Actual source code: blackscholes.c

petsc-master 2014-10-20
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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*/


102: /*
103:   User-defined application context - contains data needed by the
104:   application-provided call-back routines, FormFunction(), and FormJacobian().
105: */

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

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

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

120:   DM        dm;
121: } AppCtx;

123: /* -------- User-defined Routines --------- */

125: PetscErrorCode FormConstraints(Tao, Vec, Vec, void *);
126: PetscErrorCode FormJacobian(Tao, Vec, Mat, Mat, void *);
127: 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, "-alpha", &user.alpha, &flg);
165:   PetscOptionsGetReal(NULL, "-delta", &user.delta, &flg);
166:   PetscOptionsGetReal(NULL, "-es", &user.es, &flg);
167:   PetscOptionsGetReal(NULL, "-expiry", &user.expiry, &flg);
168:   PetscOptionsGetInt(NULL, "-ms", &user.ms, &flg);
169:   PetscOptionsGetInt(NULL, "-mt", &user.mt, &flg);
170:   PetscOptionsGetReal(NULL, "-rate", &user.rate, &flg);
171:   PetscOptionsGetReal(NULL, "-sigma", &user.sigma, &flg);
172:   PetscOptionsGetReal(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:   /* Create appropriate vectors and matrices */

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

183:   DMCreateGlobalVector(user.dm,&x);
184:   /*
185:      Finish filling in the user-defined context with the values for
186:      dS, dt, and allocating space for the constants
187:   */
188:   user.ds = user.es / (user.ms-1);
189:   user.dt = user.expiry / user.mt;

191:   PetscMalloc1(gxm,&(user.Vt1));
192:   PetscMalloc1(gxm,&(user.c));
193:   PetscMalloc1(gxm,&(user.d));

195:   /*
196:      Calculate the values for the constant.  Vt1 begins with the ending
197:      boundary condition.
198:   */
199:   for (i=0; i<gxm; i++) {
200:     sval = (gxs+i)*user.ds;
201:     user.Vt1[i] = PetscMax(user.strike - sval, 0);
202:     user.c[i] = (user.delta - user.rate)*sval;
203:     user.d[i] = -0.5*user.sigma*user.sigma*PetscPowReal(sval, user.alpha);
204:   }
205:   if (gxs+gxm==user.ms){
206:     user.Vt1[gxm-1] = 0;
207:   }
208:   VecDuplicate(x, &c);

210:   /*
211:      Allocate the matrix used by TAO for the Jacobian.  Each row of
212:      the Jacobian matrix will have at most three elements.
213:   */
214:   DMCreateMatrix(user.dm,&J);

216:   /* The TAO code begins here */

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

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

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

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

237:   /* Set routines for function and Jacobian evaluation */
238:   TaoSetFromOptions(tao);

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

243:     /* Solve the current version */
244:     TaoSolve(tao);

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

258:   /* Free TAO data structures */
259:   TaoDestroy(&tao);

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

271:   PetscFinalize();
272:   return 0;
273: }

275: /* -------------------------------------------------------------------- */
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: /* -------------------------------------------------------------------- */

315: /*
316:     FormFunction - Evaluates gradient of f.

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

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

339:   DMGetLocalVector(user->dm,&localX);
340:   DMGetLocalVector(user->dm,&localF);
341:   DMGlobalToLocalBegin(user->dm,X,INSERT_VALUES,localX);
342:   DMGlobalToLocalEnd(user->dm,X,INSERT_VALUES,localX);
343:   DMDAGetCorners(user->dm,&xs,NULL,NULL,&xm,NULL,NULL);
344:   DMDAGetGhostCorners(user->dm,&gxs,NULL,NULL,&gxm,NULL,NULL);
345:   VecSet(F, zero);
346:   /*
347:      The problem size is smaller than the discretization because of the
348:      two fixed elements (V(0,T) = E and V(Send,T) = 0.
349:   */

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

355:   /* Left Boundary */
356:   if (gxs==0){
357:     f[0] = x[0]-user->strike;
358:   } else {
359:     f[0] = 0;
360:   }

362:   /* All points in the interior */
363:   /*  for (i=gxs+1;i<gxm-1;i++){ */
364:   for (i=1;i<gxm-1;i++){
365:     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]) +
366:            (d[i]/(2*ds*ds))*(x[i+1] -2*x[i] + x[i-1] + Vt1[i+1] - 2*Vt1[i] + Vt1[i-1]);
367:   }

369:   /* Right boundary */
370:   if (gxs+gxm==ms){
371:     f[xm-1]=x[gxm-1];
372:   } else {
373:     f[xm-1]=0;
374:   }

376:   /* Restore vectors */
377:   VecRestoreArray(localX, &x);
378:   VecRestoreArray(localF, &f);
379:   DMLocalToGlobalBegin(user->dm,localF,INSERT_VALUES,F);
380:   DMLocalToGlobalEnd(user->dm,localF,INSERT_VALUES,F);
381:   DMRestoreLocalVector(user->dm,&localX);
382:   DMRestoreLocalVector(user->dm,&localF);
383:   PetscLogFlops(24*(gxm-2));
384:   /*
385:   info=VecView(F,PETSC_VIEWER_STDOUT_WORLD);
386:   */
387:   return 0;
388: }

390: /* ------------------------------------------------------------------- */
393: /*
394:    FormJacobian - Evaluates Jacobian matrix.

396:    Input Parameters:
397: .  tao  - the Tao context
398: .  X    - input vector
399: .  ptr  - optional user-defined context, as set by TaoSetJacobian()

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

418:   /* Set various matrix options */
419:   MatSetOption(J,MAT_IGNORE_OFF_PROC_ENTRIES,PETSC_TRUE);
420:   MatAssembled(J,&assembled);
421:   if (assembled){MatZeroEntries(J); }

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

425:   if (gxs==0){
426:     i = 0;
427:     col[0] = 0;
428:     val[0]=1.0;
429:     MatSetValues(J,1,&i,1,col,val,INSERT_VALUES);
430:   }
431:   for (i=1; i < gxm-1; i++) {
432:     col[0] = gxs + i - 1;
433:     col[1] = gxs + i;
434:     col[2] = gxs + i + 1;
435:     val[0] = -c[i]/(4*ds) + d[i]/(2*ds*ds);
436:     val[1] = 1.0/dt + rate - d[i]/(ds*ds);
437:     val[2] =  c[i]/(4*ds) + d[i]/(2*ds*ds);
438:     MatSetValues(J,1,&col[1],3,col,val,INSERT_VALUES);
439:   }
440:   if (gxs+gxm==ms){
441:     i = ms-1;
442:     col[0] = i;
443:     val[0]=1.0;
444:     MatSetValues(J,1,&i,1,col,val,INSERT_VALUES);
445:   }

447:   /* Assemble the Jacobian matrix */
448:   MatAssemblyBegin(J,MAT_FINAL_ASSEMBLY);
449:   MatAssemblyEnd(J,MAT_FINAL_ASSEMBLY);
450:   PetscLogFlops(18*(gxm)+5);
451:   return 0;
452: }