return variable when checking for user options */
138:   Tao            tao;          /* Tao solver context */
139:   AppCtx         user;            /* user-defined work context */
140:   PetscInt       i, j;
141:   PetscInt       xs,xm,gxs,gxm;
142:   PetscReal      sval = 0;
143:   PetscReal      *x_array;
144:   Vec            localX;

146:   /* Initialize PETSc, TAO */
147:   PetscInitialize(&argc, &argv, (char *)0, help);

149:   /*
150:      Initialize the user-defined application context with reasonable
151:      values for the American option to price
152:   */
153:   user.rate = 0.04;
154:   user.sigma = 0.40;
155:   user.alpha = 2.00;
156:   user.delta = 0.01;
157:   user.strike = 10.0;
158:   user.expiry = 1.0;
159:   user.mt = 10;
160:   user.ms = 150;
161:   user.es = 100.0;

163:   /* Read in alternative values for the American option to price */
164:   PetscOptionsGetReal(NULL,NULL, "-alpha", &user.alpha, &flg);
165:   PetscOptionsGetReal(NULL,NULL, "-delta", &user.delta, &flg);
166:   PetscOptionsGetReal(NULL,NULL, "-es", &user.es, &flg);
167:   PetscOptionsGetReal(NULL,NULL, "-expiry", &user.expiry, &flg);
168:   PetscOptionsGetInt(NULL,NULL, "-ms", &user.ms, &flg);
169:   PetscOptionsGetInt(NULL,NULL, "-mt", &user.mt, &flg);
170:   PetscOptionsGetReal(NULL,NULL, "-rate", &user.rate, &flg);
171:   PetscOptionsGetReal(NULL,NULL, "-sigma", &user.sigma, &flg);
172:   PetscOptionsGetReal(NULL,NULL, "-strike", &user.strike, &flg);

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

176:   user.ms++;
177:   DMDACreate1d(PETSC_COMM_WORLD,DM_BOUNDARY_NONE,user.ms,1,1,NULL,&user.dm);
178:   DMSetFromOptions(user.dm);
179:   DMSetUp(user.dm);
180:   /* Create appropriate vectors and matrices */

182:   DMDAGetCorners(user.dm,&xs,NULL,NULL,&xm,NULL,NULL);
183:   DMDAGetGhostCorners(user.dm,&gxs,NULL,NULL,&gxm,NULL,NULL);

185:   DMCreateGlobalVector(user.dm,&x);
186:   /*
187:      Finish filling in the user-defined context with the values for
188:      dS, dt, and allocating space for the constants
189:   */
190:   user.ds = user.es / (user.ms-1);
191:   user.dt = user.expiry / user.mt;

193:   PetscMalloc1(gxm,&(user.Vt1));
194:   PetscMalloc1(gxm,&(user.c));
195:   PetscMalloc1(gxm,&(user.d));

197:   /*
198:      Calculate the values for the constant.  Vt1 begins with the ending
199:      boundary condition.
200:   */
201:   for (i=0; i<gxm; i++) {
202:     sval = (gxs+i)*user.ds;
203:     user.Vt1[i] = PetscMax(user.strike - sval, 0);
204:     user.c[i] = (user.delta - user.rate)*sval;
205:     user.d[i] = -0.5*user.sigma*user.sigma*PetscPowReal(sval, user.alpha);
206:   }
207:   if (gxs+gxm==user.ms){
208:     user.Vt1[gxm-1] = 0;
209:   }
210:   VecDuplicate(x, &c);

212:   /*
213:      Allocate the matrix used by TAO for the Jacobian.  Each row of
214:      the Jacobian matrix will have at most three elements.
215:   */
216:   DMCreateMatrix(user.dm,&J);

218:   /* The TAO code begins here */

220:   /* Create TAO solver and set desired solution method  */
221:   TaoCreate(PETSC_COMM_WORLD, &tao);
222:   TaoSetType(tao,TAOSSILS);

224:   /* Set routines for constraints function and Jacobian evaluation */
225:   TaoSetConstraintsRoutine(tao, c, FormConstraints, (void *)&user);
226:   TaoSetJacobianRoutine(tao, J, J, FormJacobian, (void *)&user);

228:   /* Set the variable bounds */
229:   TaoSetVariableBoundsRoutine(tao,ComputeVariableBounds,(void*)&user);

231:   /* Set initial solution guess */
232:   VecGetArray(x,&x_array);
233:   for (i=0; i< xm; i++)
234:     x_array[i] = user.Vt1[i-gxs+xs];
235:   VecRestoreArray(x,&x_array);
236:   /* Set data structure */
237:   TaoSetInitialVector(tao, x);

239:   /* Set routines for function and Jacobian evaluation */
240:   TaoSetFromOptions(tao);

242:   /* Iteratively solve the linear complementarity problems  */
243:   for (i = 1; i < user.mt; i++) {

245:     /* Solve the current version */
246:     TaoSolve(tao);

248:     /* Update Vt1 with the solution */
249:     DMGetLocalVector(user.dm,&localX);
250:     DMGlobalToLocalBegin(user.dm,x,INSERT_VALUES,localX);
251:     DMGlobalToLocalEnd(user.dm,x,INSERT_VALUES,localX);
252:     VecGetArray(localX,&x_array);
253:     for (j = 0; j < gxm; j++) {
254:       user.Vt1[j] = x_array[j];
255:     }
256:     VecRestoreArray(x,&x_array);
257:     DMRestoreLocalVector(user.dm,&localX);
258:   }

260:   /* Free TAO data structures */
261:   TaoDestroy(&tao);

263:   /* Free PETSc data structures */
264:   VecDestroy(&x);
265:   VecDestroy(&c);
266:   MatDestroy(&J);
267:   DMDestroy(&user.dm);
268:   /* Free user-defined workspace */
269:   PetscFree(user.Vt1);
270:   PetscFree(user.c);
271:   PetscFree(user.d);

273:   PetscFinalize();
274:   return ierr;
275: }

277: /* -------------------------------------------------------------------- */
278: PetscErrorCode ComputeVariableBounds(Tao tao, Vec xl, Vec xu, void*ctx)
279: {
280:   AppCtx         *user = (AppCtx *) ctx;
282:   PetscInt       i;
283:   PetscInt       xs,xm;
284:   PetscInt       ms = user->ms;
285:   PetscReal      sval=0.0,*xl_array,ub= PETSC_INFINITY;

287:   /* Set the variable bounds */
288:   VecSet(xu, ub);
289:   DMDAGetCorners(user->dm,&xs,NULL,NULL,&xm,NULL,NULL);

291:   VecGetArray(xl,&xl_array);
292:   for (i = 0; i < xm; i++){
293:     sval = (xs+i)*user->ds;
294:     xl_array[i] = PetscMax(user->strike - sval, 0);
295:   }
296:   VecRestoreArray(xl,&xl_array);

298:   if (xs==0){
299:     VecGetArray(xu,&xl_array);
300:     xl_array[0] = PetscMax(user->strike, 0);
301:     VecRestoreArray(xu,&xl_array);
302:   }
303:   if (xs+xm==ms){
304:     VecGetArray(xu,&xl_array);
305:     xl_array[xm-1] = 0;
306:     VecRestoreArray(xu,&xl_array);
307:   }

309:   return 0;
310: }
311: /* -------------------------------------------------------------------- */

313: /*
314:     FormFunction - Evaluates gradient of f.

316:     Input Parameters:
317: .   tao  - the Tao context
318: .   X    - input vector
319: .   ptr  - optional user-defined context, as set by TaoAppSetConstraintRoutine()

321:     Output Parameters:
322: .   F - vector containing the newly evaluated gradient
323: */
324: PetscErrorCode FormConstraints(Tao tao, Vec X, Vec F, void *ptr)
325: {
326:   AppCtx         *user = (AppCtx *) ptr;
327:   PetscReal      *x, *f;
328:   PetscReal      *Vt1 = user->Vt1, *c = user->c, *d = user->d;
329:   PetscReal      rate = user->rate;
330:   PetscReal      dt = user->dt, ds = user->ds;
331:   PetscInt       ms = user->ms;
333:   PetscInt       i, xs,xm,gxs,gxm;
334:   Vec            localX,localF;
335:   PetscReal      zero=0.0;

337:   DMGetLocalVector(user->dm,&localX);
338:   DMGetLocalVector(user->dm,&localF);
339:   DMGlobalToLocalBegin(user->dm,X,INSERT_VALUES,localX);
340:   DMGlobalToLocalEnd(user->dm,X,INSERT_VALUES,localX);
341:   DMDAGetCorners(user->dm,&xs,NULL,NULL,&xm,NULL,NULL);
342:   DMDAGetGhostCorners(user->dm,&gxs,NULL,NULL,&gxm,NULL,NULL);
343:   VecSet(F, zero);
344:   /*
345:      The problem size is smaller than the discretization because of the
346:      two fixed elements (V(0,T) = E and V(Send,T) = 0.
347:   */

349:   /* Get pointers to the vector data */
350:   VecGetArray(localX, &x);
351:   VecGetArray(localF, &f);

353:   /* Left Boundary */
354:   if (gxs==0){
355:     f[0] = x[0]-user->strike;
356:   } else {
357:     f[0] = 0;
358:   }

360:   /* All points in the interior */
361:   /*  for (i=gxs+1;i<gxm-1;i++){ */
362:   for (i=1;i<gxm-1;i++){
363:     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]) +
364:            (d[i]/(2*ds*ds))*(x[i+1] -2*x[i] + x[i-1] + Vt1[i+1] - 2*Vt1[i] + Vt1[i-1]);
365:   }

367:   /* Right boundary */
368:   if (gxs+gxm==ms){
369:     f[xm-1]=x[gxm-1];
370:   } else {
371:     f[xm-1]=0;
372:   }

374:   /* Restore vectors */
375:   VecRestoreArray(localX, &x);
376:   VecRestoreArray(localF, &f);
377:   DMLocalToGlobalBegin(user->dm,localF,INSERT_VALUES,F);
378:   DMLocalToGlobalEnd(user->dm,localF,INSERT_VALUES,F);
379:   DMRestoreLocalVector(user->dm,&localX);
380:   DMRestoreLocalVector(user->dm,&localF);
381:   PetscLogFlops(24*(gxm-2));
382:   /*
383:   info=VecView(F,PETSC_VIEWER_STDOUT_WORLD);
384:   */
385:   return 0;
386: }

388: /* ------------------------------------------------------------------- */
389: /*
390:    FormJacobian - Evaluates Jacobian matrix.

392:    Input Parameters:
393: .  tao  - the Tao context
394: .  X    - input vector
395: .  ptr  - optional user-defined context, as set by TaoSetJacobian()

397:    Output Parameters:
398: .  J    - Jacobian matrix
399: */
400: PetscErrorCode FormJacobian(Tao tao, Vec X, Mat J, Mat tJPre, void *ptr)
401: {
402:   AppCtx         *user = (AppCtx *) ptr;
403:   PetscReal      *c = user->c, *d = user->d;
404:   PetscReal      rate = user->rate;
405:   PetscReal      dt = user->dt, ds = user->ds;
406:   PetscInt       ms = user->ms;
407:   PetscReal      val[3];
409:   PetscInt       col[3];
410:   PetscInt       i;
411:   PetscInt       gxs,gxm;
412:   PetscBool      assembled;

414:   /* Set various matrix options */
415:   MatSetOption(J,MAT_IGNORE_OFF_PROC_ENTRIES,PETSC_TRUE);
416:   MatAssembled(J,&assembled);
417:   if (assembled){MatZeroEntries(J); }

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

421:   if (gxs==0){
422:     i = 0;
423:     col[0] = 0;
424:     val[0]=1.0;
425:     MatSetValues(J,1,&i,1,col,val,INSERT_VALUES);
426:   }
427:   for (i=1; i < gxm-1; i++) {
428:     col[0] = gxs + i - 1;
429:     col[1] = gxs + i;
430:     col[2] = gxs + i + 1;
431:     val[0] = -c[i]/(4*ds) + d[i]/(2*ds*ds);
432:     val[1] = 1.0/dt + rate - d[i]/(ds*ds);
433:     val[2] =  c[i]/(4*ds) + d[i]/(2*ds*ds);
434:     MatSetValues(J,1,&col[1],3,col,val,INSERT_VALUES);
435:   }
436:   if (gxs+gxm==ms){
437:     i = ms-1;
438:     col[0] = i;
439:     val[0]=1.0;
440:     MatSetValues(J,1,&i,1,col,val,INSERT_VALUES);
441:   }

443:   /* Assemble the Jacobian matrix */
444:   MatAssemblyBegin(J,MAT_FINAL_ASSEMBLY);
445:   MatAssemblyEnd(J,MAT_FINAL_ASSEMBLY);
446:   PetscLogFlops(18*(gxm)+5);
447:   return 0;
448: }

452: /*TEST

454:    build:
455:       requires: !complex

457:    test:
458:       args: -tao_monitor -tao_type ssils
459:       requires: !single

461:    test:
462:       suffix: 2
463:       args: -tao_monitor -tao_type ssfls -tao_max_it 9
464:       requires: !single

466:    test:
467:       suffix: 3
468:       args: -tao_monitor -tao_type asils -tao_subset_type subvec
469:       requires: !single

471:    test:
472:       suffix: 4
473:       args: -tao_monitor -tao_type asils -tao_subset_type mask
474:       requires: !single

476:    test:
477:       suffix: 5
478:       args: -tao_monitor -tao_type asils -tao_subset_type matrixfree -pc_type jacobi -tao_max_it 5
479:       requires: !single

481:    test:
482:       suffix: 6
483:       args: -tao_monitor -tao_type asfls -tao_subset_type subvec -tao_max_it 9
484:       requires: !single

486:    test:
487:       suffix: 7
488:       args: -tao_monitor -tao_type asfls -tao_subset_type mask -tao_max_it 9
489:       requires: !single

491: TEST*/